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Keywords:

  • aggregated wind turbines;
  • equivalent model;
  • doubly fed induction generation (DFIG);
  • wind farm

ABSTRACT

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 COMPLETE WIND FARM MODEL
  5. 3 EQUIVALENT WIND FARM MODELS
  6. 4 SIMULATION RESULTS AND DISCUSSION
  7. 5 CONCLUSIONS
  8. APPENDIX: Parameters
  9. REFERENCES

For the representation of wind farms in transient stability studies of electrical power systems, reduced models based on aggregating identical wind turbines are commonly used. In the case of a wind farm with different wind turbines coupled to the same grid connection point, it is usual to aggregate identical wind turbines operating in similar conditions into an equivalent one. However, in the existing literature, there are not any references to the aggregation of different wind turbines (same wind turbine technology but different rated power or components) into a single one. This paper presents a comparative study of four reduced models for aggregating different DFIG wind turbines, experiencing different incoming winds, into an equivalent model. The first of them is the classical clustering model, in which each equivalent model experiences an equivalent wind. The other reduced models have the same equivalent generation system but different equivalent mechanical systems. Thus, the second and third ones are compound models with a clustering aggregated mechanical system and individual simplified models, respectively, to approximate the individual mechanical power according to the incoming wind speeds. The fourth is a mixed model that uses an equivalent wind speed, which is applied to an equivalent mechanical system (equivalent rotor and drive train) in order to approximate the mechanical power of the aggregated wind turbines. The equivalent models are validated by means of comparison with the complete model of the wind farm when simulated under wind fluctuations and grid disturbances. Finally, recommendations with regard to the applicability of models are established. Copyright © 2014 John Wiley & Sons, Ltd.

1 INTRODUCTION

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 COMPLETE WIND FARM MODEL
  5. 3 EQUIVALENT WIND FARM MODELS
  6. 4 SIMULATION RESULTS AND DISCUSSION
  7. 5 CONCLUSIONS
  8. APPENDIX: Parameters
  9. REFERENCES

In the last decades, wind power has significantly increased to the extent that large wind farms connected to transmission networks are considered (from the power system operators' point of view) as a single wind power plant with operational capabilities similar to a conventional power plant. This high degree of wind penetration has influence on the transient stability of the electrical system, because wind fluctuations and grid disturbances affect wind farm generation.[1]

Currently, a variable-speed wind turbine is the most used type of wind turbine in wind farms.[2]

Among the variable-speed wind turbines, the doubly fed induction generator (DFIG) is today the most used wind turbine. It represents an attractive wind turbine concept from an economical point of view since a partial-scale power converter (25–30% of the generator rated power) connects the rotor winding to grid[3-5] thus enabling variable-speed operation and high power control capability.

The impact analysis of DFIG wind farms on power system stability requires the development of suitable models for use in power system simulation tools, such as PSS/E, DIgSILENT, SimPowerSystems of MATLAB and others.[6] Researchers and network operators have contributed to the development of accurate dynamic models of wind farms adapted to their needs. Wind farm models have been built with different detail levels depending on the scope of the study. The complete (detailed) model is a one-to-one modeling approach, in which each wind turbine, the internal grid of the wind farm and their controls are represented. Nevertheless, the complete model presents, as the main problem, that the model order increases quickly if the wind farm has a high number of wind turbines, and thus the simulation time is high. Hence, depending on the scope of the study, the complexity of the model and computation time can be reduced by means of alternative models based on different aggregation techniques. The simplest method consists of aggregating all the wind turbines of the wind farm into a single equivalent one, which presents the same model as individual ones but with a rated power given by the addition of the rated powers of all the wind turbines of the farm. This wind turbine receives an equivalent wind that can be, as shown in,[7] the average wind of those incidents on the aggregated wind turbines (in case of similar wind speeds) or an equivalent wind obtained by using the power curve and the individual wind of each turbine. In,[7] two wind farms with different wind turbines technologies were considered: one composed of squirrel cage induction generator (SCIG) wind turbines and the other of DFIG wind turbines. In that work, an equivalent wind turbine was used for each wind farm: one for the SCIG wind turbines and the other for the DFIG wind turbines. A step further—useful when the incoming wind speeds are very different or the wind farm is composed of different wind turbines—is the cluster representation, in which groups of identical wind turbines receiving similar wind speeds are replaced by single wind turbines.[8] It enables the reduction of the model order and simulation time, achieving an adequate approximation of the wind farm's response in transient stabilities studies, when the mechanical behavior has no major impact on voltages and power flows.[9]

In case of long-term dynamic simulations with wind turbines receiving different winds, the single or cluster equivalent model cannot predict, with enough accuracy, the behavior of the wind farm, because of the high non-linearity of the mechanical system of the wind turbines. A compound equivalent model can be a good compromise between model accuracy and simulation time. This model aggregates the generation systems (generator, power converters and controls) into a single equivalent generation system but keeps the mechanical systems of the individual wind turbines. In this case, it is necessary to obtain a suitable approximation of the equivalent mechanical power from the individual ones. In,[9] the generation systems are aggregated into an equivalent one to obtain the total equivalent power, and this is shared (according to the power reference) on each individual mechanical system. In[10] and[11], a simplified model of each individual wind turbine is used. In,[10] the simplified models are used to calculate the electrical power of each individual wind turbine, which are aggregated to obtain the equivalent electric power delivered to the grid. In,[11] the simplified models obtain the approximated mechanical torques that are aggregated and applied to an equivalent generation system, which is represented by the same model than the individual generation system.

The equivalent models mentioned previously have been applied to wind farms with identical DFIG wind turbines or ones with clusters of identical wind turbines (with so many clusters as different wind turbines are in the wind farm). However, the aggregation of different wind turbines (same wind turbine technology but different rated power or components) into a single machine has not been treated in the existing literature.

The scope of this paper focuses on the comparison of reduced models for variable-speed wind farms composed of different DFIG wind turbines, experiencing different incoming wind speeds. Four reduced wind farm models have been considered in this study: (i) a cluster equivalent model in which identical wind turbines are grouped into one cluster and replaced by a single equivalent wind turbine that experiences an equivalent wind obtained by the power curve of the aggregated wind turbines,[7] (ii) a compound equivalent model with as many clustering mechanical systems as groups of identical wind turbines that are in a wind farm and an aggregated generation system that receives the mechanical power from all groups of mechanical systems, (iii) a compound equivalent model that only considers the aggregation of the generation systems but not the mechanical systems; the aggregated mechanical power is calculated in a similar way as performed in,[11] by adding the individual mechanical powers obtained from a dynamic simplified model of each individual wind turbine and (iv) a single equivalent model that represents the wind farm as a whole equivalent wind turbine, including an aggregated mechanical system composed of an equivalent rotor model with the same output power as the sum of the individual ones and an aggregated generation system that receives the mechanical power.

In this paper, the reduced models are evaluated for two wind farms with different DFIG wind turbines (same wind turbine technology but different rated power and components), one composed of 660 kW wind turbines and the other of 2 MW, each one connected through a feeder to the same point of common couple (PCC). The reduced models are validated by means of comparison with the complete model of the wind farm, when simulated under wind fluctuations and grid disturbances, in accordance with the recommendations of the International Energy Agency Wind Task 21.[12] Furthermore, the comparison between the reduced models allows establishing the application limits of each model and establishing which of them are more appropriate for a specific wind farm.

2 COMPLETE WIND FARM MODEL

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 COMPLETE WIND FARM MODEL
  5. 3 EQUIVALENT WIND FARM MODELS
  6. 4 SIMULATION RESULTS AND DISCUSSION
  7. 5 CONCLUSIONS
  8. APPENDIX: Parameters
  9. REFERENCES

For transient stabilities studies, complete models of DFIG wind farms are always composed of all the wind turbine model—with the necessary detail level depending on the scope of study—and the model of the internal grid of the farm, in which the electromagnetic transients are neglected as is normal for power system simulations.[11]

Figure 1 shows the wind farm considered in this study. It has a radial structure with 12 wind turbines in two clusters, connected to the electrical network at the same PCC. The first cluster presents 6 × 660 kW DFIG wind turbines in three branches, with two machines for each one, connected to a medium voltage (MV)/high voltage (HV) transformer (substation of the farm) and a cluster feeder to the PCC. The second one has a similar configuration but with two branches of 3 × 2 MW DFIG wind turbines on each one, connected to a common transformer and a feeder.

image

Figure 1. DIFG wind farm structure.

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In the wind farm, the wind turbines are identified by three indexes: (i) the first index is the cluster, (ii) the second one represents the number of the branch in the cluster and (iii) the third one refers to the position of the wind turbine in the branch.

In this work, the complete model of the DIFG wind farm is the reference used to compare its response at PCC with those obtained from the proposed reduced models.

2.1 Wind turbine model

The DFIG wind turbine model contains the following: (i) the mechanical system: rotor and drive train and (ii) the generation system: DFIG and partial-load power converter [a bidirectional back-to-back insulated-gate bipolar transistor frequency converter with a direct current (DC) bus] and their controls. Figure 2 depicts the configuration and basic control structure of the wind turbine under study.

image

Figure 2. DFIG wind turbine and basic control structure.

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Some modeling assumptions, widely used in electromechanical transient simulations (fundamental frequency simulations), are considered in this paper[13]:

  • Higher harmonics of voltage and currents are neglected (fundamental frequency simulations).
  • Some of the differential equations and short time constant included in the generation system model are canceled (larger time step).
  • The actuator disk theory (quasi-static approach) describes the turbine behavior.
  • The drive train is represented by the two-mass model.
  • The stator transients are neglected in the asynchronous generator (third-order model).
  • The power converter is considered ideal (internal dynamics lacks interest).

Taking in mind these assumptions, the following equations express the mechanical model of the wind turbine:

  • display math(1)
  • display math(2)
  • display math(3)
  • display math(4)

where Tw is the turbine mechanical torque; ρ is the density of air; A is the area of rotor disk; R is the length of the rotor blades; v is the incoming wind speed; cp is the power coefficient of the turbine rotor, which is a function of the tip speed ratio λ (ratio between blade tip speed and wind speed) and the pitch angle of the rotor blades θ; ωr is the rotational speed and Jr is the inertia of the turbine rotor; ωg is the rotational speed and Jg is the inertia of the generator; Tm is the generator mechanical torque; Te is the generator electromechanical torque; K is the stiffness and D is the damping, both of mechanical coupling.

The electrical system of the wind turbine is composed of the asynchronous generator and the power converter.

The behavior of the asynchronous generator is expressed by the following equations:

  • display math(5)
  • display math(6)
  • display math(7)
  • display math(8)
  • display math(9)

where (uds, uqs) and (udr, uqr) are the stator and rotor voltages, respectively; (ed', eq') represents the internal voltage; (ids, iqs) and (idr, iqr) are the stator and rotor currents, respectively; Rs and Xs are the resistance and reactance of the stator winding, respectively; Xr is the reactance of the rotor winding; Xm is the magnetizing reactance; s is the generator slip; and ωs is the synchronous speed.

As can be seen in Figure 2, the power converter consists of two converters coupled through a DC bus. The rotor side converter (RSC) enables the decoupled control of active and reactive powers acting on the rotor voltage, whereas the grid side converter (GSC) allows the power exchange with the grid. In this paper, as is normal for transient stability studies, ideal converters and constant DC bus voltage between the converters are assumed. In fact, a controlled voltage source models the RSC, in which uqr (udr) is used for controlling the rotor speed/active power (reactive power).

The control system applied to the wind turbines is the same one as that presented in.[13] During normal operation, the aims of the control system are the following:

  • Power optimization, which consists of maximizing the power extracted from the wind for a wide range of wind speeds.
  • Power limitation, which is based on limiting the output power to rated power for high winds.
  • Power regulation, which consists of adjusting the active or/and reactive power to a desirable set point.

To achieve these aims, the power converter must be controlled in collaboration with the blade pitch angle.

The controllers applied to the RSC are the following:

  • The rotational speed controller (Figure 3(a)) controls the active power by acting on uqr. It uses the power–speed curve of the wind turbine to determine the power reference according to the rotational speed. Thus, the turbine mechanical power is maximized with below rated speed and is limited with above rated speed.
  • The reactive power controller uses three control loops to define udr by using the control scheme depicted in Figure 3(b): (i) outer control loop, which controls the reactive power and determines the generation voltage reference Ug (ii)subordinated voltage control loop, which regulates idr and assures that the generation voltage is maintained between the limits while trying to reach the reactive power reference and (iii) inner control loop, which controls idr and defines udr.
image

Figure 3. Controllers applied to the DFIG: a) rotational speed controller, b) reactive power controller and c) pitch angle controller.

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The pitch angle controller (Figure 3(c)) is responsible for adjusting the blade pitch angle reducing the power extracted from the wind when the rotational speed increases up to the rated speed. It includes an actuator with pitch angle saturation and rate limiter.

On the other hand, the GSC provides the exchange of power to the grid with a certain power factor. It has been modeled as a controlled current source, where the direct and quadrature components of current source are calculated from the exchange power from the converter to the grid.

This wind turbine includes an external rotor resistance (Rext) coupled via the slip-rings to the generator rotor instead of the converter. This external rotor resistance works as crowbar protection so that it bypasses the RSC in case of over-current (during grid faults) and thus protecting the rotor and power converter. When the crowbar protection is switched on, the DFIG is turned into SCIG with an increased rotor resistance, and the independent controllability of active and reactive powers through the RSC gets lost.[14] Then, the rotor voltage is defined as follows.

  • display math(10)
  • display math(11)

Since the GSC is decoupled from the rotor windings through the DC bus, it can be used to generate reactive power. All modern grid codes are required to provide a certain amount of reactive power in order to support voltage recovery. In this work, the wind turbine is required to fulfill the Spanish grid code,[15] in which the reactive current to be provided is expressed as a function of the grid voltage. Therefore, the reactive power to be provided by the GSC during grid faults is defined from the grid voltage.

When the fault disappears, the crowbar is switched off and then the independent controllability of active and reactive powers through the RSC is recovered.

2.2 Wind farm network model and main control system

In the electrical network model of the wind farm, the electric lines and transformers are represented by constant impedances (static models), as is normal for dynamic simulations of electrical power systems.

The main control system completes the wind farm model. It tries to assure the wind farm production in a similar way to a conventional power plant, that is, controlling the power production. In this paper, the main controller presents independent proportional-integral controllers for the active and reactive powers.[16] Figure 4 shows the main control system including a dispatch center to manage the power references to each wind turbine. The power references for each wind turbine (Pwti,ref, Qwti,ref) are computed, taking into account a proportional distribution of the available powers (Pava,i, Qava,i) and the references ordered by the system operator (Pwf,so, Qwf,so).

image

Figure 4. Main control system of wind farm.

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3 EQUIVALENT WIND FARM MODELS

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 COMPLETE WIND FARM MODEL
  5. 3 EQUIVALENT WIND FARM MODELS
  6. 4 SIMULATION RESULTS AND DISCUSSION
  7. 5 CONCLUSIONS
  8. APPENDIX: Parameters
  9. REFERENCES

In this section, four reduced models of DFIG wind farms are described. The aggregation models consider different wind turbines experiencing different incoming winds. These requirements hinder the use of a single equivalent model and force the use of a clustering or a compound equivalent model.

The clustering equivalent model presents the same model as individual ones but with a rated power given by the addition of the rated powers of all the wind turbines and receives an equivalent wind speed, obtained from the individual wind speed that each wind turbine of the cluster experiences.

Two compound models are developed, which aggregate the generation systems into a single equivalent generation system. One of them keeps the cluster structure in the aggregated mechanical systems, whereas the other one considers the individual mechanical system of each wind turbine (and its non-linearity) with a simplified model of the generation system to obtain an approximation of the mechanical power.

The last model is a mixed equivalent model, between a single and a compound reduced model. It uses the simplified model of each wind turbine to obtain an aggregated rotor model, taking into account the wind speed incident on each turbine. The approximate mechanical power is applied to an equivalent generation system.

3.1 Cluster equivalent model: equivalent wind in each cluster

This equivalent model is based on aggregating wind turbines into as many equivalent wind turbines as clusters considered. Each equivalent has the power capacity of the whole cluster that it represents and experiences an equivalent wind obtained from the wind incident on the aggregated wind turbines. Figure 5 shows the cluster representation applied to the proposed wind farm: one cluster of 6 × 660 kW wind turbines and other cluster of 6 × 2 MW wind turbines. In this case, the applied clustering criterion is the simplest one—different rated powers—forcing to an efficient equivalent wind model. This criterion could be modified considering more clusters as a function on the distribution of wind patterns (wind farm effect), e.g. wind turbines operating with similar winds.

image

Figure 5. Cluster representation of the proposed wind farm.

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The equivalent model of each cluster is the same model as the individual wind turbines but with a re-scaled rated power, the same mechanical and generation systems and active and reactive power references obtained from the sum of the individual references. The equivalent electrical network of each cluster is easily obtained assuming that the short-circuit impedance of the aggregated cluster must be equal to the short-circuit impedance of the cluster.

The equivalent wind speed of each cluster structure is obtained as proposed in,[7] where different winds incident on each wind turbine are considered. The scope of the method is to obtain an approximation of the power generated by the wind farm from the sum of the individual power that each individual wind turbine of the cluster generates, according to its incoming wind speed and power curve. A re-scaled equivalent power curve (identical to the power curve of the individual wind turbines) allows for the calculation of the equivalent wind speed.

Finally, the active and reactive power responses of the equivalent model are obtained by adding the active and reactive output powers of each cluster.

3.2 Compound equivalent model with equivalent wind in each cluster

In this case, the equivalent mechanical system model presents a cluster structure, whereas the generation system model is obtained by the aggregation of the individual generation systems into a single equivalent model. Thus, following the same structure as in Section 3.1, two approximated mechanical powers are aggregated and the resulting equivalent power is applied to an equivalent generator system.

The mechanical system model of each cluster receives the incoming winds that experience all the wind turbines of the cluster, and it obtains an equivalent incoming wind. This incoming wind serves as input to a simplified model resulting in the aggregation of the mechanical systems of the cluster.[11] The simplified model (Figure 6) presents the following components: rotor, drive train, simplified model of the induction generator (represented by the first order model), simplified rotor speed controller (represented by the power–speed control curve) and a blade pitch angle controller.

image

Figure 6. Simplified mechanical model of the cluster.

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A re-scaled model of the individual generation systems cannot represent the equivalent generation system, because in this case, the clusters have DFIG wind turbines with different rated powers. Thus, the aggregation of different generation systems into an equivalent one requires the use of specific techniques, as shown in some studies about the dynamic representation of groups of induction motors.[16, 17] In order to determine the steady-state induction generator parameters used in the third-order model of the equivalent generator, the assumptions considered in this paper are the following:

  1. The equivalent magnetizing reactance (Xm,e) is obtained from the parallel of the individual ones (Xm,i).
  • display math(12)
  1. In the same way, the right branch impedances of each equivalent circuit (Zs,i + Zr,i + Zl,i) are parallel combined into equivalent impedance (Zpe) , assuming constant slip operation at rated power (si = so,i).
  • display math(13)

where Zs,i = Rs,i + jXσs,i is the stator impedance; Zr,i = Rr,i + j + Xσr,i is the rotor impedance; and Zl,i = Rr,i(si-1-1) is the load impedance.

  1. The active power equivalence between the aggregated generation system and all the individual generation systems is assumed.[17]
  2. It is considered that the leakage reactances, as well as stator and rotor resistances, are related by constant values,[18] which are obtained from the weighted average of the rated powers of the individual generation systems.

The equivalent control system keeps the same rotor speed and reactive power controllers as the individual wind turbines. However, it does not include the pitch angle controller, since it has already been considered in the equivalent mechanical system model. All the parameters of the control system have been re-tuned to the equivalent generation system.

The aggregated internal electrical network is represented by equivalent impedance, considering that the short-circuit impedance of the compound model must be equal to the short-circuit impedance of the complete model.

Figure 7 depicts the structure of the proposed compound model, integrated by the simplified model of each cluster and the aggregated generation system.

image

Figure 7. Structure of the compound model with equivalent wind and simplified model of each cluster.

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3.3 Compound equivalent model without aggregation of mechanical systems

The aggregated model, including equivalent wind of each cluster, does not have enough accuracy for wind turbines experiencing different wind speeds. The reason is the non-linearity of its model, when the operating conditions of each wind turbine are not similar.[9] This leads to the use of a mechanical model that reflects the non-linearity of each wind turbine of the wind farm, keeping the individual rotors, drive trains and pitch angle controllers. Therefore, a simplified model, similar to that shown in Figure 6, represents the mechanical system of each wind turbine in this compound equivalent model. In this case, the simplified models receive the individual incoming wind speeds, and there are as many simplified models as there are wind turbines in the farm.

The equivalent mechanical power is calculated by aggregating the approximated mechanical powers obtained from the simplified models. This equivalent serves as input to an aggregated generation system, operating in an internal electrical network, identical to that described in Section 3.2. Figure 8 depicts the structure of this compound model.

image

Figure 8. Structure of the compound model without aggregated mechanical system.

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3.4 Mixed equivalent model with aggregation of mechanical systems

The concept of a mixed model involves the use of an equivalent model on the basis of the cluster and compound models. It presents an aggregated generation model as presented in Section 3.2 and uses the simplified models of the DFIG wind turbines presented in Section 3.3. Furthermore, it can obtain an equivalent wind as input to a single-aggregated mechanical system model.

The scope is to obtain an equivalent rotor model with the same output power as the sum of the individual ones:

  • display math(14)

where the equivalent rotor area (Ae) and the equivalent power coefficient (cp,e) are calculated from the aggregation of the individual ones:

  • display math(15)
  • display math(16)

The equivalent incoming wind (ue), used as input to the rotor of the single-machine equivalent model, can easily be obtained as follows:

  • display math(17)

To obtain the power coefficient of each wind turbine (cp,i), it is necessary to calculate the tip speed ratio, which depends on the incoming wind and the rotor speed:

  • display math(18)

In this work, the power coefficient of each wind turbine is estimated on line by using the simplified model of the turbine described in Section 3.2.

The single-aggregated mechanical system needs to approximate the drive train parameters. In this case, it is assumed a parallel combination of the individual drive trains, and thus, the equivalent drive train parameters are defined by

  • display math(19)
  • display math(20)
  • display math(21)
  • display math(22)

where N is the gearbox ratio.

In these equations, it is necessary to obtain the equivalent rotor speed in both shafts of the drive train and the equivalent gearbox ratio. An easy way to obtain the equivalent rotational speed can be seen in,[19] where the equivalent slip at full load (so,e) is the average value of the individuals (Pn,i), weighted by its rated powers:

  • display math(23)

Thus, the equivalent generator speed at rated power gn,e) is calculated as

  • display math(24)

where ωs,e is the equivalent synchronous speed.

In a similar way, the equivalent rotor speed is assumed to be constant and equal to the value at the full load point so that it can be easily calculated if the equivalent gearbox ratio is determined. This value can be arbitrarily chosen, but in this case, it is considered as the average of the individuals, rated by its rated powers:

  • display math(25)

where the equivalent rotor speed at rated power is

  • display math(26)

Finally, assuming nominal operation, the parameters of the equivalent gearbox can be obtained by

  • display math(27)
  • display math(28)
  • display math(29)
  • display math(30)

Figure 9 shows the structure of this equivalent model. As previously mentioned, it presents an equivalent mechanical system, which is composed of the rotor model experiencing the equivalent wind and the equivalent drive train previously described. It receives the equivalent incoming wind speed obtained from the approximation of the power coefficients and is calculated by a simplified model of each wind turbine. This reduced model determines the approximation of the generator mechanical power, which is applied to an aggregated generation system, operating in the internal electrical network.

image

Figure 9. Structure of the mixed equivalent model with single-aggregated mechanical system model.

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4 SIMULATION RESULTS AND DISCUSSION

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 COMPLETE WIND FARM MODEL
  5. 3 EQUIVALENT WIND FARM MODELS
  6. 4 SIMULATION RESULTS AND DISCUSSION
  7. 5 CONCLUSIONS
  8. APPENDIX: Parameters
  9. REFERENCES

The equivalent models proposed in this paper are verified by comparing their responses with those of the complete wind farm. These wind farm models are implemented and simulated in Simulink®.

As mentioned in Section 2, Figure 1 shows the wind farm structure considered in this study. It presents a 6 × 660 kW DFIG cluster and a 6 × 2 MW DFIG cluster, both with radial structure. The numeric parameters of this wind farm are included in the Appendix. This configuration can correspond to an old wind farm, in which the newest wind turbines with higher rated power have replaced some old ones. Furthermore, this configuration can represent two wind farms connected to the same PCC but built with a delay of several years.

The comparisons between the complete and reduced models have been performed under two different operation conditions: normal conditions (wind fluctuations) and fault conditions (grid disturbances).

4.1 Dynamic operation during normal condition

In this study, the reduced models are tested by taking into account that the wind turbines operate with variable wind (wind fluctuations). Figure 10 shows the four different operational conditions considered:

  • Case a: all the incoming winds are below rated wind speed, with time series between 8 and 12.5 m s−1. It could correspond to wind farms located on topographically simplex sittings (smooth land or offshore) and operating with low wind speeds.
  • Case b: all the incoming winds are above rated wind speed, with time series between 13 and 16.5 m s−1. It is the same type of wind farms as case a but operating with high wind speeds.
  • Case c: the first cluster (660 kW wind turbines) experiences low incoming winds between 9 and 10 m s−1, whereas the second cluster (2 MW wind turbines) receives high wind speeds between 17.5 and 20 m s−1. There are great differences between the incident winds on the clusters, as is normal for wind farms on topographically complex sittings with widely separated clusters (mountain ladder).
  • Case d: the wind speeds incident on all the wind turbines present a great disparity, from 9.5 to 20 m s−1. This case corresponds to steep sittings with gusty winds.
image

Figure 10. Wind speed series for the four cases under study.

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In the cases a, c and d, the active power reference of the whole wind park is set to a rated value at PCC, whereas the reactive power reference is set to unity power factor. Meanwhile, case b is used to evaluate the reduced models with different power settings. Thus, the active power reference changes from 100% to 50% of rated value at 20 s with unity power factor, and at 40 s, the reactive power reference changes to 20% of the rated power.

The variables used in the comparison are the active and reactive power at PCC, represented in per unit (p.u.) (base power of 2 MW).

Case a:

All the incoming winds are below rated wind speed. The power extracted from the wind is less than that desired so that the control system must maximize the mechanical power extracted from the wind in order to achieve an active power as close as possible to the active power reference. The rotor speed controllers of all the wind turbines works by maximizing the power extracted, the reactive power controllers allow the unity power factor operation and the blade pitch controllers adjust the pitch angle to 0º.

Figure 11 shows the responses of the active and reactive power at the PCC of the complete and reduced models. The cluster equivalent model presents an active power response very close to the complete model, whereas the rest present little vertical offset with respect to the complete model response. This offset is caused by the aggregation of the generation systems in the compound model with equivalent wind and by the use of the simplified models of each wind turbine in the other two models. The reactive power responses of the reduced models are very similar to that of the complete model.

image

Figure 11. Powers at the PCC (case a).

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Case b:

All the incoming winds are around rated wind speed. In this case, the winds are above rated speed during the majority of the simulation, and the wind turbines operate controlling the output power to their power settings. At the same time, the pitch angle controller limits the rotational speed to the rated value. Figure 12 shows that the reduced models achieve an accurate approximation of the active and reactive powers of the complete wind farm.

image

Figure 12. Powers at the PCC (case b).

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When some of the wind turbines receive below rated wind speed, the active power response of the complete model is under the wind farm rated power, as can be seen in zoom of Figure 12. However, the powers of the reduced models are always at rated value. The reason for this difference is simple. In the models with equivalent wind, an approach based on the use of an equivalent power curve cannot reflect these small deviations. On the other hand, in the case of the equivalent models without mechanical aggregation, the approach based on the sum of the mechanical powers (obtained from simplified models of each wind turbine) prevents the fluctuations and flattens the response of the equivalent model.

The simulation results show an adequate performance of the equivalent models so that the desired power references are achieved. As can be seen, all the models present certain reactive power variation when the active power reference is changed from 100% to 50% by a slope of 0.3. This variation is more pronounced in the case of the compound model with equivalent wind.

Case c:

Below rated wind speed in the first cluster and above rated wind speed in the second cluster. In this simulation, the 660 kW wind turbines operate maximizing the power extracted from the wind, whereas the 2 MW wind turbines work limiting the mechanical power extracted from the wind.

As shown in Figure 13, the active powers generated by the equivalent models of the wind farm are very close to that of the complete model but with a small offset in the responses of the equivalent models with aggregation of the mechanical systems.

image

Figure 13. Powers at the PCC (case c).

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The reactive power responses of the cluster equivalent models are quite similar to that of the complete model. In the rest of the reduced models, in which the aggregation of the generation systems is applied, the differences are smaller. This is due to the fact that, in these cases, the reduced models have only a reactive power controller and there is not an accumulation of errors in the controllers.

Case d:

Great disparity in the incoming wind speeds. The aim of this case is to evaluate the ability of the reduced models to approximate the wind farm response with great disparity in the incoming winds. As can be seen in Figure 14, the responses of active power present large differences, whereas the responses of reactive power are quite similar to that of the complete model. In fact, the reduced models with equivalent wind present measurable offsets over the complete model. Meanwhile, the equivalent models without mechanical aggregation present a response quite similar to that of the complete model.

image

Figure 14. Powers at the PCC (case d).

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Summarizing, Table 1 depicts the range (maximum and minimum values), mean, standard deviation of the powers (at p.u.) obtained by each model and the integral of the absolute error (IAE) between the value obtained by the complete model and that of the equivalent model during the four cases simulated. These values are used to compare the models.

Table 1. Comparison of the results obtained for the active and reactive powers under wind fluctuation tests.
  Case aCase bCase cCase d
 ModelRange (Max, Min)MeanSTDIAERange (Max, Min)MeanSTDIAERange (Max, Min)MeanSTDIAERange (Max, Min)MeanSTDIAE
  1. STD = standard deviation.

PComplete model6.307, 6.0626.1850.066 7.981, 3.9885.2551.836 7.525, 7.4407.4850.018 7.345, 7.1527.2620.041 
Cluster equivalent model6.304, 6.0556.1770.0670.5307.981, 3.9895.5691.9320.0097.525, 7.4397.4840.0180.1197.515, 7.3257.4390.0389.554
Error−0.100, 0.0300.008−0.001 0.000, −0.001−0.314−0.096 0.000, 0.0010.0010.000 −0.170, −0.173−0.1770.003 
Compound model equivalent wind6.433, 6.1256.2630.0804.6177.980, 3.9905.1061.7750.0307.548, 7.4517.4810.0220.7307.630, 7.4247.5390.04315.82
Error−0.126, −0.063−0.078−0.014 0.001, −0.0020.1490.061 −0.023, −0.0110.004−0.004 −0.285, −0.272−0.277−0.002 
Compound model without mechanical aggregation6.407, 6.0326.2310.0853.0057.981, 3.9905.0731.7590.0297.554 7.4367.4830.0260.9317.399, 7.1437.2760.0530.001
Error0.003, 0.007−0.046−0.019 0.000, −0.0020.1820.077 −0.029, 0.0040.002−0.008 −0.054, 0.009−0.014−0.012 
Mixed equivalent model6.407, 6.0326.2310.0853.0057.981, 3.9905.5311.9230.0177.554, 7.4367.4830.0260.9327.399, 7.1437.2760.0530.001
Error0.003, 0.007−0.046−0.019 0.000, −0.002−0.276−0.087 −0.029, 0.0040.002−0.008 −0.054, 0.009−0.014−0.012 
QComplete model0.014, −0.0060.0060.004 1.599, −0.0420.5500.749 0.008, 0.0030.0069.1.10−4 0.016, −0.0050.0060.002 
Cluster equivalent model0.015, −0.0050.0060.0040.0761.598, −0.0800.4880.7250.3030.008, 0.0040.0067.5.10−40.0240.011, 0.0020.0060.0020.059
Error−0.001, −0.0010.0000.000 0.001, 0.0380.0620.024 0.000, −0.0010.0001.6.10−4 0.005, −0.0070.0000.000 
Compound model equivalent wind0.004,−0.0051.0.10−50.0020.3611.597, −0.2340.6480.7770.8600.002, −0.003−5.9.10−60.0010.3610.003, −0.0032.0.10−50.0010.363
Error0.010, −0.0010.0060.002 0.002, 0.192−0.098−0.028 0.006, 0.0060.006−9.0.10−5 0.013, 0.0020.0060.001 
Compound model without mechanical aggregation0.005,−0.0078.6.10−60.0020.3611.597, −0.1060.6710.7800.5050.003,−0.003−5.3.10−60.0010.3610.005, −0.0042.1.10−50.0020.362
Error0.009, 0.0010.0060.002 0.002, 0.064−0.121−0.031 0.005, 0.0060.006−9.0.10−5 0.011, 0.0010.0060.000 
Mixed equivalent model0.005, −0.0071.1.10−50.0020.3611.596, −0.0540.4890.7290.5220.003, −0.003−5.9.10−60.0010.3610.005, −0.0042.1.10−50.0020.362
Error0.008, 0.0010.0060.002 0.003, 0.0120.0610.020 0.005, 0.0060.006−9.0.10−5 0.011, 0.0010.0060.000 

By analyzing the results, it can be concluded that the equivalent models allow an accurate approximation of the active and reactive powers of the complete wind farm. When the four equivalent models are compared, the following conclusions can be established:

  • The cluster equivalent model achieves better approximation of active power when the control systems are optimizing the power extracted from the wind and the differences between the wind speed incident on each machine are small (case a).
  • All the equivalent models present similar response when the control systems are limiting the power extracted from the wind. Nevertheless, they have the same problem of reflecting small deviations when some of the wind turbines receive below rated wind speeds (case b).
  • The four equivalent models achieve an adequate response to active and reactive power reference changes, although the compound model with equivalent wind presents the most pronounced reactive power variation when the wind farm receives an abrupt change in the active power setting (case b).
  • When the differences between the incoming wind speeds at each cluster are greater, the equivalent models without mechanical aggregation present active power responses closer to that of the cluster equivalent model, which achieves the best approximation (case c).
  • If the incoming winds on each cluster are gusty winds, with large speed differences, the equivalent's models without mechanical aggregation have a distinct advantage in the approximation of the active power when compared with the equivalent models based on equivalent incoming wind (case d).
  • The reactive power responses of all the equivalent models are quite similar. The aggregation of control systems reduces the accumulated errors from the reactive power controllers and achieves small differences with respect to the response of the complete model.

4.2 Dynamic operation under fault conditions

To verify adequately the robustness of the equivalent models under fault conditions, two voltage dips at the PCC are studied:

  • A slow voltage dip of 0.8 p.u. with a duration of 1 s.
  • A fast voltage dip of 0.5 p.u. with a duration of 0.5 s.

In both cases, as is normal for grid disturbance simulations, constant incoming winds are assumed (with the average values of the wind speed time series used in case b).

Figure 15 shows the power responses in the case of the slow voltage dip. As can be seen during the fault, the responses of the complete and equivalent models are similar. In general, the active and reactive powers obtained by the equivalent models are a little smaller than those of the complete model, except for the compound model without mechanical aggregation and the mixed equivalent model that achieves higher active power at the end of the voltage dip. Note that the wind turbines provide reactive power during the fault.

image

Figure 15. Powers at the PCC during slow voltage dip.

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Just after fault clearance, the crowbar continues to switch on, and as the generator is operating as a SCIG, it starts to absorb reactive power for its magnetization. Once the crowbar is switched off, the RSC starts actively to control the active and reactive power, and the wind turbine starts to recover the pre-fault values. In all the equivalent models except for the cluster equivalent model, the pre-fault active power recovery is a little faster, whereas the pre-fault reactive power recovery is slightly slower. The cluster equivalent model presents a slightly oversized response in the pre-fault values recovery, although its response is the most similar to that of the complete model.

Figure 16 shows the wind farm voltage at the PCC and the voltage at the MV side of the MV/HV transformer. In this case, the voltage recovery of the complete model is nearly the same as those obtained with the equivalent models.

image

Figure 16. a) Wind farm voltage at the PCC and b) voltage at the MV side of the MV/HV transformer, during the slow voltage dip.

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Figures 17 and 18 show the response of the models during the fast voltage dip. During the fault, the active powers obtained by the equivalent models with generation system aggregation are slightly smaller than that of the complete model. However, the active power is a little higher in the case of the cluster equivalent model. The reactive power provided by the equivalent models are similar to that of the complete model, although this power is a little higher in the compound model with equivalent wind and mixed equivalent model. After fault clearance, the behavior of the equivalent models is similar to that of the slow voltage dip.

image

Figure 17. Powers at the PCC during the fast voltage dip.

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image

Figure 18. a) Wind farm voltage at the PCC and b) voltage at the MV side of the MV/HV transformer, during the fast voltage dip.

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4.3 Discussion

A summary of the simulation results obtained by each equivalent model for both wind fluctuations and grid disturbances is depicted in Table 2. The computational time reduction achieved by each equivalent model with respect to the computational time of the complete model is also provided. These results were achieved by using an integration step of 2 ms in all the simulations. The results show that the computational time reduction is similar in all the equivalent models, over 99%.

Table 2. Summary of the simulation results obtained by the equivalent models.
ModelWind fluctuationsGrid disturbancesTime simulation reduction
Cluster equivalent modelBetter active power response under rated wind speedSmall differences in the active and reactive power during the grid fault.99.08%
Impoverished response with wind speed differences in the cluster
Small deviations in case of incoming winds around rated speedSlightly oversized response after fault clearance
Reactive power adjournment with great variability of wind speeds
Compound model equivalent windActive power displacement under rated wind speedSmall differences in the active and reactive power during the grid fault.99.75%
Impoverished response with wind speed differences in the cluster
Small deviations in case of incoming winds around rated speedFast pre-fault active power recovery and slow pre-fault active power recovery after fault clearance
Small displacements on reactive power response
Compound model without mechanical aggregationAdequate for different rated powerSmall differences in the active and reactive power during the grid fault.99.58%
Small damping in active power response
Small deviations in case of incoming winds around rated speedFast pre-fault active power recovery and slow pre-fault active power recovery after fault clearance
Small displacements on reactive power response
Mixed equivalent modelAdequate for different rated powerSmall differences in the active and reactive power during the grid fault.99.50%
Small damping in active power response
Small deviations in case of incoming winds around rated speedFast pre-fault active power recovery and slow pre-fault active power recovery after fault clearance
Small displacements on reactive power response

As seen, the cluster equivalent model presents better approximation of the active and reactive power when the differences between the incoming wind speeds are short. This model achieves the lowest computational time reduction. On the other hand, when the wind speed differences increase, the equivalent models with simplified models of the mechanical systems present the best results. The compound model with equivalent wind achieves the highest computational time reduction, whereas the reduction is quite similar in the case of the compound model without mechanical aggregation and the mixed equivalent model.

During grid faults, the responses of the equivalent models present small differences with respect to that of the complete model. After the fault clearance, the cluster equivalent model presents a slightly oversized response in the pre-fault values recovery, although its response is the most similar to that of the complete model. In the rest of equivalent models, the pre-fault active power recovery is a little faster, whereas the pre-fault reactive power recovery is a little slower.

In conclusion, Table 3 shows the possibilities of the application of the equivalent models under study. The cluster equivalent model and compound model with equivalent wind achieve better results when the differences between the wind speeds incidents on each wind turbine are small. Therefore, these models are recommended for application to wind farms on topographically simplex terrains (smooth or offshore), in which the wind differences are small. However, the cluster equivalent model is suitable for wind farms with the same wind turbines in a cluster, whereas the compound model with equivalent wind could be used in wind farms composed by wind turbines with similar rated power.

Table 3. Application possibilities of the equivalent models.
ModelApplicability
Cluster equivalent modelWind farms with the same wind turbines in a cluster on topographically simplex terrains (smooth or offshore)
Compound model equivalent windWind farms with similar wind turbines on topographically simplex terrains (smooth or offshore)
Compound model without mechanical aggregationWind farms with different wind turbines on any type of terrain
Mixed equivalent modelWind farms with different wind turbines on any type of terrain

On the contrary, the compound model without mechanical aggregation and the mixed equivalent model achieve good results with great disparity in the incoming wind speeds, and therefore, they are recommended for application to wind farms with different wind turbines on any type of terrain, in which the wind differences could be significant.

5 CONCLUSIONS

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 COMPLETE WIND FARM MODEL
  5. 3 EQUIVALENT WIND FARM MODELS
  6. 4 SIMULATION RESULTS AND DISCUSSION
  7. 5 CONCLUSIONS
  8. APPENDIX: Parameters
  9. REFERENCES

This paper has presented four equivalent models of variable-speed wind farms equipped with DFIG wind turbines for dynamic studies, from the point of view of the transient stability of the electrical power system. The proposed equivalent models have the particularity that they consider different wind turbines in the wind farm layout, experiencing different incoming winds.

The first model considered is a classical clustering model, which uses the simplest criterion based on aggregating identical wind turbines. Each reduced model experiences an equivalent wind speed according to the incoming winds and power curves of the wind turbines aggregated in the cluster. The second model is a compound model that preserves the clustering structure in the mechanical system, whereas it aggregates the generation systems into a single one. Trying to solve the problem of non-linearity of the wind turbine, the third reduced model replaces the clustering mechanical model by a simplified model of each wind turbine in the farm, in order to approximate the mechanical power of the aggregated wind turbines. The latest model is a mix of previous ones that use an equivalent incoming wind, obtained from the simplified models, as input to a single-aggregated mechanical system, which is coupled to an aggregated generation system model.

To evaluate the proposed equivalent models, a wind farm composed of wind turbines with different rated power was considered. The results obtained by the equivalent models were compared with those of the complete model.

The results showed that the main advantage of using aggregated models is to reduce the computational time of the simulations while achieving an approximation of the whole response of the wind farm.

In conclusion, the cluster equivalent model is recommended for application to wind farms with the same wind turbines in a cluster on topographically simplex terrains (smooth or offshore), the compound model with equivalent wind for wind farms composed by wind turbines with similar rated power on topographically simplex terrains (smooth or offshore) and the compound model without mechanical aggregation and the mixed equivalent model for those with different wind turbines on any type of terrain.

APPENDIX: Parameters

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 COMPLETE WIND FARM MODEL
  5. 3 EQUIVALENT WIND FARM MODELS
  6. 4 SIMULATION RESULTS AND DISCUSSION
  7. 5 CONCLUSIONS
  8. APPENDIX: Parameters
  9. REFERENCES

660 kW Wind turbine

Rated voltage: 690 V, Area = 1734.94 m2, Hr = 3 s., gearbox ratio: 1:52.5, K = 100 p.u., D = 30 p.u., Hg = 0.5 s., Rs = Rr' = 0.0 p.u., Xσs = 0.04 p.u., Xσr' = 0.05 p.u., Xm = 2.9 p.u.

2 MW Wind turbine

Rated voltage: 690 V, Area = 4417.86 m2, Hr = 4.3 s., gearbox ratio: 1:89, K = 95 p.u., D = 40 p.u., Hg = 0.9 s., Rs = Rr' = 0.01 p.u., Xσs = 0.1 p.u., Xσr' = 0.08 p.u., Xm = 3 p.u.

Wind farms electrical network

LV/MV transformers:

660 kW groups (800 kVA, 20/0.69 kV, εcc = 5.0%); 2 MW groups (2500 kVA, 20/0.69 kV, εcc = 6.0%).

MV/HV transformers:

660 kW group (4 MVA, 66/20 kV, εcc = 8%); 2 MW group (16 MVA, 66/20 kV, εcc = 8.5%).

Electrical lines:

Individual MV lines: r = 0.3 Ω/km, x = 0.1 Ω/km, length = 200 m; MV lines of groups of 660 kW: r = 0.15 Ω/km, x = 0.05 Ω/km, length1 = 0.5 km, length2 = 1 km, length3 = 2 km; MV lines of groups of 2 MW: r = 0.15 Ω/km, x = 0.1 Ω/km, length1 = 0.5 km, length2 = 1 km.

Feeders:

660 kW group (r = 0.16 Ω/km, x = 0.35 Ω/km, length = 10 km); 2 MW group (r = 0.12 Ω/km, x = 0.3 Ω/km, length = 20 km).

Grid:

Short circuit power at PCC: 500 MVA. X/R ratio: 15.

REFERENCES

  1. Top of page
  2. ABSTRACT
  3. 1 INTRODUCTION
  4. 2 COMPLETE WIND FARM MODEL
  5. 3 EQUIVALENT WIND FARM MODELS
  6. 4 SIMULATION RESULTS AND DISCUSSION
  7. 5 CONCLUSIONS
  8. APPENDIX: Parameters
  9. REFERENCES