Optimal Control for a Variable-Speed Wind Turbine

Cosmin KOCH-CIOBOTARU, Radu BORACI, Octavian PROSTEAN, Nicolae BUDISAN “Politehnica” University/Department of Automation and Applied Informatics, Timisoara, Romania

cosmin.koch@aut.upt.ro

Abstract— The paper presents a control strategy for a wind energy conversion system (WECS) which tracks the maximum power point of the wind turbine. The topology of the system consists of a permanent magnet synchronous generator (PMSG), a diode bridge rectifier, a hybrid dc-dc converter which lowers the dc voltage, a supercapacitor and a battery charger. This strategy uses the measured wind speed and imposes the optimal rotation speed of the wind turbine. The control method is simulated in Matlab/ Simulink and the results are presented and discussed.

Keywords: PMSG, variable speed drive, optimal control

I. INTRODUCTION Over the last years the wind systems became the fastest

developing renewable energy technology. From the constant speed to variable speed high performance strategies, assisted with power electronic converters, connected to the grid or in insulate systems, this technology, developed in various structures, enhanced power extraction from the wind[1].

The variable speed wind energy conversion systems (WECS) offer many advantages over the constant speed operation WECS. One of such advantage is the capability of the system to track the maximum power point and harvest, in each moment, the maximum amount of energy from the wind [2]. This is due to the fact that the extracted power from the wind is influenced by the wind speed and by the rotation speed of the blades [3].

The present paper studies the tracking of the maximum power point in a WECS equipped with a permanent magnet synchronous generator (PMSG). The novelty of this topology, as presented in Fig. 1, is the usage of supercapacitors. The supercapacitor has the role to separate the control on the generator side and the control on the load/grid side, acting as an energy buffer [3]. The focus of this paper is the generator control, on the maximum power tracking, goal realized by controlling the hybrid dc-dc converter shown in Fig. 3.

II. SIMULATION MODEL The simulation model, design in Matlab/Simulink

comprises the wind generator, the aerodynamic model of the wind turbine, the generator, the diode bridge rectifier, the HDC dc-dc converter, the supercapacitor and the load.

The data for all the electrical components are taken from a real test bench and for the wind turbine the data

from the manufacturer were implemented as a relation between wind speed, rotation speed and extracted power.

The extracted amount of power, for a wind turbine is given by the equation:

(1)

where the tip-speed ratio is defined as: (2)

and: A – blade swept area [m2] – specific air density [kg/m3]

v – wind speed [m/s] R – turbine radius [m]

r – rotation speed [rad/s] Cp – power coefficient

Cp is specific to each turbine and the mathematic

relation is given by the manufacturer after mechanical analysis, wind tunnel experiments and tests. In our case, the relation is given in [5].

(3)

This Cp can be considered the efficiency of the turbine.

Theoretical demonstrated is the fact that no Cp can exceed the limit of 0.59, called the Betz coefficient.

Another characteristic of each turbine is the torque coefficient CT. The relation between the two coefficients is:

(4)

For two turbine model the following parameter table is given, for a rated optimum tip speed ratio of 3, respectively of 4:

TABLE I. CONSTRUCTIVE PARAMETERS OF TWO TURBINES MODELS

TSRo( 0) CM0 CPmax a b 4 0.0125 0,4650 0.0626 0.0046 2 3.5 3 0.0222 0,4281 0.0986 0.0113 2 3.5

As it can be seen from Fig. 2, the two curves have

different maximum points, for the values of shown in Table I. This is an important observation which reveals that the optimal torque and the optimal extracted power are reached at different tip-speed ratios.

( ) 35.0 vCAP p λρ=

vRrωλ =

( ) baMp baCC λλλλ −+= 0

λp

T

CC =

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6th IEEE International Symposium on Applied Computational Intelligence and Informatics • May 19–21, 2011 • Timi oara, Romania

978-1-4244-9109-4/11/$26.00 ©2011 IEEE

Figure. 1 Simulation model of the wind generator structure

Figure. 2 Power and torque coefficient characteristic

Usually, in lower winds sites, the wind turbines are designed for low values of .

In this paper, the model used in simulations is that of =3.

The considered electric mashine is a permanent magnet synchronous generator, having the parameters shown in Table II.

TABLE II. ELECTRIC GENERATOR DATA

Rs 1.5 Ld=Lq 29.4 mH

1.44 Wb Jg 0.6 Kgm2 2p 32

The HDC dc-dc converter is a modified buck converter

(see Figure 3a). It is used to lower the voltage at the supercapacitor.

The HDC transfer ratio between the output voltage and the input voltage is given in relation 5:

inVDDV−

=20

(5)

As for the conventional Buck converter, where the ratio is given by

inDVV =0 (6)

where D is the duty cycle and has values in [0…1]

For the switch extinction period, the two inductances,

L1 and L2, change polarity to support the declining current and they inject current in the circuit, opposing the change. Each pair of inductance and diode acts as a parallel source of current for the load.

On each inductance circuit, the current waveform is identical, and the variation of each is equal to the variation of the current in the conducting mode.

The corner frequency fc of the low-pass filter formed by L1 and Cout is [6]:

(7) The voltage ripple at the converter output can be

minimized by selecting a frequency fc of the low pass filter such that fc << fs(=1/TP).

Vin

S

L1

L2

SC D2 D1

Vo

(a) HDC dc-dc converter

(b) Buck converter topology

Figure 3 HDC and Buck dc-dc converter topologies

outc LCf

π21=

C. Koch-Ciobotaru et al. • Optimal Control for a Variable-Speed Wind Turbine

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Figure 4 The output/input voltage ratio for a HDC and a Buck dc-dc

converter

There is an economic case for reducing the size of the inductor, but at a given frequency this will increase the current ripple and associated losses and the ripple voltages (and currents) on the input and output capacitors implying a need for larger capacitors.

Increase of the switching frequency fs results in lower ripple magnitude V0. This requires a smaller inductor and lower rated smoothing capacitors [6][7].

The comparison between the two ratios is given in Figure 4.

III. CONTROL STRATEGY AND CONTROLLER DESIGN The maximum power extracted by the wind turbine can

be obtained if the Cp is also maximum, having the value Cp_max, given by the manufacturer.

This Cp_max is obtained for a certain value of the tip-speed ratio, in the two considered cases presented in Fig. 2 can be seen that this two values are 3 and 4, according to the turbine model.

This means that the optimum power extracted by the wind turbine can be written as:

(8)

or (9)

where 3

_5.0=opt

optpoptRACk

λρ (10)

From (8), it can be seen that a direct relation between the rotation speed of the turbine and the maximum extracted power is found.

vRopt

optm

λω =_

(11)

If the wind speed is measured, the optimal rotation speed of the wind turbine is accessible by multiplying the wind speed with a coefficient; opt and R are known, equation 10.

The dc-dc converter has by its design an inner controller to limit the input current. This is hardware realized by using operational amplifiers. The resulting controller is of type I – only integrator. This is used both to control and to limit the inner current loop.

For the software designed controller, the reference value is the rotation speed of the turbine. The dc-dc

Figure 5 Bode characteristics of the controller

converter must impose the current in the electric generator which affects the rotation speed of the wind turbine in order to track the maximum power point in an optimal regime.

Due to the complex transfer function of the converter and the presence of the hardware implemented controller, the transfer function of the rotation controller has a complicated form:

( )( )2

11.0sT

sTksTkH

i

ipipR ⋅

+⋅+⋅⋅⋅= (12)

The Bode plot of the transfer function is depicted in Figure 5.

For implementing this transfer function on a DSP device, the discrete transfer function is needed. Using a step time of 0.001 seconds and the advanced rectangle method, the following discrete transfer function is obtained:

360000072000003600000324000064801983240198

2

2

+−+−=ssssHRz

(13)

Implementing this controller and considering the

converter transfer function and the hardware realized integrator at the converter’s input, a fast response is obtained for the tracking algorithm.

IV. CASE STUDY

For the case study, the considered simulation model was used, with a wind speed sequence around the value of 7 m/s for a time of 10 seconds, as shown in Figure 1a.

The controller imposes the reference rotation speed which is tracked by the wind turbine due to the HDC dc-dc converter and generator load current.

As it may be seen from Figure 6.b, the wind turbine follows the wind speed in order to obtain the maximum power operation. This operation mode is emphasized in Figure 6d, depicting the power coefficient Cp which is kept to the maximum value during the entire operating regime.

In Figure 6c,e are presented the geometric place of the wind turbine output power versus rotation speed – which is on the maximum power curve, and the geometric place of the power coefficient Cp function of lambda.

3

__5.0=

opt

optmoptpopt

RACP

λω

ρ

( )3_ optmoptopt kP ω=

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6th IEEE International Symposium on Applied Computational Intelligence and Informatics • May 19–21, 2011 • Timi oara, Romania

Figure 6 Simulation results

For the considered wind turbine, the entire Cp characteristic is presented in Figure 2 and in Figure 6e the tracked values are plotted.

In Figure 6f, the output voltage of the HDC is plotted. This voltage on the supercapacitor is kept constant by the resistance R1.

V. CONCLUSIONS This strategy uses the measured wind speed and

imposes the optimal rotation speed of the wind turbine. The control method is simulated in Matlab/ Simulink and the results confirm the good evolution of the system, the controller imposing the maximum power point of operation.

By measuring the wind speed and adopting a simple method of control, good results are obtained.

ACKNOWLEDGMENT This work was supported by the European Economic

Area (EEA) project “RO 018 Improvement of the Structures and Efficiency of Small Horizontal Axis Wind Generators with Non-Regulated Blades”.

This work was partially supported by the strategic grant POSDRU 2009 project ID 50783 of the Ministry of Labor,

Family and Social Protection, Romania, co-financed by the European Social Fund – Investing in People.

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[3] Muntean N., Cornea O., Petrila D., “A New Conversion and Control System for a Small Off-Grid Wind Turbine”, 12th International Conference on Optimization of Electrical and Electronic Equipment, OPTIM 2010, may 2010, Brasov, Romania, pp 1167-1173

[4] Haque Md. E., Negnevitsky M., Muttaqi K. M. “A Novel Control Strategy for a Variable-Speed Wind Turbine with a Permanent Magnet Synchronous Generator”, IEEE Transactions on Industry Applications, vol. 46, No. 1, Jan/Feb 2010, pp 331-339

[5] Arjan M. van Voorden, Laura M. Ramirez Elizondo, Gerard C. Paap, Jody Verboomen, Lou van der Sluis “The Application of Super Capacitors to relieve Battery-storage systems in Autonomous Renewable Energy Systems”,

[6] Rolan, A., Luna, A., Vazquez, G., Aguilar, D., Azevedo, G., Modeling of a variable speed wind turbine with a Permanent Magnet Synchronous Generator, Industrial Electronics, 2009. ISIE 2009. IEEE International Symposium on, 2009 , pp 734 - 739

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