Electrical Oscillations in Wind Farm Systems: Analysis and Insight Based on Detailed Modeling

This paper presents modeling and analysis of electrical oscillations in a wind farm system. The detailed modeling and modal analysis of a wind farm system are presented in this paper. The approach to modeling uses detailed representation of a wind turbine generator and collection system including high-voltage direct-current (HVDC) power converter system control, facilitating a comprehensive analysis of the wind farm system. Various modes are classified according to the frequency of oscillation. The detailed modal analysis is used to characterize the critical modes. Time-domain simulation also confirms the presence of these modes. The effect of wind farm operating conditions and voltage source converter control tuning on critical oscillatory modes are also assessed and discussed in detail.


I. INTRODUCTION
Wind energy is the fastest growing renewable energy resource for electricity generation in recent times.Increasing concern for energy security, improvements in wind turbine technology and reduction in cost are expected to maintain this trend for the foreseeable future.Several countries such as the United Kingdom, Germany, Spain, Ireland are already meeting a significant proportion of their energy demand from wind.Several large offshore and onshore wind farms are in construction or are in the planning stage.Voltage Source Converter (VSC) based High Voltage Direct Current (HVDC) technology is employed to carry power from remote offshore wind farms where potential for wind energy extraction is high.
There have been operating difficulties reported in wind farms connected to the shore using VSC HVDC links for certain operating circumstances [1], [2].The operation and control of the offshore wind farm network is difficult due to its distance from shore, weather conditions, accessibility, reactive power issues etc.It is very important to understand the possible problems through accurate modelling and simulation of the system.
Studies and analysis carried out in this topic generally concentrate on the grid side problems of power system such as stability, voltage control, and the operational aspects of the power system.Such studies use an aggregate or semi aggregated model of wind farm due to simulation time constraint L. P. Kunjumuhammed and B. C. Pal are with the Electrical and Electronic Engineering Department, Imperial College, London SW7 2AZ, U.K. (e-mail: linash.p.k@imperial.ac.uk; b.pal@imperial.ac.uk).This work was supported by EPSRC, U.K. under Grants PV2025, EP/K02227X/1.Data supporting this publication can be obtained on request from cap-publications@imperial.ac.uk C. Oates and K. Dyke are with Alstom Grid Stafford, United Kingdom.(e-mail: colin.oates@alstom.com;kevin.dyke@alstom.com)where several wind turbine generators (WTGs) are aggregated and represented as one WTG [3], [4], [5], [6], [7], [8].While there is a large volume of literatures on small signal modeling of individual or groups of WTGs within small network, there has been no research into large arrays of WTGs interconnected by AC cables.
Large offshore wind farms have complicated electrical networks containing many WTGs, networks of medium voltage cables, long high voltage cables, and a HVDC converter control and link [9], [10], [11].The dynamic characteristics of such a system, if not controlled correctly, can threaten the stability of the wind farm to grid interconnection.In this paper, the dynamic behaviour of a wind farm system is investigated using detailed modelling and small signal stability or modal analysis.The frequency domain results are further validated through time domain simulations.The impact of various operating conditions and control parameters on oscillatory modes of the system are assessed and presented.The study will help to identify additional control requirements and specify the control design for the wind farm operation.The effect of aggregation on the modes are detailed in the companion paper (Part II).
A practical wind farm cluster containing two separate wind farms is used for the modelling and analysis.Doubly fed induction generator (DFIG) based WTGs have been used throughout the windfarms and the system connects to the onshore grid using a HVDC link.The voltage source converter (VSC) at the wind farm side provides wind farm voltage and frequency control.Section II explains the layout of the wind farm system and various test conditions.Following the description of wind farm, detailed modelling of the WTGs, collector system components and VSC, and the development of a simulation program using MATLAB/Simulink software are presented.Modal analysis is presented in Section IV where the oscillatory modes of wind farm are classified according to the frequency of oscillation.Relevant characteristics of critical modes and their sensitivity to operating conditions of the wind farm and VSC controller tuning are also discussed.The modal analysis results are validated using non-linear dynamic simulation presented in Section V in which a step change has been applied to the WTG and the VSC reference inputs to excite various oscillatory modes.Voltage and power flow at different locations are plotted to show the presence of these oscillatory modes.

II. LAYOUT OF WIND FARM
A reasonably large wind farm (500-800 MW) will contain many hundreds of WTGs spread over an area covering many square kilometers.WTGs are connected to a medium voltage network called the 'collector system'.Wind farm transformers are used to step up the voltage level before transmitting power to the grid.For offshore wind farms where the transmission distance is 100 km or more, VSC HVDC is cost effective for grid connection.
The example wind farm system used in the present work contains two wind farms of capacity 465 MW (93x5 MW) and 165 MW (33x5 MW).They are named as Wind Farm 1 (WF1) and Wind Farm 2 (WF2), respectively.Each WTG unit in the wind farm system is formed of a wind turbine, a doubly fed induction generator (DFIG), and a pad mounted transformer.A schematic diagram of the 5 MW DFIG based WTG used throughout the wind farm system is shown in Fig. 1.The DFIG rotor is energised through the rotor side converter (RSC) and the grid side converter (GSC) as shown in the figure.The transformer steps up the 0.6 kV voltage produced by the DFIG to 33 kV.The WTG parameters for the 5 MW machine are adopted from [8] and are given in Appendix A. The 33 kV section within the wind farm system consists of many strings of wind turbines having between 5 and 10 WTGs as shown in Fig. 2. The main interconnection is by 33kV cabling with 600V/33kV step up transformers (indicated by dotted lines) distributed regularly at 1km separation.The low voltage 600V connection going to individual wind turbines (indicated by triangles).Each string is connected to a wind farm transformer that convert the voltage to 132 kV.In WF1, a 33 kV cable connecting a string to the wind farm transformer has a higher capacity compared to the other 33 kV cables.Similarly WF2 also contains two types of 33 kV cables.Lower capacity cables carry power from, at most, three WTGs.The cable distance between any two WTGs is typically 1km so this value has been assumed throughout.Fig. 3 shows the high voltage network of the wind farm system and the interconnection between the medium voltage strings.A 132 kV cable connects the wind farm transformer to a point of common coupling (PCC) with the VSC and a HVDC link connects the PCC with the main ac grid.The cable distance between the PCC and the wind farm transformer terminals (132 kV cables) is assumed to be 30 km.Both of the wind farms are divided into two areas, Area-1, and Area-2.The voltage and frequency at the PCC are regulated using the VSC.For this purpose, a VSC controller compares the PCC voltage with a reference value and regulates converter voltage.Other devices that are likely to be present in a wind farm such as auxiliary transformers, auxiliary loads, and reactive power supporting devices are not considered here.Parameters of all the system components are given in Appendix A.

A. Wind farm operating conditions
A WTG produces useful output at wind speeds above the "cut-in" wind speed (≈3.5 m/s) and below the "cut-out" wind speed (≈25 m/s).The WTG rated output is available only above rated wind speed (≈13 m/s).The number of operating WTGs and their output in a wind farm vary depending on the prevailing wind speed and wake effect [12], [13], [14], [15].Hence, selecting the probable operating conditions of a wind farm is complex compared to that for the synchronous generator which is driven by a controllable energy source.
In this paper, four test cases are considered for analysis.In all the cases, the WTGs are assumed to be operating above the rated wind speed and have a non-zero pitch angle.The test cases are; Test-1: The base case where all the WTGs of both the wind farms are in service.
Test-2: WF2 is partially shut down.Only five WTGs in WF2 are working, which are located at the end of the strings.They are selected such that the entire 33 kV collector cables remain energised.
Test-3: WF1 A2 is partially shut down.Only eleven WTGs in WF1 A2 are working which are located at the end of the strings.All WTGs in WF1 A1, and WF2 are producing rated output.
Test-4: WF1 A1 and WF1 A2, are partially shut down.Only ten WTGs in WF1 A1 and eleven WTGs in WF1 A2 are working which are located at the end of the strings.WTGs in WF2 are producing rated output.

III. MODELLING OF THE WIND FARM
Wind farms are generally modelled using aggregated or semi aggregated representations [3], [4], [5], [6], [7], [8].In a fully aggregated model, a wind farm is represented using one WTG and a transformer and/or series impedance.The rating of the aggregated WTG is equal to the sum of the outputs of all WTGs [16].Since the generator parameters are in a per unit system, the aggregated generators adopt the same parameters, and the capacity of generator and converter is re-scaled [17], [18].The rated output of the aggregated WTG is equal to the rated output of one WTG multiplied by the number of WTGs being aggregated.Consequently the collector system network is also reduced to a single equivalent impedance [4], [16], [19], [20], [21].
However, in order to identify and characterise various frequency modes in a wind farm, this paper adopts a detailed modelling strategy of the wind farm components.The model has a representation of all the WTGs, cables, transformers, VSC converter and control.Although the description of the programming given in this paper is based on the wind farm system topology presented earlier, the method is generic and can be used to model other wind farm systems.The modelling approach for each of the individual blocks are described below.
A two axis rotating reference frame (d-q axes) is used for voltage and current in the generator where the q-axis is aligned with stator voltage and the d-axis leads the q-axis.Each generator has individual d-q components.For the collector system, the entire network is transformed using a single transformation with reference to a common synchronously rotating reference frame.The three phase variables (abc) of the network components are transformed to two phase variables (DQ) such that v Q = |v a |, v D = 0, where the D-axis leads the Q-axis.It is assumed that wind speed is constant during the course of the simulation and the WTGs are producing full rated output.

A. Wind Turbine Generator (WTG)
The modelling of a WTG employing a DFIG is presented in several papers [22], [23].The modelling approach used in this paper has been adopted from [22].An internal block diagram description of the model is shown in Fig. 4. A detailed description of the blocks is beyond the scope of the paper and only the equations required to build the WTG model are presented here.

1) Equations of the turbine Algebraic block:
The block computes the mechanical torque produced by the wind turbines using inputs of pitch angle for the blades, wind speed, and rotor speed.The performance coefficient of the turbine is given as C p .For accurate results manufacturers supplied curves should be used to provide this value; however, the expression ( 3) is commonly adopted for academic research purposes [8].The symbols R, ρ, λ, ω t and v w represent blade length, air density, tip speed ratio, turbine speed and wind velocity respectively.
2) Equations for the turbine generator block: The turbine generator model represents the torque-angle loop of the turbine generator system.A two mass representation (5-7) is used in this work.
where, p = d/dt, Te is electrical torque, Tt is turbine torque, θ tw is the equivalent twist angle of drive train shaft, H g is WTG generator inertia, H t is WTG turbine inertia, ω elB is WTG base speed, K is the drive train shaft stiffness, and c is the drive train damping coefficient.
3) Pitch angle controller block: The internal block diagram for the pitch angle controller and actuator are shown in Fig 5, where, ω r and β represent rotor speed and pitch angle respectively.The block is active when the wind speed is greater than the rated wind speed.

4) Equations for the optimum power point tracking (OPPT)
Control block: This block is active during operation below the rated speed.The reference torque is related to rotor speed by (8).
mrr + R r , and K 2 mrr = L m /L rr .Subscripts s, r, q, and d represents stator, rotor, quadrature axis and direct axis, respectively.
6) Rotor side converter block: The internal block diagram of the RSC block is shown in Fig. 6.The slower outer loop controls electrical torque and reactive power and produces a current set point for the faster inner loop control.The rotor side converter block controls both the reactive power and the terminal generator voltage.7. It is assumed that the reactive power being supplied from the rotor side through the GSC at the WTG transformer is zero and the GSC reactive power set point calculation is based on the reactive power requirement of the filter circuit.

B. Cable, transformer, WTG filter, and VSC impedance
Components such as cables, transformers, WTG GSC side filter, and VSC converter impedance are distinctly different in terms of their construction, size and use.However, for the dynamic simulation of a balanced system, they are to be modelled as an R-L-C 'Π' section.When connected together it is equivalent to connecting several Γ sections as in Fig. 8, where the vertical line represents the effective capacitance at  18)-( 21).The subscripts, s, r, D, and Q in ( 18)-( 21) represents the sending end, receiving end, D-axis, and Q-axis, respectively.

C. VSC Control block
The HVDC line connecting the wind farm to the grid is not modelled in this work as any disturbance from either side is blocked by the asynchronous link.Since the VSC control in the wind farm side regulates voltage and frequency at the PCC by regulating the converter voltage, it will play a significant role in defining the dynamic response of the wind farm.A controller transfer function is defined such that the transfer function between the reference voltage and the PCC voltage has a gain cross over frequency of 100 Hz.

D. Wind farm simulation program
The wind farm simulation program is organised by merging the models of the WTG, transformer, cable, VSC converter and VSC control.For clarity of presentation a detailed view of the WF1, A1 is shown in  The output of the WTG block is fed to the WTGtr block representing 126 WTG transformers.The block is modelled using ( 18)- (21).Input to the block are V rD , V rQ : the voltage at the 33 kV bus or receiving end bus and i D , i Q : the current injected from the WTG.State variables are V sD , V sQ : the voltage at the 0.6 kV bus or sending end bus and i LD , i LQ : the current through the transformer.
The remaining blocks S1 to S6 of the WF1 A1 33 kV collector system are programmed using ( 18)-( 21).The procedure used for the WTGtr block is repeated for the building blocks for the other 33 kV collector systems of the wind farm system.
The blocks WFT, 132 kV, and VSC represent the four wind farm transformers, three 132 kV cables and the VSC converter impedance.The VSC control block takes the PCC voltage as input and outputs the converter bus voltage.

IV. MODAL ANALYSIS OF WIND FARM
The wind farm simulation model so far discussed contains a number of differential and algebraic equations and it can be summarized in the following form, ẋ = f (x, z, u), 0 = g(x, z, u) where f and g are functions of differential and algebraic equations, and h is a vector function of the output equations.The notations x, z, u, and y represent vectors of the state variables, algebraic variables, inputs and outputs, respectively.
By Linearising (22) and eliminating the vector of algebraic variables z, we can obtain a state space representation of the system as, where A is the state matrix, B is the input matrix, C is the output matrix, and D is the feedthrough matrix.
A linearised model of the wind farm system is obtained from the MATLAB/Simulink program using the command linmod [24] which returns the state space matrices.Eigenvalues {λ i = σ i ± jω i } n 1 and eigenvectors, φ i : right eigenvector and ψ i : left eigenvector, are obtained using the command eig [24].The frequency and damping ratio of a mode are found using f = ω/2π Hz and ζ = −100σ/ √ σ 2 + ω 2 %, respectively.The relative participation of kth state variable in ith mode pf ki is calculated as [25], In the following sections, vector pf i is normalized using the maximum value of the vector.

A. Oscillatory modes in the wind farm system
Modal analysis of the wind farm model reveals 1273 oscillatory modes in the system.The modes are classified into following five categories based on their frequency.
1) Very high frequency mode: Modes having a frequency of more than 2 kHz are categorized into very high frequency modes.There are 378 modes with frequencies in this range.Participation factor analysis shows that these modes are related to the electrical dynamics of the collector system cables.Considering their very high frequency these modes are considered to be too high to have anything more than a minor disturbance effect on the system and no further analysis of the modes are performed.
2) High frequency mode: These modes have frequency in the range of 500 Hz and 2 kHz.There are 14 modes in this frequency range.The participation factor analysis shows that these modes are related to wind farm transformer, 132 kV cable, and HVDC converter sections.The modes have low damping but their frequencies are generally high.The lowest frequency in this group is 893 Hz.The modes are plotted in Fig. 11(a).It can be inferred that these modes are important but not critical for stable operation of the wind farm system.4) Low frequency mode: These modes have frequencies between 1 Hz and 50 Hz and damping ratios of less than 50 %.There are 385 low frequency modes identified in the system model which appear as distinct groups in Fig. 12.They represent the WTG stator electrical dynamics, the WTG rotor electrical dynamics , the WTG mechanical dynamics and the WTG GSC electrical dynamics [8].The frequency, damping ratio and state participation for each of these modes are listed in the Table.II.The WTG stator electrical dynamic modes (G1) have poor damping and are very close to the power frequency (50Hz).There are 126 such modes related to 126 machines in the wind farm system.Most of the stator modes have a participation from the stator current and voltage of one of the WTGs.However, it is found that a few of the modes have a participation from the states of the machines located in different strings of a wind farm.
Table III shows the participating factor analysis result for one such stator mode.The mode has a participation from the stator current and stator voltage states of three of the WTGs,  with the WTGs participating in the mode being located in three different strings of WF1 A1.Some of the modes in group G3 represent the turbinegenerator mechanical dynamics of the WTG.The modes have a participation only from the ω r and θ tω states of the WTG.It is important to notice that the mechanical modes are isolated from states located on the electrical side of the WTG and collector systems.

B. Analysis of medium frequency modes
The medium frequency modes observed in the system deserve special attention due to their poor damping and closeness of frequency to harmonic frequencies such as 2 nd , 5 th and 7 th harmonics, generally observed in power system.
1) Medium frequency mode -1 (MF1): Table IV details various system states participating in the mode and their normalized participation factor.It is observed that the mode has a participation from the voltage and current states located between the wind farm transformer buses and the PCC.The participation from the VSC controller state indicates the possible effect of the control parameters on damping.
132kV /v sQ 1 0.27 V real at WF1 A1 tr HV bus 132kV /v sQ 2 0.27 V real at WF1 A2 tr HV bus 132kV /v sQ 3 0.27 V real at WF2 tr HV bus 132kV /v sD 1 0.16 V imag at WF1 A1 tr HV bus 132kV /v sD 2 0.16 V imag at WF1 A2 tr HV bus 132kV /v sD 3 0.16 V imag at WF2 tr HV bus 2) Medium frequency mode -2 (MF2): Table V shows the participation factor analysis result corresponding to MF2.Similar to MF1, this mode also shows a participation from the voltage and current states located between the PCC to wind farm transformers.Poor damping and closeness to the 5 th harmonic could potentially create instability in the wind farm.Similar to MF1, MF2 is also influenced by the VSC controller states.
3) Medium frequency mode -3 (MF3): MF3 has a participation from current states of the VSC model and the VSC controller state as listed in Table VI.Its frequency is close to 100 Hz.Since these modes have frequencies very close to harmonics of the power frequency and have poor damping they require further analysis.It is very important to tune the controller such that the VSC provides a dynamically stable waveform at the PCC.
Analysis of the medium frequency modes show clear dynamic interaction between VSC-HVDC and wind farm (WF1) which is one of the primary reasons that may lead to a loss of synchronization between VSC and wind farm.

C. Effect of operating condition on medium frequency modes
The modal analysis is repeated for the three other test conditions for the wind farm as listed in Section II-A.Table TABLE VII shows the frequency and damping ratio of the medium frequency mode for the selected cases.The variation in damping of MF-1 and MF-2 are in opposite directions.MF-1 becomes more stable as some of the WTGs are taken out of service.MF-2 is better for Test-2 and Test-3, but for Test-4, where the output of WF-1 is reduced significantly, the damping of MF2 is reduced.In the case of MF3, the damping is reduced when the wind farm output is reduced.It is to be noted that, for the given set of parameters, the damping of the medium frequency modes is affected by the operating condition.This could be a serious issue for a wind farm as its operating condition will continuously vary based on prevailing wind condition.

D. Effect of VSC controller bandwidth on the MFM
The modal analysis results presented so far show a significant influence of the VSC controller on the damping of the medium frequency modes.The analysis is performed so as to understand the effect of the controller on the modes.Also, the controller is designed such that the closed loop transfer function between the VSC servo reference and the PCC voltage provides a gain cross over frequency of 100 Hz.The analysis is helpful to answer the question: is there a connection between the bandwidth and the frequency of MF3?
Retuning the VSC control to give servo bandwidths ranging from 75 Hz to 175 Hz and repeating the modal analysis for Test-1 give a change in damping and frequency for the medium frequency modes as listed in Table VIII.The frequency of MF1 and MF2 remain more or less constant for the selected range of controller gain but the damping ratios of both the modes reduce with the increase in controller gain.The change has a significant impact on the frequency of MF3.The frequency of MF3 when connected to an infinite bus was 62.65 Hz with 43.60 % damping.When the VSC is connected and the gain is increased the frequency of MF3 increases and its damping ratio reduces.However, the change in frequency of MF3 is not proportional to the change in the tuned frequency of the controller.For 100 Hz tuned frequency, MF3 is 99.0 Hz.For 75 Hz and 175 Hz tuned frequency, MF3 is 92 Hz and 118 Hz, respectively.It can be concluded that MF3 exists in the system independent of VSC controller, however, the controller has significant impact on the stability of the mode.A non-linear dynamic simulation is carried out using the operating condition Test-1 where all the WTGs are working at rated operating condition.They are set to control reactive power output at their terminal and the VSC regulates voltage at the PCC.During the simulation, one of the following two disturbances is applied to the system at time t = 1 sec.(a) A 10% increase in the PCC reference voltage, and (b) the reactive power reference input of all the WTGs set to zero.

A. 10% increase in the PCC reference voltage
Fig. 13-16 show the responses obtained from the wind farm system following a 10% increase in the VSC reference input.The voltage at the PCC settles to a new value within a second as shown in Fig. 13.However, immediately after the disturbance a high frequency oscillation in the range of MF modes is observed in the PCC voltage waveform.A magnified view of the oscillation is shown in the inset.Fig. 14 shows voltages at the low voltage side of the wind farm transformers.The high frequency oscillations observed in the PCC voltage is not visible at this point.This is consistent with the participation factor analysis where no states from the 33kV side have participation in the MF modes.However, the settling time of the oscillations in the voltage waveforms range of 1 sec to 3 sec, which is higher compared to the settling time Time (sec)  Voltage (pu) Fig. 16 shows the active and reactive power output of wind farms measured at the 132kV cables.A damped oscillation for 1 to 3 sec is observed in all the waveforms.The response from WF2 settles down faster that the WF1 response.This is due to higher cable resistance in the WF2 collector system which helps to damp the oscillations.

B. Reactive power reference input of all the WTGs set to zero.
Fig. 17-19 show response of the wind farm system following a step change in the reactive power reference set point of all the WTGs.At time t=1sec, the reference is set to zero forcing the WTGs to operate at unity power factor.The magnitude of change in the PCC voltage for this case is relatively small when compared to previous case.However, the oscillations generated in this case take more time to settle as shown in Fig. 17.The inset in the figure shows the magnified view of the oscillations that contain components of themedium frequency modes.The PCC voltage settles to the previous value due to the control effect of the VSC controller.Voltage at the 33kV side of wind farm transformer and 0.6kV terminal of WTGs are plotted in Fig. 18 and Fig. 19, respectively.The effect of the disturbance on the stator mode is explained in the following subsection.

C. Effect of stator mode
In the previous case, though the disturbance excited the stator mode, the oscillations were not visible in waveforms as the stator modes have sufficient damping.In order to highlight the effect of the stator modes, the rotor controller parameters of the WTGs have been varied to reduce the damping of the stator modes.The simulation study is repeated by changing the reactive power set point to zero at time t=1sec.Fig. 20 shows the reactive power output of a WTG following the disturbance.In this case, the oscillations set up following the disturbance take longer time to settle.A magnified view of the signal in the inset clearly shows the stator modes close

VI. DISCUSSIONS
The nonlinear dynamic model of the wind farm system developed in this work contains 3436 ordinary differential equations and its linearised model has 1273 pairs of complex eigenvalues.The results both in the frequency and time domain show that the three medium frequency modes and the WTG stator modes are critical for the stable operation of wind farms.A study using an aggregated wind farm system model may not provide these insights; however the size of the system model is too big to be considered for detailed analysis of the many possible operating scenarios of a wind farm.Also, many of the offshore wind farms being proposed are of higher capacity than the one discussed in this paper which will increase the dimensionality of the model.It is inevitable that some aggregation is required and this being performed using various industry grade software, however it is important to understand whether an aggregated model contains the critical modal characteristics.From that perspective this detailed modelling and simulation provide a novel and deeper insight into the problem of understanding the requirement of effective control design.Future publication will cover the effect of the model aggregation on the critical modes identified in this analysis.Voltage (pu)

VII. CONCLUSIONS
The paper describes the modelling of a wind farm system connected to a VSC and explains various oscillatory modes present in the system.The oscillatory modes are classified into different categories based on the frequency and damping ratio.Three oscillatory modes in the collector system are identified which have low damping, and frequencies in the range of 100 Hz to 500 Hz.They are important because of their closeness to harmonics of the power frequency, particularly because of VSC-HVDC harmonic generation.The damping of the medium frequency modes depends on the operating condition of wind farm.The result is very important as the wind farm operating conditions will continuously vary depending on the prevailing wind conditions.Also, it is found that the VSC controller parameters have a large influence on the damping of the medium frequency modes.The time domain results confirm the presence of the modes.
The stator modes of the WTGs are found to have poor damping and frequencies close to the power frequency (50 Hz).Some of the stator modes have a participation from states of WTGs located in different strings of a wind farm and reflect interaction between the WTGs in different strings through part of the AC network.Because of such close coupling, disturbance in one WTG could disturb the synchronised stable Voltage (pu)

Fig. 2 .
Fig. 2. Structure of strings in the wind farm system

Fig. 3 .
Fig. 3. Single line diagram showing high voltage side of wind farm collector system

Fig. 6 .
Fig. 6.Schematic diagram of RSC of WTG 7) Grid side converter block: The capacitor voltage dynamics and GSC control are shown in Fig.7.It is assumed that the reactive power being supplied from the rotor side through the GSC at the WTG transformer is zero and the GSC reactive power set point calculation is based on the reactive power requirement of the filter circuit.

Fig. 7 .
Fig. 7. Block diagram showing capacitor dynamics and GSC control.Vc is back to back capacitor voltage, I gq,I gd current through filter inductor, Vq, V d voltage at WTG terminal, and V gd , Vgq voltage at gsc converter terminal a node and the horizontal line represents the series impedance of a section.Differential equations representing a Γ section are given in (18)-(21).The subscripts, s, r, D, and Q in (18)-(21) represents the sending end, receiving end, D-axis, and Q-axis, respectively.

Fig. 9 .
Fig. 9. Collector system structure of wind farm-1 Area-1Fig.10showsthe structure of a simulation program developed for the wind farm.The extreme left block represents the WTG model which includes the differential-algebraic equations presented in Sec.III-A.The input to the block is the terminal voltage and the output is injected current at WTG bus.Wind speed, which is an input to the wind turbine is held constant during the course of simulation.Hence it is represented using a constant inside the WTG block rather than showing as an input to the block.The number at the top of the block represents the size of the state vector inside the block, which, in effect, equals the number of WTGs (126).The output of the WTG block is fed to the WTGtr block representing 126 WTG transformers.The block is modelled using (18)-(21).Input to the block are V rD , V rQ : the voltage at the 33 kV bus or receiving end bus and i D , i Q : the current injected from the WTG.State variables are V sD , V sQ : the voltage at the 0.6 kV bus or sending end bus and i LD , i LQ : the current through the transformer.The remaining blocks S1 to S6 of the WF1 A1 33 kV collector system are programmed using (18)-(21).The procedure used for the WTGtr block is repeated for the building blocks for the other 33 kV collector systems of the wind farm system.

Fig. 13 .
Fig. 13.PCC Voltage following a 10% increase in PCC reference voltage.Inset shows magnified view of the plot of the PCC voltage.A similar pattern is observed in the voltage waveforms at the terminal of WTGs as shown in Fig. 15.The stator mode is not observed in the WTG voltage waveform as the mode is not excited by the change in the PCC reference voltage.The participation factor analysis of the stator modes shows zero participation from states close to PCC.

Fig. 18 .
Fig. 18.Voltage at the low voltage side of the wind farm transformers following a change in reactive power set point of WTGs to 50Hz.The poor damping of stator modes and subsequent changes in output of WTG have significant impact on the PCC voltage as shown in the Fig. 21.

Fig. 20 .
Fig. 20.Reactive power output of WTG1 following a change in the reactive power set point for the case where stator mode damping is poor.Inset picture shows the magnified view.

Fig. 21 .
Fig. 21.PCC Voltage following a change in reactive power set point of WTGs for the case where stator mode damping is poor.Inset shows magnified view of the plot operation of other WTGs.
qs 3 0.39 WTG-3 stator i qs W T G/i ds 1 0.65 WTG-1 stator i ds W T G/i ds 3 0.71 WTG-3 stator i ds W T G/i ds 4 0.38 WTG-4 stator i ds W T G/e ds 1 0.90 WTG-1 stator e ds W T G/e ds 3 1.00 WTG-3 stator e ds W T G/e ds 4 0.53 WTG-4 stator e ds LQ 3 0.25 I real through WF2 A1 tr W F T /i LQ 4 0.20 I real through WF2 A2 tr W F T /i LD 3 0.11 I imag through WF2 A1 tr V SC/i LD 1 0.19 Fig. 19.Voltage at the terminal of four WTGs following a change in reactive power set point of WTGs.