Author: National Renewable Energy Laboratory^{[1]}
The full converter wind turbine (FCWT) employs a permanent magnet alternator (PMA). This technology has a number of significant advantages^{[2]}. It effectively decouples the generator from the grid, improving fault response. It allows the turbine to operate over a wide speed range, leading to improved power extraction from the wind. The converter interfacing the turbine to the grid has to handle the entire output of the generator (unlike in a DFIG turbine where the converter handles only 30% to 40% of the generator output) and hence is more costly and lossy, but also provides more headroom to supply reactive power to the grid. The permanent magnet alternator (PMA) itself has no rotor windings, reducing excitation losses and reducing the size of the generating unit with respect to competing technologies. Absence of rotor slip rings reduces maintenance requirements. This combination of factors is driving the increasing penetration of FCWTs, especially for offshore wind power plants.
The popularity of FCWTs has led to a search for reliable models to evaluate the impacts of integrating these FCWTs into the existing grid. The model presented in this article is a generic, manufacturerindependent model for a PMAbased FCWT, with no restrictions on its use. The converter topology of the model described below is a popular one; the PMA is interfaced to the grid through an ACDCAC conversion system. The ACDC converter is comprised of a diodebridge rectifier and a buckboost converter which controls the DC link voltage. The DCAC conversion is accomplished using a currentcontrolled inverter which controls the real and reactive output power. Although the focus in this article is on the specific topology mentioned, various converter topologies can be modeled with simple modifications using the same framework. In the past FCWT models with an entirely different topology using this framework have been developed^{[3]}. The mechanical and aerodynamic components of the model presented here are identical to those presented in^{[3]}.
The complete model has been implemented in PSCAD/EMTDC for the purposes of this article. However, the model is straightforward to implement using other popular simulation packages such as MATLAB/Simulink. The model is based on parameters from an Enercon E82 2MW turbine.
Contents
 1 Model Development
 1.1 Aerodynamic Block
 1.2 Mechanical DriveTrain
 1.3 Reference Power Calculation from Wind Speed
 1.4 Pitch Control Block
 1.5 Permanent Magnet Alternator
 1.6 Rectifier and Buck/Boost Converter for DCLink Voltage Control
 1.7 Inverter
 1.8 Unit Transformer and Grid Representation
 1.9 Complete Model Implemented in PSCAD/EMTDC
 2 Model Testing
 3 Dynamic Response
 4 References
Model Development
Turbine Properties  

Turbine make  Enercon E82 2MW 
Regulation method  Pitch control (enabled) 
Rotor diameter  82 m 
Hub height  78 m 
Number of blades  3 
Cutin wind speed  4 m/s 
Cutout wind speed  28 m/s 
Rated wind speed  15 m/s (14 m/s used) 
Rotor speed  6/18 rpm 
From a modeling standpoint, a full converter PMA wind turbine consists of the following mechanical and electrical subsystems:
 Aerodynamic model for rotor;
 Mechanical twomass model for drivetrain;
 Reference power calculation block;
 Pitch controller;
 Permanent magnet alternator (PMA) model;
 Rectifier and buck/boost converter models (for DClink voltage control);
 Inverter model (currentcontrolled VSI);
 Unit transformer and grid representation.
The interaction between each of the components listed above determines the wind turbine model’s steadystate and dynamic response. Each of these subsystems presents a unique modeling challenge. Modeling of the aerodynamics and mechanical drivetrain is based on the differential and algebraic equations that describe their operation. Reference power is currently calculated based on wind speed, though in the future it will be calculated based on rotor speed. The pitch controller currently utilizes both power and rotor speed inputs, though it too will be modified in the future to only use rotor speed as input. A general PMA model is available in PSCAD/EMTDC which can be modified to serve the purposes of this model. The rectifier, buck/boost converter and inverter are modeled explicitly using diode and IGBT models supplied with PSCAD/EMTDC. Details for each of the subsystems are presented in the following subsections.
Aerodynamic Block
The aerodynamic block consists of three subsystems: tipspeed ratio calculation, rotor power coefficient (C_{P}) calculation, and aerodynamic torque calculation. Wind speed and pitch angle are userdefined inputs. Since the model is intended to study the dynamic response of wind turbines to grid events, the assumption is usually made that the wind speed stays constant during the grid event. However, this model allows the wind speed input signal to be set to any value at the start of the simulation runtime and also to be modified during the run. It is also possible to use a timeseries of actual wind speed data.
TipSpeed Ratio Calculations
The tipspeed ratio or TSR, denoted by λ, is the ratio of the bladetip linear speed to the wind speed^{[4]}. The TSR determines the fraction of available power extracted from the wind by the wind turbine rotor. The TSR can be calculated as follows
\(\lambda=\frac{\omega_{rotor} · R_{rotor}}{V_{wind}}\)where,
ω_{rotor} = rotor angular speed [rad/s]
R_{rotor} = rotor radius [m]
V_{wind} = wind speed [m/s]
Rotor Power Coefficient (C_{p}) Calculation
The TSR, together with the userdefined blade pitch angle β, are used to calculate the rotor power coefficient, denoted by C_{P}. The rotor power coefficient is a measure of the rotor efficiency and is defined as
\(\mathrm{C}_P=\frac{Extracted\ Power}{Power\ in\ Wind}=\frac{P_{rotor}}{P_{wind}}\)There is a constant value of λ which, if maintained for all wind speeds, will result in an optimal C_{P} curve and optimal power extraction from the wind. Variablespeed wind turbines are equipped with a pitchchange mechanism to adjust the blade pitch angle and obtain a better power coefficient profile. In the model, a set of generic C_{P} curves are used to calculate the value of C_{P}^{[5]}.
Aerodynamic Torque Calculation
The kinetic energy E (in J) of an air mass m (in kg) moving at a speed V_{wind} (in m/s) is given by^{[4]}
\(E=\frac{1}{2}mV_{wind}^2\)
If the air density is ρ (kg/m^{3}), mass flow through an area A is given by
\(\dot{m}=ρAV_{wind}\)
Thus, an equation for the power (in W) through a crosssectional area A normal to the wind is
\(P_{wind}=\frac{1}{2}ρAV_{wind}^3\)
In the case of a wind turbine, area A is the area swept by the rotor blades. Only a part of this power may be captured due to the nonideal nature of the rotor, hence the need for the coefficient C_{P}.
\(P_{rotor}=\frac{1}{2}ρ\cdot\mathrm{C}_p\cdotπ\cdot\mathrm{R}_{rotor}^2\cdot\mathrm{V}_{wind}^3\)
The aerodynamic torque developed (in Nm) can then be calculated
\(\Gamma_{rotor}=\frac{P_{rotor}}{ω_{rotor}}=\frac{\frac{1}{2}\cdotρ\cdot\mathrm{C}_p\cdotπ\cdot\mathrm{R}_{rotor}^2\cdot\mathrm{V}_{wind}^3}{ω_{rotor}}\)
Mechanical DriveTrain
The mechanical block consists of the rotor shaft, generator shaft, and a gearbox. The shafts and the gearbox are modeled using a twomass inertia representation. For a rotational system^{[6]}, consisting of a disk with a moment of inertia J mounted on a shaft fixed at one end, let us assume that the viscous friction coefficient (damping) is D and that the shaft torsional spring constant (stiffness) is K. The torque acting on the disk can be calculated from the freebody diagram of the disk, as follows
\(\Gamma(t)=J\frac{d^2\theta(t)}{dt^2}+D\frac{d\theta(t)}{dt}+K\theta(t)\)
A more complex rotational system consists of two such systems. The two systems are coupled through a gear train, and Γ is the external torque applied to the disk of System 1. Γ_{1}, Γ_{2} are transmitted torques. N_{1}, N_{2} are the numbers of teeth of Gear 1 and Gear 2. J_{1}, J_{2}, D_{1}, D_{2}, K_{1}, K_{2} are the moments of inertia, damping, and stiffness of System 1 and System 2, respectively. The system is still timedependent, but the notation t is dropped for the sake of clarity.
The torque equation at J_{1} can then be calculated as
\(\Gamma_1=J_1\frac{d^2\theta_1}{dt^2}+D_1\frac{d\theta_1}{dt}+K_1\theta_1\)
The torque equation at J_{2} can then be calculated as:
\(\Gamma_2=J_2\frac{d^2\theta_2}{dt^2}+D_2\frac{d\theta_2}{dt}+K_2\theta_2\)
Since Γ_{1} = (N1/N2)Γ_{2} and Θ_{2} = (N1/N2)Θ_{1}, the quantities on Gear 2 side can be referred to the Gear 1 side^{[6]}.
\(\Gamma_1=\left (\frac{N_1}{N_2} \right)\left (J_2\frac{d^2\theta_2}{dt^2}+D_2\frac{d\theta_2}{dt}+K_2\theta_2 \right)\)
\(\Gamma_1=\left (\frac{N_1}{N_2} \right)\left [J_2\left (\frac{N_1}{N_2} \right)\frac{d^2\theta_1}{dt^2}+D_2\left (\frac{N_1}{N_2} \right)\frac{d\theta_1}{dt}+K_2\left (\frac{N_1}{N_2} \right)\theta_1 \right]\)
\(\Gamma_1=J_{refl}\frac{d^2\theta_1}{dt^2}+D_{refl}\frac{d\theta_1}{dt}+K_{refl}\theta_1\)
where J_{refl}, D_{refl}, and K_{refl} are the quantities reflected on the Gear 1 side. Substituting the above equations and rearranging them, we obtain the equation for the applied torque. The gear train is eliminated in the equivalent system.
\(\Gamma=J_{equiv}\frac{d^2\theta_1}{dt^2}+D_{equiv}\frac{d\theta_1}{dt}+K_{equiv}\theta_1\)
where,
\(J_{equiv}=J_1+J_2\left (\frac{N_1}{N_2} \right)^2=J_1+J_{refl}\)
\(D_{equiv}=D_1+D_2\left (\frac{N_1}{N_2} \right)^2=D_1+D_{refl}\)
\(K_{equiv}=K_1+K_2\left (\frac{N_1}{N_2} \right)^2=K_1+K_{refl}\)
The wind turbine drivetrain can therefore be modeled as a twomass system coupled through a gear train. The quantities on the wind turbine rotor side of the gearbox can be reflected to the generator side. This eliminates the gear ratio and results in a twomass representation of the wind turbine. Neglecting the effects of the gearbox moment of inertia, damping, and stiffness is justifiable since the moment of inertia of the wind turbine rotor is comparatively very high.
Torque equations representing the mechanical behavior of the wind turbine are derived, based on the twomass model. The aerodynamic torque from the wind turbine rotor and the electromechanical torque from the directconnect induction generator act in opposition to each other. Torque equations with all quantities referred to the generator side are
\(J_T\ddot {\theta_T}+D(ω_Tω_G)+K(\theta_T\theta_G)=\Gamma_T\)
\(J_G\ddot {\theta_G}+D(ω_Gω_T)+K(\theta_G\theta_T)=\Gamma_G\)
where,
J_{T}, J_{G} = moments of inertia of the wind turbine rotor and the generator kgmm
Γ_{T}, Γ_{G} = wind turbine aerodynamic and generator electromagnetic torque Nm
ω_{T}, ω_{G} = wind turbine rotor and the generator speed rad/s
Θ_{T}, Θ_{G} = angular position of the rotor and the generator rad
D, K = equivalent damping and stiffness Nms/rad, Nm/rad
Speeds and torques of the turbine rotor and the generator can be determined for each simulation time step by solving the above equations using a statespace approach. The statespace equations are:
\(\frac{d}{dt}(\theta_G\theta_T)=(ω_Tω_G)\):
\(\dot {\omega_T}=\left (\frac{1}{J_T} \right)\left [\Gamma_TD(\omega_T\omega_G)K(\theta_T\theta_G) \right]\):
\(\dot {\omega_G}=\left (\frac{1}{J_G} \right)\left [D(\omega_T\omega_G)+K(\theta_T\theta_G)\Gamma_G \right]\)
Reference Power Calculation from Wind Speed
The reference power calculation is based on userdefined wind speed. Wind speed is per unitized based on rated wind speed, cubed and multiplied by rated power (2 MW) to get output power. If userdefined wind speed exceeds rated wind speed (13 m/s), output power is fixed to 2 MW. A tablebased characteristic is used to remove sharp changes from the power output curve. This “softening” of the power curve is discussed in the model testing section.
Pitch Control Block
The pitch control block changes blade pitch angle at higher than rated wind speeds to spill excess power. Thus power output is maintained at rated value even though wind speed exceeds rated wind speed. In this particular implementation, reference power is per unitized and converted to reference speed based on lookup table. A multiplier after the lookup table is included to maintain stability. The reference speed and actual speed are compared, and the error drives the upper PI controller. Reference power and actual power are also compared and the error drives the lower PI controller. PI controller outputs are summed and hardlimited to generate the pitch angle signal. The pitch angle signal is active only when wind speed is close to rated, and otherwise is fixed at zero.
Permanent Magnet Alternator
The permanent magnet alternator is modeled using a builtin PSCAD/EMTDC PMA block.

Rectifier and Buck/Boost Converter for DCLink Voltage Control
The rectifier and buck/boost converter models are responsible for converting the AC output of the PMSG to a fixed DC voltage. An example of the use of buck/boost converters for DC link control for PMA wind turbines is provided in literature^{[7]}. A 3phase diode bridge converts PMA output to a variable DC voltage. The buckboost converter maintains the DC link at a constant 3.6 kV. The DC link capacitor is modeled as two identical capacitors with ground in between them due to PSCAD/EMTDC ground reference issues. This controller is based on PI control; any error between the desired voltage setpoint (3.6 kV here) and the actual voltage drives the PI controller and generates a duty signal output. The duty signal is compared to a triangle wave to generate firing signals for the IGBT in the buck/boost converter.
Inverter
The inverter implemented here is a current controlled voltagesource inverter. It is capable of decoupling real and reactive power control, since the controller design for this inverter is based on fluxvector theory. Real and reactive power reference signals are compared with actual values and the error is used to drive two independent PI controllers. The real power error drives the Iq signal, while reactive power error drives the Id signal. These dq0 domain values are converted to reference Iabc values (note that the angle signal phis is calculated from the voltage phasor). The reference Iabc currents are compared with actual currents and a hysteresis controller switches the inverter IGBTs such that actual current follows the reference current. When the reference currents are achieved, reference real and reactive power is also achieved. The actual current can be seen to be closely following the reference current.
Unit Transformer and Grid Representation
The unit transformer and grid are both modeled using inbuilt blocks supplied by PSCAD/EMTDC. The unit transformer is a wyedelta 2MVA transformer with a primary voltage of 34.5 kV and a secondary voltage of 0.6 kV, and a per unit leakage reactance of 0.1 p.u. During the development and testing phase, the grid is represented by a 34.5kV voltage source.
Complete Model Implemented in PSCAD/EMTDC
Model Testing
The model testing phase is essential to evaluate the capabilities of the model. The model testing phase has three main objectives:
 To verify that desired wind turbine power curve is achieved;
 To demonstrate independent real and reactive power control;
 To demonstrate pitch controller action.
If the model is able to meet these objectives, we can use it with confidence as a platform for modeling more advanced controls, such as providing inertia and frequency response.
Power Curve
The desired power curve is a cubic function of per unit wind speed (up to rated wind speed). Rated power is achieved at rated wind speed. Beyond rated wind speed, the desired power is flat at rated power. Thus, the curve has sharp edge at rated wind speed. This edge needs to be smoothed out, or else the model will have stability issues around the rated wind speed. The smoothed curve is the one actually implemented in the reference power calculation block discussed earlier. Rated power is thus achieved at 14 m/s rather than 13 m/s. The obtained results show that the power output tracks the desired (softened) curve closely. The plot is obtained from a PSCAD/EMTDC simulation multirun to ensure that the power output measured for each wind speed is the steadystate value. Thus, we can claim that that the first objective, that of achieving the desired power curve, is met.

Independent Real and Reactive Power Control
To test if independent real and reactive power control has been achieved, four tests were carried out: real power drop, reactive power drop, real power rise, and reactive power rise. Each of these is modeled as a step change, i.e., at a particular instant, the simulation was paused and a step change was made to either the wind speed or reactive power demand. In the first test, a real power drop was simulated. The wind speed was changed from 13 m/s to 11 m/s at t=12s. The reactive power demand was set at 0.4 MVAR. The real power output drops and settles to the new value. Reactive power drops initially, but recovers to the original value. In the second test, a reactive power drop was simulated. The wind speed remains at 13 m/s throughout the run, but reactive power demand was changed from 0.5 MVAR to 0.2 MVAR at t=12s. The reactive power dropped, as expected. The real power output shows a slight perturbation, but recovers to the original value. In the third test, a real power rise was simulated. The wind speed was changed from 11 m/s to 15 m/s at t=12s. The reactive power demand was set at 0.4 MVAR. The real power output rises and settles to the new value. Reactive power rises initially but recovers to the original value. In the fourth test, a reactive power rise was simulated. The wind speed remains at 13 m/s throughout the run, but reactive power demand was changed from 0.5 MVAR to 0.7 MVAR at t=12s. The reactive power rose, as expected. The real power output once again shows a slight perturbation but recovers to the original value.
Consider the real power drop. In this case, when wind speed drops suddenly, the pitch controller is deactivated, and the pitch angle moves quickly to zero. This is likely the cause for the overshoot observed in the power waveform. Within the wind turbine control system, the change in wind speed changes the P_{genref} value. At this instant, the error between P_{genref} and the actual power output becomes large, driving a change (reduction) in the value of qaxis current Iq. The daxis current does not change since the reactive power setpoint is not changed. This change in Iq leads to a corresponding change in the threephase currents, leading to the desired change in the real power output. The reactive power output stays constant at the setpoint. It should be noted that the decoupling effect observed here for Type 4 turbines is even more pronounced than in the Type 3 turbine, especially under the transient conditions. The reason for this is that the Type 4 turbine is completely decoupled from the grid, unlike the Type 3 turbine in which flux linkages in the stator can be momentarily affected by transients on the grid side. The results here conclusively show that a change in either real power or reactive power demand does not affect the other quantity. Based on the results of the testing, we can claim that the objective of independent real and reactive power control has been achieved.
Pitch Control
A test was devised to evaluate pitch controller action. The wind speed changed from 11 m/s to 15 m/s at t=25s. The pitch angle was initially at 0 degrees (i.e., the pitch controller was inactive). From the obtained results, it can be seen that the pitch controller becomes active when wind speed change occurs. This occurs due to new wind speed (15 m/s) being higher than rated. Eventually, the pitch angle settles close to 8 degrees, effectively spilling some excess power. It should be noted that the pitch angle values are inverted (negative rather than positive) due to the C_{P} lookup table characteristics; 8 degrees here thus corresponds to 8 degrees in the real world. The test shows that the pitch controller does indeed work in a stable fashion.
Dynamic Response
To demonstrate the model’s ability to reproduce wind turbine dynamics, a test was created. The wind turbine was operated with a constant wind speed (15 m/s). This wind speed was higher than rated wind speed and hence, the pitch controller was active. A voltage sag on the grid was simulated, and the real and reactive power response of the wind turbine was observed. Note that this is not an implementation of lowvoltage ride through (LVRT), but rather a test of dynamic response. The grid voltage drops from 1 p.u. to 0.6 p.u. at t=30s, and the sag persists for 18 cycles (0.3 seconds). The intent of the test is to show that the model does indeed respond to events occurring in the dynamic timescale and that the response of the machine to this event is realistic. The obtained results show that the model does indeed respond to the grid event as expected. The grid voltage, converter current demand, real power and reactive power, rotor speed, and pitch angle during the event are shown.
From the obtained results it can be seen that real and reactive power show a sharp increase when the event starts. This is due to the gridside converter attempting to maintain real power output at 2 MW despite the voltage drop occurring. The converter greatly increases the current demand from the generator, which reflects as a jump in power. The pitch controller also seeks to maintain real power output at 2 MW, and hence, begins to move to a lower angle to counter the effects of the converter action. Based on these results, we can say that the model is behaving as expected, and realistic explanations for the response can be offered. The transients when the voltage sag occurs and when the sag ends are also visible. We can thus claim that the model described here offers good resolution and detail. This event is similar to the event used for testing purposes in literature^{[8]}, which offers a useful comparison: the voltage drop was from 1 p.u. to 0.5 p.u. and the sag persisted for 0.5 seconds. The power electronic converters are not explicitly modeled in literature^{[8]}, and the topology and controls are also different from those presented here, leading to a considerable difference in results.
References
 ↑ NREL, Dynamic Models for Wind Turbines and Wind Power Plants (NREL/SR550052780), October 2011, [Online]. Available: http://www.nrel.gov/docs/fy12osti/52780.pdf. [Accessed February 2013].
 ↑ A. Perdana and O. Carlson, “Aggregated Models of a Large Wind Farm Consisting of Variable Speed Wind Turbines for Power System Stability Studies,” in Proceedings of 8th International Workshop on LargeScale Integration of Wind Power into Power Systems as well as on Transmission Networks for Offshore Wind Farms, (Bremen, Germany), October 2009.
 ↑ ^{3.0} ^{3.1} M. Singh, M. Vyas, and S. Santoso, “Using generic wind turbine models to compare inertial response of wind turbine technologies,” in IEEE Power and Energy Society General Meeting, (Minneapolis, MN), July 2010.
 ↑ ^{4.0} ^{4.1} J. Manwell and J. McGowan, Wind energy explained: theory, design and application. Wiley Chichester, 2003.
 ↑ R. Delmerico, N. Miller, W. Price, and J. SanchezGasca, “Dynamic Modeling of GE 1.5 and 3.6 MW Wind TurbineGenerators for Stability Simulations,” in IEEE Power Engineering Society General Meeting, (Toronto, ON).
 ↑ ^{6.0} ^{6.1} B. Kuo, Automatic control systems. Prentice Hall, 1995.
 ↑ T. Tafticht, K. Agbossou, and A. Cheriti, “DC bus control of variable speed wind turbine using a buckboost converter,” in IEEE Power Engineering Society General Meeting, (Montreal, QC), July 2006
 ↑ ^{8.0} ^{8.1} J. Morren, J. Pierik, and S. de Haan, “Voltage dip ridethrough control of directdrive wind turbines,” in 39th International Universities Power Engineering Conference, vol. 3, (Bristol, UK), pp. 934 – 938, 2004.