Assessment and Mitigation of DC Breaker Impacts on VSC-MTDC Grid Equipped With Power Flow Controller

DC circuit breakers are required to interrupt dc faults in multi-terminal HVDC (MTDC) grids. Consequently, their current limiting inductors can substantially impact the dynamic performance and stability of MTDC grids. However, the literature does not address the assessment and mitigation of dc breaker inductance impacts on the stability of an MTDC grid equipped with a power flow controller (PFC). To fill in this gap, this paper evaluates the effects of dc breaker inductance on the stability of a PFC-equipped MTDC grid. Furthermore, the paper expands the capability of the PFC via the development of a PFC-based damping controller to improve the dynamic performance of the MTDC grid and enhance its damping. First, a comprehensive small-signal model of the PFC-equipped MTDC grid is developed. Then, eigenvalue and frequency response analyses are employed to assess the dynamic performance and stability of the MTDC grid, considering the effects of breakers inductances, PFC and converter station control parameters, and network parameter uncertainty. Finally, time-domain simulations and hardware-in-the-loop real-time simulations are carried out to evaluate the performance of the proposed damping controller and verify the theoretical analysis.


I. INTRODUCTION
In recent years, the deployment of high-voltage direct current (HVDC) links based on voltage-source converter (VSC) technology has increased significantly to enable the transmission of bulk power and facilitate the grid-interconnection of renewable energy resources [1], [2]. In light of the growing number of HVDC link installations, the multi-terminal HVDC (MTDC) grid concept is proposed to enhance the reliability and security of the dc grid [3]. An MTDC grid consists of multiple VSCs linked through a network of transmission lines which continues to function under contingency. In Europe, a large-scale MTDC grid called "SuperGrid" is under implementation to interconnect the European countries [4]. However, the MTDC grid concept raises several technical challenges related to the grid's control, operation, and protection.
To assure the safety of an MTDC grid, rapid isolation of dc faults is crucial. However, the MTDC grid's low impedance and the absence of the current zero crossing complicate the fault interruption task [5]. In point-to-point HVDC links, fault clearance is carried out by the ac-side breakers. This practice is not feasible for large-scale MTDC grids as it causes the disconnection of the whole system. Thus, dc circuit breakers (DCCBs) have been developed to interrupt the faulty dc lines. In the literature, various DCCB structures have been presented and are categorized as mechanical DCCB, solid-state DCCB, and hybrid DCCB (HCB) [6], [7]. Among these types, the HCB is more favorable as it incorporates the merits of the mechanical and solid-state DCCBs, offering a fast operation speed and low power loss [8]. The DCCBs rely on large current-limiting inductors to suppress fault current increase. These inductors have values in the hundreds of microhenries range. For example, a 150 mH DCCB inductor is used for a 500 kV dc system [9]. The deployment of large inductors across the MTDC grid can significantly impact its stability and dynamic performance. In [10], it was shown that DCCB inductors significantly degrade the controllability of the MTDC grid and might cause unwanted dc voltage oscillations and, eventually, instability.
Recently, power flow controllers (PFCs) have been proposed to enable power flow management and mitigate potential transmission bottlenecks in MTDC grids; they are classified as resistive PFCs (R-PFCs) and capacitive PFCs (C-PFCs) [11], [12]. The former regulates the power flow by connecting a variable series resistance to the dc line. In addition to the significant losses of the R-PFCs due to the power dissipated in the series resistor, this type cannot achieve power reversal as it only adjusts the power flow in one direction. These problems can be addressed using C-PFCs which rely on inserting an adjustable dc voltage into the dc line. C-PFCs can be classified into parallel-connected and series-connected PFCs [13], [14]. Series-connected PFCs are more favorable due to their compact size and lower cost compared to parallel PFCs, which must sustain the rated MTDC grid voltage. The interline PFC is one of the most efficient PFCs topologies, which controls the power flow by facilitating power transfer among dc transmission lines [15]. Multiport topologies are developed to regulate current flow among multiple dc lines, as in [11], [16]. Studies [9], [17] proposed an HCB that combines the functionality of the PFC and DCCB into a single device to save cost and footprint.
In addition to current flow control, PFCs are anticipated to provide additional functions to the MTDC grid, such as oscillation mitigation and stability enhancement [18]. Nevertheless, few works investigated the prospective functionalities of PFCs other than power flow management. In [19], the damping capability of interline PFC to mitigate oscillations caused by resonance in the dc grid was assessed. Active damping controllers incorporated with the PFC control system were proposed, and their damping performance was compared. In [20], the functionality of the PFC was extended to include power oscillations suppression caused by ac-side faults. A damping controller was developed based on a proportional-resonant compensator integrated with the PFC control structure. Oscillation damping in a dc grid was attempted through individual VSC control [21], [22]; this approach increases the control burden and might not ensure sufficient damping. A PFC, by contrast, has a faster response and ability to control the line currents. Therefore, in addition to fully utilizing PFC capabilities, a PFC-based damping controller can release control stress on VSCs and achieve improved damping performance. However, the literature has not reported a PFC-based damping control approach to mitigate low-frequency oscillations caused by the deployment of DCCB inductors. This paper comprehensively analyzes the impacts of DCCB inductors on the dynamic performance and stability of a PFCequipped MTDC grid. Moreover, a PFC-based dc system stabilizer is developed to mitigate low-frequency dc current and voltage oscillations to extend the PFC functionality. First, a comprehensive small-signal model of a PFC-equipped MTDC grid is developed. Then, using the derived smallsignal model, the system stability and dynamic performance are assessed using eigenvalue and frequency response analyses, considering the effects of DCCB inductances, PFC and converter station control parameters, and network parameter uncertainty. Finally, time-domain simulations and hardwarein-the-loop tests were carried out to evaluate the performance of the proposed damping controller and verify the theoretical analysis. The main contributions of this paper include the following: r Analyzing the impacts of DCCB inductors on the dynamic performance and stability of a PFC-equipped MTDC grid using a comprehensive small-signal model, r Developing a PFC-based dc system stabilizer (PFCS) to improve the dynamic performance of the MTDC grid with DCCB inductors and suppress dc voltage and current oscillations, and r Assessing the damping capability of the proposed PFCS and comparing its performance with the interline PFCbased damping controller developed in [19]. The rest of this paper is organized as follows. Mathematical modeling of the MTDC grid is presented in Section II. Section III discusses the stability of the MTDC grid. The stability enhancement of the MTDC grid is presented in Section IV. Simulation results and discussions are given in Section V. In Section VI, hardware-in-the-loop real-time simulation studies are presented. Finally, conclusions are summarized in Section VII.

II. SYSTEM MODELING
A schematic diagram of a four-terminal dc grid equipped with a PFC is shown in Fig. 1. Practical converter and network parameters are used [26], [27] and given in the Appendix. The grid is comprised of five transmission lines, with a dc breaker inductor (L dc ) is connected at each end of the lines to limit the increase of fault current. As shown in Fig. 1(b), each VSC is interfaced with an ac system via an LC filter (R f − L f − C f ) to mitigate the current and voltage harmonics generated from the switching action of the VSC. The strength of the ac system is modeled by a stiff voltage (V g ) and a series grid impedance (R g + jωL g ). On the dc side, each VSC is connected to the dc grid through a dc-link capacitor ( C dc ).

A. AC SYSTEM MODEL
The dynamics of the ac system can be represented in the ac grid d-q reference frame, where the d-axis is aligned with the ac grid voltage (V g ). The ac currents and voltages of the VSC can be expressed as where i sdq and i gdq are the d-q components of the ac filter and grid current, respectively. v tdq , v sdq and v gdq are the d-q components of the VSC terminal voltage, point of common coupling (PCC) voltage, and ac grid voltage, respectively. Fig. 2 illustrates the control system of the VSC. The control structure is composed of two control loops: outer and inner control. Two outer control loops are used to adjust the VSC dc voltage and PCC ac voltage, whereas the inner control loop tracks the d-q ac current references.

1) OUTER DC-LINK VOLTAGE CONTROL
Voltage droop control is adopted to regulate the MTDC grid voltage. The dc-link voltage V dc is compared to the reference value V * dc and processed through a droop gain k d to generate the reference power P * s . A proportional-integral (PI) compensator, G p (s) is used to regulate the active power P s to the reference value by generating the reference current i * sd as

2) OUTER PCC VOLTAGE CONTROL
A PI compensator, G ac (s), is used to adjust the PCC voltage by generating the reference current i * sq as Quantities in the converter d-q reference frame are signified using the superscript "c."

3) INNER CURRENT CONTROL
Two PI compensators, G c (s), are utilized to adjust the d-q components of the converter currents. The current control can be represented in the converter frame as (9) where m c dq are duty ratios of the VSC.

4) PHASE LOCKED LOOP (PLL) CONTROL
The PLL is required to synchronous the VSC with the ac system frame. It uses a PI compensator, G pll (s), to extract the PCC voltage angle (δ) as The angle δ is used to transform variables between the ac grid d-q reference frame and converter frame based on where f dq and f c dq represent electrical variables in the ac system and converter reference frame, respectively.
The rate of change of the energy in the VSC dc-link capacitor is governed by the power balance between the power injection of the VSC into the ac system and the power exchanged between the dc-link capacitor and the dc grid. Therefore, the dynamics of the VSC capacitor voltage can be formulated as where P s is the real power that the VSC exchanges with the ac system, and it can be described as The dynamics of the four VSCs can be derived based on (4)- (12) and are combined as C. PFC MODEL Fig. 3(a) shows the topology of the PFC, which has been installed between lines L12 and L14 to control their current flow. The PFC comprises three half-bridges (HBs) and a partially power-rated auxiliary dc/dc converter based on an input-series output parallel (ISOP) topology. The auxiliary converter generates the PFC internal voltage (V p f c ), while the half-bridges are controlled to regulate the line currents by inserting a series dc voltage into each line based on the duty ratio of each HB. Unlike the interline type PFC, where the state of line currents impacts the PFC voltage, this topology provides a decoupled PFC voltage by using an auxiliary dc/dc converter. A detailed analysis of the PFC and its operating modes can be found in [23]. For the current configuration depicted in Fig. 3(a), keeping the switch Q b in the on-state and adjusting the on/off state of switches S 1a and S 2a , the dc voltages inserted by the PFC into the transmission lines can be expressed as where V L12 and V L14 are the dc voltages applied to lines L12 and L14 by the PFC, respectively. D 1 and D 2 are the duty ratios of HB1 and HB2, respectively. Fig. 3(b) illustrates the schematic diagram of the PFC control structure. HB0 is controlled based on the direction of the total current I T . If I T is positive, the lower switch Q b will be in the on state, while for a negative I T , the upper switch Q a will be in the on-state. In the case of n transmission lines, it is possible to regulate the current flow in all of them by only actively controlling n − 1 lines. One of the lines can be designated as a slack line, and the switches in its connected half-bridge only need to function at a constant duty ratio. Based on Fig. 3(b), line L12 is selected as a slack line; thus, its half-bridge HB1 is operated at a constant duty ratio. To actively regulate I 14 , HB2 is controlled using a PI controller, as shown in Fig. 3(b). The line current I 14 is compared to the reference value I * 14 , and the error is processed by a PI compensator, G p f c (s), to generate the duty ratio D 2 . The dynamics of the PFC control can be expressed as

D. DC NETWORK MODEL
The equivalent circuit of the dc network is depicted in Fig. 4.
The T-type model is utilized to represent the dc transmission lines.
The T-type model is preferable for an MTDC grid equipped with PFC because it is more accurate for modeling and analysis than the π -type model [24]. This is because, in the case of the π -model, the switching action of the PFC appears on the π -model capacitors. But, for a T-model, the inductance can mitigate the switching effect of the PFC. Based on Fig. 4, the dynamics of the transmission line between Similarly, the dynamics of other lines can be derived based on Fig. 4 and are combined as where To assess the dynamic performance and stability of the entire system, (1)- (20) are linearized, and the augmented linearized state-space model of the MTDC grid is obtained as shown in (21) and (22), where the details of system matrixes are omitted due to space limitations.

III. STABILITY ASSESSMENT A. MODAL ANALYSIS
Based on the developed small-signal model (21), the eigenvalues of the system matrix A are analyzed under the rated operating conditions of VSCs, as given in the Appendix. The matrix A comprises 68 eigenvalues, all with negative real parts indicating a stable system. The identified eigenvalues consist of 28 real eigenvalues and 20 pairs of complex-conjugate eigenvalues. Table 1 provides the 20 oscillatory modes of the system. The eigenmodes λ 6 , λ 7 and λ 8 have relatively low frequency and poor damping. Therefore, these eigenmodes dominate the system's transient response, and any attempt to enhance the system's damping must consider these modes. The participation factor analysis indicates that the system dc voltages are the primary contributing states to these modes. Moreover, Table 1 shows that the eigenmodes λ 1 to λ 5 and λ 9 to λ 16 correspond to fast transients and are mainly affected by the dc transmission lines and VSC ac-side parameters, respectively. Fig. 5 compares the influence of the PFC's proportional gain, k p , on the system eigenvalues with and without including the DCCB inductance. k p is varied from 0 to 0.505, while the integral gain, k i , is fixed at 0.127. As Fig. 5 shows, the dominant eigenvalues migrate toward the LHP as k p increases from 0 to 0.006 (without DCCB inductors), indicating improved system stability. Further increase in k p shifts the dominant eigenvalues toward the RHP, indicating a reduced stability margin. Fig. 5 also shows that with the variation of k p , the system remains stable with and without the DCCB inductance.  However, the stability margin is greatly reduced when the DCCB inductance is considered. The influence of the PFC's integral gain k i on the system eigenvalues is depicted in Fig. 6, whereas k i is varied from 0 to 22. 5, while k p is fixed at 0.006. As k i increases from 0 to 0.156, the dominant eigenvalues move toward the LHP, indicating improved system damping. When k i >0.156, the dominant modes migrate toward the RHP. Based on Figs. 5 and 6, it can be concluded that the small-signal stability of the system is maintained under the variation of the PFC control parameters. However, introducing the DCCB inductance reduces the system stability margin greatly. Moreover, higher PFC proportional and integral gains are needed to achieve satisfactory damping of the dominant eigenvalues when DCCB inductors are considered. Fig. 7 illustrates the trajectories of the dominant eigenvalues, as the reference power of VSC1 is varied from 400 MW to −400 MW for two different settings of the DCCB inductance. It is assumed that VSC2 injects 100 MW into the dc grid, whereas VSC2 and VSC4 are operating in the droop voltage control mode. As Fig. 7 shows, irrespective of the DCCB inductance, the MTDC grid and its dynamics are sensitive to power flow variation. As VSC1's rectifying power rises, the dominant eigenvalues migrate toward the RHP. This study demonstrates the well-known robustness and destabilizing issue of the so-called constant-power loads. However, a larger dc grid inductance significantly amplifies the problem as the system poles become more sensitive to power variation. Fig. 7 also indicates that the maximum amount of power that  VSC1 can exchange with the ac system in the rectification mode before the system becomes unstable decreases as the DCCB inductance increases. Fig. 8 shows the dominant eigenvalues loci when the droop gain is increased from 0.15 to 5 (kW/V) for two settings of the DCCB inductance. From Fig. 8, as the inductance increases, the dominant eigenvalue moves toward the RHP, and a high droop gain may be required to achieve a stable system. This analysis implies that the dc system inductance imposes a lower bound on the droop control gain. For a higher dc inductor size (i.e., 200 mH), the damping of the dominant eigenvalues will be significantly low, and achieving a stable closed-loop system with satisfactory performance will be very challenging. Studies [10], [25] also reported this low gain instability issue imposed by larger dc inductance. Fig. 9 illustrates the Bode diagram of the transfer function (TF) V dc1 /P * 1 considering three settings of the DCCB inductor size. As the system inductance increases, the resonance peak becomes larger at a lower frequency. A high resonance magnitude of the frequency response implies large and more severe dc voltage oscillations under power disturbances in the system. The simulation result presented in Fig. 10 demonstrates the frequency response analysis. In Fig. 10, the response of the voltage V dc1 is shown when P * 1 is increased by 50 MW  at t = 2 s. It is evident that a larger dc inductor size causes a higher poorly damped dc voltage oscillation and significantly deteriorates the dc voltage dynamic response. The dynamic response of the dc voltages in an MTDC grid may differ based on whether the associated VSC performs active power control or droop voltage control. Fig. 11 illustrates the Bode plot of the TFs relating the dc grid voltages and the reference power of VSC1 considering a dc inductor of 100 mH. VSC2 and VSC4 use droop voltage control, while VSC1 and VSC3 perform active power control. As shown in Fig. 11, the transfer functions related to VSC1 and VSC3 demonstrate significantly larger resonance peaks than those associated with VSC2 and VSC4. This analysis shows that VSCs performing active power control would have poorly damped dc-link voltages, indicating a poor dynamic performance under power disturbances. This performance and robustness issue of the dc voltages associated with VSCs operating in the power control mode exists even without considering the dc breaker inductance. However, a larger system inductance significantly worsens this problem. The transient simulation depicted in Fig. 12 affirms the frequency response analysis. Fig. 12 shows the response of the dc grid voltages when P * 1 is increased by 50 MW at t = 2 s. The responses of the voltages V dc1 and V dc3 experience higher and lightly damped oscillations compared to those of V dc2 and V dc4 , which have a significantly damped response.

IV. STABILITY ENHANCEMENT
A dc damping scheme to enhance the dynamic response and stability of the MTDC grid called a PFC-based stabilizer (PFCS), is proposed in this paper. The PFCS is an internal model-based active compensator that modifies the PFC current control using a supplementary damping signal. As depicted in Fig. 13, the PFCS is composed of a high-pass filter and a stabilizer gain. The PFCS injects a damping signal which modifies the PFC reference current. A PFC equipped with the proposed compensator functions like a regular PFC  to regulate dc currents. During disturbances, the PFCS injects a damping signal to reject disturbances in the dc grid. The transfer function of the PFCS is where k c is the stabilizer gain, and ω c is the cut-off frequency. Eigenvalue analysis has been utilized to evaluate the performance and damping capabilities of the proposed PFCS, cross-verified with frequency response analysis. The parameterization for the PFCS is carried out by modifying the stabilizer gain k c and cut-off frequency ω c to realize sufficient damping and minimize the dynamic interaction of the compensator with the current controller of the PFC. The system small-signal model in (21) is modified to include the dynamics of the PFCS based on (23). Fig. 14 shows the impact of altering the PFCS gain and cut-off frequency on the eigenvalues of the dc system. In Fig. 14, for the three cut-off frequency settings, increasing the PFCS gain causes the poorly damped modes of the system to migrate toward the LHP, implying improved damping and stability margin. Moreover, the analysis shows that at a lower cut-off frequency, the dominant modes move further toward the LHP; hence, better damping can be achieved. However, A PFCS with a lower cut-off frequency or a much higher gain may yield undesired dynamic coupling with the current controller of the PFC and deteriorate its current tracking capability. Fig. 15 compares the Bode plots of the PFC current tracking closed loop TF under the three settings of cut-off frequency and considering a PFCS gain of k c = 6.5.  The bandwidth of the PFC current controller is 30 rad/s. In Fig. 15, as the cut-off frequency decreases, the bandwidth of the PFC current loop is reduced. For cut-off frequencies of 480, 240, and 130 rad/s, the bandwidth is reduced to 23, 17, and 10 rad/s, respectively. Therefore, there is a tradeoff between the damping performance of the PFCS and dynamic coupling with the PFC current controller when designing the parameters of the damping controller. For a PFCS with ω c = 240 rad/s and k c = 6.5, Table 2 demonstrates the damping enhancement of the system's poorly damped modes. Frequency response analysis is conducted to assess the proposed PFCS's performance further. A PFCS with gain k c = 6.5, and a cut-off frequency of 240 rad/s is considered for this analysis. Fig. 16(a) and (b) show the Bode plots of the transfer functions V dc1 /P * 3 and V dc1 /P * 1 , respectively, when the proposed PFCS is adopted. As depicted in Fig. 16(a), the PFCS mitigates the resonance of the transfer function V dc1 /P * 3 , and significantly improves its damping. This result demonstrates the effectiveness of the PFCS in rejecting disturbances coming from other terminals to affect the dc voltage of VSC1. This can be explained as follows. As dc grid disturbances propagate through the transmission lines, a PFCS implemented at a certain VSC will damp oscillations in the line currents going into the VSC dc-link capacitor. Hence, improving the dc-link voltage dynamic response. In Fig. 16(b), although the PFCS considerably reduced the resonance peak of the TF V dc1 /P * 1 , the damping improvement is slightly limited. This is because when P * 1 is changed, V dc1 will be disturbed, and the PFCS will not act until the disturbance is reflected on the transmission line currents. Based on Fig. 16, it can be concluded that a PFCS applied at a certain VSC station can effectively reject disturbances coming from other terminals to affect the dc voltage of that VSC. Moreover, it considerably reduces the impact of disturbances on the dc voltage dynamic response of that VSC (at which the PFCS is installed), as depicted in Fig. 16(b).
The Bode plots of the TFs V dc3 /P * 1 and V dc3 /P * 3 , with the PFCS adopted, is shown in Fig. 17(a) and (b), respectively. Fig. 17(a) illustrates the capability of the PFCS to mitigate the resonance of the transfer function V dc3 /P * 1 , which implies the effectiveness of the PFCS in rejecting disturbances coming from VSC1 to affect the dc voltage of VSC3. In Fig. 17(b), although the PFCS reduced the transfer function V dc3 /P * 3 resonance amplitude, its damping improvement is limited because of the considerable electrical distance between the installed PFC and VSC3. Based on the frequency response analysis of Figs. 16 and 17, it can be concluded that the PFCS is significantly effective in rejecting disturbances coming from converters (VSC2, 3, and 4) to affect the dc voltage of VSC1.  Meanwhile, it can substantially reject disturbances coming from VSC1 to affect other dc voltage terminals.

V. SIMULATION RESULTS AND DISCUSSION
Time-domain simulation studies are carried out to assess the effectiveness of the proposed PFCS. The MATLAB/Simulink is used to implement the meshed MTDC grid shown in Fig. 1. Droop voltage control is adopted in VSC2 and VSC4, while VSC1 and VSC3 perform active power control. The complete system and controller parameters are given in the Appendix. The following simulation scenarios are considered:  r Operation under fault conditions, and r PFCS performance in a 5-terminal HVDC grid using a 4-port PFC.

A. PFCS PERFORMANCE WITHOUT DCCB INDUCTANCE
Without including the dc breaker inductance, the responses of the selected dc currents and voltages are depicted in Figs. 18 and 19, respectively, with and without the proposed damping controller. At first, the dc system operates in a steady state, with VSC1 and VSC3 delivering 350 and −100 MW to the MTDC grid, respectively. Meanwhile, the PFC adjusts the current I 14 to a reference value of 500 A. VSC3's power injection stepped to −200 MW at t = 3 s, then reduced to −100 MW at t = 4 s. The power delivered by VSC1 raised to 400 MW at t = 3.5 s. As shown in Fig. 18, the PFC keeps the current I 14 regulated at its setpoint for the uncompensated and compensated responses. However, as it can be seen at t = 3.5 s, the proposed damping controller slightly slows the response of the PFC current controller, which meets the frequency response analysis depicted in Fig. 15. The results show that even without including the DCCB inductance, the dc grid currents and voltages exhibit poorly damped transient oscillations due to the resonance between the transmission line inductances and VSCs dc-link capacitances. With the application of the PFCS, the dc grid voltage and current damping is significantly improved. However, the damping improvement of V dc3 is slightly limited, as shown in Fig. 19. As the PFC is installed beside VSC1, the large electrical distance between the PFC and VSC3 limits the damping capability of the PFCS. The results verify the frequency response analysis presented in Section IV.

B. PFCS PERFORMANCE INCLUDING DCCB INDUCTANCE
Herein, the performance of the proposed PFCS is evaluated when a 100 mH dc breaker inductor is installed at each end of the MTDC grid transmission lines. The responses of the selected dc currents and voltages are shown in Fig. 20 and  Fig. 21, respectively. Compared to the first simulation scenario, the dc grid currents and voltages demonstrate more significant oscillations at lower frequencies due to increased system inductance. Moreover, the oscillations are poorly damped as the dominant eigenvalues are shifted closer to the RHP when the system inductance increases.
Compared to the uncompensated response, the proposed PFCS provides a significant damping improvement of the MTDC grid. As a result, the response of the dc currents and voltages is largely enhanced with the magnitude and duration of the oscillations significantly decreased. As shown in Fig. 21, at t = 3 s when the power injection of VSC3 is increased, the dc voltage V dc1 stabilizes after 50 ms when the PFCS is activated, compared to 500 ms for the uncompensated system. Fig. 21 also shows, despite the improvement of V dc3 response, the damping is slightly limited because of the large electrical distance between the PFC and the dc-link of VSC3.

C. COMPARISON WITH INTERLINE PFC DAMPING CONTROLLER
To further evaluate the effectiveness of the proposed PFCS, its response is compared with the interline PFC-based stabilizer (IPFCS) discussed in [19]. Figs. 22 and 23 compare the responses of the selected dc currents and voltages with two types of damping control applied. It can be noted that both controllers enhance the MTDC grid damping. However, the damping improvement of the proposed PFCS is considerably better compared to the IPFCS, as demonstrated in Figs. 22 and 23. When the power injection of VSC3 is increased at t = 3 s, the dc voltage V dc1 is completely damped after 50 ms and 150 ms with the proposed PFCS and IPFCS, respectively. The damping capability of the IPFCS is limited compared to the proposed PFCS because the capacitor voltage of an interline power flow controller is influenced by the line currents state, and its magnitude varies depending on the power flow management signal. On the other hand, the PFC utilized in this paper uses a partially rated auxiliary dc/dc converter, ensuring  that the PFC capacitor voltage is maintained constant and not influenced by the line currents state.

D. OPERATION UNDER FAULT CONDITION
This study assesses the MTDC grid performance under ac system fault with the PFCS adopted. Fig. 24 shows the response of the selected dc voltages when a single-line-to-ground fault occurred at the PCC of VSC1. The fault that causes a 25% voltage sag of the ac voltage occurred at t = 3 s and is interrupted after 200 ms. As depicted in Fig. 24, the dc voltage of VSC1 exhibits severe oscillations during the fault at the double-fundamental frequency for both the uncompensated and compensated system. However, after clearing the fault, the compensated system demonstrates a better response with reduced oscillations and settling time due to improved system damping. Fig. 24 also shows that during the fault, the dc voltage of VSC3 does not exhibit double-frequency oscillations due to the large inductance of the dc system.

E. PFCS PERFORMANCE IN A FIVE-TERMINAL HVDC GRID USING A FOUR-PORT PFC
The present study aims to assess the effectiveness of the proposed PFCS under different PFC and MTDC grid configurations. To achieve this goal, a four-port PFC equipped with the proposed damping controller was integrated into a five-terminal HVDC grid, as shown in Fig. 25. The PFC, installed at Terminal 1, is tasked to regulate the line currents I 12 , I 14 , and I 15 . The responses of the selected currents and  Initially, the system was in a steady state, with the power injections from VSCs 1, 3, and 5 set to 400 MW, −100 MW, and −200 MW, respectively. The PFC regulates the currents I 12 and I 14 to reference values of 300 A and 250 A, respectively. The dc system was then subjected to various disturbances to assess the performance of the proposed PFCS. At t = 3 s, the power injection from VSC3 was increased to −200 MW. Then, at t = 3.5 s, VSC5 experienced a step change in power of −50 MW, and finally, a terminal outage at VSC3 occurred at t = 4 s.
As illustrated in Figs. 26 and 27, the proposed PFCS demonstrated a significant improvement in the dynamic response and reduced oscillation levels of the dc grid currents and voltages. This improvement was achieved through the damping enhancement introduced by the PFCS. In Fig. 26, it can be noted that the PFC is capable of maintaining the currents I 12 and I 14 at their reference values even in the presence of disturbances. Fig. 23 reveals that the damping improvement of V dc3 and V dc5 was slightly limited due to the substantial electrical distance between the PFC and Terminals 3 and 5.
The tripping of transmission lines in an MTDC grid caused by faults can result in significant disruptions, which may challenge the secure operation of the grid. Consequently, evaluating the effectiveness of the proposed PFCS in such a scenario is crucial. The responses of the selected dc grid currents and voltages are depicted in Figs. 28 and 29 when the transmission line between VSC3 and VSC4 is disconnected at t = 3 s and then reconnected at t = 3.7 s. The results demonstrate that disconnecting line L34 leads to significant currents and voltage oscillations caused by the sudden change in system impedance. The system currents and voltages are minimally impacted when the line is reconnected. In this case, the system impedance changes gradually, thus enabling the system to adjust and reduce the risk of disruption. Compared to the uncompensated system, as depicted in Figs. 28 and 29, the proposed PFCS effectively reduces the current and voltage oscillations resulting from the line tripping.

VI. VALIDATION RESULTS
To demonstrate the effectiveness of the proposed PFCS, a hardware-in-the-loop (HIL) real-time simulation setup has been implemented, as depicted in Fig. 30. The HIL setup uses the OPAL-RT OP5600 real-time simulation platform. The platform, which is fully integrated with MATLAB/Simulink, uses a Virtex-6 FPGA board with a time step of 290 ns for rabid control prototyping applications. The MTDC grid is simulated in real-time, and the proposed PFCS 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. The performance of the MTDC grid with the proposed PFCS is evaluated during the step change of VSC1 active power and under ac-side fault conditions at PCC1, as shown in Figs. 31 and 32, respectively.
The results depicted in Figs. 31 and 32 demonstrate a close agreement with the transient simulation results given in Section V. As shown in Fig. 31(a), due to the large inductance of the DCCB, the dc grid currents and voltages exhibit poorly damped oscillatory response as the system dominant modes are shifted closer to the RHP. Using the proposed PFCS, the MTDC grid currents and voltages demonstrate a well-damped response with significantly reduced oscillation levels, as depicted in Fig. 31(c). In Fig. 31(b), despite the enhancement of the dc grid currents and voltages responses using the IPFCS, the damping improvement is limited compared to the proposed PFCS as the capacitor voltage of an IPFC is influenced by the line currents state. The response of the selected MTDC grid currents and voltages under ac system fault at PCC1 is depicted in Fig. 32 with and without the proposed PFCS. It can be noted that when the proposed PFCS is adopted, the dc currents and voltages demonstrate improved response after clearing the fault due to the enhancement of the system damping.

VII. CONCLUSION
This paper has investigated the influence of the DCCB inductance on the dynamic performance and stability of a PFC-equipped MTDC grid using eigenvalue and frequency response analyses. The DCCB inductance imposes a lower bound on the PFC control parameters and VSC droop control gains. The dominant eigenvalues of the system shift closer to the RHP, and higher PFC and droop control gains are needed to achieve a system with satisfactory dynamic performance. Moreover, a large DCCB inductance causes a significant resonance at a lower frequency. As a result, the dc voltages and currents exhibit poorly damped oscillations. A PFCS is proposed to mitigate dc current and voltage oscillations and improve the dynamic performance of the MTDC grid. The PFCS modifies the PFC current control scheme using a supplementary damping signal based on a band-limited derivative compensator. In addition to extending the PFC functionality, the proposed damping controller also relieves control stress on VSC terminals. Compared to an interline PFC-based damping controller, the proposed PFCS achieves better damping capability as its input voltage is not dependent on the state of the lines' currents and does not vary with the current regulation command. Simulation and HIL tests, based on a four-terminal HVDC MTDC grid, demonstrate the efficacy of the proposed PFCS and confirm the theoretical analysis.