Interaction Dynamics and Active Suppression of Instability in Parallel Photovoltaic Voltage-and Current-Source Converters Connected to a Weak Grid

Parallel operation of power electronic converters is becoming popular in utility-scale photovoltaic (PV) systems. However, the literature does not cover the interaction dynamics and stabilization of parallel PV-based voltage- and current-source converters (VSC and CSC) connected to a weak grid. To fill in this gap, this article characterizes the dynamic interactions among the parallel PV-based VSC and CSC systems considering the effects of the PV source dynamics, grid strength, operating point variation, and control parameters. A detailed small-signal model of the parallel system is developed and used to characterize the dynamic interactions using eigenvalue and input/output impedance-based analyses. The study showed that undesirable converters interactions are yielded, the dc-links stability is reduced, and the parallel system cannot inject 1.0 p.u. of active power at unity short-circuit ratio (SCR). Therefore, an active stabilization approach is proposed for the parallel VSC-CSC system to reduce the interactions, improve stability, and facilitate 1.0 p.u. of active power injection at unity SCR. Detailed nonlinear time-domain simulations and real-time simulation results verified the accuracy of the analytical results and the effectiveness of the proposed stabilization method under a wide range of operating conditions.


I. INTRODUCTION
Due to the adverse consequences of fossil fuel-based energy sources, renewable energy sources are getting consideration worldwide, with a dominant spotlight on photovoltaic (PV) systems [1]. Furthermore, advancements in power electronic gears contribute to permissive technology integrating voltageand current-source converters (VSCs and CSCs) into the power grid [2]. However, the long transmission lines between the converter and the power grid lead to a weak grid (i.e., a large grid impedance), which intimidates the stability of large-scale PV systems. A short-circuit ratio (SCR) outlines the gird rigidity (weak when 2 ≤ SCR ≤ 3) as viewed from the point-of-common-coupling (PCC), where the short-circuit capacity of the grid at the PCC to the rated power of the converter defines the SCR. The interdependent system dynamics become more complex when SCR = 1 and, in recent times, become a hot subject [3].
A comprehensive analysis has been noted on a single VSC connected to a weak grid [1], [2], [3], [4], [5], [6]. Furthermore, improvement in semiconductor technology allows CSCs for renewable energy (PV and wind farms) and motor drive applications [7]. However, CSCs connected to a weak grid are not reported broadly [7], [8], [9], [10]. For PV integration, VSCs are prominent; however, CSCs have been considered a substitute option for the following criterion [7], [8], [9], [10], [11]. 1) Although the power losses in the CSC dc choke are higher than the VSC dc-link capacitor, the dc-current is smooth for CSCs, which is suitable for PV applications. Moreover, the CSC dc choke is more reliable than the capacitor in the VSC. 2) The controlled dc choke current in the CSC provides additional short-circuit protection compared to the VSC. 3) A VSC is a buck-type inverter; however, a CSC is a boost inverter, an asset in PV grid integration. Due to the potential benefits of CSCs, VSCs and CSCs can be connected in parallel to meet the increasing capacity demand in PV systems. However, if the parallel converter system is connected to a weak grid, PV dynamics are coupled with each other due to the sensitive PCC voltage. Therefore, changes in the PV generators' (PVG) source dynamics (i.e., changes in the dynamic resistance of a PVG [7], [27]) in one converter or PV output power variations can impact the dynamics and stability of the other converter through the coupled network. Moreover, the effect of the parallel VSC-CSC systems interaction on converters' dc-link stability is an important problem that has not been addressed previously up to the authors' insight.
From a scholarly and technical point of view, the singlestage VSC tied with long transmission lines leading to a weak grid has drawn much interest [12], [13], [14], [15]. The authors in [12] proposed a feedback linearization method to minimize the negative impacts of the phase-lock-loop (PLL) under a weak grid by regulating the power output and the converters' output voltage. However, the study did not consider the source dynamics to investigate the interaction among the dc-and ac-side of the converter. Wen et al. [13] recorded that the impedance of the q-q channel (Z qq ) acts as a negative additive resistor resulting from the PLL and current injection. However, the study is limited to small-signal-based analysis and a strong grid. Reference [14] reported that a frequencybased synchronization control mitigates the low-frequency oscillations and instability introduced by the PLL when a VSC-based high-voltage direct current (HVDC) station is connected to a weak grid. In [15], a modified outer loop was proposed for a weak-grid-tied VSC, considering a scheduling controller, contributing to an extensive working range for lower SCR levels. However, the study did not consider the source dynamics to analyze the dynamic interactions.
The PLL plays a decisive part in the weak grid stability study. In [2], considering the PLL dynamics, a robust control strategy was reported to inject 1.0 p.u. of active power at unity SCR. According to [5], proper PLL tuning can lower the adverse effect of the PLL on the current control when connected to a weak grid. However, a very low PLL bandwidth would be needed to reduce the PLL effect on the system stability. The authors in [8] proposed a compensation method for a CSC to inject 1.0 p.u. of active power at SCR = 1. In [9], ignoring the PLL and PVG source dynamics, a power synchronization control topology was proposed for a CSC system when tied to a weak grid. Reference [16] reported that a VSC system's high-frequency stability margin decreases with increasing grid impedance. However, the authors did not consider the rated power injection when SCR = 1. Zhou et al. [17] showed that PLL gains at low SCR significantly affect the VSC-HVDC systems' dynamic and steady-state behavior. It is, furthermore, reducing the maximum power transfer capability. Reference [18] showed the effects of the varying operating point on the dc-link voltage control stability of wind-based VSC systems when connected to a weak grid. However, the study did not consider the dynamics of a wind turbine. In [19], for a grid-tied VSC, the authors included the PLL dynamics in the dq-frame impedance model, where the advanced PLLs have an alternative representation. It is clear from the above survey that the weak-grid-connected converter system studies considered a single converter system, ignored the effects of the source dynamics, and did not develop a compensation method enabling rated active power injection at unity SCR.
Converters can be connected in parallel to meet the need for increased capacity. However, simultaneous dc-links management and power distribution are required for parallel operation, which is challenging. In [20], the dc-link voltage control stability of muti-VSCs when integrated into a weak grid was carried out considering the effects of different operating points, control bandwidth, and SCR levels. The authors reported that under a weak grid, the interactions among the VSCs become severe due to the strong coupling between the wind turbines' control and grid dynamics. However, the authors did not consider the impacts of changing wind turbines' dynamic on dc-link stability. Moreover, the study lacks a compensation method to reduce the interactions and improve the stability under faults and uncertainties (e.g., grid-voltage phase angle or frequency change). The authors in [21] reported an improved notch filter for parallel grid-connected VSCs, which ensures sufficient system stability in the weak grid by changing the impedance properties. Although the study reported stability improvement, it did not show how the proposed compensation method impacted the parallel converters' dynamic interaction and improved power injection levels with changing SCR. In [22], PV-based parallel converters stability was conducted based on multi-input multi-output (MIMO) methodology to reveal stability issues related to the converters. However, the authors did not report the dynamic interactions between converters linked to the PVG dynamics and grid strength. Reference [23] addressed the resonance problem of the multi-VSC integrated weak grid system, which threatens the power quality and system stability. However, no compensation was proposed to improve the low-and highfrequency oscillations when connected to a weak grid. In [24], the authors reported that dynamic interactions among the paralleled CSC introduce multiple resonances affecting the stability and power quality. Therefore, a damping method was proposed to mitigate the low-order harmonic resonance problem in parallel CSCs. However, the authors did not show how parallel CSCs interact due to source or grid strength variation disturbances. In [25], the authors proposed a compensation method to damp subsynchronous torsional oscillations in a series-compensated system by reshaping the output admittance of a VSC interfacing an HVDC or wind farm. However, the study did not consider the weak grid system. The authors in [26] proposed a state feedback reshaping method by adding full state feedback to the VSC current reference when connected to a weak grid, resulting in a 50% higher damping capability than the virtual admittance-based solution. However, the source dynamics and the effect of the developed method on the power injection level were not considered. Reference [27] proposed an adaptive power-sharing control scheme for paralleled inverters in microgrid applications that ensures the stability of each paralleled inverter at different load conditions. However, the study focused on isolated microgrid applications and did not address weak grid integration problems.
It is apparent from the above survey that the current literature did not address the following: 1) dynamic interactions in a parallel PV-based VSC and CSC system connected to a weak grid, and 2) active stabilization methods to facilitate reduced interaction, stability, and rated power injection [18], [19], [20], [21], [22], [23], [24], [25], [26]. Motivated by these shortcomings, this article provides, for the first time, a dc-links interactions and stability study of a parallel PVbased VSC and CSC system connected to a weak grid. The interaction dynamics and their effect on the dc-link control stability are characterized when the SCR changes from 4 to 1 considering the following: changes in the PVG source dynamics, external disturbances from the grid side, and control parameters variations (e.g., changes in the PLL bandwidth). Furthermore, a compensation technique is proposed for the parallel VSC-CSC system to reduce undesirable interactions between the converters, improve dc-link stability, and ensure a highly damped system at weak and very weak grid conditions (SCR = 1).
This article shows that if a parallel VSC-CSC system is connected to a weak grid, the interaction between the converters increases with the changing PVG source dynamics and operating points as the PVG dynamics are coupled through the sensitive PCC. The VSC input impedance investigation shows that if the SCR decreases from 4 to 1, the resonance peak shifts to a lower frequency. In contrast, it moves to a higher frequency in the CSC input impedance. However, in the d-d and q-q components of the VSC output impedance (Z ddvsc and Z qqvsc ), the magnitude in the frequency domain increases with the decreasing SCR, implying a higher coupling of the PV-VSC system with the PCC. The d-d component magnitude of the CSC output impedance decreases, affecting the output power injection and increasing dynamic interactions among converters' dc-links via the coupled PCC. Finally, a compensator is proposed to suppress undesirable dc-link interactions and alleviate parallel VSC-CSC system instability in weak grids. It is shown in this article that the compensated parallel system can inject 1.0 p.u. of active power at SCR = 1. In contrast, an uncompensated system could inject 0.7-0.75 p.u. of active power into the grid under the same conditions. The contributions of this article to the research field are as follows: 1) Developing a detailed small-signal model and inputoutput converter impedances for a parallel PV-based VSC and CSC system connected to a weak grid considering the PVG source dynamics and complete control loops, including PLLs, 2) Analyzing the interaction dynamics among parallel converters considering the effects of the PV source dynamics, SCR, power injection levels, and control parameters, and 3) Developing an active compensator assuring the parallel system stability, rated power injection at unity SCR, and improved damping in a weak grid. The rest of the article is organized as follows. Section II introduces the parallel PV-based VSC and CSC system and the models of the components. Section III presents the VSC and CSC systems control, including the PLL dynamics and frame transformation. The parallel VSC and CSC systems interactions when connected to a weak grid are assessed in Section IV. Section V presents a compensator for the parallel VSC-CSC system, ensuring reduced interactions and better damping. Detailed nonlinear time-domain simulation results are presented in Section VI to evaluate the parallel system's interactions and show the effectiveness of the proposed compensator. Section VII presents the real-time simulation results using the OPAL-RT OP5600 platform. Finally, the conclusions are drawn in Section VIII. Fig. 1 shows the circuit topology of a parallel utility-scale VSC-CSC system interfacing PVGs. The parameters of the VSC and CSC are typical, and their vector controllers are designed following the standard guidelines of grid-connected converters [2], [7], [29]. The complete system and control parameters are given in Appendix A. It should be noted that practical parallel converters can have different ratings, switching frequencies, and control parameters; therefore, a detailed dynamic analysis of the parallel VSC-CSC is essential to correctly evaluate the interaction dynamics and the parallel systems' stability.

A. PV-VSC MODEL
In the PV-VSC system, the PVG is connected to the VSC via a dc-link capacitor (C dc ), as shown in Fig. 1. The filter reactor has a resistance (R f ) and inductance (L f ). The ac-side filter capacitance (C f ) attenuates the switching harmonics. Additionally, the VSC is connected to the PCC through a feeder having the inductance of L 1 . The mathematical model of the PV-VSC system in the d-q frame is given by (v pccd , v pccq ) are the inverter side d-q voltage, the voltage of the shunt capacitor (v), reactor current, the current through the inductor L 1 (i L1 ), and voltage at the PCC (v pcc ), respectively. The VSC dc-side voltage is V dc , PV current is I pv , s is the Laplace operator, and ω is the angular speed of frame rotation.

B. PV-CSC MODEL
In the PV-CSC system, the PVG is connected to the CSC via an inductor (L dc ) to maintain a smooth dc-side current (I dc ). A capacitive filter (C s ) attenuates the switching harmonics of the CSC output current (i w ). A feeder with inductance (L 2 ) connects the CSC to the PCC. The mathematical model of the PV-CSC system in the d-q frame is given by where (i wd , i wq ), (v sd , v sq ) and (i sd , i sq ) are the d-q components of the inverter current, the capacitor voltage, and the current through the inductor L 2 , respectively.

C. GRID DYNAMICS AND THE CHARACTERISTICS OF A WEAK GRID
The grid-side dynamics in the d-q frame are given by where (i gd , i gq ) and (v gd , v gq ) are the d-q components of the grid-side current and voltage, respectively. The transformer ratio is N, and L g is the grid-side inductance. Due to the long transmission lines between the converter and the grid, a large grid impedance (X g ) results in a weak grid. Therefore, the PCC voltage becomes sensitive to the PV output power variations and other disturbances. The SCR defines the grid strength, which is dependent on X g , as [8], [20] where, VA sc is the grid short-circuit capacity [in VA], P pv is the rated PV power, and v g is the rms three-phase grid voltage. From (6d), the grid becomes weak when 2 ≤ SCR ≤ 3 and very weak when SCR < 2.

D. PVG MODEL AND EFFECT OF SOURCE DYNAMICS
The PV array is structured with N p parallel strings with N s series-connected PV modules having n s series-connected PV cells. The PV array current I pv can be expressed as where PV module series and shunt resistance are R s and R sh , respectively. The photon-generated current is I ph ; the reverse saturation current is I rs ; the electric charge is q; the Boltzmann constant is k, T is the temperature in K; the diode ideality factor is A; and V pv is the terminal PV voltage. The nonlinear PVG characteristics in (7) result in an operating point-dependent dynamic resistance (r d ), which can be expressed as the negative reciprocal of (dI pv /dV pv ) as Due to the changing weather condition, the PVG operating point moves from the right side of the maximum power point (MPP), known as the constant-voltage region (CVR), to the left of the MPP, known as the constant-current region (CCR) [7]. Changes in the PVG operating point change r d from a minimum to a maximum value, affecting the system's stability [28], [30], [31]. Therefore, the effect of r d is considered in this article to investigate its impact on paralleled systems coupling, interactions, and stability when connected to a weak grid.
The authors in [7] reported that the movement of the PVG operating point from the CVR (i.e., low r d ) to the CCR (i.e., high r d ) results in resonant roots and introduces high-frequency instability in the PV-CSC system. However, the system remains stable if it operates at the CVR. Reference [28] reported that operating the PV-VSC system at the CCR affects the dc-link voltage control stability and injected power levels into the grid. However, the study was limited to a single-stage strong grid system. In [30], a metaheuristic algorithm-based r d determination method was proposed. The authors concluded that reduced dc-link capacitance and movement of the PVG operating point from the CVR to CCR introduce low-and high-frequency instability in the dc-link voltage control. Nousianen et al. in [31] showed that operating point-dependent dynamic resistance affects the interfacing converter's dynamics. However, the study did not consider the converter's dynamic interaction when connected to a weak grid. The authors in [32] reported that the PV-VSC system's stability degrades at the MPP and CCR with a higher series resistance in the dc-link capacitance. However, the study did not show the VSC dc-and ac-side interactions when the PVG operating point changes. The above survey shows that the movement of the PVG from the CVR to the CCR affects the dc-link stability for both VSC and CSC systems. However, up to the authors' insight, how the changing PVG dynamics and grid strength affect the stability of parallel VSC and CSC systems is not reported [7], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]. This shortcoming is addressed in this article.

III. VSC-CSC CONTROL SYSTEM
For the parallel system shown in Fig. 1, the dc-link voltage control of PV-VSC and dc-link current control of PV-CSC systems architecture is shown in Figs. 2 and 3, respectively. The effect of the PLL of each converter is considered in the overall system dynamics.

A. CONTROL OF THE PV-VSC SYSTEM
As shown in Fig. 2(a), the maximum power point tracking (MPPT) controller generates the reference dc-link voltage (V * dc ) via the Perturb and Observe (P&O) algorithm. As shown in (9a), the reference d-axis current (i * d ) is generated by processing the error (V 2 dc − V * 2 dc ) via a proportional-integral (PI) controller (G vdc (s) = K pvdc + K ivdc /s) and a feed-forward term (ηP pv /1.5v 0 d ) to enhance the robustness of the dc-link voltage control, where η is a feed-forward gain, and superscript "0" indicates a steady-state value at the linearization  point.
The bandwidth (BW) of the dc-link voltage control loop is usually selected as 10-20% of the BW of the ac current control loop (ω b ) [28]. The current tracking is achieved via a PI current controller (G i (s) = K pi + K ii /s ), which regulates the currents i d and i q to their reference values i * d and i * q to regulate the active (P) and reactive (Q) power injection. The converter control voltages in the converter's d-q frame (v c id and v c iq ) in (9b) and (9c) is obtained to generate the modulating signals (m d and m q ) for the VSC, where m d = 2v id /V dc and m q = 2v iq /V dc . To mitigate the effect of dq coupling terms in (1a) and (1b), the decoupling ( jωL f i c dq ) and feed-forward terms (v c dq ) are implemented in the VSC control loops, which results in two decoupled first-order linear systems. Therefore, the dq-current controller can be expressed as The superscript "c" in (9b) and (9c) represents the quantities in the converter frame. Section III-B describes the principle of frame transformation. The BW of G i (s) (in rad/s) is usually 0.1-0.2 times the switching frequency (ω sw ) of the VSC for faster response and higher BW. The actual and reference current vectors i and i * relationship can be given as where, The BW of PLL is within the BW of the dc-link voltage control loop [33]. By integrating the angular frequency, the PLL generates the synchronization angle (ε) and transforms the grid-frame signals to the converter frames using f c dq = e − jε f g dq , and vice versa, where the superscript "g" donates the signal in the grid frame. To consider the influence of the PLL on the system dynamics, the measured quantities (i.e., v and i) should be transformed to the converter frame, whereas the VSC output voltage (i.e., v i ) should be retransformed to the grid frame. The PLL output angle ε oscillates in transient conditions; however, in steady-state becomes zero [8], [33]. The transformed vectors can be described by (11a)-(11c) using the Taylor series expansion as Detailed design guidelines for the VSC control system can be found in [28], [29], [33].

C. CONTROL OF THE PV-CSC SYSTEM
The MPPT controller generates the reference dccurrent (I * dc ). As shown in Fig. 3(a), the PI controller (G idc (s) = K pidc + K iidc /s) processes the error (I 2 dc − I * 2 dc ) to generate the reference d-axis current (i * sd ) needed to balance the input-output active power across the converter. The BW of the dc-link current control loop is in the range of 0.1-0.2 of the BW of the CSC current control (BW cc CSC ). Furthermore, the inner current control loop adopts a PI controller (G ic (s) = K pic + K iic /s ) for reference current tracking (i.e., i * sd and i * sq ). Therefore, from Fig. 3(a) and (b), the CSC current control dynamics can be expressed as Considering, K pic ≈ 0, i s / i * s = 1 1+s/K ii is a first-order lowpass filter with BW cc CSC = K iic in rad/s, which is 0.1-0.2 times the switching frequency of the CSC [7]. Fig. 3(c) shows the CSC PLL control structure, where the terminal voltage (v s ) is injected to synthesize the q-component of the terminal voltage to zero using a PI controller (G δ (s) = K pδ + K iδ /s). The CSC PLL generates the synchronization angle (δ), which transforms the measured quantities (i.e., v s and i s ) to the converter frame and the converter output current (i w ) to the grid frame to model the impact of the PLL on systems dynamics [8]. For the CSC, the frame transformation is described by (13a)-(13c) using the Taylor series expansion as

IV. PARALLEL VSC-CSC INTERACTIONS AND STABILITY ANALYSIS WHEN CONNECTED TO A WEAK GRID
This section explains the impact of parallel VSC-CSC interactions on dc-link stability, considering PV or grid disturbances. Furthermore, small-signal stability studies at different operating conditions are described in detail, considering the PVG source dynamics. Fig. 4 depicts the interaction mechanism among parallel PV-VSC and PV-CSC systems. As connected to a weak grid, disturbances from either the PVG or grid affect the PCC voltage. Therefore, one converter influences the other converter's synchronization performance, and instability in one converter leads to stability problems in another [21], [22]. Changes in the PVG source dynamics or disturbances in a VSC system affect its dc-link stability leading to output power and impedance changes. These affect the terminal voltage and the PCC voltage, which further affects the terminal voltage of CSC [18]. In addition, changes in the CSC terminal voltage affect the PLL operation, impacting the CSC output power and impedance. As a result, the dc-link current control of the CSC is also affected, further affecting the CSC input impedance. In turn, instability in the CSC dc-link affects the dc-link stability of the VSC through the coupling network, and so forth.

B. SMALL-SIGNAL ANALYSIS OF PARALLEL VSC-CSC SYSTEM
In this part, the small-signal stability of the parallel VSC-CSC system in Fig. 1 is investigated under different operating conditions to show the impact of one converter system on another at different operating conditions. Initially, the statespace model is developed by linearizing the system model (1a)-(13c) and considering the control systems in Figs. 2 and 3. The linearized matrices of the parallel VSC-CSC system are given in Appendix B, where " " stands for a perturbed variable and superscript "0" indicates a steadystate value. From the complete systems state matrices A, B, C, and D, the Laplace domain output variables can be expressed as where G Y is the transfer function matrix, U (s) is the input matrix, and Y (s) is the output matrix. In the linearized model, variables ( ϕ id ϕ iq ), ( ε ϕ ε ), and ϕ vdc is the integral states of the VSC d-q current, PLL, and dc-link voltage control loops. For the CSC, ( ϕ icd ϕ icq ), ϕ idc , and ( δ ϕ δ ) are the d-q current, dc-current, and PLL control loops integral terms. The following scenarios are considered to analyze the parallel VSC-CSC systems interaction study.

1) EFFECT OF PVG DYNAMICS ON VSC-CSC INTERACTION
The transfer function between the CSC control variable (d d ) and the d-axis output current (i sd ) is the control-to-output current transfer function, G codcsc (G codcsc = i sd /d d ). However, G codvsc is the transfer function between the control variable (m d ) and VSC's d-axis output current (i d ) (G codvsc = i d /m d ). Fig. 5(a) and (b) show the effect of the decreasing SCR from 4 to 1 on the frequency response of G codcsc and G codvsc ,  codcsc , (b) G codvsc , (c) G coqcsc , and (d) G coqvsc . respectively, where both converters operate at the MPP. As shown in Fig. 5(a), the magnitude and phase of G codcsc increase with decreasing SCR, which changes the CSC terminal voltage and influences the PCC voltage, and further increases the interaction with the VSC dc-link, leading to an increase in magnitude and phase of G codvsc , as shown in Fig. 5(b). In addition, a resonance peak for G codcsc is formed from the filter capacitance and grid-side inductance combination (lower SCR levels result in higher peaks), affecting the CSC terminal voltage and the VSC dc-link stability. Therefore, the controlto-output current transfer function (G cod ) indicates that grid disturbance influences the PCC voltage, which affects the VSC-CSC terminal voltages, changes PLL operation, and affects the dc-link stability. Furthermore, the impact of the SCR on the q-axis control-to-output current transfer function (G coq ) for the CSC (G coqcsc = i sq /d q ) and VSC (G coqvsc = i q /m q ) is shown in Fig. 5(c) and (d), respectively. The q-axis control variable of the CSC and VSC are d q and m q , respectively. To show the converters' interactions considering the PVG dynamics, in this case, the CSC operates at the CCR (i.e., higher r d ); however, the VSC operates at the MPP (i.e., lower r d ). The determinant of G coqcsc has resonant roots, as r d is higher, which changes the system's damping by migrating a pole to the right side of the s-plane. As a result, the magnitude increases for G coqcsc with decreasing SCR, as shown in Fig. 5(c), resulting in increased interactions with the PCC, which increases the interactions with the VSC dc-link leading to an increase of G coqvsc magnitude and decreases the stability margin, as shown in Fig. 5(d). However, for G coqvsc , the phase leads and then lags when SCR = 1. Therefore, G coq indicates that the changes in the converters' output voltage affect the PLL operation and the dc-link current and voltage control stability.

2) EFFECT OF CSC ACTIVE POWER INJECTION
The transfer function between the CSC input (I pv ) and the d-axis output current (i sd ) is the CSC forward transfer function, G iodcsc (G iodcsc = i sd /I pv ). Fig. 6 shows the frequency response characteristics of G iodcsc , where the impact of the CSC changing active power injections (P s ) is investigated when SCR = 1. As shown in Fig. 6, if P s increases from 0.75 to 1.25 p.u., the magnitude of G iodcsc increases with a decrease in the gain margin from 34.8 to −30.5 dB. However, the phase margin reduces from 11.1 to −48.1°. The CSC remains stable till P s = 0.75 p.u.; however, a further increase in P s results in an unstable response, which affects the converter's output and the PCC voltage. Eventually, it affects the VSC terminal and dc-link voltage through the coupled network.

3) EFFECT OF SCR ON CONVERTERS INPUT IMPEDANCE
In this case, the effect of the SCR on converters' input impedances is investigated. Applying small-signal linearization to (2a) and the grid dynamics results where Z g is the grid impedance (Z g = R g + sL g ). Applying small perturbations to (1a) and solving with (14a) results in Considering a lossless power converter, the input power delivered from the dc-link (V dc I inv ) equals the power supplied by the VSC terminal (P = 1.5(v id i d + v iq i q )), where I inv is the input current to the VSC. Applying small-signal linearization to the dc-ac power balance results in Using (14a), (14b), and (15a) From (14c) and (15b), the input impedance of the VSC Z invsc (s) ≈ V dc I inv is determined. Similarly, the input impedance of the CSC, Z incsc (≈ V pv I dc ), is computed to analyze the dc-side interactions of the CSC and its coupling with the VSC. Applying linearization to V pv = sL dc I dc + 3 2 d d v sd + 3 2 d q V sq , (6b) and (12b) results in Z incsc (s) as The impact of changing SCR on the small-signal input impedance of VSC ( Z invsc ) and CSC ( Z incsc ) is shown in Fig. 7(a) and (b), respectively. As shown in Fig. 7(a), if the SCR decreases from 4 to 1, the resonance peak shifts to the lower frequency range for Z invsc , where the magnitude of the resonance peak increases. However, the phase increases and leads when SCR = 1 compared to SCR = 4, contributing to the instability in the VSC dc-link, which affects the power injection and output voltage. On the other hand, if the SCR decreases from 4 to 1, the magnitude of Z incsc increases while the phase remains unchanged. As a result, more interactions are seen between the CSC dc-and ac-sides than the VSC. The analysis shows that disturbances in the PV-CSC system highly impact the interactions and threaten the stability of the interconnected system.

4) EFFECT OF SCR ON CONVERTERS' OUTPUT IMPEDANCE
The effect of changing grid strength on the converters' output impedances is shown in Fig. 8. Fig. 8(a) shows the VSC d-axis output impedance ( Z ddvsc ), which is the ratio between the d-axis output current to the d-axis output voltage. If the SCR decreases from 4 to 1, the magnitude in the frequency domain increases, which is the indicator of increasing interaction between the converter's output current and terminal voltage and affects the PCC voltage. From Fig. 8(b), where Z qqvsc is the ratio between the VSC q-axis output current to the q-axis output voltage, the magnitude increases with the decreasing SCR, indicating that changes in v q affect the PLL operation and synchronization. Therefore, when SCR becomes unity, the PLL introduces instability in the dc-link voltage control, which affects the dc-link current control stability due to the increased interaction. In Fig. 8(c), where Z ddcsc is the ratio between the CSC d-axis output current to the d-axis output voltage, the magnitude is high at SCR = 4, indicating a stable operation for 1.0 p.u. active power injection. However, the magnitude decreases with the decreasing SCR, resulting in instability in the CSC system.

C. STABILITY ANALYSIS
In this section, the stability of the parallel VSC-CSC system is carried out using the placement of the eigenvalues in the s-plane with the changing operating point (i.e., PVG source dynamics, converters active power injection, grid strength, and PLL bandwidth) [26], [34]. The steady-state values of the converter inputs define the operating points. Initially, the state-space model of the system shown in Fig. 1 is derived using (1a)-(13c) and introducing the integral terms of the controllers as new variables can be expressed aṡ The steady-state value of the states X in (17a) is the equilibrium point. However, for increased power injection or during low voltage grid conditions, an equilibrium point does not exist [37]. By setting all the differential terms to zero, the equilibrium point of (17a) can be obtained as (17b) [35], [36]. (17b) Then, at the equilibrium point, the Jacobian A can be derived as A = ∂F ∂X | X =X e . Therefore, to show the effect of one converter on another one's parallel operation, the eigenvalues of the Jacobian are used, which can be calculated as [36] det [λI − A] = 0 (17c) where I is the identity matrix. As shown in Fig. 9(a), the dc-link current and voltage eigenvalues (λ I dc and λ V dc ) depend on the SCR level. The equilibrium point and the equilibrium point-dependent Jacobian matrix are evaluated at each SCR level using MATLAB symbolic toolbox to show the impact of one converter on another. With the changes in the SCR from 4 to 1, λ I dc moves towards the right side of the s-plane, which impacts the placement of λ V dc . The latter moves to the unstable region, as shown in Fig. 9(a), verifying that instability in one converter's dc-link impacts another converter's dc-link stability through the coupled network. The effect of changing the VSC output power (P vsc ) on the PLL eigenvalue (λ ε ) is shown in Fig. 9(b). When SCR = 4 and P vsc = 1.0 p.u., λ ε is located in a stable region. However, if the SCR decreases to 1, with the increase of P vsc , λ ε moves toward the right side of the s-plane; however, λ ε is in the stable region with P vsc ≤ 0.7 p.u., as shown in Fig. 9(b). This implies in a weak grid, changes in the power injection affect the converter output voltage, which affects the synchronization operation. The effect of the CSC PVG source dynamics on the dominant eigenvalues related to the parallel VSC-CSC operation is shown in Fig. 9(c). The eigenvalues in black denote that the PVG operates at the MPP; however, the blue one denotes the eigenvalues in the CCR.
As shown in Fig. 9(c), the eigenvalue of (λ I dc ) moves with the changing PVG source dynamics in the CSC, which shifts the terminal voltage eigenvalues (λ v sd and λ v sq ) from the stable to the unstable plane. In addition, as coupled through the PCC, it shifts the VSC terminal voltage eigenvalue (λ v d ) to an unstable plane, affecting the movement of (λ V dc ). Fig. 9(d) shows the effect of varied CSC PLL bandwidth from 5 to 30 Hz on the dominant eigenvalue. When the CSC PLL bandwidth exceeds the VSC PLL bandwidth, the dominant eigenvalues move toward the right side of the s-plane, demonstrating increased interactions, as shown in Fig. 9(d).

V. PROPOSED COMPENSATOR OF THE VSC-CSC SYSTEM
The frequency response and equilibrium point-dependent stability analyses, presented in Section IV, showed that significant undesirable interaction dynamics exist in the parallel weak-grid connected PV VSC-CSC system. Therefore, a compensation technique (shown in Fig. 10) is proposed in this section to improve the dc-link stability, reduce the interactions among the VSC-CSC systems, and facilitate rated power injection under different operating conditions when parallel converters are connected to a weak grid. As the PVG or grid disturbances change the PCC voltage and affect the PLL synchronization operation, the PLL output frequency (ω) is considered input to the compensator and injected in the d-axis control structure of the VSC and CSC system to control the active power injection. The signal is processed through a band-pass filter with a damping ratio, cut-off frequency, and scaling gain for the VSC and CSC systems ε f , ω f , K f and ε c , ω c , and K c , respectively (i.e., x dvsc (s) = K f s+ω q v c sq ) is added to the q-axis current control of the CSC with a cut-off frequency and scaling gain of ω d , K d and ω q , K q , respectively. Therefore, the modified control dynamics with the proposed compensator

FIGURE 11. Effect of different compensator gains and cut-off frequencies: (a) design of y q (s), (b) y dcsc (s), (c) y d (s), and (d) x dvsc (s).
can be expressed as

A. COMPENSATORS PARAMETER DESIGN
The gain and cut-off frequency of the proposed compensator is determined in three steps for the CSC and one step for the VSC, considering 1.0 p.u. active power injection at SCR = 1.

1) DESIGN OF y q (s)
Initially, the scaled version of v c sq is considered for the compensator design as it is the input of the PLL. The parameters of y q (s) (i.e., K q and ω q ) are designed by varying K q , and ω q , and their impact on the dc-link current control eigenvalue (λ Idc ) is shown in Fig. 11(a). If K q increases from 0 to 2.0, with an increase of 0.4 for ω q = 50 rad/s, λ Idc moves towards the right side of the s-plane but remains stable (i.e., λ Idc moves from −1305 to −1093.9). However, if ω q decreases from 50 to 10 rad/s for K q = 2.0, λ Idc moves further in the left side of the s-plane (i.e., λ Idc moves from −1144 to −1693), showing a more stable system. Therefore, K q = 1.5 and ω q = 50 rad/s are selected for optimal performance and better damping.

2) DESIGN OF y dcsc (s)
Next, the scaled version of the PLL output (ω) is considered in the CSC control loop to verify the system's stability, and the effect of varying K c and ω c is shown in Fig. 11(b). As shown in Fig. 11(b), if K c increases from 0 to 150 with an increment of 10, the dominant dc-link current control eigenvalue (λ ϕ idc ) moves toward the left side of the s-plane. However, if ω c increases from 50 to 650 rad/s with an increment of 100 rad/s for K c = 150, λ ϕ idc moves toward the unstable region. Therefore, K c = 150 and ω c = 550 rad/s are selected to ensure rated power injection, where ε c = 1 is selected.

3) DESIGN OF y d (s)
Finally, in addition to y q (s) and y dcsc (s), a scaled version of v c sd is injected in the CSC q-axis current control to add additional disturbance rejection capability, and its effect is shown in Fig. 11(c). If K d increases from 0.1 to 0.6, with ω d = 50 rad/s, λ Idc moves further to the left side of the s-plane when SCR = 1 compared to λ Idc shown in Fig. 11(a). However, if ω d increases, λ Idc moves toward the right. Therefore, K d = 0.5 and ω d = 50 rad/s are selected, significantly enhancing the system's damping.

4) DESIGN OF x dvsc (s)
In this step, the compensator parameters related to the VSC are determined to ensure reduced dynamic interactions among the VSC and CSC systems for SCR = 1. The locus of the dc-link voltage control eigenvalue of the VSC (λ V dc ) is shown in Fig. 11(d). In addition to the previously injected three signals, a scaled version of the PLL output is injected in the d-channel VSC structure. As shown in Fig. 11(d), if K f increases from 2 to 10 for ω f = 100 rad/s, λ V dc is relocated from −1.94 ± j4.33 to −2.15 ± j4.20, increasing the damping from 0.4085 to 0.4556. However, if ω f increases, the damping remains the same, but the compensated eigenvalues move further to the left side of the s-plane, which verifies the stability improvement with the proposed compensator for a weak grid system. Fig. 12(a) and (b) show the effect of the proposed compensator on the frequency response of G codcsc and G codvsc , respectively, when SCR = 1. As shown in Fig. 12(a), compared to the uncompensated case, the proposed compensator reduces the magnitude of G codcsc in the low-frequency range, which reduces changes in the CSC terminal voltage and interactions with the dc-link current control. However, the phase slightly decreases in the low-frequency and then remains almost identical. Therefore, as the CSC dc-link and PCC interaction decrease, the interaction of the PCC with the VSC terminal voltage decreases, resulting in reduced interactions with the dc-link voltage control. This is verified through the reduction in the magnitude of G codvsc , as shown in Fig. 12(b). Therefore, the proposed compensator adds damping to the system, reduces interactions among the VSC and CSC, increases the dc-link control stability, and injects rated power when connected to a weak grid. The results verify the compensator parameters design and the eigenvalue analysis shown in Fig. 11.

VI. SIMULATION RESULTS
Time-domain simulation results are performed in the MAT-LAB/Simulink environment for the PV-based parallel converters system connected to a weak grid, shown in Fig. 1. The simulation parameters are listed in Appendix A. A discretetype simulation is used with a sampling time of 20 μs. Simulation results are used to verify the analytical results and the effectiveness of the proposed compensation method for the parallel VSC-CSC system. The following simulation scenarios are considered: 1) Operation under changing grid strength 2) Impact of the interconnection impedances 3) Operation under varying VSC operating points 4) Operation under varying CSC operating points  Fig. 13 shows the dc-links (I dc , V dc ) and power injected into the grid (P grid ) responses under changing SCR, where the solid blue and black lines represent the responses with SCR = 4 and 1, respectively. Fig. 13(a) shows that if SCR decreases from 4 to 1, the low-frequency oscillation increases in I dc , and becomes unstable, affecting V dc stability, as shown in Fig. 13(b). The oscillations and instability in the dc-links are reflected in the P grid response when SCR = 1, as shown in Fig. 13(c). However, it remains stable when SCR = 4 and the rated power (2.0 MW from the VSC and 1.0 MW from the CSC) is injected, as shown in Fig. 13(a)-(c). Therefore, it is clear that the grid strength affects the dc-links stability in parallel converters by affecting the PCC voltage, verifying the accuracy of the frequency domain and the eigenvalue analyses shown in Section IV.

B. IMPACT OF THE INTERCONNECTION IMPEDANCES
As shown in Fig. 1, the VSC and CSC are connected to the PCC by feeders having inductances L 1 and L 2 , respectively. Fig. 14 shows the impact of the feeders on dc-link control stability when SCR = 4. When the feeders connect the VSC and CSC to the PCC has the same length (L 1 = L 2 = 10 μH), and the dc-link current and voltage remain stable, as shown in Fig. 14(a) and (b), respectively. With L 1 = 20 μH and L 2 = 10 μH, I dc and V dc responses remain stable, as shown in Fig. 14(c) and (d), even though the feeders connecting the PCC have different lengths.

C. OPERATION UNDER VARYING VSC OPERATING POINTS
This case study demonstrates the effects of changing VSC operating points on I dc , V dc , and P grid . At t = 0.5 s, the solar radiation (G) is increased from 0.5 to 0.75 kW/m 2 and from 0.75 to 1 kW/m 2 at t = 1.5 s. As shown in Fig. 15(a)-(c), I dc and V dc have well-damped stable responses, and the rated power is injected into the grid when SCR = 4. However, if SCR = 1, the system remains stable till G = 0.75 kW/m 2 .
Then, an increase in G yields oscillations in I dc and V dc . Both become unstable, affecting the active power injection capability, as shown in Fig. 15(a)-(c). The results match the frequency-domain analysis shown in Fig. 5 and demonstrate that changes in one converter's operating point affect the PCC voltage, which impacts the stability of the other converter and overall parallel system power injection levels under a weak grid.

D. OPERATION UNDER VARYING CSC OPERATING POINTS
The effect of changing the PVG operating point of the CSC (i.e., changes in the reference dc-link current, I * dc ) on I dc , V pccc , and three-phase VSC output current (I abc ) responses are shown in Fig. 16(a), (b), and (c), respectively, when the SCR changes from 4 to 1. In Fig. 16(b) and (c), the dotted lines represent the response with SCR = 4, and the solid lines represent the SCR = 1 case. As shown in Fig. 16(a), with SCR = 4, I dc experiences an overshoot of 1.08 p.u. for a duration of 0.06 s upon increasing reference current; however, the dc-link current remains stable, resulting in a balanced PCC voltage, as shown in Fig. 16(b). The threephase VSC reactor current remains stable, as depicted in Fig. 16(c). However, with SCR = 1, I dc remains stable until the operating point changes. Once the operating point is changed, the overshoot reaches 1.12 p.u., oscillations are yielded, and it becomes unstable with high-frequency oscillations. The oscillations in I dc change the PCC voltage, which becomes distorted, as shown in Fig. 16(b). Furthermore, the coupling nature impacts I abc , as shown in Fig. 16(c). Therefore, the PVG source dynamics in the overall system stability study are essential for accurate dynamic interaction results.

E. OPERATION UNDER VARYING PLL BANDWIDTH OF THE CSC
The effect of the CSC PLL BW (i.e., from 5 Hz to 40 Hz) is shown in Fig. 17 when SCR = 1. If the BW of the CSC PLL (i.e., 5 Hz) is lower than that of the VSC (i.e., 10 Hz), I dc in Fig. 17(a) and V dc in Fig. 17(b) show a highly damped response. When both BWs are equal, I dc suffers from oscillations, transferred to V dc due to the coupled network. However, if it exceeds the BW of VSC PLL (i.e., 20 Hz and 30 Hz), there are more oscillations in I dc and V dc , resulting in instability in the dc-link operation. Therefore, the BW of the CSC PLL, which is lower or close to that of the VSC, ensures enhanced systems performance.

F. OPERATION WITH THE PROPOSED COMPENSATOR
The operation of the parallel VSC-CSC system at SCR = 1, with the proposed compensator in (18), is shown in Fig. 18. The compensated I dc response experiences a moderate undershoot at t = 4 s but reaches a steady state in 0.3 s, as shown in Fig. 18(a). The effect of the highly damped I dc response is reflected in the V dc response, as shown in Fig. 18(b). The stability of I dc and V dc is improved compared to the uncompensated responses, shown in Fig. 13(a) and (b). The compensated system remains stable, and 1.0 p.u. active power is injected into the grid, as shown in Fig. 18(c), demonstrating the proposed compensator's effectiveness under very weak grid conditions.

G. OPERATION UNDER FAULT
The performance of the proposed compensator is tested under a phase-to-ground fault at the secondary side of the transformer. The fault occurs at t = 1.0 s with a duration of 0.06 s at SCR = 1.3, and its influence on the dc-link stability is shown in Fig. 19. As shown in Fig. 19(a), the uncompensated I dc suffers from overshoot and undershoots of 1.16 and 0.89 p.u., respectively, after the fault clearance. However, the proposed compensator reduces the overshoot to 1.01 p.u. and has no undershoot, demonstrating a highly damped response after fault clearance. During the fault, voltage loss is reflected in the PCC voltage, which is also affected by the grid disturbance, as shown in Fig. 19(b). In addition, the disturbance and loss of PCC voltage affect V dc , as shown in Fig. 19(c). As a result, the uncompensated responses suffer from undershooting after the fault clearance and do not reach a steady state. However, the compensated V dc reaches the steady state without undershooting, demonstrating the proposed compensator's effectiveness under disturbances caused by grid faults.

H. OPERATION UNDER GRID-VOLTAGE PHASE ANGLE SHIFT
The accuracy of the proposed compensator is tested under a 30 0 phase angle shift in grid-voltage phase A between t = 0.65 s and t = 1.15 s, and its influence on the dc-link control stability of the parallel converters is shown in Fig. 20. As shown in Fig. 20(a), the uncompensated I dc suffers an overshoot of 1.06 p.u. and an undershoot of 0.98 p.u. and 0.94 p.u. between t = 0.7-0.86 s and t = 1.16 s, respectively. However, the compensated I dc shows a highly damped response, reducing the oscillations. The uncompensated V dc in Fig. 20(b) suffers from increased oscillations between t = 0.65 s and t = 1.15 s. However, the compensated V dc is faster. This case study demonstrates that the grid disturbances affect the PCC voltage, which impacts each converter's terminal characteristics and further affects the dc-link operation of both converters due to direct coupling.

I. OPERATION UNDER GRID-VOLTAGE FREQUENCY CHANGE
The influence of the proposed compensator is tested for varying the grid voltage frequency by 2 Hz between t = 0.65 and 0.80 s with SCR = 1.2. The uncompensated I dc suffers from overshoot/undershoot for the frequency change, takes longer to reach the steady state, and oscillates compared to the compensated case, as shown in Fig. 21(a). The uncompensated V dc in Fig. 21(b) takes 1.5 s to reach the steady state and oscillates. However, the compensated response demonstrates a highly damped system and reaches a steady state faster. Therefore, the impact of grid frequency disturbances on parallel converters' operation is essential to analyze the dc-link dynamics properly.
The presented case studies showed that the proposed compensator improves the parallel system performance and reduces the interactions under several grid disturbances.

VII. VALIDATION RESULTS
The interaction between the dc-links of a parallel VSC-CSC system and the effectiveness of the proposed compensation method are tested in real-time using the OPAL-RT OP5600  platform. The setup shown in Fig. 22 uses the OPAL-RT OP5600 real-time simulation platform [38]. The platform is fully integrated with MATLAB/Simulink and uses a Virtex-6 FPGA board with a time step of 290 ns for rapid control prototyping applications. A 3.0 MW parallel VSC-CSC PV system is simulated in real-time with a sampling time of 20 μs. The proposed control system is implemented in real-time to assess its real-time performance and implementation aspects. The results are accessed through the I/O interface units and displayed on a 500 MHz oscilloscope. In addition, the host computer connected to the real-time simulator provides input data (i.e., solar radiation and changes in the operating points. Fig. 23 shows the I dc and V dc real-time simulation responses when the SCR changes from 4 to 1. As shown in Fig. 23, the dc-link responses are stable for SCR = 4. However, if the SCR decreases to 1, I dc oscillates and becomes unstable. The instability in I dc affects the CSC output power injection, which affects the PCC voltage. Due to coupling through the PCC, V dc becomes unstable with SCR = 1, as shown in Fig. 23, which verifies the time-domain simulation results shown in Fig. 13. The effect of changing the CSC operating point on I dc and V dc responses with SCR = 1 is shown in Fig. 24, where G is varied from 0.75 to 1.0 kW/m 2 . The dc-link responses are stable with G = 0.75 kW/m 2 . Then,  an increase in G yields oscillations in I dc and becomes unstable, which affects the CSC terminal and PCC voltage. Furthermore, it affects the VSC terminal voltage and results in an unstable V dc response. This validates the time-domain simulation results and verifies that instability in one converter leads to stability problems in another with increased dynamic interactions through the coupled network. In the next test, the effects of the CSC PVG source dynamics on I dc and VSC three-phase output current (I abc ) are evaluated under changing SCR and are shown in Fig. 25. As shown in Fig. 25(a), the system remains stable with SCR = 4. However, if the SCR reduces to 1, and the operating point of the PVG moves to the CCR, I dc oscillates and becomes unstable. Furthermore, the instability in the CSC dc-link current affects the stability of the VSC output current (I abc ), which becomes unstable, as shown in Fig. 25(b). Therefore, it is clear that disturbances from the PVG affect the converters' terminal and PCC voltage. Therefore, parallel VSC-CSC systems interaction study considering the PVG source dynamic is essential.
A single-line-to-ground fault is introduced at the secondary side of the transformer. Its effect on interactions between the converters' dc-links is shown in Fig. 26. As shown in Fig. 26(a), during the fault interval, there is a voltage loss in phase A, which verifies the simulation results shown in Fig. 19(b). In addition, there is an undershoot, overshoot, and oscillation in I dc during the fault interval, as shown in Fig. 26(b). The variations in I dc affect V dc , which also suffers  from under/over-shooting during the fault interval. However, once the fault is cleared, damped responses are reflected.
The effect of the proposed compensator when the CSC operating point changes with SCR = 1 is shown in Fig. 27. The increase of G from 0.75 to 1.0 kW/m 2 for the PV-CSC system results in a highly damped stable I dc response. As a  result, V dc remains stable even if the CSC operating point varies at a very weak grid condition. Compared to the unstable I dc and V dc responses shown in Fig. 24 with SCR = 1, the proposed compensated parallel VSC-CSC yields a highly damped stable response, demonstrating the validity of the proposed compensator in a weak grid.
The impact of the proposed compensator with changing SCR is shown in Fig. 28. As shown in Fig. 28, if the SCR changes from 4 to 1, I dc experiences a slight undershooting. However, it overcomes the undershooting, reaches a steady state, and remains stable compared to the uncompensated I dc response, shown in Fig. 23. Furthermore, V dc remains stable, verifying the proposed compensator's effectiveness at very weak grid conditions.

VIII. CONCLUSION
This article has analyzed the interaction dynamics and stabilization of parallel utility-scale PV-VSC and PV-CSC systems connected to a weak grid. A detailed small-signal model of the parallel system is developed and used for comprehensive frequency-domain and eigenvalue analyses under different operating conditions. The impacts of the PV source dynamics, SCR, and changing operating conditions on parallel converters' stability and interactions are evaluated. Furthermore, a compensator has been proposed to reduce undesirable interactions, maintain stability, and facilitate rated active power injection at very weak grid conditions (SCR = 1). Detailed nonlinear time-domain simulations and real-time simulation results verify the analytical findings and the effectiveness of the proposed compensator in stabilizing the parallel VSC-CSC system under various operating conditions. The key findings are summarized below.
1) Changes in the PV source dynamics significantly affect the VSC or CSC stability under weak grid conditions. Therefore, PV weak-grid integration studies should consider the PV source dynamics. 2) The resonance peak of the input impedance of the VSC shifts to a lower frequency when the SCR decreases. However, it moves to a higher frequency for the CSC, reducing the dc-links stability margins for both converters.
3) The magnitude of the output impedances of the VSC ( Z ddvsc and Z qqvsc ) increases, and that of the CSC ( Z ddcsc ) decreases with reduced SCR, resulting in increased dynamic interactions via the PCC voltage. 4) The BWs of the converters' PLLs play a significant role in the overall system stability. For example, in the system understudy, the parallel system becomes unstable if the BW of the CSC PLL exceeds that of the VSC. 5) The proposed compensator reduces the interaction between the VSC and CSC and maintains overall system stability and rated power injection at SCR = 1. Additionally, it offers better damping under faults and grid parameter variation (phase angle and frequency shifts). Strong Grid: 36 MVA (rated), 12.47 kV (ph-ph rms), 60 Hz, X/R = 7.0.