Small-Signal Stability Analysis of the Interactions Between Voltage Source Converters and DC Current Flow Controllers

Small-signal stability analysis of the voltage source converter (VSC) based multi-terminal DC (MTDC) grids has been mainly focused on the dynamics of AC/DC converters while there is a lack of consideration of DC networks with DC power-flow controllers. In this paper, an integrated small-signal model is established to investigate the interactions between VSC and DC current flow controllers (CFC). The impacts of the control system of CFC on small-signal stability of the integrated system and the impacts of the control system of VSC on the DC networks with CFC are discussed via eigenvalue analysis. The above interactions are demonstrated through dynamic simulations against both small and large disturbances using real time digital simulator (RTDS).


I. INTRODUCTION
V OLTAGE source converter (VSC) based multi-terminal DC (MTDC) system is the preferred solution for grid connection of large-scale offshore wind farms [1], [2], as it is the feasible option for long distance submarine interconnection of passive and weak systems [3], [4].For the sake of discussions, Fig. 1 shows a generic topology of MTDC/AC system, which consists of the CFC-integrated DC network and the VSC based AC/DC converter integrated AC network.The AC network includes multiple synchronous generators (G 1 to G n ), multiple VSC converters (VSC 1 to VSC n ), active or passive loads, and/or transmission lines.The DC network consists of DC power-flow controller and DC cables.Previous publications on small-signal stability analysis of VSC MTDC have been focused on the dynamics of VSC (top blue area in Fig. 1) while some other papers have been focused on DC networks with DC power flow controllers (bottom green area in Fig. 1).However, there is a lack of a considerations of these two systems integrated simultaneously using detailed models and hence the potential interactions between them.
With the increasing installations of VSC HVDC systems in power grids, the investigation of controllability and stability of MTDC systems becomes important for it benefits the existing AC systems with a means of increased energy trading between interconnected system operators [5], [6].The generalized dynamic model of VSC MTDC was derived in [7]- [9].Advanced control extensions with distributed DC voltage control were developed in [10], and a supplementary decentralized control structure to damp interarea oscillations was proposed in [11].Frequency support function for the surrounding interconnected AC systems and adaptive droop control for effective power sharing through MTDC system were presented and well analyzed in [12], [13].Electromechanical transient modelling of MMC MTDC was studied in [14] to provide the theoretical foundation for future MMC MTDC applications.
Small-signal stability analysis has been widely used for the study of system dynamics and controller design for VSC [15]- [19].In [8] the small-signal stability analysis has been carried out for a symmetric bipolar MTDC grid where different modes and interactions between AC systems and VSC following various disturbances were characterized.In [9] small-signal stability analysis was carried out to define the stable operating ranges for the gains of VSC controllers that ensure the dynamic stability of MTDC system.Wind farm generators were considered and a method to calculate converter controller gains was provided.In [12], an adaptive frequency droop control was proposed by investigating the sensitivity of eigenvalues with respect to different droop coefficients, control parameters and changes in operating conditions.In [13], small-signal stability analysis was used for the design of an adaptive droop control scheme.In [20], a method was introduced for systematically identifying interactions of modes in a VSC MTDC system by analyzing the participation pattern from different system elements.
A DC power-flow controller with the use of full-bridge DC-DC converter, which is a relatively small, inexpensive and low-loss device for DC power flow management was proposed in [21], [22].In [23] a DC power-flow controller was presented to regulate the power flow.In [24], the design of a CFC, along with its control strategy, was proposed which is capable of controlling the DC branch currents to transfer power from an overloaded branch to an underutilized branch to minimize the power loss.Small-signal stability of DC network with CFC was discussed in [25].A parametric analysis scheme for optimization of control gains of CFC is proposed in [26].The characteristics, modelling, control, operational performance of different configurations of series-connected CFCs are studied, including dual-thyristor converter CFC, cascaded VSC dc chopper based CFC, and dual H-bridge CFC (2B-CFC) in [27], and resistive based CFC, RC circuit Based CFC, and capacitive based interline CFC in [28].Distributed CFC, which is based on series interline CFC with extended and simplified topology proposed in [29], is presented in [30] to reduce overload in DC grid with lower CFC voltage, and increase operational area compared to 2B-CFC, when properly located.A coordinated droop control strategy and droop gain evaluation of multi-line CFC is proposed in [31], and coordinated operation control between multiple CFCs is presented in [32].Small-signal stability of DC network with CFC was discussed in [25], [26], [31].
However, in the work in [25], [26], [31], VSCs are represented as constant current injections and hence the interactions between CFCs and VSCs cannot be studied.The published papers on small-signal stability of VSC MTDC systems have mainly been focused on the dynamics of VSCs while there is a lack of considerations of DC networks with CFC.And there is also a lack of analysis on the potential impacts of DC power-flow controller on existing MTDC system in terms of network stability and dynamic performance under disturbances.Hence there is a real need to fill in the gap of the interactions between DC networks with CFC and AC/DC converters (even more broadly, the detailed representation of connected AC networks).
The aim of this paper is to fill in this gap of the understanding of the interactions between VSC and CFC and the main contributions of this paper are summarized as follows: 1) In the studies, the detailed mathematical models of the integrated AC/DC system consisting of CFC, VSC and AC synchronous machines have been fully implemented.
2) The modal analysis is used to characterize modes, via participation factor matrix, of the integrated AC/DC systems without/with CFC, investigate the impacts of CFC on the integrated AC/DC systems, and design the proposed methodology to derive stable operating range of CFC.
3) The interactions between CFC and VSC are investigated via eigenvalue trajectory studies and dynamic stability analysis.4) The small-signal stability analysis results are validated in RTDS against small and large disturbances.

II. SMALL-SIGNAL STABILITY MODEL OF THE INTEGRATED AC/DC SYSTEMS WITHOUT/WITH CFC
Fig. 1 shows the generic topology of a generalized AC/DC system, which consists of CFC-integrated DC networks, VSC and AC networks.The CFC integrated AC/DC system model is represented in differential-algebraic equations (DAEs) and is linearized around an equilibrium point in state-space form as: where x and u are the incremental change of the state variable vector and input vector, A and B are state matrix and input matrix.The left and right eigenvectors of matrix A form participation factor matrix which provides insight into the relationship between states and modes.

A. CFC AND INTEGRATED DC NETWORK MODEL
The CFC model proposed in [25] is used as an example of DC power-flow controller to investigate its impacts on the existing AC/DC system.For example, the topology of a meshed three-terminal DC network with CFC is shown in Fig. 2. 1) CFC MODEL [25] The CFC model in [25] is composed of two identical fullbridge DC-DC converters sharing a common capacitor as shown in Fig. 2. By considering the charging and discharging state of the capacitor within a switching cycle and applying Kirchhoff's voltage law to the DC network, the dynamic equations of three-terminal DC network with CFC can be derived as: where u c is the voltage across capacitor; i

2) CFC CONTROL SYSTEM MODEL
The control system presented in [25], which is to generate gate signals G SC1 , and G SC2 , is shown in Fig. 3.In this paper, based on the CFC control system in Fig. 3, the duty cycles d c1 , and d c2 are calculated as: where i * 12 is the reference of current through DC branch 12; u * c is the reference of voltage across the capacitor; k pc1 , k ic1 , k pc2 , and k ic2 are the proportional and integral gains of the CFC PI control loop; y 1 and y 2 are the auxiliary state variables to represent the integral terms of the PI control loop.
The final dynamic equations can be derived by substituting (3) into (2).

B. SYNCHRONOUS GENERATOR MODEL
Each synchronous generator is modeled with a sixth-order model consisting of six state variables as follows: Then the differential and algebraic equations for synchronous generator i are represented as: where all the state variables, state matrices, input vectors and input matrices can be found in [18].First-order IEEE type 2 exciter and third-order power system stabilizer controllers are used with synchronous generators.

C. VSC MODEL
Fig. 4 shows the single-line diagram of a VSC with its control system.dq decoupling control is used.DC voltage or active power is controlled in d-axis, whilst AC voltage at point of common coupling is regulated in q-axis.The dynamic equations of the converter in dq reference frame can be expressed as: The dynamic equation for DC side capacitor can be expressed as: where R c and L c represents total impedance of the phase reactor and converter transformer.1) OUTER CONTROL LOOP: As the architecture of V dc -V ac control shown in Fig. 5, the current references in dq frame can be obtained from the outer control loop by: where k pdc , k idc , k pac , and k iac are the proportional and integral gains of DC voltage and AC voltage controllers.The auxiliary state variables of x 1 and x 2 are used to represent the integral terms of PI controllers.
The AC voltage at point of common coupling V ac1 can be represented as: 2) OUTER CONTROL LOOP: P -V AC CONTROL As the architecture of P-V ac control mode also shown in Fig. 5, the current reference in dq frame can be obtained as: where k pdc , k idc , k pac , and k iac are the proportional and integral gains of DC voltage and AC voltage controllers.The auxiliary state variable x 2 represents the integral terms of PI control loop.

3) INNER CONTROL LOOP OF VSC
As the converter inner control loop shown in Fig. 5, the reference of VSC converter voltage can be obtained as: where i td and i tq are current components through phase reactor in dq frame; k pid , k iid , k piq , and k iiq are the proportional and integral gains of the current controller; x 3 and x 4 are auxiliary state variables to represent the integral terms of PI control loop.

D. AC NETWORK
The buses of multi-bus AC network can be classified as generator-connected buses and non-generator buses.The power balance equation of a generator connected bus i is: where n is the total bus number, P Li and Q Li are the active power and reactive power injection, respectively, to bus i.
Similarly, the power balance equation of a non-generator bus i can be obtained as: The values of G ik and B ik can be determined by system network admittance matrix Y.

E. FORMULATION OF THE OVERALL LINEARIZED SYSTEM
The complete state-space representation of the multi-model test system is formulated by combining the linearized subsystem models of synchronous generators, VSC, DC network where B is the input matrix, and u is the input vector with references of operating point, submatrix A ACSys and A MTDC represent state matrix of AC systems and MTDC system, x ACSys and x MTDC represent state variables of AC systems and MTDC system.

III. INTERACTIONS BETWEEN CFC AND VSC
In Section III, the test system is introduced.Then dynamic analysis of the integrated AC/DC system without and with CFC is studied to investigate the impacts of the incorporation of CFC.Finally, the interactions between CFC and VSC are investigated.

A. TEST SYSTEM
Fig. 6 shows the test system topology of the CFC integrated AC/DC system, which was modified based on the Kundur's 4-generator 2-area system proposed in [15] where parameters can be found.The test system consists of two areas connected by a weak tie (220km transmission line) and a three-terminal AC/DC network.The pole-to-pole DC voltage is 120 kV.
A CFC is integrated into the DC network to control DC currents.VSC 2 is operating in V dc -V ac control, controlling the DC voltage and the AC voltage.VSC 1 and VSC 3 are operating in P-V ac control, controlling the active power through converters and the AC voltages.A two-line CFC is installed at the DC side of VSC1 to control current flow from VSC 1 to VSC 3 and VSC 1 to VSC 2 .The parameters of the DC network and CFC are shown in Table 1.The control parameters for control system of CFC are from [25] and the control parameters for control system of VSC is based on the approach proposed in [9].

B. DYNAMIC ANALYSIS OF THE INTEGRATED SYSTEM WITHOUT/WITH CFC
In order to study the impacts of the incorporation of CFC on small-signal stability of the integrated AC/DC system, eigenvalue analysis is carried out for the system without CFC and the system with CFC.The small-signal stability of the AC/DC system shown in Fig. 6 is to be assessed by modal analysis.The complete state-space matrix of the integrated system without CFC has a dimension of 52 × 52 and complete state-space matrix A of CFC integrated system has a dimension of 55 × 55.The calculated eigenvalues of system with CFC are given in Table 2.

TABLE 2. Eigenvalue results.
There are 52 eigenvalues for the system without CFC, and 55 eigenvalues for the system with CFC.The additional three states are resulting from the CFC capacitor and 2 integral controllers of CFC ( u c , y 1 , y 2 ).From Table 2, it can be seen that all 55 eigenvalues have non-positive real part.This indicates a stable operating condition around the operating point.There are two zero eigenvalues λ 48,49 , as there is no infinite bus bar in the testing system.One zero eigenvalue arises from a redundancy in synchronous generator angle.The second zero eigenvalue results from that the generator torque is independent of machine speed deviations, and this is caused by the fact that mechanical damping is neglected and governor action is not represented [17].
The eigenvalues λ 11,12 , λ 14,15 , λ 18,19 to λ 24,25 , λ 29,30 , and λ 35,36 are related to the dominant state variables of synchronous generators according to their participation factors.The damping of these modes is mainly affected by synchronous generator settings, associated exciter settings, and corresponding PSS parameters.The incorporation of CFC has no significant impacts on these synchronous generator related eigenvalues.The participation matrix also indicates that controller of VSC 3 has dominant influence on λ 39,40 .Based on participation matrix, it should be noted that state variables i 12 and i 13 of the DC network have dominant influence on the oscillation modes of λ 5,6 and λ 7,8 , while auxiliary state variable y 1 of CFC controller dominantly influences λ 31,32 .The results of modal analysis show that the incorporation of CFC and its controller parameters affect the small-signal stability of multi-machine AC/DC system.Fig. 7 shows a plot of eigenvalues of the system without CFC while Fig. 8 shows a plot of eigenvalues of the system with CFC.The dominant states and the associated pairs of eigenvalues are highlighted in Fig. 7 (b) for the system without CFC and in Fig. 8 (b) for the system with CFC.Compared to the system without CFC as shown in Fig. 7 (b), it can be seen that the system with CFC (as shown in Fig. 8 (b)) has 3 additional conjugate pairs of eigenvalues, which are dominated by state variables of DC branch currents i 12 , i 13 , and CFC integral controller auxiliary state y 2 .Also, the system with CFC has very slight impacts on the generator-related eigenvalues as there are no significant changes in the placements of those eigenvalues.However, the system with CFC leads to the changing of VSC controller related eigenvalues from the pure negative real values to the conjugate pair of eigenvalues with negative real part, as highlighted by VSC 2 x 3 in Fig. 8 (b).This change results in VSC controller related system oscillations for the system with CFC.This is due to the potential interactions between CFC and adjacent VSC.
From Fig. 8, the critical modes which have major impacts small signal performance of the system are associated with DC CFC auxiliary state y 2 VSC 2 auxiliary state x 3 , and the angle rotating speed of Synchronous Generator 1.
The AC/DC test system as shown in Fig. 6 has been established in RSCAD to validate the results of modal analysis.The power flow data, AC voltages, and impedance values are based on the 4-generator 2-area benchmark system model in [15], while the parameters of the DC network and CFC are shown in Table 1.The max power loss of CFC, considering conduction and switching loss, is 1.6 MW (1.6% of the system power rating), which is much lower than losses due to the resistances in the DC lines.
1) SIMULATIONS UNDER SMALL DISTURBANCES Fig. 9 shows the dynamic behavior of the system (controlled DC branch current i 12 , and voltage across CFC bridge capacitor u c ) in response to a step reference change applied to CFC (i 12ref changes from 0.55 kA to 0.50 kA at 0.1s).It can be seen from Fig. 9 that the controlled current of DC branch 12 decreases from 0.55 kA to 0.50 kA.The voltage of DC capacitor returns to its reference value after the small disturbance.Stable dynamic performance can be seen from the simulation results, which is in line with the modal analysis results.

2) SIMULATIONS UNDER LARGE DISTURBANCES (AC-SYSTEM FAULT)
In this subsection, a large disturbance is applied to test the stability under large A 10 single line to ground is simulated at the point of common coupling of VSC 3 (primary side of coupling transformer) at 0.3s lasting for 7 cycles (116 ms).Fig. 10 shows system dynamics of MTDC system, including active power transfer of VSC 1 and VSC 3 (these two converters are in P -V ac control), and current of DC branch 12.The system recovers to steady-state in a short period of time after the fault is cleared.It demonstrates that the system can endure large AC disturbance and recover quickly.

3) SIMULATIONS UNDER LARGE DISTURBANCES (DC-CABLE FAULT)
For the testing of large disturbance at DC side, a 25 lineto-ground DC-cable fault is simulated at dc-side of VSC 3 at 0.3s lasting for 7 cycles (116 ms).
Fig. 11 shows the system dynamic response results of controlled DC branch current i 12 , and bridge capacitor voltage u c .The system recovers to its former state after the fault is cleared.The simulation results show that the system can endure DC side large disturbances.

C. IMPACTS OF CONTROL PARAMETERS OF CFC ON VSC
The effects of the control parameters of CFC on VSC are studied in this section.
It has be found from modal analysis that different values of the CFC controller gains k pc1 , k ic1 , k pc2 , and k ic2 can affect dynamic stability of the integrated AC/DC system, especially on those VSC state variable related modes.The stable/unstable setting ranges of k pc1 , k ic1 , k pc2 , and k ic2 are obtained by the procedure shown in Fig. 12.For the given test system described in Section II, the stable/unstable setting ranges of k pc2 is listed as shown in Table 3, which is used as an example to investigate system dynamic stability when CFC controller gains are changed.Table 4 lists three different values of k pc2 (k pc2 = 0.01, 0.1, and 1) and relevant modes.The corresponding values of λ 31,32 are 0.16 ±j1.07, 0.09 ±j1.08, and −0.55±j0.93.It can be found that the system is stable only when k pc2 ≥ 1.The time-domain simulations in Fig. 13 are in line with the results shown in TABLE 4 when a small change of the reference is applied (i 12ref changes from 0.55 kA to 0.50 kA at 0.5s).Stable system dynamics can be achieved only when k pc2 = 1.Unstable oscillations are observed for k pc2 = 0.1 and k pc2 = 0.01.It can also be seen that large k pc2 decreases the oscillation frequency and the settling time.The value of k pc2 has significant effects on state variables i 12 , and y 2 .These results demonstrate that the control parameters of CFC may cause instability of the system if they are not carefully-tuned.
As shown in Fig. 14, eigenvalue trajectories study is used to plot the movements of critical conjugate pairs so that the impacts of changing control parameters of CFC on VSC related modes can be analyzed.In Fig. 14, the control gain k pc2 varies from 0 (p.u.) to 1000 (p.u.) while the values of other control gains remain same.The purple arrows indicate the movements of conjugate pairs resulting from this variation of k pc2 .As shown in Fig. 14 (a), the imaginary parts of CFC related conjugate pairs (i 12 and i 13 ) decrease as k pc2 increases, which indicates greater damping ratios of CFC related states.As shown in Fig. 14(a), two new VSC related conjugate pairs V dc2 and V dc1 (highlighted in blue) emerge as k pc2 increases.The movements of VSC related pairs (V dc2 , V dc1 , and VSCx 32 ) as shown in Fig. 14 (a)-(b) clearly present the impacts of CFC on VSC.The two conjugate pairs of V dc2 and V dc1 move towards left half of splane which indicates larger damping ratios of VSC related states V dc2 and V dc1 .The conjugate pairs of VSCx 32 move towards right hand side indicating less damping ratios of VSCx 32 .As shown in Fig. 14 (a)-(b), there are no significant movements of generator related poles, which indicates CFC has no significant impacts on synchronous machine related states.Based on eigenvalue trajectory studies of CFC control gains, it can be summarized that the variation of CFC controller can affect dynamic stability of CFC and VSC related modes.These impacts of CFC on VSC can result in different damping characteristics of VSC related states, as CFC controller changes.
As listed in Table 5, three cases of different CFC control parameters are designed based on eigenvalue trajectories of various CFC control gains, to investigate their effects on the VSC controller related modes.Case 1: the control system of CFC is disabled as the base case to represent without CFC control system.
Case 2: the gains of CFC control system are set the same as those in [25] (k pc1 = 1, k ic1 = 1, k pc2 = 1, k ic2 = 1), to represent the normal settings.The oscillations of VSC 2 controller related modes are seen from Table 5.The AC/DC system is stable for the first two cases.
Case 3: the gains of CFC control system are changed (k pc1 = 0.01, k ic1 = 1000, k pc2 = 1, k ic2 = 1000), to represent extreme or possible fault conditions of CFC control system.One conjugate pair of eigenvalues with positive real part is seen, which indicates small-signal instability.
Fig. 15 shows time-domain simulation results of three cases.As shown in Fig. 15, a small perturbation (DC branch current reference i * 12 changes from 0.55 kA to 0.5 kA) is applied to CFC at 0.15s.For Case 1, no significant change of VSC 2 auxiliary state variable x 3 is observed because the CFC control system is disabled.For Case 2, a small increase of x 3 at 0.2s can be seen.Then the value of x 3 recovers to its pre-disturbance level after a period time of oscillations.For Case 3, large unstable oscillations are seen.Simulation results are in line with the modal analysis results in Table 5, and illustrate that the CFC control system can affect dynamic performance of VSC converter controller.This effect can be severe and cause instability to the AC/DC system, for extreme DC CFC operating conditions.
The possible reason of these interactions between CFC control system and VSC controller can be the dynamical changing of the interfacing variables that relate CFC controlled variable and adjacent VSC controlled variable.For instance, CFC controls DC branch current i 12 , and VSC converter 2 controls DC voltage of VSC 2 V dc2 .And the relation between these two controlled state variables is given in equation 5a.

D. IMPACTS OF CONTROL PARAMETERS OF VSC ON CFC
The effects of the control parameters of VSC on CFC are studied in this section.Eigenvalue trajectory study is implemented to investigate potential impacts of different values of the control gains of VSC controller (k pdc , k idc , k pid , k iid , k pac , k iac , k piq and k iiq ) on system dynamic stability of the integrated AC/DC system, especially on CFC state variable related critical modes.The stable setting ranges of the control parameters of VSC controller are obtained using the approaches presented in [9].
As eigenvalue trajectories shown in Fig. 16, the movements of critical conjugate pairs present the impacts of changing control parameters of VSC on CFC related modes.In Fig. 16 related states.It should be noted that positive eigenvalues are seen in Fig. 16(d).This is due to the variation range of k idc covers all stable and part of unstable operating range.As shown in Fig. 16(b) and (d), the movements of CFC related pairs (i 12 , i 13 , and CFCy 2 ) clearly present the impacts of VSC on CFC.The two conjugate pairs of i 12 and i 13 move towards right half of s-plane which indicates reduced damping ratios of CFC related states i 12 and i 13 .The conjugate pair of CFCy 2 moves towards left hand side indicating increased damping ratios of CFCy 2 .
Based on eigenvalue trajectory studies of VSC control gains, it can be summarized that the variation of VSC controller can affect dynamic stability of synchronous generaors, VSC, and CFC related modes.These impacts of VSC on CFC can result in different damping characteristics of CFC related states, as VSC controller changes.And improper VSC controller tuning may cause instability to the integrated system.
As listed in Table 6, two cases with different VSC control parameters are designed based on eigenvalue trajectory studies of various VSC control gains, to investigate their effects on the CFC related system stability modes.Simulation results demonstrate that the VSC controller can affect the dynamic performance of CFC.And this effect can be severe and cause instability of the CFC integrated DC network, for extreme VSC operating conditions.

IV. CONCLUSION
An integrated small-signal stability model for the study of interactions between CFC and VSC has been established.Modal analysis results have been verified by the dynamic simulations in RTDS under both small and large AC/DC disturbances (the AC system grounding fault applied in III.B.2) and DC system grounding fault applied in III.B.3)), which illustrates the incorporation of CFC has been shown to affect dynamic stability of the existing MTDC/AC system especially the adjacent VSC converters.The impacts of control parameters of CFC on the integrated AC/DC system with VSC and the impacts of VSC controller on the DC network with CFC have been investigated using both modal analysis and time-domain simulations.It has been found that the negative interactions between CFC and VSC could be severe and may cause instability to the integrated AC/DC system.The proposed modelling approach and methodology of analysis in this paper has provided a useful foundation that can be expanded for research of coordinated design of CFC control system and control system of VSC to avoid their possible negative interactions, as future work.

FIGURE 2 .
FIGURE 2. DC network with DC CFC.

FIGURE 6 .
FIGURE 6. Topology of the test system.

FIGURE 7 .
FIGURE 7. Eigenvalues of the integrated AC/DC system without CFC.

FIGURE 8 .
FIGURE 8. Eigenvalues of the integrated AC/DC system with CFC.

FIGURE 9 .
FIGURE 9. Dynamics of DC network during small disturbance.

FIGURE 10 .
FIGURE 10.Dynamics of MTDC system against AC system fault.

FIGURE 11 .
FIGURE 11.Dynamics of MTDC system against DC cable fault.

FIGURE 12 .
FIGURE 12. Flowchart to obtain stable operating range of CFC control gains.

FIGURE 13 .
FIGURE 13.Dynamics of DC current i 12 under different CFC control parameters.

FIGURE 14 .
FIGURE 14. Eigenvalue trajectories of various CFC control parameter k pc2 .

FIGURE 15 .
FIGURE 15.Effects of CFC control parameters on dynamics of VSC 2 .
FIGURE 16.Eigenvalue trajectories of various VSC control parameter k idc .

FIGURE 17 .
FIGURE 17.Effects of control parameters of VSC 2 on CFC.

NA
DENG received the B.Eng. degree from the Huazhong University of Science and Technology (HUST), China, in 2011, and the Ph.D. degree from the University of Birmingham, Birmingham, U.K., in 2015.She is currently with the Jiangsu Industrial Technology Research Institute (JITRI).Her research interest includes the applications of dc/dc and ac/dc converters in HVDC systems and microgrids.YING XUE received the B.Eng. degree in electrical engineering from the Huazhong University of Science and Technology (HUST), Wuhan, China, and the University of Birmingham, Birmingham, U.K., in 2012, and the Ph.D. degree in electrical engineering from the University of Birmingham in 2016.He is currently a Lecturer with the University of Birmingham.His main research interest includes HVDC modeling and control.XIAO-PING ZHANG (M'95-SM'06) is currently a Professor of electrical power systems with the University of Birmingham, U.K., and he is also the Director of the Smart Grid, Birmingham Energy Institute and the Co-Director of the Birmingham Energy Storage Center.He has coauthored the first and second edition of the monograph Flexible AC Transmission Systems: Modeling and Control, (Springer, 2006 and 2012).He has coauthored the book Restructured Electric Power Systems: Analysis of Electricity Markets With Equilibrium Models, (IEEE Press/Wiley, 2010).

12 , i 13 and i 23 are
VOLUME 7, 2020with CFC, and AC networks.The synchronous generator models are linked with multi-bus AC network model through power flow equations.The VSC models and DC network with CFC model are linked by DC current/voltage relationships.The VSC models are linked to AC network model via power flow equations.