Abstract
This paper puts forward a new practical voltage source converter (VSC) based AC-DC converter model suitable for conducting power flow assessment of multi-terminal VSC-based high-voltage direct current (VSC-MTDC) systems. The model uses an advanced method to handle the operational limits and control modes of VSCs into the power flow formulation. The new model is incorporated into a unified framework encompassing AC and DC power grids and is solved by using the Newton-Raphson method to enable quadratically convergent iterative solutions. The use of complementarity constraints, together with the Fischer-Burmeister function, is proposed to enable the seamless incorporation of operational control modes of VSC and automatic enforcement of any converter’s operational limits that become violated during the iterative solution process. Thus, a dedicated process for checking limits is no longer required. Furthermore, all existing relationships between the VSC control laws and their operational limits are considered directly during the solution of the power flow problem. The applicability of the new model is demonstrated with numerical examples using various multi-terminal AC-DC transmission networks, one of which is a utility-sized power system.
Set of AC nodes
Set of nonregulated AC nodes
Set of PV nodes in AC grids
Set of slack nodes in AC grids
Set of all DC nodes
Set of nodes at DC side of converter
Set of load nodes in DC grids
Representation of all nodes j adjacent to node i
The minimum and maximum values
Specified value
Complex number
The upper and lower limits
Slack converter control mode of AC grid
Susceptance from nodes i to j
Imaginary part of
Imaginary part of
Imaginary part of
Conductance from nodes i to j
Real part of
Real part of
Real part of
Modulation factor of voltage source converter(VSC)
Active power generated at AC node i
Active power consumed at AC node i
Specified active power flow from nodes i to j
Nominal power capacity of VSC
Reactive power generated at AC node i
Reactive power consumed at AC node i
Specified reactive power flow from nodes i to j
Conducting resistance of DC-DC converter
Conducting resistance of insulated gate bipolar transistor
DC voltage droop control reference at node i
Specified voltage magnitude at node i
Voltage-power (V-P) droop control
V-P droop control with dead band
Vector of state variables
Phase reactor admittance
Shunt filter admittance
Transformer admittance
Voltage phase angle of VSC
Reference voltage phase angle
Voltage phase angle at node ac of VSC
Voltage phase angle at AC node i
D Duty cycle of DC-DC converter
Conductance of switching losses
VSC positive sequence current magnitude
d- and q-axis components VSC current before current limiter
d- and q-axis components VSC current after current limiter
Amplitude modulation index of VSC
Modulation index used to control variable
Active power flow from nodes i to j
Power loss of VSC
Reactive power flow from nodes i to j
Reactive power generated by VSC
Voltage magnitude at DC node i
Voltage magnitude at node ac of VSC
Voltage magnitude at node dc of VSC
Nodal complex voltage at AC node i
Voltage magnitude at AC node i
, Complementary constraint auxiliary variables related to the lower and upper limits
VOLTAGE source converters (VSCs) that comprise high-voltage direct current (HVDC) links represent the best solution to integrating large blocks of renewable energy into AC power grids, particularly offshore wind power [
To quantify the manner in which VSC-based converter technology enables the flexible operation of MTDC systems, it is necessary to develop suitable mathematical VSC models,where the converter control characteristics and operational limits are duly incorporated. In this context, several proposals have aimed at modeling the steady-state characteristics of these systems. However, thus far, the control modes and operational limits of the converters have not been addressed in sufficient depth. This study fills this void and introduces a new VSC model, where the VSC operational limits and control modes are suitably combined in a unified framework and used to achieve quadratically convergent power flow solutions.
From the perspective of modeling, one approach has been to represent each converter station of the VSC-HVDC link by a controllable AC voltage source behind impedance, with the interaction of both converters represented through their common DC link by an active power flow constraint [
The generic model reported in [
An alternative VSC model, represented by a controllable AC voltage source and a controllable DC current source, is proposed in [
In a VSC, the AC and DC voltages are related by the modulation index. Moreover, the DC-AC power conversion process may be represented by an ideal complex tap-changing transformer [
Based on the previous discussion and with the aim of filling the research void, this paper proposes a new VSC model in which a realistic representation of the steady-state operational characteristics and control operation modes is considered from the outset, bearing in mind the current limits of the converter stations. Within this context, the specific contributions of the VSC model presented in this study compared with existing models are given as follows.
When the VSC model is represented by a controllable AC voltage source, it is impossible to know the values of parameters that determine the VSC operational limits. This drawback is overcome in our model by including the VSC DC voltage, voltage phase angle at the AC side of converter, and the VSC modulation index as state variables.
The new VSC model includes the control modes of operation and operational limits within a unified formulation framework. In this case, the AC terminals of the VSC are explicitly modeled by separating its corresponding power flow mismatch equations from those associated with the common point of coupling. This enables the explicit representation of the VSC reactive power injected into/from the network together with its operational constraints. This reactive power internally generated by the VSC is modeled as a reactive power source and solved as a state variable to correctly constrain the solution to the current-based reactive power limits of the VSC. These limits are derived from the converter AC voltage magnitude and inner current controller of the VSC. Note that the representation of the VSC reactive power by variable shunt susceptance [
Unlike the approaches presented in [
The VSC limit revision presented in [
Unlike [
The complementarity constraint concept together with the Fischer-Burmeister merit function (FBMF) has been envisioned as an attractive approach for directly including the operational limits of equipment into the power flow problem [
The stated individual contributions form the basis of an extended formulation to encompass multi-terminal VSC-based AC-DC power flow solutions in which all the VSC control variables such as the modulation index, converter phase angle, and VSC control modes of operation are combined with the state variables associated with the AC and DC grids for unified iterative solutions.
The remainder of the paper is organized as follows. The steady-state VSC model is elaborated in Section II. Section III details the representation of the VSC operational limits by means of equality constraints. The formulation and solution method of the multi-terminal AC-DC power flow problem is detailed in Section IV. Illustrative examples are presented in Section V, which emphasizes the performance of the new VSC-MTDC model. Section VI provides concluding remarks.
The derivation of the mathematical model representing the VSC steady-state operation is also based on the concept of an ideal complex tap-changing transformer [

Fig. 1 AC-DC converter station and transformer. (a) Schematic representation. (b) Positive-sequence equivalent circuit model.
Based on the equivalent circuit of VSC shown in
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
(8) |
The value of is used to represent the relationship between the DC voltage and line-to-line root-mean-square (RMS) voltage at the VSC ends.
In the context of converter power losses, the series resistance represents the ohmic losses, and the conductance represents switching losses. Note that when a more accurate equation for obtaining the VSC power losses is available, i.e., floss, this equation can be directly integrated into the proposed formulation by iteratively modifying the value of as .
The VSC operation involves the direct control of two state variables, i.e., the amplitude modulation index and VSC phase angle , which allows the control of the magnitude and phase angle of the VSC voltage at the AC side , respectively. Finally, since at the power flow solution, is followed from (2) and (4).
The control system of a VSC-based converter is based on an inner current control loop that controls the AC current and a set of outer controllers that supply the AC current reference values in the dq-frame for the inner current controller.
The choice of outer controllers depends on the desired control modes of operation for the VSC. In this context, the control targets are met by regulating the VSC voltage output with respect to the nodal voltage through the independent control of and . Thus, several control modes can be set as follows.
The magnitude of is adjusted through the modulation index to achieve one of the following control actions: ① to maintain the voltage magnitude of at a given set point by injecting or extracting the necessary reactive power; ② to ensure that the reactive power is exchanged with the AC grid to a specified value . These two control modes are performed as long as the modulation index is within its limit values. If a limit violation occurs, then the modulation index is fixed at that limit, and the controlled variable is freed. In this situation, the possible values of freed variable depend on the reactive power limits of VSC.
Three VSC control strategies can be specified by regulating the phase angle of through the phase angle . The first control action consists of maintaining a specified active power flow through the converter, where this power flow can be set to a constant value (i.e., ), or to a value that depends on the type of DC voltage droop control used for the VSC (i.e., ). In this latter case, three control approaches are considered: ① voltage-power (V-P) droop; ② voltage-current (V-I) droop; ③ V-P droop with a dead band. The corresponding equations for are given in Section III of [
The second control mode is associated with the voltage regulation at the DC side of converter to a specified value of by injecting or extracting the required active power . In this case, the converter becomes a DC grid slack converter because it will maintain the power balance and the capacitor charge of the DC grid.
In the last control strategy, the converter transfers the necessary active power to achieve an active power balance in the AC grid. In this case, the VSC can be regarded as an AC grid slack converter. In order to avoid convergence problems, these two slack control modes are not limited in the proposed formulation.
To guarantee that the power flow solution corresponds to a feasible steady-state operation of VSC, three limiting factors must be considered in the formulation: ① the upper and lower charge limits of DC capacitor; ② the operating range of modulation index;③ the maximum continuous current that the switching elements of converter can handle [
The current control scheme applied to the VSC is shown in

Fig. 2 Current control scheme.
The outer control of converter defines the desired d- and q-axis components of the AC current, denoted as and , respectively, based on the values of active and reactive power required to achieve the control targets. A current limiter control module receives and adjusts both d- and q-axis components of the desired current to ensure that the converter will not operate in an overcurrent condition. Finally, the current limiter supplies the reference values of both d- and q-axis currents to the inner current control to adjust the modulation index and phase angle . How these controllers operate dynamically is out of the scope of this study but can be found in [
By considering a voltage invariant transformation between the frames of reference abc and dq0, a phase-locked loop connected to phase A of node vi, and a balanced three-phase system, the constant voltages and currents in the dq framework can be directly related to the positive-sequence active and reactive power. In this case, the active and reactive power and as shown in
Based on this transformation of coordinates, the d- and q-axis components of the desired current at the iteration of the power flow solution process are calculated as:
(9) |
(10) |
For the sake of simplicity, the upper index k is removed under the assumption that the magnitudes of all currents are calculated at each Newton-Raphson iteration.
The desired current of the VSC is always constrained to by the current limiter control module. The current limiter technique can be in a vector form to satisfy this constraint, as detailed in [
(11) |
(12) |

Fig. 3 Current limiter techniques. (a) Vector limiter. (b) Active power priority.
Another type of current limiter technique gives priority to over , which translates into a priority for the control of active power [
Once the values of and have been obtained, it is possible to specify the converter operational limits in terms of active and reactive power instead of currents. In this case, the allowable range of values for is defined as . When the reactor resistance is neglected, it results in , which means that , i.e., . Based on the aforementioned, the reactive power limits of converter are computed by:
(13) |
(14) |
Finally, the converter losses are computed as such that the limits of the active power entering the DC node are given by:
(15) |
(16) |
The double-sided inequality constraints representing the VSC operational limits are included in the proposed formulation of the power flow problem by using complementarity constraints and the FBMF. To achieve this goal, inequality constraints are transformed into equality constraints as follows. First, the double-sided inequality constraints are expressed mathematically by two complementarity constraints, each of which is then transformed into a nonlinear equality constraint using the FBMF. Then, the two equality constraints are directly included in the power flow problem.
The complementarity constraint states that the product of the two variables and must be 0 (i.e., ) while satisfying the conditions and . This condition can be expressed as an equality constraint by using the FBMF [
(17) |
The manner in which is formulated depends on the physical limits that it represents.
A VSC can adjust one of its state variables within limits to maintain the value of another variable fixed at a given set point . However, when reaches one of its limits, is fixed at the violated limit, and the controlled variable is no longer regulated, and therefore . This behavior can be captured by the set of complementarity constraints of (18) under the assumption that when (), ().
(18) |
However, (18) does not entirely satisfy the complementarity constraint conditions when the violation of one limit occurs. This is due to the set of equations associated with a double-sided inequality constraint with only one controlled variable, where only one of the equations in (18) satisfies those conditions. This drawback is overcome by defining two independent complementarity auxiliary variables: and . Thus, (18) can be expressed as (19), which can then be transformed into a set of two equality constraints (20) by using the FBMF.
(19) |
(20) |
The pair of equality constraints in (20) representing the double-sided inequality constraint is directly incorporated into the set of power flow mismatch equations. The resultant set of nonlinear equations is entirely determined by adding and to the set of state variables to be solved in the power flow problem. In this case, and are initialized at 0 such that the complementarity constraint conditions are satisfied when is within limits.
Finally, a value bigger than 0 of one of these auxiliary variables indicates the extent to which the controlled variable deviates from its target value. This also indicates that the adjusted variable has been set at the corresponding limit. Since only one limit can be violated, one of the following two conditions are satisfied. If , then , , and ; if , then , , and . Note that in both cases, the complementary constraint conditions are satisfied.
In this proposal, the operating state of each VSC is constrained to and . On the other hand, all VSCs using the phase angle for controlling the active power , excluding those working in the or control mode, are also constrained to satisfy and .
Based on the information mentioned above, the set of equality constraints representing the double-sided inequality constraints associated with , , and in , and are given by (21)-(24), respectively.
(21) |
(22) |
(23) |
(24) |
A generator connected to the
(25) |
The control of a state variable xi or a function of state variables is mathematically represented by complementarity-based control constraints, which must consider the existing relationship between the outer controls and inner current control in the case of VSCs. This means that the constraints must consider the proper limits of the state variables that are adjusted for achieving the specified control target together with the limits associated with the VSC operational condition.
When the VSC modulation index is used to achieve the control mode, the voltage magnitude to be controlled is solved as a state variable in the power flow formulation, i.e., , subjected to the following complementarity-based control constraint:
(26) |
In this case, if ma has a value above its maximum limit at a given iteration of the solution process, its value steers towards during the remaining iterations required to solve the power flow problem. Thus, at the power flow solution, the final values of the variables composing (26) are , , , and .
A similar reasoning applies when during the solution of the power flow problem. In this solution, .
Since the tuning of ma modifies the amount of reactive power produced by the VSC to achieve the control mode, this reactive power could violate one of its bounds because of an overcurrent condition. In this case, the target control is not achieved even though . This operating condition is mathematically represented by including the auxiliary variables and associated with the VSC reactive power limits in (26). Thus, if (), the corresponding auxiliary variable has a value of (), which results in a voltage magnitude of ().
The aforementioned operating rationale is also applied when ma is tuned to control at a specified value , resulting in the following control constraint:
(27) |
The control mode performed by modulating the VSC phase angle is represented by:
(28) |
The amount of active power injected from the DC grid into the DC side of VSC is composed of two terms that directly depend on the value of the converter DC voltage . The first term causes to increase linearly with , whereas the second term causes to decrease quadratically with . Thus, an increment (decrement) in the value of will reduce (increase) the amount of active power exchanged between the DC grid and VSC. Within the context of this control mode, this implies that if violates one of its limits because of the network operating conditions, the active power will have a lower or higher value with respect to the target control. Thus, if (), the value of will satisfy (). This is why the auxiliary variables and are included in (28). However, the inner current control can limit the amount of active power flowing through the converter to avoid an overcurrent condition such that the auxiliary variables and must also be included in (28).
Finally, when the converter phase angle is used to perform the or control mode, (28) is replaced by (29) or (30), respectively.
(29) |
(30) |
The generalized power flow model for a multi-terminal AC-DC system can be categorized into AC grids and DC grids.
In the proposed formulation, the magnitudes and phase angles of voltages associated with the set of all AC nodes are considered with unknown state variables to be solved, that is, and in ,.
However, the pair of nodal equality constraints used in the power flow formulation depends on the manner in which each node is defined. In this context, the conventional active and reactive power mismatch equations (
(32) |
(33) |
The active power mismatch
(34) |
On the other hand, the reactive power mismatch
The pair of constraints associated with the set of slack nodes is given by:
(35) |
(36) |
The power mismatch constraints for the set of AC nodes of converters are given by:
(37) |
(38) |
where , , and are given by (3), (7), and (8), respectively. In addition, in is a state variable for the
Finally, the constraint that ensures the null exchange of reactive power between the DC and AC networks is given by:
(39) |
Assuming that the network is composed of a set of nodes .All nodal voltage magnitudes are state variables to be solved in the power flow problem: in , ; in ,,. Depending on the type of node, one single nodal constraint is defined as follows.
The active power mismatch constraint that must be satisfied is given by:
(40) |
where is computed from (1) for the
For the remaining nodes composing the set , the active power mismatch equations are given by:
(41) |
Finally, the duty cycle in of the DC-DC converters is associated with the converter capacity for controlling the active power injected into one of its terminals [
(42) |
Note that some vectors of x can be expressed as: ;; ;;; and . The solution of the power flow problem is obtained by iteratively solving for in the linearized problem , where J is the Jacobian matrix. From a given set of initial conditions , all state variables are updated at each iteration k, i.e., , until the maximum absolute value of is less than a specified tolerance tol or until the maximum number of iterations

Fig. 4 Unified power flow solution.
The effectiveness of the proposed approach is numerically demonstrated by using the CIGRE test system [

Fig. 5 CIGRE test system.

Fig. 6 Modified RTS-96 system.
In the proposed approach, the power flow study of the CIGRE test system is performed by considering the same current capacity reported in [
The converters connected to dc nodes 37 and 46 operate under the V-P droop control with a dead band from 0.98 p.u. to 1 p.u., reference voltages of 0.99 p.u., and set points of active power control of 800 MW and 1500 MW, respectively. In addition, the converter connected to dc node 42 uses a V-I droop with a reference voltage of 1.01 p.u. and a reference current of -5.9406 p.u.. For all droop controls, the slope is . In addition, the power losses of six VSCs are given by , where a is 1% of the converter nominal power , b is 0.3% of the converter nominal voltage , and [
The power flow solution converges in four iterations and 0.06 s to a tolerance of , with all converters operating within the specified limits listed in
In this case study of the CIGRE test system, the operating conditions of three converter stations are modified with respect to the base case used to demonstrate numerically the manner in which limit violations are correctly detected and enforced using complementarity constraints. The limits of these converters are changed as shown in
The power flow solution is obtained in seven iterations to a tolerance of 1×1
Finally, this case study has been newly solved by the proposed approach but by using an alternative iterative solution method based on a robust projected Levenberg-Marquardt (PLM) algorithm [
This section presents numerical comparisons of the results obtained by the proposed approach and a similar unified model proposed in [
The results obtained with the proposed model and [
By comparing both solutions, it clearly shows that the results are very similar in the modulation indices and active power flows but with differences in the amount of reactive power , which are caused by the manner in which each model represents the reactive power generation of converter. Note that corresponds to the reactive power that flows through the transformer in converter station, as shown in
The performance of the proposed approach is assessed by analyzing the IEEE RTS-96 system with the inclusion of two MTDC systems [
The power flow solution is obtained by the proposed approach in 10 iterations and 0.2 s to a convergence tolerance of 1
The proposed approach has been applied to a 7791-bus model of a large-scale interconnected power system consisting of seven regional transmission control areas. This system is composed of 7747 AC buses, 44 DC buses, 482 generator buses, 3493 load buses, 4182 transmission lines, and 4524 transformers. In this case, the 21 tie-lines interconnecting the seven regional control areas plus the 22 most important transmission lines of the system, are replaced by 11 point-to-point HVDC links and six DC multi-terminal networks. The solution for this system converges in seven iterations and 55 s to meet a tolerance of . In this case, four converters violate one of their operational limits according to the specified control modes, with one converter set at its lower DC voltage limit and one fixed at its maximum modulation index, while the iterative solution process enforces the current limits for the other two converters.
A comprehensive and flexible model of a VSC converter suitable for VSC-MTDC power flow studies is introduced in this paper. The model incorporates the outer and inner current control loops of the VSC for correct representation of its steady-state operating point. Several operational control modes are defined using the outer control schemes, which supply the current references to the inner current control loop. The inner current control loop is used to prevent overcurrent operating conditions. This is accomplished by constraining the current of AC converter using a vector-form control or by giving priority to the control of active power.
The concept of complementarity constraints and the FBMF are combined to represent directly the operating and physical limits of synchronous generators and VSCs in the power flow formulation. The formulation enables the automatic enforcement of violated limits during the iterative process of the power flow solution. In addition, complementarity-based control constraints are introduced to represent the specified VSC control modes of operation as well as the interactions between the various controls and their limits. The functionality of the VSC model and complementary constraints is demonstrated by numerical examples.
References
B. Mitra, B. Chowdhury, and M. Manjrekar, “HVDC transmission for access to off-shore renewable energy: a review of technology and fault detection techniques,” IET Renewable Power Generation, vol. 12, no.13, pp. 1563-1571, Sept. 2018. [Baidu Scholar]
C. Angeles-Camacho, O. L. Tortelli, E. Acha et al., “Inclusion of a high voltage DC-voltage source converter model in a Newton-Raphson power flow algorithm,” IEE Proceedings–Generation, Transmission and Distribution, vol. 150, no. 6, pp. 691-696, Nov. 2003. [Baidu Scholar]
X. -P. Zhang, “Multiterminal voltage-sourced converter-based HVDC models for power flow analysis,” IEEE Transactions on Power Systems, vol. 19, no. 4, pp. 1877-1884, Nov. 2004. [Baidu Scholar]
A. Pizano-Martinez, C. R. Fuerte-Esquivel, H. Ambriz-Perez et al., “Modeling of VSC-based HVDC systems for a Newton-Raphson OPF algorithm,” IEEE Transactions on Power Systems, vol. 22, no. 4, pp. 1794-1803, Nov. 2007. [Baidu Scholar]
J. Beerten, S. Cole, and R. Belmans, “Generalized steady-state VSC MTDC model for sequential AC/DC power flow algorithms,” IEEE Transactions on Power Systems, vol. 27, no. 2, pp. 821-829, May 2012. [Baidu Scholar]
M. Z. Kamh and R. Iravani, “Steady-state model and power-flow analysis of single-phase electronically coupled distributed energy resources,” IEEE Transactions on Power Delivery, vol. 27, no. 1, pp. 131-139, Jan. 2012. [Baidu Scholar]
Q. H. Nguyen, G. Todeschini, and S. Santoso, “Power flow in a multi-frequency HVac and HVdc system: formulation, solution, and validation,” IEEE Transactions on Power Systems, vol. 34, no. 4, pp. 2487-2497, Jul. 2019. [Baidu Scholar]
S. Khan and S. Bhowmick, “A generalized power-flow model of VSC-based hybrid AC-DC systems integrated with offshore wind farms,” IEEE Transactions on Sustainable Energy, vol. 10, no. 4, pp. 1775-1783, Oct. 2019. [Baidu Scholar]
Y. Ye, Y. Qiao, L. Xie et al., “A comprehensive power flow approach for multi-terminal VSC-HVDC system considering cross-regional primary frequency responses,” Journal of Modern Power Systems and Clean Energy, vol. 8, no. 2, pp. 238-248, Mar. 2020. [Baidu Scholar]
E. Karami, G.B. Gharehpetian, H. Mohammadpour et al., “Generalised representation of multi-terminal VSC-HVDC systems for AC-DC power flow studies,” IET Energy Systems Integration, vol. 2, no. 1, pp. 50-58, Mar. 2020. [Baidu Scholar]
E. Acha and B. Kazemtabrizi, “A new STATCOM model for power flows using the Newton-Raphson method,” IEEE Transactions on Power Systems, vol. 28, no. 3, pp. 2455-2465, Aug. 2013. [Baidu Scholar]
E. Acha, B. Kazemtabrizi, and L. M. Castro, “A new VSC-HVDC model for power flows using the Newton-Raphson method,” IEEE Transactions on Power Systems, vol. 28, no. 3, pp. 2602-2612, Aug. 2013. [Baidu Scholar]
E. Acha and L. M. Castro, “A generalized frame of reference for the incorporation of, multi-terminal VSC-HVDC systems in power flow solutions,” Electric Power Systems Research, vol. 136, pp. 415-424, Jul. 2016. [Baidu Scholar]
G. Díaz and C. González-Morán, “Fischer-Burmeister-based method for calculating equilibrium points of droop-regulated microgrids,” IEEE Transactions on Power Systems, vol. 27, no. 2, pp. 959-967, May 2012. [Baidu Scholar]
L. Sundaresh and P. N. Rao, “A modified Newton-Raphson load flow scheme for directly including generator reactive power limits using complementarity framework,” Electric Power Systems Research, vol. 109, pp. 45-53, Apr. 2014. [Baidu Scholar]
Y. Ju, J. Wang, Z. Zhang et al., “A calculation method for three-phase power flow in micro-grid based on smooth function,” IEEE Transactions on Power Systems, vol. 35, no. 6, pp. 4896-4903, Nov. 2020. [Baidu Scholar]
R. Tapia-Juárez, C.R. Fuerte-Esquivel, E. Espinosa-Juárez et al., “Steady-state model of grid-connected photovoltaic generation for power flow analysis,” IEEE Transactions on Power Systems, vol. 33, no. 5, pp. 5727-5737, Sept. 2018. [Baidu Scholar]
T. M. Haileselassie, “Control, dynamics and operation of multi-terminal VSC-HVDC transmission systems,” Ph.D. dissertation, Department of Electric Power Engineering, Norwegian University of Science and Technology, Trondheim, Norway, 2012. [Baidu Scholar]
A. Yazdani and R. Iravani, Voltage-sourced Converters in Power Systems: Modeling, Control, and Applications. New Jersey: Wiley, 2010. [Baidu Scholar]
T. K. Vrana, Y. Yang, D. Jovcic et al., “The CIGRE B4 DC grid test system,” Electra, vol. 270, no. 1, pp. 10-19, Jan. 2013. [Baidu Scholar]
A. Fischer, “A special newton-type optimization method,” Optimization, vol. 24, no. 3-4, pp. 269-284, Mar. 1992. [Baidu Scholar]
Y. Ju, J. Wang, and Z. Zhang, “An improved power flow method to cope with non-smooth constraints of integrated energy systems,” CSEE Journal of Power and Energy Systems, Early Access, pp. 1-10, Aug. 2020. [Baidu Scholar]
J. Beerten, MatACDC. (2021, Feb.). [Online]. Available: http://www.esat.kuleuven.be/electa/teaching/matacdc/ [Baidu Scholar]