Abstract
In this study, a coordinated voltage control strategy based on model predictive control (MPC) is proposed for offshore radial DC-connected wind farms. Two control modes are designed in this strategy. In the economic operation mode, the wind farm controller generates optimal active power references as well as bus voltage references of medium-voltage collector for DC-connected wind turbine (DCWT) systems and high-voltage DC/DC converters, where the goal is to minimize power losses inside the wind farm and ensure that voltages are within a feasible range, all while tracking the power references. In the voltage control mode, the main control objective for the wind farm controller is to minimize voltage deviations from the rated voltage. With the MPC, the control objective and operation constraints can be explicitly represented in the optimization problem while considering the dynamic response of the DCWT system. In addition, a sensitivity coefficient calculation method for radial DC-connected wind farms is developed to improve computational efficiency. Finally, DC-connected wind farms with 20 wind turbines are used to demonstrate the performance of the proposed strategy.
OFFSHORE wind power generation has become a significant power generation method because of its excellent continuity and wind energy capture capabilities [
To overcome these disadvantages, wind farms with DC collection and transmission have become the focus of recent studies [
To address these issues, DC offshore wind farms have motivated several recent studies. In [
Several configurations of DC-connected wind farms have been investigated [
In recent years, coordinated or optimal voltage control methods for wind farms have attracted considerable attention as means of improving the operational performances of wind farms. In [
Most previous studies have focused on the optimal voltage control of conventional AC wind farms. For a DC wind farm, the DC/DC converter topology and control scheme as well as the novel structure of an offshore DC-connected wind farm were studied. Few studies have focused on the optimal operation of radial DC-connected wind farms to achieve special control objectives. For a large-scale radial DC-connected wind farm, the terminal voltage of WTs may violate the voltage threshold as the active power output increases, significantly affecting secure operations of wind farms. In addition, the economic operation of DC-connected wind farms should also be addressed.
Therefore, coordinated voltage control strategies based on MPC have been proposed for offshore radial DC-connected wind farms. The MPC is used in multi-input multi-output systems to formulate an optimal voltage control problem that considers the dynamics of DC-connected wind turbines (DCWTs) while obtaining an optimal solution for a multi-objective control. The MPC has the following advantages for offshore radial DC-connected wind farms: ① the control objective and operation constraints with multi-objective control periods can be explicitly represented in the optimization problem; ② the dynamic response of DCWTs can be formulated as an optimization problem to obtain a more effective solution; ③ it can optimally coordinate the DCWTs inside the wind farm with different control parameters.
A DCWT system and high-voltage DC/DC converter models are presented in this study. Two control modes i.e., economic operation and voltage control modes are designed. If all terminal bus voltages are maintained within a safe range, the wind farm operates in the economical operation mode. The main control objective is to minimize the power losses inside the wind farm and maintain voltages within a safe range while tracking the power command from the transmission system operator (TSO). Once any terminal bus voltage violates the voltage threshold, the wind farm controller switches to the voltage control mode. The main control objective for the voltage control mode is to mitigate all terminal bus voltage deviations of DCWTs from the rated voltage.
The main contribution of this study is the proposal of an MPC-based coordinated voltage control strategy for a radial DC-connected wind farm. The high-voltage DC/DC converter and DCWTs are optimally coordinated in this control strategy to maintain DCWT voltages within a safe range, to ensure the economic operations of the wind farm, and to track the dispatch command from the TSO simultaneously. The wind farm controller that combines the economic operation and voltage control modes can provide more flexibility for the control of radial DC-connected wind farms. In addition, a sensitivity coefficient calculation method for radial DC-connected wind farms is developed to improve computational efficiency.
The remainder of this paper is organized as follows. Section II provides the configuration of an offshore radial DC-connected wind farm and an overview of the proposed control strategy. Section III presents a sensitivity coefficient calculation method for radial DC-connected wind farms, and Section IV presents the predictive modeling of DC-connected wind farms. The MPC-based coordinated voltage control strategy is proposed in Section V. Simulation results and a case study are presented and discussed in Section VI, and conclusions are drawn in Section VII.
II. Configuration of Offshore Radial DC-connected Wind Farm and Overview of Proposed Control Strategy
The configuration of an offshore radial DC-connected wind farm is shown in

Fig. 1 Configuration of an offshore radial DC-connected wind farm.
The radial DC-connected wind farm has a simpler power-conversion chain and is routine. This type of MVDC collection system will have greater reliability and is more efficient because of the use of fewer power conversion sections and no AC cable losses.

Fig. 2 Structure of proposed control strategy.
Two control modes are designed for different operation conditions. If all bus voltages are within a feasible range, the radial DC-connected wind farm operates in the economic operation mode. The wind farm controller coordinates the high-voltage DC/DC converter and DCWTs to minimize network losses while regulating all node bus voltages within a safe range. If any bus voltage violates the limits, the wind farm operates in the voltage control mode. The control objective is to minimize bus voltage deviations from the rated voltage.
In conventional optimization problems, the voltage sensitivity coefficients are obtained by iteratively updating the Jacobian matrix. The calculation involves non-trivial computational constraints for implementation in real-time control problems and cannot calculate the sensitivity coefficients with respect to the slack bus voltage [

Fig. 3 Circuit of radial DC-connected wind farm.
As each feeder is connected to a common slack bus, the voltages in each feeder can be controlled independently. Let us consider a single feeder inside the wind farm as an example to calculate the sensitivity coefficients with respect to the slack bus voltage. The
(1) |
where is the resistance of the line segment between the (
(2) |
can be treated as the power reference for the
(3) |
During each control period, the power reference is constant. Then, we can obtain:
(4) |
Since the power losses are small compared with the power flows themselves, the third item in (2) is much smaller than the first and second items, which can be ignored. Then, we can obtain:
(5) |
Given the remaining partial derivatives with respect to the voltage magnitude , can be obtained using the same process. Once all are obtained, the partial derivatives of with respect to the voltage magnitude of the slack bus can be expressed as:
(6) |
We denote S and M as sets of slack buses and buses with injections, respectively. The link between the bus voltages and power injections can be expressed as:
(7) |
where is the admittance. The partial derivatives of with respect to the active power can be calculated as:
(8) |
In [
The power losses in the radial DC-connected wind farm can be expressed as:
(9) |
The partial derivatives of the power losses with respect to the voltage magnitude can be calculated by:
(10) |
Then, the sensitivity with respect to the power output of WTGs and terminal bus voltage can be obtained as:
(11) |
where y consists of the active power output of DCWTs and the slack bus voltage.
In this study, the high-voltage DC/DC converter adopts a dual active bridge (DAB) topology, which is one of the most attractive topologies for high-power DC/DC applications. DAB consists of two full-bridge converters and a medium-frequency transformer. The primary bridge converter converts the DC voltage to AC voltage, and the secondary bridge converter regulates the DC voltage of the secondary terminals by rectifying the secondary side of the transformer. With the DAB topology, bidirectional power flow is possible.
Because of the voltage limit of the DAB converter, it is difficult to meet the HVDC transmission voltage level when using a single DAB converter. In [

Fig. 4 Topology of DAB converter system.
The wind farm control period (in seconds) is considerably longer than the dynamic response of the DC/DC converter (approximately 100 ms). Then, when the effects of the time delay of the communication system and dynamic response of the high-voltage DC/DC control system are considered, the dynamic behavior of the power control loops of the DC/DC control system can be described by a first-order lag function as:
(12) |
where denotes the change in a variable during the control periods; and is the voltage control time constant of DC/DC converter. Then, we can obtain:
(13) |
A DCWT is composed of a WTG, an AC/DC converter, and an MV DC/DC converter. The AC/DC converter is used to regulate the active power output of the WTG by controlling the q-axis current of the generator stator. The DC/DC converter is used to provide a constant DC voltage for the AC/DC converter and to transfer power to the MV collection system. The MV DC/DC converter adopts a DAB topology, which is similar to that of the high-voltage DC/DC converter.
A permanent magnet synchronous generator (PMSG) is adopted in this study. The control scheme of the AC/DC converter is shown in

Fig. 5 Control scheme of AC/DC converter.
The control scheme implemented in the AC/DC converter realizes the decoupled control of active and reactive power, which is achieved by q-axis current regulation in a stator flux-oriented synchronously rotating reference frame. The d-axis current reference is set to be zero. Since the DC/DC converter is used to control the DC-link voltage stability, the power flowing inside the DC/DC converter can be described by a first-order lag function. Ignoring the power loss inside the generator and converter, the active power output of the DCWT can be written as:
(14) |
where is the electromagnetic torque of the PMSG; is the mechanical speed of the generator; and is the number of pole pairs. Thus, the active power control loop of the DCWT can be obtained, as shown in

Fig. 6 Active power control loop of DCWT.
Here, and are the proportional and integral gains of the PI controller, respectively; is the time constant of the current loop; is the time constant of the filter; is the generator output power; and is the output power time constant of the DC/DC converter. Thus, the incremental mode of the active power control loop of the DCWT can be obtained as:
(15) |
(16) |
(17) |
(18) |
where is the auxiliary variable of active power control loop. The incremental mode of the DCWT can be expressed in matrix form as:
(19) |
where
Accordingly, the continuous state space model of a radial DC-connected wind farm with DCWTs and a high-voltage DC/DC converter can be formulated as:
(20) |
where
where and are the state variable and control variable, respectively; and and are the coefficients of the incremental mode of the DCWT1. Based on the continuous time model, the discrete time state space model with sampling time can be expressed as:
(21) |
where ; and .
Two control modes based on MPC are designed in this study for a radial DC-connected wind farm, i.e., economical operation and voltage control modes. The control principle of the two modes is illustrated in

Fig. 7 Control principle of economic operation and voltage control modes.
The wind farm controller should maintain all terminal bus voltages within a safe range while tracking the power command from the TSO. The upper/lower mode decision voltages are set to determine the wind farm control mode. The maximum/minimum controlled voltages are set for the economic operation mode to maintain all terminal bus voltages within a safe range. A deadband is set between the upper/lower mode decision voltages and the maximum/minimum controlled voltages. If all terminal bus voltages are maintained in the safe range, the radial DC-connected wind farm operates in an economic operation mode. The main control objective is to minimize the power losses inside the wind farm and maintain voltages within the safe range while tracking the dispatch command. Once any terminal bus voltage hits the upper/lower-mode decision voltages, the wind farm changes the control mode to voltage control mode. The voltage control mode continues to operate several control periods after the voltage hits the upper/lower-mode decision voltages. The main control objective for the voltage control mode is to minimize the DCWT terminal bus voltage deviation from the rated voltage.
1) Objective 1: the first objective for the economic operation mode is to minimize the loss, which is given by:
(22) |
(23) |
(24) |
(25) |
where is the output power of the
2) Objective 2: to achieve fair active power sharing among DCWTs, the second control objective for the economic operation mode is to minimize the active power output deviation from the proportional distribution (PD) based reference.
(26) |
The predictive value can be calculated by:
(27) |
where is the initial output power of the
3) Objective 3: to smooth the active power output of the wind farm, the third control objective for the economic operation mode is to minimize the active power variations of the DCWTs.
(28) |
Then, the cost function can be expressed by:
(29) |
where , , and are the weighting coefficients.
In the voltage control mode, the main control objective is to minimize the voltage deviation from the rated voltage as:
(30) |
Then, the voltage deviation can be expressed as:
(31) |
where is the measured value at the initial moment of DCWT; and is the rated value of DCWT terminal voltage.
The economic operation and fair active power sharing among DCWTs are also considered in this mode. Then, the cost function is obtained as:
(32) |
where is the weight required to minimize voltage deviation. Since voltage control is the main control objective in this mode, is set to be larger than and .
Each DCWT terminal bus must be maintained within a safe range according to its predictive voltage .
(33) |
where and are the maximum and minimum controlled voltages, respectively. can be obtained by:
(34) |
where and are the reference and initial voltages, respectively. These constraints are used only in the economic operation mode.
The voltage reference at the DC collector bus is constrained by:
(35) |
where and are the minimum and maximum voltage limits, respectively; is the initial voltage; and is the maximum ramp rate. The active power of the DCWT is constrained as:
(36) |
where is the available wind power.
The radial DC-connected wind farm must track the power reference from the system operators, which can be expressed as:
(37) |
where is the active power reference of the DCWT.
The DC-connected wind farm depicted in
Scenario | Control mode |
---|---|
1 | Combined economic operation mode and voltage control mode |
2 | Voltage control mode |
3 | PD control mode |
In the PD control mode, referring to the available active power, the active power reference value of
(38) |
where is the utilization ratio.

Fig. 8 Total available wind power and dispatch command.
The performance of the controlled MV collector bus voltage in different scenarios is shown in

Fig. 9 Performance of controlled MV collector bus voltage in different scenarios.
Figures

Fig. 10 Terminal bus voltage of DCWT8 in different scenarios.

Fig. 11 Terminal bus voltage of DCWT14 in different scenarios.

Fig. 12 Active power output of radial DC-connected wind farm.

Fig. 13 Active power output of DCWT8 in different scenarios.

Fig. 14 Active power output of DCWT14 in different scenarios.

Fig. 15 Power losses of radial DC-connected wind farm.
The total voltage deviations of wind farm, average power losses of wind farm, and total active power deviations of the WTs from their PD references are listed in
Scenario | Total voltage deviation (p.u.) | |||
---|---|---|---|---|
0-150 s | 150-300 s | 300-450 s | 450-550 s | |
1 | 0.26695 | 0.28558 | 0.30472 | 0.36947 |
2 | 0.15843 | 0.20184 | 0.30472 | 0.28296 |
3 | 0.24259 | 0.31243 | 0.54260 | 0.73401 |
Scenario | Average power loss (MW) | |||
---|---|---|---|---|
0-150 s | 150-300 s | 300-450 s | 450-550 s | |
1 | 1.2188 | 2.5412 | 3.0121 | 2.6952 |
2 | 1.1103 | 2.3184 | 3.0121 | 2.8183 |
3 | 1.2832 | 3.6733 | 2.8162 | 2.5129 |
Scenario | Total active power deviation (MW) | |||
---|---|---|---|---|
0-150 s | 150-300 s | 300-450 s | 450-550 s | |
1 | 6.8234 | 4.2112 | 0 | 0 |
2 | 17.1260 | 14.3860 | 0 | 0 |
3 | 0 | 0 | 0 | 0 |
The terminal bus voltages of DCWT8 and power losses of the wind farm with different weights are shown in Figs.

Fig. 16 Terminal bus voltages of DCWT8 under different weights.

Fig. 17 Power losses of wind farm with different weights.
Weight | |||
---|---|---|---|
1 | 0.1 | 50 | 20 |
2 | 3.0 | 2 | 20 |
3 | 1.0 | 30 | 20 |
A coordinated voltage control strategy based on MPC is proposed in this study for a radial DC-connected wind farm. Two control modes are designed for a radial DC-connected wind farm to maintain DCWT terminal bus voltages within a feasible range while considering wind farm economic operation while tracking the power command from the TSO. An analytical sensitivity coefficient calculation method is developed for a radial DC-connected wind farm to improve computational efficiency. In a case study, three scenarios are compared in a simulation. In terms of wind farm secure operations, the wind farm in the voltage control mode proves to be better than that in the PD control mode or in the mode that combines the economic and voltage control modes. From an economic point of view, combining the economic and voltage control modes is better, as it provides more flexibility for radial DC-connected wind farm control.
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