Implement phasor model of variable speed doubly-fed induction generator driven by wind turbine

Renewables/Wind Generation

The wind turbine and the doubly-fed induction generator (WTDFIG)
are shown in the figure called The Wind Turbine and the Doubly-Fed Induction
Generator System.
The AC/DC/AC converter is divided into two components: the rotor-side
converter (C_{rotor}) and the grid-side converter
(C_{grid}). C_{rotor} and
C_{grid} are Voltage-Sourced Converters that use
forced-commutated power electronic devices (IGBTs) to synthesize an
AC voltage from a DC voltage source. A capacitor connected on the
DC side acts as the DC voltage source. A coupling inductor L is used
to connect C_{grid }to the grid. The three-phase
rotor winding is connected to C_{rotor} by slip
rings and brushes and the three-phase stator winding is directly connected
to the grid. The power captured by the wind turbine is converted into
electrical power by the induction generator and it is transmitted
to the grid by the stator and the rotor windings. The control system
generates the pitch angle command and the voltage command signals
V_{r} and V_{gc} for C_{rotor} and
C_{grid} respectively in order to control the
power of the wind turbine, the DC bus voltage and the reactive power
or the voltage at the grid terminals.

**The Wind Turbine and the Doubly-Fed Induction
Generator System**

Operating Principle of the Wind Turbine Doubly-Fed Induction Generator

The power flow, illustrated in the figure called The Power Flow, is used to describe the operating principle. In this figure the followings parameters are used:

P | Mechanical power captured by the wind turbine and transmitted to the rotor |

P | Stator electrical power output |

P | Rotor electrical power output |

P | C |

Q | Stator reactive power output |

Q | Rotor reactive power output |

Q | C |

T | Mechanical torque applied to rotor |

T | Electromagnetic torque applied to the rotor by the generator |

ω | Rotational speed of rotor |

ω | Rotational speed of the magnetic flux in the air-gap of the generator, this speed is named synchronous speed. It is proportional to the frequency of the grid voltage and to the number of generator poles. |

J | Combined rotor and wind turbine inertia coefficient |

The mechanical power and the stator electric power output are computed as follows:

*P _{m}* =

For a lossless generator the mechanical equation is:

$$J\frac{d{\omega}_{r}}{dt}={T}_{m}-{T}_{em}.$$

In steady-state at fixed speed for a lossless generator *T _{m}* =

It follows that:

$${P}_{r}={P}_{m}-{P}_{s}={T}_{m}{\omega}_{r}-{T}_{em}{\omega}_{s}=-{T}_{m}\frac{{\omega}_{s}-{\omega}_{r}}{{\omega}_{s}}{\omega}_{s}=-s{T}_{m}{\omega}_{s}=-s{P}_{s},$$

where *s* is defined as the slip of the generator: s = (*ω _{s}*–

**The Power Flow**

Generally the absolute value of slip is much lower than 1 and,
consequently, *P*_{r }is
only a fraction of *P*_{s}.
Since *T*_{m} is
positive for power generation and since ω_{s} is
positive and constant for a constant frequency grid voltage, the sign
of *P*_{r} is
a function of the slip sign. *P*_{r} is positive for negative slip (speed greater
than synchronous speed) and it is negative for positive slip (speed
lower than synchronous speed). For super-synchronous speed operation, *P*_{r }is transmitted to DC bus capacitor and
tends to rise the DC voltage. For sub-synchronous speed operation, *P*_{r} is taken out of DC bus capacitor and tends
to decrease the DC voltage. C_{grid} is used to
generate or absorb the power *P*_{gc} in order to keep the DC voltage constant.
In steady-state for a lossless AC/DC/AC converter *P*_{gc} is equal to *P*_{r} and the speed of the wind turbine is determined
by the power *P*_{r} absorbed
or generated by C_{rotor}. The power control will
be explained below.

The phase-sequence of the AC voltage generated by C_{rotor} is
positive for sub-synchronous speed and negative for super-synchronous
speed. The frequency of this voltage is equal to the product of the
grid frequency and the absolute value of the slip.

C_{rotor} and C_{grid} have
the capability of generating or absorbing reactive power and could
be used to control the reactive power or the voltage at the grid terminals.

The rotor-side converter is used to control the wind turbine output power and the voltage (or reactive power) measured at the grid terminals.

Power Control

The power is controlled in order to follow a pre-defined power-speed
characteristic, named tracking characteristic. An example of such
a characteristic is illustrated in the figure called Turbine Characteristics and Tracking Characteristic, by the
ABCD curve superimposed to the mechanical power characteristics of
the turbine obtained at different wind speeds. The actual speed of
the turbine ω_{r} is measured and the corresponding
mechanical power of the tracking characteristic is used as the reference
power for the power control loop. The tracking characteristic is defined
by four points: A, B, C and D. From zero speed to speed of point A
the reference power is zero. Between point A and point B the tracking
characteristic is a straight line, the speed of point B must be greater
than the speed of point A. Between point B and point C the tracking
characteristic is the locus of the maximum power of the turbine (maxima
of the turbine power vs turbine speed curves). The tracking characteristic
is a straight line from point C and point D. The power at point D
is one per unit (1 pu) and the speed of the point D must be greater
than the speed of point C. Beyond point D the reference power is a
constant equal to one per unit (1 pu).

**Turbine Characteristics and Tracking Characteristic**

The generic power control loop is illustrated in the figure
called Rotor-Side Converter Control System. The actual electrical
output power, measured at the grid terminals of the wind turbine,
is added to the total power losses (mechanical and electrical) and
is compared with the reference power obtained from the tracking characteristic.
A Proportional-Integral (PI) regulator is used to reduce the power
error to zero. The output of this regulator is the reference rotor
current Iqr_ref that must be injected in the rotor by converter C_{rotor}.
This is the current component that produce the electromagnetic torque
T_{em}. The actual Iqr component of positive-sequence
current is compared to Iqr_ref and the error is reduced to zero by
a current regulator (PI). The output of this current controller is
the voltage Vqr generated by C_{rotor}. The current
regulator is assisted by feed forward terms which predict Vqr.

**Rotor-Side Converter Control System**

Voltage Control and Reactive Power Control

The voltage or the reactive power at grid terminals is controlled
by the reactive current flowing in the converter C_{rotor}.
The generic control loop is illustrated in the figure called Rotor-Side Converter Control System.

When the wind turbine is operated in voltage regulation mode, it implements the following V-I characteristic.

**Wind Turbine V-I Characteristic**

As long as the reactive current stays within the maximum current values (-Imax, Imax) imposed by the converter rating, the voltage is regulated at the reference voltage Vref. However, a voltage droop is normally used (usually between 1% and 4% at maximum reactive power output), and the V-I characteristic has the slope indicated in the figure called Wind Turbine V-I Characteristic. In the voltage regulation mode, the V-I characteristic is described by the following equation:

*V* = *V*_{ref} + *X _{s}I*

where

| Positive sequence voltage (pu) |

| Reactive current (pu/Pnom) (I > 0 indicates an inductive current) |

| Slope or droop reactance (pu/Pnom) |

Pnom | Three-phase nominal power of the converter specified in the block dialog box |

When the wind turbine is operated in var regulation mode the reactive power at grid terminals is kept constant by a var regulator.

The output of the voltage regulator or the var regulator is
the reference d-axis current Idr_ref that must be injected in the
rotor by converter C_{rotor}. The same current
regulator as for the power control is used to regulate the actual
Idr component of positive-sequence current to its reference value.
The output of this regulator is the d-axis voltage Vdr generated by
C_{rotor}. The current regulator is assisted by
feed forward terms which predict Vdr.

Vdr and Vqr are respectively the d-axis and q-axis of the voltage Vr.

Note:

for C

_{rotor }control system and measurements the d-axis of the d-q rotating reference frame is locked on the generator mutual flux by a PLL which is assumed to be ideal in this phasor model.the magnitude of the reference rotor current Ir_ref is equal to $$\sqrt{{I}_{dr\text{\_ref}}^{2}+{I}_{qr\text{\_ref}}^{2}}$$. The maximum value of this current is limited to 1 pu. When Idr_ref and Iqr_ref are such that the magnitude is higher than 1 pu the Iqr_ref component is reduced in order to bring back the magnitude to 1 pu.

The converter C_{grid }is used to regulate
the voltage of the DC bus capacitor. In addition, this model allows
using C_{grid }converter to generate or absorb
reactive power.

The control system, illustrated in the figure called Grid-Side Converter Control System, consists of:

Measurement systems measuring the d and q components of AC positive-sequence currents to be controlled as well as the DC voltage Vdc.

An outer regulation loop consisting of a DC voltage regulator. The output of the DC voltage regulator is the reference current Idgc_ref for the current regulator (Idgc = current in phase with grid voltage which controls active power flow).

An inner current regulation loop consisting of a current regulator. The current regulator controls the magnitude and phase of the voltage generated by converter C

_{grid }(Vgc) from the Idgc_ref produced by the DC voltage regulator and specified Iq_ref reference. The current regulator is assisted by feed forward terms which predict the C_{grid }output voltage.

The magnitude of the reference grid converter current Igc_ref is equal to

$$\sqrt{{I}_{dgc\text{\_ref}}^{2}+{I}_{qr\text{\_ref}}^{2}}$$

. The maximum value of this current is limited to a value defined by the converter maximum power at nominal voltage. When Idgc_ref and Iq_ref are such that the magnitude is higher than this maximum value the Iq_ref component is reduced in order to bring back the magnitude to its maximum value.

**Grid-Side Converter Control System**

The pitch angle is kept constant at zero degree until the speed reaches point D speed of the tracking characteristic. Beyond point D the pitch angle is proportional to the speed deviation from point D speed. The control system is illustrated in the following figure.

**Pitch Control System**

Turbine Model

The turbine model uses the Wind Turbine bloc of the Renewables/Wind Generation library. See documentation of this model for more details.

Induction Generator

The doubly-fed induction generator phasor model is the same as the wound rotor asynchronous machine (see the Machines library) with the following two points of difference:

Only the positive-sequence is taken into account, the negative-sequence has been eliminated.

A trip input has been added. When this input is high the induction generator is disconnected from the grid and from C

_{rotor}.

The WTDFIG parameters are grouped in four categories: Generator data, Converters data, Turbine data, and Control parameters. Use the Display listbox to select which group of parameters you want to visualize.

**External turbine (Tm mechanical torque input)**When you select this parameter, the

**Turbine**tab is not visible, and a Simulink^{®}input named Tm appears on the block, allowing to use an external signal for the generator input mechanical torque. This external torque must be in pu based on the nominal electric power and synchronous speed. For example, the external torque may come from a user defined turbine model. Following the convention used in the induction machine, the torque must be negative for power generation.**Nominal power, line-to-line voltage and frequency**The nominal power in VA, the nominal line-to-line voltage in Vrms and the nominal system frequency in hertz.

**Stator**The stator resistance Rs and leakage inductance Lls in pu based on the generator rating.

**Rotor**The rotor resistance Rr' and leakage inductance Llr', both referred to the stator, in pu based on the generator rating.

**Magnetizing inductance**The magnetizing inductance Lm in pu based on the generator rating.

**Inertia constant, friction factor and pairs of poles**Combined generator and turbine inertia constant H in seconds, combined viscous friction factor F in pu based on the generator rating and number of pole pairs p.

You may need to use your own turbine model, in order for example, to implement different power characteristics or to implement the shaft stiffness. Your model must then output the mechanical torque applied to the generator shaft. If the inertia and the friction factor of the turbine are implemented inside the turbine model you specify only the generator inertia constant H and the generator friction factor F.

**Initial conditions**The initial slip s, electrical angle Θ in degrees, stator phasor current magnitude in pu, stator phasor current phase angle in degrees, rotor phasor current magnitude in pu and rotor phasor current phase angle in degrees.

**Display wind turbine power characteristics**Click to plot the turbine power characteristics at zero degree of pitch angle for different wind speeds. The tracking characteristic is also displayed on the same figure.

**Electric Power-Speed characteristic**This parameter is visible only when the

**External mechanical torque****Nominal wind turbine mechanical output power**This parameter is not visible when the

**External mechanical torque**The nominal turbine mechanical output power in watts.

**Tracking characteristic speeds**This parameter is not visible when the

**External mechanical torque**Specify the speeds of point A to point D of the tracking characteristic in pu of the synchronous speed. speed_B must be greater than speed_A and speed_D must be greater than speed_C.

**Power at point C**This parameter is not visible when the

**External mechanical torque**Specify the power of point C of the tracking characteristic in pu of the

**Nominal wind turbine mechanical output power****.****Wind speed at point C**This parameter is not visible when the

**External mechanical torque**Specify wind speed in m/s for point C. The power at point C is the maximum turbine output power for the specified wind speed.

**Pitch angle controller gain [Kp]**This parameter is not visible when the

**External mechanical torque**Proportional gain Kp of the pitch controller. Specify Kp in degrees/(speed deviation pu). The speed deviation is the difference between actual speed and speed of point D in pu of synchronous speed.

**Maximum pitch angle**This parameter is not visible when the

**External mechanical torque**The maximum pitch angle in degrees.

**Maximum rate of change of pitch angle**This parameter is not visible when the

**External mechanical torque**The maximum rate of change of the pitch angle in degrees/s.

**Converter maximum power**The maximum power of both C

_{grid}and C_{rotor}in pu of the nominal power. This parameter is used to compute the maximum current at 1 pu of voltage for C_{grid}. The maximum current for C_{rotor}is 1 pu.**Grid-side coupling inductor**The coupling inductance L and its resistance R in pu based on the generator rating.

**Coupling inductor initial currents**The coupling inductor initial phasor current in positive-sequence. Enter magnitude IL in pu and phase ph_IL in degrees. If you know the initial value of the current corresponding to the WTDFIG operating point you may specify it in order to start simulation in steady state. If you don't know this value, you can leave [0 0]. The system will reach steady-state after a short transient.

**Nominal DC bus voltage**The nominal DC bus voltage in volts.

**DC bus capacitor**The total capacitance of the DC link in farads. This capacitance value is related to the WTDFIG rating and to the DC link nominal voltage. The energy stored in the capacitance (in joules) divided by the WTDFIG rating (in VA) is a time duration which is usually a fraction of a cycle at nominal frequency. For example, for the default parameters, (C=10000 µF, Vdc=1200 V, Pn=1.67 MVA) this ratio $$1/2\cdot C\cdot {V}_{\text{dc}}^{2}/{P}_{n}$$ is 4.3 ms, which represents 0.26 cycle for a 60 Hz frequency. If you change the default values of the nominal power rating and DC voltage, you should change the capacitance value accordingly.

**Mode**Specifies the

**WTDFIG**mode of operation. Select either`Voltage regulation`

or`Var regulation`

.**Reference grid voltage Vref**This parameter is not visible when the

**Mode**parameter is set to`Var regulation`

.Reference voltage, in pu, used by the voltage regulator. When

**External**is selected, a Simulink input named Vref appears on the block, allowing you to control the reference voltage from an external signal (in pu). The**Reference grid voltage**parameter is therefore unavailable.**Generated reactive power Qref**This parameter is not visible when the

**Mode**parameter is set to`Voltage regulation`

.Reference generated reactive power at grid terminals, in pu, used by the var regulator. When

**External**is selected, a Simulink input named Qref appears on the block, allowing you to control the reference reactive power from an external signal (in pu). The**Generated reactive power Qref**parameter is therefore unavailable**Grid-side converter generated reactive current reference (Iq_ref)**Reference grid-side converter reactive current, in pu, used by the current regulator. Specify a positive value of Iq_ref for generated reactive power. When

**External**is selected, a Simulink input named Iq_ref appears on the block, allowing you to control the grid-side converter reactive current from an external signal (in pu). The**Grid-side converter generated reactive current reference**parameter is therefore unavailable.**Grid voltage regulator gains [Kp Ki]**This parameter is not visible when the

**Mode**parameter is set to`Var regulation`

.Gains of the AC voltage regulator. Specify proportional gain Kp in (pu of I)/(pu of V), and integral gain Ki, in (pu of I)/(pu of V)/s, where V is the AC voltage error and I is the output of the voltage regulator.

**Droop Xs**This parameter is not visible when the

**Mode**parameter is set to`Var regulation`

.Droop reactance, in pu/nominal power, defining the slope of the V-I characteristic.

**Reactive power regulator gains [Kp Ki]**This parameter is not visible when the

**Mode**parameter is set to`Voltage regulation`

.Gains of the var regulator. Specify proportional gain Kp in (pu of I)/(pu of Q), and integral gain Ki, in (pu of I)/(pu of Q)/s, where Q is the reactive power error and I is the output of the var regulator.

**Power regulator gains [Kp Ki]**Gains of the power regulator. Specify proportional gain Kp in (pu of I)/(pu of P), and integral gain Ki, in (pu of I)/(pu of P)/s, where P is the power error and I is the output of the power regulator.

**DC bus voltage regulator gains [Kp Ki]**Gains of the DC voltage regulator which controls the voltage across the DC bus capacitor. Specify proportional gain Kp in (pu of I)/(Vdc), and integral gain Ki, in (pu of I)/(Vdc)/s, where Vdc is the DC voltage error and I is the output of the voltage regulator.

**Grid-side converter current regulator gains [Kp Ki]**Gains of the grid-side converter current regulator.

Specify proportional gain Kp in (pu of V)/(pu of I) and integral gain Ki, in (pu of V)/(pu of I)/s, where V is the output Vgc of the current regulator and I is the current error.

**Rotor-side converter current regulator gains [Kp Ki]**Gains of the rotor-side converter current regulator.

Specify proportional gain Kp in (pu of V)/(pu of I) and integral gain Ki, in (pu of V)/(pu of I)/s, where V is the output Vr of the current regulator and I is the current error.

**Maximum rate of change of reference grid voltage**This parameter is not visible when the

**Mode**parameter is set to`Var regulation`

.Maximum rate of change of the reference voltage, in pu/s, when an external reference voltage is used.

**Maximum rate of change of reference reactive power**This parameter is not visible when the

**Mode**parameter is set to`Voltage regulation`

.Maximum rate of change of the reference reactive power, in pu/s, when an external reference reactive power is used.

**Maximum rate of change of reference power**Maximum rate of change of the reference power in pu/s.

**Maximum rate of change of converters reference current**Maximum rate of change of the reference current in pu/s for both the rotor-side and the grid-side converters.

`A B C`

The three terminals of the WTDFIG.

`Trip`

Apply a Simulink logical signal (0 or 1) to this input. When this input is high the WTDFIG is disconnected and its control system is disabled. Use this input to implement a simplified version of the protection system.

`Wind (m/s)`

This input is not visible when the

**External mechanical torque**parameter is checked.Simulink input of the wind speed in m/s.

`Tm`

This input is visible only when the

**External mechanical torque**parameter is checked.Simulink input of the mechanical torque. Tm must be negative for power generation. Use this input when using an external turbine model.

`Vref`

This input is visible only when the

**Mode of operation**parameter is set to`Voltage regulation`

and the**External grid voltage reference**parameter is checked.Simulink input of the external reference voltage signal.

`Qref`

This input is visible only when the

**Mode of operation**parameter is set to`Var regulation`

and the**External generated reactive power reference**parameter is checked.Simulink input of the external reference generated reactive power signal at grid terminals.

`Iq_ref`

This input is visible only when the

**External reactive current Iq_ref for grid-side converter**parameter is checked.Simulink input of the external reference grid-side converter reactive current signal.

`m`

Simulink output vector containing 29 WTDFIG internal signals. These signals can be individually accessed by using the Bus Selector block. They are, in order:

Signal

Signal Group

Signal Names

Definition

1-3

Iabc (cmplx)

(pu)Ia (pu)

Ib (pu)

Ic (pu)Phasor currents Ia, Ib, Ic flowing into the WTDFIG terminals in pu based on the generator rating.

4-6

Vabc (cmplx)

(pu)Va (pu)

Vb (pu)

Vc (pu)Phasor voltages (phase to ground) Va, Vb, Vc at the WTDFIG terminals in pu based on the generator rating.

7-8

Vdq_stator

(pu)Vd_stator (pu)

Vq_stator (pu)Direct-axis and quadrature-axis component of stator voltage in pu based on the generator rating. Vd_stator and Vq_stator are respectively the real and imaginary parts of the positive-sequence stator phasor voltage.

9-11

Iabc_stator (cmplx)

(pu)Ia_stator (pu)

Ib_stator (pu)

Ic_stator (pu)Phasor currents Ia, Ib, Ic flowing into the stator in pu based on the generator rating.

12-13

Idq_stator

(pu)Id_stator (pu)

Iq_stator (pu)Direct-axis and quadrature-axis component of stator current in pu based on the generator rating. Id_stator and Iq_stator are respectively the real and imaginary parts of the positive-sequence stator phasor current.

14-15

Vdq_rotor

(pu)Vd_rotor (pu)

Vq_rotor (pu)Direct-axis and quadrature-axis component of rotor voltage in pu based on the generator rating. Vd_rotor and Vq_rotor are respectively the real and imaginary parts of the positive-sequence rotor phasor voltage.

16-17

Idq_rotor

(pu)Id_rotor (pu)

Iq_rotor (pu)Direct-axis and quadrature-axis component of currents flowing into the rotor in pu based on the generator rating. Id_rotor and Iq_rotor are respectively the real and imaginary parts of the positive-sequence rotor phasor current.

18

wr (pu)

Generator rotor speed (pu)

19

Tm (pu)

Mechanical torque applied to the generator (pu)

20

Te (pu)

Electromagnetic torque in pu based on the generator rating.

21-22

Vdq_grid_conv

(pu)Vd_grid_conv (pu)

Vq_grid_conv (pu)Direct-axis and quadrature-axis component of grid-side converter voltage in pu based on the generator rating. Vd_grid_conv and Vq_grid_conv are respectively the real and imaginary parts of the grid-side converter phasor voltage.

23-25

Iabc_grid_conv

(cmplx)

(pu)Ia_grid_conv (pu)

Ib_grid_conv (pu)

Ic_grid_conv (pu)Phasor currents Ia, Ib, Ic flowing into the grid-side converter in pu based on the generator rating.

26

P (pu)

WTDFIG output power. A positive value indicates power generation.

27

Q (pu)

WTDFIG output reactive power. A positive value indicates reactive power generation.

28

Vdc (V)

DC voltage (V).

29

Pitch_angle (deg)

Blade pitch angle in degrees.

See the `power_wind_dfig`

example,
which illustrates the steady-state and dynamic performance of the
WTDFIG in a 9 MW Wind Farm connected on a 25 kV, 60 Hz, system.

[1] R. Pena, J.C. Clare, G.M. Asher, "Doubly fed induction generator using back-to-back PWM converters and its application to variable-speed wind-energy generation," IEEE Proc.-Electr. Power Appl., Vol. 143, No. 3, May 1996

[2] Vladislav Akhmatov, "Variable-Speed Wind Turbines with Doubly-Fed Induction Generators, Part I: Modelling in Dynamic Simulation Tools," Wind Engineering Volume 26, No. 2, 2002

[3] Nicholas W. Miller, Juan J. Sanchez-Gasca, William W. Price, Robert W. Delmerico, "DYNAMIC MODELING OF GE 1.5 AND 3.6 MW WIND TURBINE-GENERATORS FOR STABILITY SIMULATIONS," GE Power Systems Energy Consulting, IEEE WTG Modeling Panel, Session July 2003

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