Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Model Predictive Control Based Coordinated Voltage Control for Offshore Radial DC-connected Wind Farms  PDF

  • Wu Liao
  • Qiuwei Wu
  • Hesong Cui
  • Sheng Huang
  • Yusheng Gong
  • Bin Zhou
the College of Electrical and Information Engineering, Hunan University, Changsha, China; Electrical Technology and Economics of Machinery Industry Institute of Beijing, Beijing, China; the Department of Electrical Engineering, Technical University of Denmark, Lyngby, Denmark; Electric Power Research Institute, State Grid Hunan Electric Power Co., Ltd., Changsha, China

Updated:2023-01-25

DOI:10.35833/MPCE.2020.000685

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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.

I. Introduction

OFFSHORE wind power generation has become a significant power generation method because of its excellent continuity and wind energy capture capabilities [

1]. With the rapid development of offshore wind farms, the researches of cost-effective collection and transmission of large-scale offshore wind power have increased. Wind turbines (WTs) with higher wind power are typically installed at a remote distance from the shore, and long-distance power transmission with submarine cables causes many problems in conventional AC wind farms such as large reactive currents, higher costs, and power losses [2], [3].

To overcome these disadvantages, wind farms with DC collection and transmission have become the focus of recent studies [

4]. Voltage source converter based high-voltage direct current (VSC-HVDC) transmission technology is preferable for use in long-distance power transmission [5], [6], as it can eliminate many of the large filters required in conventional high-voltage direct current (HVDC) technology. The costs of transportation, installation, and maintenance are reduced. However, in conventional offshore wind farms, the collection system must still be operated using AC networks. Heavy AC transformers must be installed in the WT towers to meet the voltage level of collection systems. In addition, a large-scale offshore AC/DC converter station is required.

To address these issues, DC offshore wind farms have motivated several recent studies. In [

7], it was shown that wind farms with DC grids are reasonable for future wind farms. Compared with existing AC collector systems with VSC-HVDC, a full DC wind farm has several advantages such as the potential to eliminate large AC transformers, increased efficiency and power density, and reduced sizes and weights of cables [8].

Several configurations of DC-connected wind farms have been investigated [

9]-[15]. In [9], an all-DC wind farm with series-connected turbines was proposed as an alternative to the classical AC parallel or radial wind farms. In [10], a series interconnecting method of two or more sets of wind turbine generators (WTGs) used in a wind farm was proposed, and the basic characteristics of the integrated WT generation system were discussed. In [11], the design of an offshore wind farm using a DC parallel-connected grid based on resonant DC/DC converters was proposed. In [12], the redundancy of an HVDC transmission system under faults was studied, and a fault-ride-through strategy for an offshore DC parallel-connected wind farm was proposed. In [13], an analytical design method was proposed for the DC collector systems of offshore wind farms, and an improved topology was introduced to overcome the limitations of the conventional DC series-parallel topology. In [16], six DC/DC converter topologies for offshore DC-connected wind farm applications were examined. A DC parallel-connected wind farm can also be classified as a star or radial DC-connected topology [17]. It should be noted that DC series and series-parallel wind farms become infeasible with high turbine output voltages because a very high turbine power rating is required.

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 [

18], a model predictive control (MPC) based optimization framework was proposed to coordinate the reactive power references among WTs, with the goal of improving the voltage profile of the buses and reducing network loss. An area-automatic voltage control scheme and a new pilot-bus selection method were proposed in [19], which considered not only the physical response of the generator but also fluctuations in wind power. In [20], particle swarm optimization (PSO) was adopted to dispatch reactive power of WTs by minimizing the total active power losses along the cables and transformers of WTs. In [21], the objectives of optimal control included the power losses of offshore wind farm collector systems, grid-side converters of WTs, and HVDC converters. MPC-based centralized optimal voltage control schemes were proposed in [22], [23] for offshore VSC-HVDC-connected AC wind farms, where the goal was to maintain the voltages within feasible ranges while considering the economic operations of wind farms. In [24], a series-connected method was proposed for two or more WTGs used in a wind farm with DC grids. In [25], the characteristics of the series-parallel DC-connected wind farm topology were analyzed, and an overvoltage limitation method was proposed for a series-parallel DC-connected wind farm to ensure its safe operation.

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

A. Configuration of Offshore Radial DC-connected Wind Farms

The configuration of an offshore radial DC-connected wind farm is shown in Fig. 1. The DCWT system is composed of a WT, an AC/DC rectifier, and a medium-voltage (MV) DC/DC converter. Several DCWTs are connected using a 33 kV DC feeder and placed at a distance of 4 km from one another. The DC feeders are connected to a common medium-voltage direct current (MVDC) collector bus. The MVDC collector bus voltage is controlled by a high-voltage DC/DC converter at an offshore station. The power collected from the MVDC collector bus is transferred to the high-voltage side by the high-voltage DC/DC converter, and then the power is sent to the onshore main grid through an 800 kV VSC-HVDC transmission system.

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.

B. Overview of Proposed Control Strategy

Figure 2 shows the structure of the proposed control strategy, where Ploss is the active power losses of the wind farm; Vs and Vsref are the terminal voltage of the high-voltage DC/DC converter and the voltage reference from the controller, respectively; Pwt and Pwtref are the active power output and reference of the DCWT, respectively; Pwfref is the active power reference of the wind farm; and V is the voltage of the DCWT. The active power reference of the wind farm is provided by the TSO and delivered to the wind farm controller. The wind farm controller generates optimal active power references and MV references for the DCWTs and high-voltage DC/DC converter.

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.

III. Sensitivity Coefficient Calculation Method

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 [

23]. Thus, inspired by [26], an efficient calculation method of the sensitivity coefficients of bus voltages is developed in this study for radial DC-connected wind farms to improve computational efficiency.

A. Voltage Sensitivity

Figure 3 shows the circuit of a radial DC-connected wind farm, which is composed of N buses and Nw DCWTs. In Fig. 3, P0i (i=1,2,,N) is the power flowing away from the ith MV slack bus and Pi-1(i=1,2,,Nw) is the power flowing away from the (i-1)th DCWT terminal bus to the ith DCWT terminal bus. Since the MV collector bus voltage is controlled by the high-voltage DC/DC converter, the MV collector is defined as the slack bus of the DC-connected wind farm. This study also characterizes the power flow from the DC/DC converter to the wind farm as having a positive direction.

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 ith DCWT terminal bus voltage Vi can be expressed as:

Vi=Vi-1-Ri-1Pi-1Vi-1 (1)

where Ri-1 is the resistance of the line segment between the (i-1)th and ith DCWT terminal buses. Then, the partial derivative Vi with respect to the voltage magnitude Vi-1 can be obtained as:

ViVi-1=1+Ri-1Pi-1Vi-12-Ri-1Vi-1Pi-1Vi-1 (2)

Pi-1 can be treated as the power reference for the ith DCWT terminal bus Pi-1ref minus the power losses in the ith DCWT terminal bus Ploss*, i.e.,

Pi-1=-(Pi-1ref-Ploss*) (3)

During each control period, the power reference is constant. Then, we can obtain:

Pi-1Vi-1=Ploss*Vi-1 (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:

ViVi-11+Ri-1Pi-1Vi-12 (5)

Given the remaining partial derivatives Vi with respect to the voltage magnitude Vi-1, Vi/Vi-1 can be obtained using the same process. Once all Vi/Vi-1 are obtained, the partial derivatives of Vi with respect to the voltage magnitude Vs of the slack bus can be expressed as:

Vi+1Vs=Vi+1ViViVi-1V3V2V2Vs (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:

Pi=VijSMYbus,ijVj (7)

where Ybus,ij is the admittance. The partial derivatives of Pi(iM) with respect to the active power Pi (iM) can be calculated as:

PiPl=ViPljSMYbus,ijVj+VijSMYbus,ijVjPl=1    i=10    i1 (8)

In [

23] and [26], jSMYbus,ijVj and VijSMYbus,ij can be calculated according to the measured voltage. The unknown variables are Vi/Pl (iM). According to [26], M equations with M unknown variables can be rewritten. Vi/Pl can then be obtained by solving the M equations.

B. Power Loss Sensitivity

The power losses in the radial DC-connected wind farm can be expressed as:

Pgrid,loss=i=1Mj=1MViVjYbus,ij (9)

The partial derivatives of the power losses with respect to the voltage magnitude can be calculated by:

Pgrid,lossVi=2j=1MVjYbus,ij (10)

Then, the sensitivity with respect to the power output of WTGs and terminal bus voltage can be obtained as:

Pgrid,lossy=Pgrid,lossViViy (11)

where y consists of the active power output of DCWTs and the slack bus voltage.

IV. Predictive Modeling of DC-connected Wind Farms

A. Modeling of High-voltage DC/DC Converter

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 [

27], a DAB converter system was adopted for a high-voltage DC/DC converter. In the DAB converter system, DAB converters are connected in series or parallel to meet the power and voltage rating of the system. The topology of the DAB converter system is illustrated in Fig. 4. Additional details can be found in [27] and [28].

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:

ΔVs=11+sTdcΔVsref (12)

where Δ denotes the change in a variable during the control periods; and Tdc is the voltage control time constant of DC/DC converter. Then, we can obtain:

ΔV˙s=-1TdcΔVs+1TdcΔVsref (13)

B. Modeling of DCWTs

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 [

29], where Pe is the actual electromagnetic power of the PMSG; Pe* is the reference electromagnetic power of the PMSG; Rs is the stator resistance of PMSG; Ld is the d-axis inductance of PMSG; Lq is the q-axis inductance of PMSG; ψm is the permanent magnet flux linkage of PMSG; udc is the DC-side bus voltage of AC/DC converter; ωs is the electrical angular velocity of the generator; θ is the electrical angle of PMSG; uα and uβ are the stator voltages of PMSG in stationary two-phase coordinate system; id and iq are the stator currents of PMSG in dq coordinate system; ud and uq are the stator voltages of PMSG in dq coordinate system; and id* and iq* are the reference values of id and iq, respectively.

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:

Pwt=TeΩ=Teωsnp=32npψmiqωsnp=32ψmωsiq (14)

where Te is the electromagnetic torque of the PMSG; Ω is the mechanical speed of the generator; and np is the number of pole pairs. Thus, the active power control loop of the DCWT can be obtained, as shown in Fig. 6.

Fig. 6  Active power control loop of DCWT.

Here, kp and ki are the proportional and integral gains of the PI controller, respectively; T is the time constant of the current loop; Tf is the time constant of the filter; Pg is the generator output power; and TDCp 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:

Δiq=11+sTkp+kis(ΔPwtref-ΔPg) (15)
ΔPg=11+sTf32ψmωsΔiq (16)
ΔPint=ΔPgref-ΔPgs (17)
ΔPwt=1sTDCp+1Pg (18)

where ΔPint is the auxiliary variable of active power control loop. The incremental mode of the DCWT can be expressed in matrix form as:

Δx˙wt=AwtΔxwt+Bwtuwt (19)

where

Δxwt=ΔPwtΔiqΔPgΔPint
uwt=ΔPwtref
Bwt=0kpT01
Awt=-1TDCp01TDCp001T-kpTkiT03ψmωs2Tf-1Tf00010

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:

Δx˙=AΔx+BΔu (20)

where

Δx=[ΔVs    Δxwt,1    Δxwt,2    ...    Δxwt,Nw]T
Δu=[ΔVsref    Δuwt,1    Δuwt,2    ...    Δuwt,Nw]T
A=diag[-1/Tdc    ΔAwt,1    ΔAwt,2    ...    ΔAwt,Nw]
B=diag[1/Tdc    ΔBwt,1    ΔBwt,2    ...    ΔBwt,Nw]

where xwt,i and uwt,i are the state variable and control variable, respectively; and Awt,i and Bwt,i 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 ΔTp can be expressed as:

Δx(k+1)=AdΔx(k)+BdΔu(k) (21)

where Ad=eAΔTP; and Bd=0ΔTPeAτBdτ.

V. MPC-based Coordinated Voltage Control Strategy

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.

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.

A. Economic Operation Mode

1) Objective 1: the first objective for the economic operation mode is to minimize the loss, which is given by:

ObjL=k=1NpPloss(k)2 (22)
Ploss(k)=PlossPwtΔPwt(k)+PlossVsΔVs(k)+Ploss0 (23)
ΔPwt=[ΔPwt,1    ΔPwt,2        ΔPwt,Np] (24)
PlossPwt=PlossPwt,1    PlossPwt,2        PlossPwt,Np (25)

where Pwt,i is the output power of the ith DCWT; Pwt is the vector composed of the output power of each DCWT; Np is the total prediction step; and Ploss0 is the initial power losses.

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.

ObjP=i=1NwΔPwt,iPD(k)2 (26)

The predictive value ΔPwt,iPD can be calculated by:

ΔPwt,iPD=Pwt,i0+ΔPwt,i(k)-Pwt,iPD,ref (27)

where Pwt,i0 is the initial output power of the ith DCWT; and Pwt,iPD,ref is the PD-based reference for the ith DCWT.

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.

Objd=i=1NwΔPwt,i(k)2 (28)

Then, the cost function can be expressed by:

min(λLObjL+λPObjP+λdObjd) (29)

where λL, λP, and λd are the weighting coefficients.

B. Voltage Control Mode

In the voltage control mode, the main control objective is to minimize the voltage deviation from the rated voltage as:

ObjV=k=1Npi=1NwΔVwt,i(k)2 (30)

Then, the voltage deviation ΔVwt,i(k) can be expressed as:

ΔVwt,i(k)=ΔV0+ΔViPwtΔPwt(k)+ViVsΔVs(k)-Vrated (31)

where V0 is the measured value at the initial moment of DCWT; and Vrated 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:

min(λVObjV+λLObjL+λPObjP+λdObjd) (32)

where λV is the weight required to minimize voltage deviation. Since voltage control is the main control objective in this mode, λV is set to be larger than λL and λP.

C. Constraints

Each DCWT terminal bus must be maintained within a safe range according to its predictive voltage Vi,kpre.

VcminVi,kpreVcmax    i=1,2,...,Nw,k=1,2,...,Np (33)

where Vcmax and Vcmin are the maximum and minimum controlled voltages, respectively. Vipre can be obtained by:

Vipre(k)=Vi0+ViPwtΔPwt(k)+ViVsΔVs(k) (34)

where Vipre and Vi0 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:

VsminΔVs0(k)+Vs0VsmaxΔVsrefΔVsmax    k=1,2,...,Nc (35)

where Vsmin and Vsmax are the minimum and maximum voltage limits, respectively; Vs0 is the initial voltage; and ΔVsmax is the maximum ramp rate. The active power of the DCWT is constrained as:

0Pwt,i(k)Pwt,iavi    i=1,2,...,Nw,k=1,2,...,Np (36)

where Pwt,iavi is the available wind power.

The radial DC-connected wind farm must track the power reference PWFref from the system operators, which can be expressed as:

i=1NwPWiref=PWFref (37)

where PWiref is the active power reference of the ith DCWT.

VI. Case Study

A. Test System

The DC-connected wind farm depicted in Fig. 1 is used to demonstrate the performance of the proposed control strategy, which is validated by fluctuating wind power output. In the test system, the radial DC-connected wind farm consists of three MVDC feeders. Eight, seven, and five 5 MW DCWTs are connected to the first, second, and third feeders, respectively. Wind farm control is conducted every 1 s. The upper- and lower-mode decision voltages are set to be 1.055 p.u. and 0.945 p.u., respectively, and the maximum and minimum controlled voltages are set to be 1.05 p.u. and 0.95 p.u., respectively. To assess the performance of the proposed control strategy, three scenarios are compared in the case study, which are listed in Table I.

TABLE I  Three Scenarios in Case Study
ScenarioControl 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 ith DWTC obtained according to the PD can be calculated by:

Pwt,iPD,ref=αpdPwt,iaviαpd=PWFrefi=1NwPwt,iavi (38)

where αpd is the utilization ratio.

B. Control Performance

Figure 8 shows the total available wind power and dispatch command sent from the TSO, where the total available wind power fluctuates between 70 and 80 MW during 0-150 s. During 150-300 s, the total available wind power gradually increases to 90 MW. After 300 s, the total available wind power decreases. The dispatch command is set to be 50 MW from 0 to 150 s. The wind farm outputs constant power. The dispatch command increases to approximately 85 MW during 150-300 s. After 300 s, the wind farm operates in maximum power point tracking (MPPT) mode.

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. The voltage in scenario 3 is maintained at the rated voltage during the entirety of the control periods. In scenario 1, the voltage is controlled at approximately 1.008 p.u. during 0-150 s and gradually decreases during 150-300 s as the dispatch command increases. At approximately 300 s, the wind farm switches from economic operation mode to voltage control mode. The voltage decreases rapidly to approximately 0.965 p.u.. At approximately 450 s, the wind farm recovers to the economic operation mode. The voltage is maintained at approximately 0.985 p.u.. In scenario 2, the voltage is kept lower than the rated voltage during the entirety of the control periods.

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

Figures 10 and 11 show the terminal bus voltages of DCWT8 and DCWT14 in different scenarios, respectively. DCWT8 is the farthest WT from the MV collector bus in this case. In scenario 1, the voltage of DCWT8 is maintained at approximately 1.049 p.u. during 0-300 s. At 300 s, the economic operation mode cannot maintain the terminal bus voltage of DCWT8 within a safe range, meaning that the voltage fluctuated. At 300 s, the terminal bus voltage of DCWT8 exceeds 1.05 p.u.. The wind farm changes to voltage control mode, and the voltage rapidly decreases. After voltage recovery, the voltage is maintained at approximately 1.05 p.u. again. The terminal bus voltage of DCWT14 is maintained within the safe range during the entirety of the control periods. In scenario 2, the terminal bus voltage of DCWT8 is maintained at approximately 1.015-1.025 p.u. during 0-300 s and reaches the highest voltage of 1.035 p.u. at approximately 300 s. The terminal bus voltage of DCWT14 fluctuates from approximately 1.008 p.u. to approximately 1.015 p.u. throughout the entirety of the control periods. In scenario 3, the terminal bus voltage of DCWT8 reaches 1.07 p.u. at approximately 300 s. This could result in DCWT8 shutdown and would affect the secure operation of the wind farm. The voltage performance in scenario 2 is better than those in scenarios 2 and 3.

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

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

Figure 12 shows the active power output of the radial DC-connected wind farm. The performances in the three scenarios are similar. The active power outputs of DCWT8 and DCWT14 in different scenarios are shown in Figs. 13 and 14. Here, the performances in the three scenarios are different during 0-300 s. During 300-550 s, as all DCWTs operate in MPPT mode, the active power outputs in the three scenarios are the same.

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.

Figure 15 shows the power losses of the radial DC-connected wind farm. During 0-300 s, the power losses in scenario 2 are lower than those in scenario 1, and the power losses in scenario 1 are lower than those in scenario 3. During 300-450 s, as the wind farm operates in voltage control mode in both scenarios 1 and 2, the performances are similar. During 450-550 s, the power losses in scenario 3 are the lowest because the highest DCWT terminal bus voltages are obtained with the PD control mode. Overall, the performance in the economic operation mode is better than that in the voltage control mode.

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 Tables II, III, and IV, respectively.

TABLE II  Total Voltage Deviations of Wind Farm
ScenarioTotal voltage deviation (p.u.)
0-150 s150-300 s300-450 s450-550 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

Table II shows that the total voltage deviation of the wind farm in scenario 2 is the lowest among the three scenarios. The voltage deviation in scenario 3 is the highest because the PD method does not consider the optimal voltage operation of the wind farm. Table III shows that during 0-300 s, the average power loss of the wind farm in scenario 2 is the lowest. When the active power output increases, the power loss in scenario 2 is less than that in scenario 1. During 300-550 s, the power loss in scenario 3 is better than those in scenarios 1 and 2. However, the terminal bus voltages in scenario 3 exceed their thresholds, which are then at risk of being tripped. Table IV shows that the total active power deviations from the PD references in scenario 1 are better than those in scenario 2.

TABLE III  Average Power Losses of Wind Farm
ScenarioAverage power loss (MW)
0-150 s150-300 s300-450 s450-550 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
TABLE IV  Total Active Power Deviations of WTs from Their PD References
ScenarioTotal active power deviation (MW)
0-150 s150-300 s300-450 s450-550 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. 16 and 17, and the parameter values of different weights are shown in Table V.

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

Fig. 17  Power losses of wind farm with different weights.

TABLE V  Parameter Values of Different Weights
WeightλVλLλP
1 0.1 50 20
2 3.0 2 20
3 1.0 30 20

Figure 16 shows that the terminal bus voltages of DCWT8 decrease as λV increases, and Fig. 17 shows that the power losses of the wind farm decrease as λL increases. The simulation results reveal that scenarios 1 and 2 perform better than scenario 3. In terms of secure operations, as the terminal bus voltage deviation of the DCWT in scenario 2 is the lowest, the voltage control mode is better. From an economic point of view, this scenario is also better. In addition, scenario 1 combines the economic and voltage control modes, thereby providing more flexibility for the control of the DC-connected wind farm.

VII. Conclusion

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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