Open access peer-reviewed chapter

Nonlinear and Sampled Data Control of Wind Turbine

By Marwa Hassan

Submitted: March 29th 2019Reviewed: January 17th 2020Published: April 1st 2020

DOI: 10.5772/intechopen.91246

Downloaded: 38

Abstract

This chapter aims to investigate the effectiveness of the nonlinear control-based model and the sampled-data design through the power system application. In particular, the study will focus on a model of a wind turbine system fed by a doubly fed induction generator (DFIG). First of all, a detail dynamical model of a DFIG-based wind-turbine grid-connected system is presented in the direct and quadratic synchronous reference frame. Afterward, mathematical modeling is performed for the nonlinear and sampled data systems. The nonlinear control will ensure the reproduction of the rotor direct and quadratic current that converge to the reference signal generated from. The proposed sampled-data system is built upon the nonlinear model and is introduced as an alternative of the classical discrete control which is known as emulation design. The simulation’s results will show that implementing the approximate feedback will yield better results than the one obtained from the mere emulation.

Keywords

  • sampled data model
  • nonlinear control
  • renewable energy
  • wind turbine
  • feedback

1. Introduction

In recent years global warming emissions have been one of the most important topics discussed by the researcher. The carbon dioxide causes harmful impacts on the environment as it acts like a blanket that traps heat. According to the latest survey done in the United States, about 29% of global warming emissions are caused by fossil fuel used in the electricity sector [1, 2, 3]. Therefore it becomes more essential to look for alternative sources. Renewable energy sources especially wind energy produce little to no global warming emissions as burning natural gas for electricity releases between 0.6 and 2 pounds of carbon dioxide equivalent per kilowatt-hour (CO2E/kWh); coal emits between 1.4 and 3.6 pounds of CO2E/kWh. Wind, on the other hand, is responsible for only 0.02–0.04 pounds of CO2E/kWh on a life-cycle basis [3]. In conclusion, Wind energy represents one of the fastest-growing energy sources in the world due to it is a major advantage in terms of cost and effectiveness [4, 5, 6, 7, 8]. The wind farm system based on the doubly fed induction generator (DFIG) will be studied in this chapter. In simple words, the DFIG is a generator that has its rotor winding connected to the grid via slip rings and back-to-back voltage source converters that control both the rotor and the grid currents. Thus, rotor frequency can freely differ from the grid frequency. DFIG has become more effective in the industry in the last few years due to its advantage compared to the permanent magnet synchronous generator as it provides better results when compared to cost, size, and weight. A lot of researches have been carried out in the area of modeling and controlling of stator/rotor flux in DFIG [9, 10, 11]. Most of these researches face some drawbacks due to the nonlinear nature of the DFIG. Examples can be found on [12, 13, 14, 15, 16, 17, 18, 19]. In this chapter, we aim to introduce a nonlinear control technique that aims to reproduce an output signal that converges to a distinct reference signal. Later on, the sampled data control technique will be applied as an alternative solution of the emulation based technique that is usually applied to the nonlinear system. A few examples of the sampled data are illustrated in [20, 21, 22]. The chapter is structured as follow: Section 2 recalls the modeling considerations of the doubly fed induction generator while Section 3 illustrates the nonlinear control approach. Section 4 presents the Grid side command model. Finally, Section 5 presents the sampled data model and the obtained results while Section 6 concludes the paper and formulates further research directions.

2. Modeling

The modeling of the doubly fed induction generator (DFIG) will be recalled in this section. These equations will be used to design the nonlinear control system.

2.1 Doubly fed induction generator model

1. Stator equations

Vsd=Rsisd+ddtλsdλsqWsE1
Vsq=Rsisq+ddtλsq+λsdWsE2
λsd=Lsisd+MirdE3
λsq=Lsisq+MirqE4

2. Rotor equations

Vrd=Rrird+ddtλrdϕrqWrE5
Vrq=Rrirq+ddtλrq+ϕrdWrE6
λrd=Lrird+MisdE7
λrq=Lrirq+MisqE8
Wr=g.WsE9

with

isd=λsdMirdLsE10
isq=MirqLsE11

where Rsand Rrare, respectively, the stator and rotor phase resistances, Ls,Lr,MStator and rotor per phase winding and magnetizing inductances and Ws,Wrare the stator and rotor speed pair pole number. The direct and quadrate stator and rotor currents are respectively represented as isd,isq,irdand irq. The electromagnetic torque is presented by the following equation:

JdWrdt+frWr=cemcrE12
Cem=pλrqirdλrdirqE13

The system now will be modeled with respect to the rotor side direct and quadratic (d, q) synchronous reference frame. The input in such case are Irdand Irq.

First the system expression w.r.t daxis frame:

vrd=Rrird+ddtLrird+MisdLrirq+MisqWrE14
=Rrird+ddtirdLrM2LsLrirqWrMWrMirqLsE15
Rrird+Lrddtird1M2LsLrLrirqWr+M2LsWrirqE16
=Rrird+Lri̇rd(1M2LsLr)LrWr1M2LsLrirqE17
=Rrird+LrΛi̇rdLrΛWrirqE18
i̇rd=1LrΛvrdRrLrΛird+wrirqE19
i̇rd=1LrΛvrd1TΛird+wrirqE20

with Λ=1M2LsLrT=RrLr.

Now consider qaxis frame:

Vrq=Rrirq+ddtλrq+λrdWrE21
=Rrirq+ddtLrirqM2Lsirq+WrLrirdM2irdLSE22
=Rrirq+Lri̇rq(1M2LsLr+LrWr1M2LsLrirdE23
=Rrirq+LrΛi̇rqLrΛWrirdE24
i̇rq=1LrΛvrq1TΛirqwrirdE25

Finally we obtain the speed from the torque equation as:

Ẇr=frJWr+pJλrqirdpJλrdirqE26

3. Nonlinear control of the DFIG

First, we start by putting the model in the standard nonlinear form. Recalling from modeling, the system is introduced in the condensed nonlinear form:

ΣC:ẋ=fx+g1xu1+g2xu2,xRn,uRny=hx.E27

where, X=x1x2x3T=irdirqWrT, U=u1u2T=vrdvrqT. The function fx,gxare smooth vector fields and the output function hxis a smooth scalar function.

fx=1TΛx1+x2x31TΛx2x2x3frJx3+pJϕrqx1pJϕrdx2E28
g1x=1TΛ00,g2x=01TΛ0.E29

Since the purpose of this study is to control the rotor side converter current, the output was chosen as hx=irdirqT.

Remark 1.According to the previous results obtained by Isidori, A multi variable nonlinear system in the form of(36)has a relative degreer1,,rmat pointx0ifLgjLfkhix=0for all1jm, for all1im, for allkri1, and for all neighbor ofx0.

Following the same definition, it can be easily verified that the system relative degree w.r.t the outputs r=2.

3.1 Control of d-axis rotor current

In order to track rotor current irqwe assume that the system is only affected by u1and u2=0.

ẋ=fx+g1xE30
y=h1x=irdE31

The system relative degree w.r.t the output r=1. Now we apply a coordinate transformation and introduce the system in to the normal form.

ϕx=z1=x1η1=x2η2=x3,Lg.η=0.E32
ż1=1TΛz1+η1η2+1TΛu1η̇1=1TΛη1z1η2η̇2=pϕrdz1pϕrdη1frJeta2E33

After applying the proper control law in the form of u=TΛ1TΛx1+x2x3+x1rc0x1where x1r,c0represents the rotor current desired value and the chosen zero, we obtain the desired output.

3.2 Control of q-axis rotor current

In this case the effect of u2is studied

fx=1TΛx1+x2x31TΛx2x2x3frJx3+pJϕrqx1pJϕrdx2g2x=01TΛ0E34
y=h2x=irq.E35

The system relative degree rq=1. The coordinate transformation and the normal take the form of

ϕx=z1=x2η1=x3η2=x1E36
ż1=1TΛz1+η1η2+1TΛu2η̇1=pϕrdη1pϕrdz1frJη1η̇2=1TΛη2z1η2.E37

The input

u=TΛ1TΛx2x1x3+x2rc0x2.E38

3.3 Simulation and results

This section presents the evaluation of the performance of the proposed technique. Two cases were developed. The first case study based on the Doubly Fed Induction Generator model while the second one studies the gird side converter command model when the power factor is set to unity. Table 1 presents the parameters of the DIFG parameters. The Bitz limit at which the maximum efficiency is obtained for the first case study is shown in Figure 1. In this case, the optimal point corresponds to Beta angle is equal to zero. Two feedbacks were applied in this stage in the sake of evaluating proposed control strategy. The primary feedback was applied in the direct axis frame with an input value

The DFIG data of a typical 3.6 MW generator
Power7 kW
Efficiency at rated speed79%
Voltage690 V
Locked rotor voltage1000 V
Operation speed range2000 rpm
Power factor0.90 cap
Rotor ResistanceΩ
Rotor Inductance0.2 mH
Stator Resistance0.5 Ω
Stator Inductance0.001 mH
Mutual inductance0.078 H
Number of poles4
Inertia moment0.3125 Nms2

Table 1.

The DFIG data sheet.

Figure 1.

Power coefficient curve.

u=TΛ1TΛx1+x2x3+x1r1000x1E39

Figure 2 shows the result of the rotor side reference signal and the generated current signals after applying the feedback. It can be noticed that the proposed control technique succeeded in reproducing a current signal that coincides with the required reference signal. As for the quadratic axis frame another feedback was designed to track the required current signal.

Figure 2.

Doubly fed induction generator ird rotor current.

u=TΛ1TΛx2x1x3+x2r1200x2E40

The applied input will produce a signal that follows the reference signal (see Figure 3). Figure 4 presents the continuous bus voltage of the DFIG regulated to the standard reference voltage fixed at 1000 V. It is clear that in spite of fluctuation of the wind the voltage remains stationary which is considered a major advantage as the system will be affected by the harmonics.

Figure 3.

Doubly fed induction generator irq rotor current.

Figure 4.

Doubly fed induction generator continuous bus voltage.

In the next half of this chapter, the sampled data design techniques and its application will be discussed. For the simplicity of the design, we choose to set the power factor to unity such that the system is converted to a SISO system. In such a case the Grid Side Converter command model is studied.

4. Grid side converter command model

vfd=Rfifd+LfddtifdWrLfifqVGdE41
vfq=Rfifq+Lfddtifq+WrLfifdVGqE42
i̇fd=1LfRfifd+vfd+WrLfifq+VGdE43
i̇fq=1LfRfifq+vfqWrLfifd+VGqE44

with VGd,VGqindicated the input voltage of the AC-DC converter in the direct and quadrature frame. The electric network components of voltage and current on the AC side for both the direct and quadrature frame are given by vfd,vfq,ifdand ifqrespectively, while the Lfreferred to the inductance of the system. The active and reactive power is expressed as:

P=VGdifd+VGqifqE45
Q=VGqifqVGdifd.E46

Remark 2.Through setting the power factor to be1and neglecting the filter losses one can get the following expressionVGd=Vfd=VG,VGq=Vfq=0, leading the active and reactive power to bePf=VGifd and Qf=VGifq.

5. Nonlinear grid side converter model

Referring to Remark 2 we know that through setting the power factor to unity we get VGd=Vfd=VG,VGq=Vfq=0. In such a case the system is converted into single input-single output system in the form:

fx=RfLfx1+x3x2RfLfx2x3x1frJx3pJϕrdx2,gx=2Lf00E47
y=x1.E48

where x: state vector = ifdifqWrTU=u1u2T.

5.1 Nonlinear modeling and control of the quadratic axis control

In this part, the asymptotic output tracking technique will be studied in the quadratic axis frame. The system has a well define the relative degree of r=1. Consequently one can apply a coordinate transformation in the form Γx=x1x3x2.

The state space description in the new coordinates

ż1=azη+bzηη̇1=frJη1pJϕrdη2η̇2=RfLfη1η2z1.E49

Remark 3. The system has a stable zero dynamics. In fact by calculating the Jacobian matrixQwhich describes the linear approximation atη=0of the zero dynamics of the original nonlinear system

Q=frJpJϕrdRfLfb0E50

we can see that the matrix is nonsingular. Hence the zero dynamics are asymptotically stable. The stability of the zero dynamics will depend on the parameters of the DFIG.

The stator of the DFIG was directly connected to the grid while its rotor was connected to it via a cascade (Rectifier, Inverter, and Filter). To evaluate the grid side model the power factor was set to one, thus only the quadratic rotor current will be produced. The voltage on the output of the inverter will suffer from disturbance signals formed by the original frequency f=50 Hz and other signals. A passive R-L filter was used to eliminate harmonics. The input in the form u=1azηbzη+c0z1ensures the reproduction of an output irqthat will track the required reference signal. Figure 5 depicts that the system nonlinear controller has reproduced an output that will converge asymptotically to the required reference signals and minimizes the effect of disturbance.

Figure 5.

Nonlinear control applied to rotor current.

5.2 Feedback design under sampling

We now address the problem of preserving under system behavior under sampling. In fact, considering utUTand yt=ykTfor tkTk+1T(T the sampling period). Now we compute the single-rate sampled data equivalent model of (43)

xk+1=FTxkukE51
yk=hxkE52

with xkxkT,ykykT,ukukT,hx=irdand FTxkuk=eTLf+ukLgxK. In this case we compute a digital control law

ud=ukT+Tw1kE53

which solve the problem.

5.3 Simulation

The wind speed and the DFIG are shown in Figure 6. The estimation of the wind speed was generated based upon the nonlinear mapping of the measured output power of the generator while taking into account the loss of power in the wind turbine. The quadratic rotor current that shall be set as the reference signal so that a better performance is provided are shown in Figure 7. A feedback that is based on the proposed technique is applied and this will yield to an output signal that will follows the rotor signal (see Figure 5). Figure 8 depicts the emulated and the sampled rotor speed after applying the feedback. Maximum Power Point Tracking technique was used to set the best conditions in order to arrive to maximum efficiency. It can be shown that sampled-data design provided better results such as the variation is smoother and the transient time is less than the emulated one. The Tip Speed Ratio TSR is illustrated in Figure 9 for both the emulated and sampled base. The results will show that the TSR has been reduced by more than design which indicates that the size of power converters is reduced. Then, the power converters can be downsized without reducing the output power.

Figure 6.

Wind speed estimation.

Figure 7.

DFIG rotor current.

Figure 8.

DFIG rotor speed for MPPT.

Figure 9.

Tip speed ratio.

6. Conclusion

This paper aims to investigate the different modern control strategies in the power system application. In particular, the study will focus on the effect of nonlinear control techniques and SampledData Model when it is applied to a Doubly Fed Induction Generator DFIG. The mechanical model was first recalled and then the nonlinear model and control techniques were discussed. In the nonlinear, the asymptotic output tracking technique was chosen where feedback is designed to ensure that the system will converge to a specific target or reference. In this case, through controlling the direct and quadratic frame we can control the active and reactive power which was proven by the results. In the second half of the chapter we choose to investigate the digital control techniques where a comparison between the emulation design and the sampled data techniques are carried out The MATLAB program was to choose to simulate and test the control strategies. It can be noted from the results that as time increased the emulation design fail to preserve the same behavior as in the continuous-time and an oscillation takes place, unlike the sampled data design. Finally, it can be concluded that applying the sampled data model over the nonlinear system provides powerful results than the classical solution. Further investigation will be carried out regarding practical cases.

Acknowledgments

All thanks to professor Salvator Monaco, my mentor.

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Marwa Hassan (April 1st 2020). Nonlinear and Sampled Data Control of Wind Turbine, Advances in Modelling and Control of Wind and Hydrogenerators, Amir Ebrahimi, IntechOpen, DOI: 10.5772/intechopen.91246. Available from:

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