Abstract
The power loss minimization and DC voltage stability of the multi-terminal direct current (MTDC) system with large-scale wind farm (WF) cluster affect the stability and power quality of the interconnected power grid. This paper proposes a distributed optimal voltage control (DOVC) strategy, which aims to optimize voltage distribution in MTDC and WF systems, reduce system power losses, and track power dispatch commands. The proposed DOVC strategy employs a bi-level distributed control architecture. At the upper level, the MTDC controller coordinates power flow, DC-side voltage of grid-side voltage source converters (GSVSCs), and WF-side voltage source converters (WFVSCs) for power loss minimization and DC voltage stabilization of the MTDC system. At the lower level, the WF controller coordinates the controlled bus voltage of WFVSC and the active and reactive power of wind turbines (WTs) to maintain WT terminal voltages within feasible range. Then, the WF controller minimizes the power loss of the WF system, while tracking the optimal command from the upper-level control strategy. Considering the computational tasks of multi-objective optimization with large-scale WF cluster, the proposed DOVC strategy is executed in a distributed manner based on the alternating direction method of multipliers (ADMM). An MTDC system with large-scale WF cluster is established in MATLAB to validate the effectiveness of the proposed DOVC strategy.
THE offshore wind power has attracted extensive attention due to its excellent wind energy capture capability and abundant offshore wind energy resources [
As large-scale WF cluster is integrated into the power grid via the MTDC system, the inherent randomness and volatility of wind power, coupled with the lower short-circuit power contribution of wind turbines (WTs) and MTDC converters, can lead to significant voltage fluctuations and even voltage violations under disturbances. Given that the voltage distribution of the power system is influenced by both the power output of WTs within each WF and the power flow among VSCs in the MTDC system, the key to maintaining all bus voltages and WT terminal voltages within a feasible range lies in developing efficient voltage and power regulation control methods.
Voltage and power controls of WFs have been extensively studied in recent years. Reference [
To ensure the stable operation of VSC-MTDC systems, maintaining DC voltage stability is crucial. Generally, control methods for DC voltage regulation in MTDC systems can be classified into two main categories: master-slave methods and voltage droop control methods [
However, as the size and number of MTDC and WFs increase, solving a global optimization problem that involves large-scale constraints from both the MTDC system and WFs in real-time becomes increasingly complex and challenging [
Most of existing voltage optimization control strategies are designed for AC WFs or VSC-HVDC-connected WFs, with a limited focus on voltage regulation for MTDC systems with large-scale WF cluster. The optimization of the VSC-MTDC system and the power tracking for GSVSC are often neglected, which can significantly impact both the economic operation and stability of the power system. The coordination between WFs and the MTDC system is essential to achieve optimal operation across the entire system. As WFs and MTDC systems expand, developing fast and efficient solutions to large-scale optimization problems becomes crucial for achieving real-time system optimization. This paper proposes a distributed optimal voltage control (DOVC) strategy for the MTDC system with large-scale WF cluster based on ADMM. The model of the MTDC system with large-scale WF cluster includes the WT, the WF-side voltage source converters (WFVSC), and the grid-side voltage source converter (GSVSC) models. The DOVC strategy aims to maintain the voltages of WTs inside each WF and VSCs of the MTDC system within a feasible range while minimizing the overall grid power losses. The global optimization problem is divided into subproblems, which are solved in parallel using the ADMM on MTDC and WF controllers, respectively. Through the proposed DOVC strategy, the MTDC controller and WF controllers solve the optimization problem in a distributed manner, ensuring global optimality without any compromise. The main contributions of this paper are summarized as follows:
1) A DOVC strategy for the MTDC system with large-scale WF cluster is proposed to achieve voltage regulation for the VSCs of MTDC and WTs within WFs while minimizing grid power losses in both the MTDC system and WF cluster. The entire system, including the WT, the WFVSC, and the GSVSC models, is established. Through the proposed DOVC strategy, WT power output and the DC-side voltage of the VSCs are coordinated to realize effective control performance for the entire system.
2) A bi-level distributed control architecture is designed. The upper-level MTDC controller solves the optimization problem related solely to the MTDC system and updates the global variables with global constraints, while each lower-level WF controller addresses the optimal problem under local variable constraints. This method reduces control complexity and ensures global optimization across the entire system.
3) To efficiently solve the large-scale multi-objective optimization problem for the MTDC system connected to the WF cluster, an ADMM-based solution method is proposed to execute the proposed DOVC strategy to distribute the computational burden of multi-objective optimizations. Each WF controller only exchanges information with the MTDC controller, and certain information between the WF controller and WT controllers is exchanged, enhancing system privacy and reducing data exchange requirements.
The rest of this paper is organized as follows. Section II provides a DOVC strategy architecture. Section III introduces the mathematical model of the MTDC system with large-scale WF cluster. Then, the framework of the DOVC strategy and the ADMM-based solution are given in Section IV. Simulation results are presented and discussed in Section V. Finally, the conclusions are drawn in Section VI.

Fig. 1 Configuration of an MTDC system with large-scale WF cluster.
The MTDC system forms a meshed grid comprising WFVSCs and GSVSCs. Each WF is connected to a WFVSC via an high-voltage (HV) or medium-voltage (MV) transformer. The WFVSC is responsible for providing stable slack bus voltage for the WF and transferring the wind power from the WF side to the MTDC system. The GSVSC converts the DC power output from WFVSC into three-phase AC power for direct connection to the AC grid, which is responsible for transmitting offshore power from the MTDC system to the onshore AC grid. The interconnected VSCs of the MTDC system are connected through HV cables, while each WFVSC is connected to its corresponding WF through 155 kV submarine cables. The WTs are interconnected via MV 33 kV collector cables, with the WTs spaced 4 km apart.
The DOVC control structure is illustrated in

Fig. 2 DOVC control structure.
According to the power balance theorem, the upper-level controller calculates the active power reference for each WF and sends it to the lower-level WF controller. The lower-level WF controller regulates and manages the active and reactive power references for WTs within each WF, aiming to keep within feasible ranges, minimize power losses for WF system, and track the active power commands from the upper-level controller.
To reduce the computational burden on the system, the global optimal control problem is divided into subproblems, which are solved in parallel by the MTDC and WF controllers using the ADMM framework. Each WF controller only exchanges limited information with the MTDC controller. The MTDC controller continuously solves for , and distributes them to the WF controllers based on and received from the WF controllers.
In this section, the model of the MTDC system with large-scale WF cluster is introduced. The simplified structure of the MTDC system with large-scale WF cluster is shown in

Fig. 3 Simplified structure of MTDC system with large-scale WF cluster.
The voltage and current dual closed-loop control structure of the WFVSC is illustrated in

Fig. 4 Voltage and current dual closed-loop control structure of WFVSC.
The three-phase AC voltage is decoupled into d-q axis components for independent control, and the simplified voltage control structure of the WFVSC is presented, as shown in

Fig. 5 Simplified voltage control structure of WFVSC.
The mathematical model of WFVSC can be described by:
(1) |
(2) |
(3) |
where is the controlled bus voltage reference increment, with superscript ref indicating the reference value and indicating the initial time; is the increment of the d-axis component of the controlled bus voltage of WFVSC; is the auxiliary variable, denoting the integral gain of ; and are the proportional and integral gains of the proportional integral (PI) controllers of the outer control loop, respectively; and is the time constant for the inner loop.
According to (1)-(3), the mathematical model of the continuous state space model of WFVSC can be formulated as:
(4a) |
(4b) |
(4c) |
where is the state variable vector of the WFVSC system; is the control variable; is the state matrix; and is the control matrix.
The voltage control structure of the GSVSC is shown in

Fig. 6 Voltage control structure of GSVSC.

Fig. 7 Simplified voltage control structure of GSVSC.
Unlike the WFVSC, the DC-side voltage of GSVSC is measured after being filtered by a capacitor. As a result, a capacitor filtering stage is incorporated before obtaining the DC-side voltage measurement for the GSVSC. The mathematical model of the GSVSC can be expressed as:
(5) |
(6) |
(7) |
(8) |
where and are the current and voltage on the DC side of GSVSC, respectively; is the DC-side voltage reference of GSVSC sent by the MTDC; is the introduced auxiliary variable denoting the integral gain of ; and C is the capacitance of the DC-side capacitor.
According to (5)-(8), the mathematical model of the continuous state space model of GSVSC can be obtained as:
(9a) |
(9b) |
(9c) |
where is the state variable vector of the GSVSC system; is the control variable of the GSVSC system; is the state matrix; and is the control matrix.
In steady-state analysis, the DC power flow is determined by the line resistances and the voltage drop differences between the interconnected DC buses. The current injection of the
(10) |
where is the conductivity between node i and node j of the MTDC system; is the DC-side voltage of the WFVSC; and is the DC-side voltage of the GSVSC. The active power of the WFVSC that injects to the MTDC system can be written as:
(11) |
Similarly, the active power that transfers to the AC grid through the GSVSC can be obtained as:
(12) |
of the MTDC system can be rewritten by combining (10)(12) as:
(13) |
where is the total number of VSCs in the MTDC system.
from the WFVSC to MTDC is obtained according to Taylor expansion of (13), which is described as:
(14) |
The active and reactive power outputs of WT can be adjusted independently by the equipped full power converter with the decoupling control of the converter. and are the active and reactive power current measurements, respectively. By defining the active and reactive power increment vectors as and , respectively, the dynamic of WTs can be represented as [
(15) |
(16) |
where and are the time vectors of active and reactive control loops, respectively.
The continuous state space model of WF can be described as:
(17a) |
(17b) |
(17c) |
(17d) |
(17e) |
where and are the state and control variable vectors of the WF system, respectively; and and are the state and control matrixes, respectively.
The WF controlled AC bus voltage increment can be affected by the terminal voltage increment of WFVSC and the WT power outputs, which is expressed as:
(18) |
where , , and are the sensitivity coefficients of the WF-controlled AC bus voltage with respect to active power vector , reactive power vector , and WFVSC terminal voltage, respectively.
For the MTDC system with largescale WF cluster, the interaction between the MTDC system and WF cluster poses significant challenges for system controllers. Several factors must be considered to enhance the overall performance of the system, including minimizing grid power losses in the MTDC system and WF cluster, maintaining the voltages of WTs within each WF close to their rated levels, and achieving optimal active power distribution among the WFs. The WF control system is designed with three key control objectives.
1) The first control objective is the power loss of the WF, which can be calculated by:
(20) |
(21) |
(22) |
where is the power loss at time ; is the prediction value of power loss at time ; , , and are the sensitivity coefficients of the power loss of WF cluster to the active power vector of WTs, reactive power vector of WTs, and WFVSC terminal voltage, respectively; is the prediction step; Gij is the conductance between nodes i and j of the WF system with N nodes; and is the phase angle difference between nodes i and j.
2) The second control objective is the WF bus voltage deviation from the rated voltage.
(23) |
where is the predictive value of voltage deviation of the bus to its reference value , which can be expressed by (24).
(24) |
where is the voltage measurement of the
3) When WT tracks active power based on the proportional distribution (PD) strategy, the maximum available active power of each WT is taken into account, and its available reactive capacity can be maximized. The active power of each WT should be as close as possible to the scheduling result of the PD strategy. The third cost function can be written as:
(25) |
The deviation between the active power output of the
(26) |
where is the active power output of the
(27) |
where is the available active power of the WT.
To reduce the power loss of the entire system and improve its economy, the control objective of MTDC focuses on the active power loss of the MTDC system:
(28) |
(29) |
where is the DC-side voltage vector of VSCs in the MTDC system; and is the DC-side control matrix.
According to (20), (23), (26), and (28), the overall cost function of the MTDC system is obtained as:
(30) |
where , , , and are the weighting coefficients for , , , and , respectively.
1) WF system constraints: the active power and reactive power of WTs in WF are constrained by:
(31) |
(32) |
where is the reactive power output of the WT; is the reactive power output increment of the WT at time ; and and are the minimum and maximum available reactive power of the WT, respectively.
The output power flow of the WF to the DC system can be expressed by:
(33) |
where is the power loss of the
The terminal voltage increment of the
(34) |
where is the terminal voltage of the
2) MTDC system constraints: the active power flow between the
(35) |
where is the rated active power of the DC cable connecting the WFVSC and the GSVSC.
The GSVSC is required to track the WF power reference from system operators. The PI-based dynamic controller is introduced to eliminate the active power output error of
(36) |
(37) |
(38) |
(39) |
where and are the active power reference and measurement transferred to the AC grid through the GSVSC, respectively; is the difference between the scheduling command for active power output of the GSVSC and the active power output measurement of GSVSC at the previous control instant; and are the scheduling command and measurement of the active power output for the
is constrained by:
(40) |
where and are the minimum and maximum DC-side voltages of the
With the continuous expansion of MTDC system and WF cluster, the optimization problem for the entire system evolves into a large-scale, multi-input, multi-output optimization problem with extensive constraints. Utilizing a centralized solution method places a significant computational burden on the central controller, making real-time control difficult to achieve.
To reduce computational burden, an ADMM-based solution method is proposed for the DOVC strategy. The DOVC strategy distributes the computational tasks between the MTDC controller and the individual WF controllers. The optimization problem is decomposed into separate parts for the MTDC and each WF, with the WFVSC active power outputs serving as the common variables. Consequently, the optimization problem (30) can be distributed to the MTDC controller and WF controllers and processed in parallel while ensuring global optimality [
(41) |
where and are the boundary active power flows of the
By defining the global variable as the active power output vector of the WFs, as the local variable vector obtained by the WF controllers, and as the global variable vector obtained by the MTDC controller, the augmented Lagrangian function can be described as:
(42) |
where is the penalty factor vector; and is the dual-variable vector.
The
(43) |
where is the
The initial values of and are set to be 0. The iteration process is as follows.
1) The MTDC controller only needs to solve the power loss optimization problem of the DC system. The DC-side voltage reference vector of VSCs and global variable vector can be updated by:
(44) |
where the superscript r denotes the iteration step.
2) The
(45) |
3) is also updated in the WF controller, which can be updated by:
(46) |
Then, the dynamic is updated. With J as the primal residual vector and h as the dual residual vector, Jk and hk are the
(47) |
(48) |
(49) |
where , , and are the penalty coefficients.
4) The convergence is checked by:
(50) |
where and are the tolerable boundaries of Jk and hk, respectively.
The flowchart of the proposed ADMM-based solution method is shown in

Fig. 8 Flowchart of proposed ADMM-based solution method.
The structure of the testbed system is shown in

Fig. 9 Structure of testbed system.
Equipment | Parameter |
---|---|
33 kV cable | Ω/km, mH/km, μF/km |
155 kV cable | Ω/km, mH/km, μF/km |
400 kV cable | Ω/km |
33 kV/155 kV transformer | MVA, p.u., p.u. |
0.9 kV/33 kV transformer | MVA, p.u., p.u. |
HVDC converter | MVA |
WT converter | MVA |
To evaluate the performance of the proposed DOVC strategy, two traditional control strategies are selected for comparison: the centralized optimal voltage control (COVC) strategy and the two-tier optimal control (TOC) strategy.
1) In the COVC strategy, all optimization problems are solved by a central controller.
2) In the TOC strategy, the optimization problem of the MTDC system with large-scale WF cluster is divided into two independent optimization problems that are solved sequentially. One focuses on optimizing the MTDC system, while the other addresses the WF optimization problems.

Fig. 10 Active power output tracking performance of GSVSC with proposed DOVC strategy. (a) With error elimination. (b) Without error elimination.
The active power losses with DOVC, COVC, and TOC strategies are shown in

Fig. 11 Active power losses with DOVC, COVC, and TOC strategies.
In

Fig. 12 Voltages of WTs with DOVC, COVC, and TOC strategies. (a) Voltage of WT12 in WF1. (b) Voltage of WT8 in WF2.

Fig. 13 DC-side voltage of GSVSC2 with DOVC, COVC, and TOC strategies.

Fig. 14 Iteration curves of and . (a) and . (b) and . (c) and .
We test the proposed DOVC, COVC, and TOC strategies using 64 WTs, 128 WTs, and 256 WTs on a personal computer (Intel Core i7-11700KF, 32 GB RAM). The comparison of computation time is presented in
Strategy | Computation time (s) | ||
---|---|---|---|
64 WTs | 128 WTs | 256 WTs | |
COVC | 0.966 | 4.881 | 32.709 |
TOC | 0.841 | 3.164 | 25.125 |
DOVC | 0.751 | 1.694 | 7.285 |
By decomposing the global optimization problem into smaller subproblems that can be solved in parallel, the proposed DOVC strategy strikes a balance between computational efficiency and optimized performance. The proposed DOVC strategy has shorter computation time than both COVC and TOC strategies, reducing computation time by 77.73% and 71%, respectively, for the system with 256 WTs. The proposed ADMM-based solution method demonstrates excellent scalability, avoiding the need for centralized computation and intensive data exchange, ensuring that the computation time remains manageable even when the size of the WF increases.
This paper proposes DOVC strategy for MTDC system with large-scale WF cluster based on ADMM. The proposed DOVC strategy optimizes system operation by minimizing power losses, reducing terminal voltage deviations of WTs, and achieving optimal active power distribution. The proposed ADMM-based solution method is used to decompose the large-scale optimization problem into several sub-optimization problems. The computational burden is reduced and the real-time control for the MTDC system with large-scale WF cluster can be ensured. Simulation results demonstrate that the proposed DOVC strategy achieves control performance compared with the COVC strategy in terms of minimizing voltage deviations and power losses while outperforming the TOC strategy. Additionally, as the scale of the WF increases, the computation time of the proposed DOVC strategy is significantly lower than that of the COVC strategy. The proposed DOVC strategy enhances the dynamic response, voltage stability, and overall efficiency of the MTDC system with large-scale WF cluster.
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