## 1. Introduction

Renewable energy systems, especially wind energy have attracted interest as a result of the increasing concern about CO_{2} emissions. Wind energy systems using a doubly fed induction generator (DFIG) have some advantages due to variable speed operation and four quadrants active and reactive power capabilities compared with fixed speed squirrel cage induction generators (Simões & Farret, 2004).

The stator of DFIG is directly connected to the grid and the rotor is connected to the grid by a bi-directional converter as shown in Figure 1. The converter connected to the rotor controls the active and the reactive power between the stator of the DFIG and ac supply or a stand-alone grid (Jain & Ranganathan, 2008).

The control of the wind turbine systems is traditionally based on either stator-flux-oriented (Chowdhury & Chellapilla, 2006) or stator-voltage-oriented (Hopfensperger et al, 2000) vector control. The scheme decouples the rotor current into active and reactive power components. The control of the active and reactive power is achieved with a rotor current controller. Some investigations using PI controllers and stator-flux-oriented have been reported by Peña et al (2008). The problem with the use of a PI controller is the tuning of gains and the cross-coupling on DFIG terms in the whole operating range.

Some investigations using predictive functional controller (Morren et al, 2005) and internal mode controller (Guo et al, 2008) have presented a satisfactory power response when compared with the power response of PI, but it is hard to implement one of them due to the predictive functional controller and internal mode controller formulation. Another way to achieve the DFIG power control is using fuzzy logic (Yao et al, 2007). The controllers calculate at each sample interval the voltage rotor to be supplied to the DFIG to guarantee that the active and the reactive power reach their desired reference values. These strategies have satisfactory power response, although the errors in parameters estimation and the fuzzy rules can degrade the system response.

The aim of this chapter is to provide the designing and the modeling of a deadbeat power control scheme for DFIG in accordance with the present state of the art. In this way, the deadbeat power control aims the stator active and reactive power control using the discretized DFIG equations in synchronous coordinate system and stator flux orientation. The deadbeat controller calculates the rotor voltages required to guarantee that the stator active and reactive power reach their desired references values at each sample period using a rotor current space vector loop. Experimental results using a TMS320F2812 plataform are presented to validate the proposed controller.

## 2. Doubly-fed induction machine model

The doubly-fed induction machine model in synchronous reference frame is given by (Leonhard, 1985).

The relationship between fluxes and currents is done by

and

Where*L* is inductance of the winding, the subscripts 1, 2, *m* denotes stator, rotor and

mutual, *NP* is the pole pairs and *ω _{mec}* is the mechanical rotor speed.

The electromagnetic torque is given by

The superscript

The mechanical dynamics of the machine is given by:

Where *J* is the load and rotor inertia moment and *T _{L}* is the load torque.

The induction machine active power *P* is

and the reactive power Q is

## 2.1 Power control principles using stator field orientation

The DFIG power control aims independent stator active P and reactive Q power control by means a rotor current regulation. For this propose, the stator field orientation (Novotny & Lipo, 1996) technique is used. Thus, the *P* and *Q* are represented as functions of each individual rotor current using the stator flux space vector position. In the synchronous system reference frame *dq*, the synchronous speed ω1 is the speed of the stator flux vector and it is given by

The stator flux space vector transformation from stationary reference frame *αβ* to the synchronous system reference frame *dq* is given by:

Therefore, the components of the direct and quadrature axis become:

and

Thus, by using the stator flux orientation, the flux space vector components become:

and

The relationship between the fluxes and currents of Equation (3) becomes, respectively:

and

In the same way of the stator flux vector components, the stator voltage vector components become:

and

Now, the active power can be calculated substituting the expression of *i _{1q}* using Equation (16), the value of

*v*of Equation (17) and the value of

_{1d}*v*using Equation (18) in the Equation (7). The new active power expression, using the rotor quadrature axis current i2q, is given by

_{1q}In the same way of the active power, the reactive power can be calculated by substituting the expression of i1d using Equation (15), the value of v1d of Equation (17) and the value of v1q using Equation (18) in the Equation (8). The new reactive power expression, using the rotor direct axis current *i _{2d}*, is given by:

From Equations (19) and (20), the stator power can be calculated using the rotor current space vector components. As the stator of the doubly-fed induction generator is directly connected to the grid, the magnitude of the stator flux space vector and the stator voltage space vector is constant. Thus, the independent stator active and reactive power control is achieved through rotor current space vector control.

## 3. Deadbeat control theory

The deadbeat control is a digital control technique that allows to calculate the required input *N* samplings intervals using a discrete equation of the continuous linear system (Franklin et al, 1994).

A linear continuous system (Ogata, 2002) can be represented by

Where *ω* denotes the perturbation vector and *A, C, B* and *G* are *nxn* matrices. In this paper *C = I,* where *I* is the identity matrix.

The Equation (21) can be discretized considering *T* as the sampling period and *k* as the sampling time. Thus, using zero-order-hold (ZOH) with no delay Equation (21) becomes

Where

The input calculation to guarantee a null steady state error (Franklin et al, 1994) is given by

Where

Substituting (24) in (22) and making

The block diagram of the deadbeat control is presented in Figure 2.

## 4. Deadbeat power control for doubly-fed induction generator

### 4.1. Rotor side equations

The stator power control of DFIG is made by the rotor current control using the stator field orientation. Thus, the rotor state space equation was necessary in the application of the deadbeat control theory in which the rotor current space vector in synchronous reference frame *dq* is the state variable. In this way, the rotor voltage space vector substituting Equation (4) in Equation (2) is given by

Which means

and

Thus, substituting the stator direct axis current i1d of Equation (15) in the derivative of Equation (27) and substituting the stator quadrature axis current *i _{1q}* of Equation (16) in the derivative of Equation (28), the rotor voltage equations are done by

and

In this work, the stator of the DFIG is directly connected to the grid and the stator voltage has constant frequency and magnitude. Thus, the magnitude of the stator flux is also constant and in this case

and

which means

Where

.In the state-space form, Equation (33) becomes:

which means

(36) |

Where

From now on, it will be assumed that the mechanical time constant is much greater than the electrical time constants. Thus, *ω _{mec}*

*=*

*constant*is a valid approximation for a sample time. Since the synchronous speed ω1 is fixed by the grid and

*ω*is also a valid approximation for a sample time (Sguarezi Filho et al., 2011).

_{sl}=*ω*_{1}-NP ω_{mec}, ω_{sl}=constant### 4.2. Deadbeat power control

The DFIG power control scheme uses a deadbeat controller to obtain rotor voltages which should be applied on induction generator in order to guarantee the active and reactive power reach their desired reference values in a few sample intervals. The sample time *T* interval is the same time of PWM modulator. The power control is made using the rotor current space vector control in the synchronous reference frame *dq*.

Equation (35) can be discretized, since the rotor applied voltage remains constant during a control period of the PWM voltage source inverter. Thus, Equation (35) can be discretized using Equations (22) and (23), considering the mentioned hypothesis above and making:

The discrete equation of the rotor voltage is shown in Equation (36).

which means

(38) |

Where

(39) |

The rotor voltage which is calculated to guarantee null steady state error using Equations (24) and (37) is given by

and

The rotor current space vector references are

For the active power control, the rotor current quadrature axis reference using Equation (19) is given by

Using Equation (20) for the reactive power control, the rotor current direct axis reference is

Thus, if the *d* and *q* axis of the rotor voltage space vector components are calculated according to Equations (39)-(42) mentioned above and they are applied to the generator, then the active and reactive power convergence to their respective commanded values will occur in a few sampling intervals. The space vector modulation using the desired rotor voltage in the rotor stationary reference frame *αβr* generates switching signals for the rotor side inverter. The transformation from *dq* reference frame to the *αβr* reference frame is done by:

In this work the back to back converter is used in the power control strategy. The rotor voltage calculated using the deadbeat control theory will allow to drive the inverter connected to the rotor of DFIG. The converter that is connected to the grid controls the DC-link voltage and this one can be controlled by a current control presented by Rodríguez et al. (2005). The Deadbeat power control block diagram is shown in Figure 3 and a detailed block diagram of the deadbeat power control implementation is shown in Figure 4.

Stator currents and voltages, rotor speed and currents are measured to stator flux position and magnitude, synchronous frequency and slip frequency estimation.

### 4.3. Estimation

The stator flux estimation in stationary reference frame *αβ* is given by

The position of stator flux is estimated by using the trigonometric function and it is given by

The synchronous speed *ω _{1}* estimation is given by

and the slip speed estimation using the rotor speed and the synchronous speed is

The angle in rotor reference frame is

## 5. Experimental results

The deadbeat power control strategy was implemented with a Texas Instruments DSP TMS320F2812 platform which also has a T = 400µs. The system consists of a three-phase voltage source inverter with insulated-gate bipolar transistors (IGBTs) and the three-phase doubly-fed induction generator and its parameters are shown in the appendix. The rotor voltage commands are modulated by using symmetrical space vector PWM, with switching frequency equal to 2.5 kHz. The DC bus voltage of the inverter is 36 V. The stator voltages and currents are sampled in the frequency of 2.5 kHz. The encoder resolution is 3800 pulses per revolution.

The algorithm of the deadbeat control was programmed on the Event Manager 1 of the Texas Instruments DSP TMS320F2812 platform and its flowchart is presented in Figure 5. The schematic of the implementation of the experimental setup is presented in Figure 6 and the experimental setup is shown in Figure 7.

Six tests were made, five in the subsynchronous operation and one in several speed operations from supersynchronous to subsynchronous operation. The first one was the response of i2d step from 0.5A to 5 A which is shown in Figure 8 (a) and the satisfactory performance of the controller can be seen due to the fact that the reference was followed. In this test the i2q is 0.5A.

The second one was the response of i2q step from 0.5A to 5 A. The satisfactory performance of the controller in this test can be seen in Figure 8 (b), due to the fact that the reference was followed. In this test *i _{2d}* is 4A.

The same test of the i2q step from 0A to 5A, as mentioned above, with rotor currents in rotor reference frame is presented in Figure 9. In this test the i2d is 5A. The satisfactory response of the controller can be seen due to the fact that the reference was followed and the amplitude of the rotor ac currents increased.

The fourth test was the response of the reactive power Qref of -300VA, 300VA and 0VA which means leg, lead and unitary power factor. The active power reference is -300W. The rotor current references were calculated using Equations (41) and (42). The satisfactory performance of the controller can be seen in Figure 10(a), due to the fact that the reference was followed. The rotor current is shown in Figure 10(b).

The fifth test was the steady state of unitary power factor and the active power was -300W. Again, the rotor current references were calculated using Equations (41) and (42). The response of stator power and rotor current are presented in Figures 11(a)and 11(b), respectively. The stator voltage (127Vrms) and the stator current (0.8Arms) are shown in Figure 12. The satisfactory performance of the controller can be seen because the angle between the stator voltage and the stator current is 180°.

In the last test, the generator operates with several speed from 1850 rpm to 1750 rpm and a constant active and reactive power reference of 0W and 0VA, respectively. The rotor current references were also calculated using Equations (41) and (42). So, i_{2dref}= 7A and i_{2qref}= 0A. In this case, this test just maintains the magnetization of the generator. The response of the active and reactive power is shown in Figure 13(a)and the rotor current is presented in Figure 13(b). The rotor speed in several operations and the rotor current of phase α are shown in Figure 14. The satisfactory performance of the controller can be seen during several speed operations, since the reference was followed.

## 6. Conclusion

This book chapter has presented a model and design of a deadbeat power control scheme for a doubly-fed induction generator using a deadbeat control theory and rotor current space vector loop. The stator field orientation technique allows the independent control of the rotor current components in synchronous reference frame *dq*, in this case, the direct and quadrature axis of the rotor current space vector. Thus, the control of the rotor current components allows controlling the active and reactive power of the generator. The deadbeat controller uses the DFIG discretized equations to calculate at each sample period the required rotor voltages, so that the active and reactive power values reach the desired reference values. Thus, the deadbeat controller does not need to tune gains as the PI controllers. This strategy constant switching frequency overcomes the drawbacks of conventional direct power control (Xu & Cartwright, 2006).

The experimental results confirm the effectiveness of the power controller during several operating conditions of generator speed. Thus, the deadbeat power control strategy is an interesting tool for doubly-fed power control in wind turbines.

## 7. Appendix

Doubly-fed induction generator parameters:

*R1 = 2.2 Ω; R2 = 1.764 Ω; Lm = 0.0829 H; Ll1 = 0.0074 H; Ll2 =0.0074H ; J = 0.05 Kg.m2; NP = 2; PN = 2.25 kW; VN = 220 V.*