BEHAVIOUR OF PMSG WIND TURBINES WITH FRACTIONAL CONTROLLERS TO A VOLTAGE

DECREASE IN THE GRID

R. Melício*,†, V.M.F. Mendes#, J.P.S. Catalão*,†

* University of Beira Interior, Covilhã, Portugal (email of corresponding author: catalao@ubi.pt) † Center for Innovation in Electrical and Energy Engineering, IST, Lisbon, Portugal

# Instituto Superior de Engenharia de Lisboa, Lisbon, Portugal

Keywords: Wind turbines, power converters, power quality, wind energy simulation.

AbstractThe effect on permanent magnet synchronous generator (PMSG) wind turbines due to grid voltage decrease and fifth harmonic is studied in this paper by computer simulations. A multilevel power converter topology and two control strategies, namely fractional and integer-order controllers, are considered in order to evaluate that effect. A comparison between the two control strategies is presented. Finally, conclusions are duly drawn.

1 Introduction Wind generation levels are growing in power systems around the world in response to increased pressure to reduce CO2levels and dependence on fossil fuels [18]. A high penetration of wind power into the grid is taking place [6], especially in European countries such as Portugal [12].

In Portugal, the total installed renewable energy capacity reached 10276 MW in September 2011, of which wind power capacity is responsible for 4261 MW, and continues growing. An overview of the Portuguese technical approaches and methodologies followed in order to plan and accommodate the ambitious wind power goals, preserving the overall quality of the power system, is given in [9].

The increasing share of wind in power generation will change considerably the dynamic behaviour of the power system [8], and may lead to a reduction of power system frequency regulation capabilities [2]. In addition, grid operators have to ensure that consumer power quality is not compromised [5]. Hence, new technical challenges emerge due to increased wind power penetration, power quality and dynamic stability, implying research of more realistic and accurate physical models for wind energy conversion systems (WECSs).

Power converters play a vital role in the integration of wind power into the grid [7]. The use of power converters allows for variable-speed operation of the wind turbine, and enhanced power extraction [3].

In a variable-speed WECS with full-power converter, the generator is completely decoupled from the grid [2]. The wind turbine is directly connected to the generator, which in this paper is a permanent magnet synchronous generator (PMSG). The matrix converter is capable of converting the variable frequency AC from the generator into constant frequency AC to the grid in one stage. Two-level converters generate an output voltage with two values (levels) with respect to the negative terminal of the capacitor, while the three-level converters generate three voltages. Multilevel converters, specifically three-level converters, are a good trade-off solution between performance and cost in high-power systems [16]. The increasing number of voltage levels lead to the production of high power quality waveforms, causing the total harmonic distortion (THD) to be lower.

In this paper, an accurate modelling of the WECS with a multilevel power converter topology and PMSG is presented, alongside a control strategy based on fractional-order controllers. As a new contribution to earlier studies, the behaviour of the WECS to a sudden voltage decrease in the grid with fifth harmonic component will be thoroughly assessed in this paper. Conclusions are duly drawn.

2 Modelling

2.1 Wind Speed

The wind speed usually varies considerably and has a stochastic character. The wind speed variation can be modelled as a sum of harmonics with the frequency range 0.1–10 Hz [19]:

nnn tAuu )(sin10 (1)

where u is the wind speed value subject to the disturbance, 0u is the average wind speed, n is the kind of the mechanical

eigenswing excited in the rotating wind turbine, nA is the magnitude of the eigenswing n , n is the eigenfrequency of the eigenswing n . Hence, the physical wind turbine model is subjected to the disturbance given by the wind speed variation model [1].

2.2 Wind Turbine

During the conversion of wind energy into mechanical energy, various forces produce various mechanical effects [19]. The mechanical eigenswings are mainly due to the following phenomena: asymmetry in the turbine, vortex tower interaction, and eigenswings in the blades. The mechanical part of the wind turbine model can be simplified by modelling the mechanical eigenswings as a set of harmonic signals added to the power extracted from the wind. Therefore, the mechanical power of the wind turbine disturbed by the mechanical eigenswings tP may be expressed by:

3

1

2

1)()(1

nn

mnmnmnttt thtgaAPP (2)

tnmnnm dttmg

0')'(sin (3)

where ttP is the mechanical power of the wind turbine, m is the order of the harmonic of an eigenswing, nmg is the distribution of the m-order harmonic in the eigenswing n ,

nma is the normalized magnitude of nmg , nh is the modulation of the eigenswing n , and nm is the phase of the m-order harmonic in the eigenswing n .

The eigenfrequency range of the wind turbine model with is from 0.1 to 10 Hz. The values used for the calculation of (1) to (3) are given in the Table 1 [1].

n Source nA n hn m nma nm

1 Asymmetry 0.01 t 11 4/5 0

2 1/5 /2

2Vortextower

interaction0.08 3 t 1

1 1/2 0

2 1/2 /2

3 Blades 0.15 9 1/2 (g11+g21) 1 1 0

Table 1: Mechanical eigenswigs excited in the wind turbine.

2.3 Mechanical Drive Train Model

The mechanical drive train considered in this paper is a two-mass model, consisting of a large mass and a small mass, corresponding to the wind turbine rotor inertia and generator rotor inertia, respectively. The model for the dynamics of the mechanical drive train for the WECS used in this paper was reported by the authors in [13].

2.4 Generator

The generator considered in this paper is a PMSG. The equations for modelling a PMSG can be found in the literature [14]. In order to avoid demagnetization of permanent magnet in the PMSG, a null stator current 0diusually is imposed [17].

2.5 Multilevel Converter

The multilevel converter is an AC-DC-AC converter, with twelve unidirectional commanded IGBTs ikS used as a rectifier, and with the same number of unidirectional commanded IGBTs used as an inverter. The rectifier is connected between the PMSG and the capacitor banks. The inverter is connected between these capacitor banks and a second order filter, which in turn is connected to the grid. The groups of four IGBTs linked to the same phase constitute a leg k of the converter. A three-phase active symmetrical circuit in series models the grid. The model for the multilevel converter used in this paper was reported by the authors in [13]. The configuration of the simulated WECS with multilevel converter is shown in Figure 1.

3 Control Strategy

3.1 Fractional-Order Controllers

A control strategy based on fractional-order PI controllers is proposed. Fractional-order controllers are based on fractional calculus theory, which is a generalization of ordinary differentiation and integration to arbitrary (non-integer) order [15]. Recently, applications of fractional calculus theory in practical control field have increased significantly [11]. The fractional-order differentiator can be denoted by a general operator ta D [4], given by:

0)(

0)(

0)(

,)(

,1

,

t

a

ta

d

dtd

D (4)

The mathematical definition of fractional derivatives and integrals has been the subject of several approaches. The most frequently encountered definition is called Riemann–Liouville definition, in which the fractional-order integrals are defined as:

t

ata dfttfD )()()(

1)( 1 (5)

while the definition of fractional-order derivatives is: t

a nn

n

ta dt

fdtd

ntfD 1)(

)()(

1)( (6)

where:

01)( dyeyx yx (7)

is the Gamma function, a and t are the limits of the operation, and is the fractional order which can be a complex number. In this paper, is assumed as a real number that satisfies the restrictions 10 . Also, a can be taken as a null value. The following convention is used:

tt DD0 .

PMSGLf Ln RnRf

Cf

k=1,2,3 k=4,5,6 Electric grid

ifk

C1

C2

ufk

ikik uk

t

Figure 1: WECS with multilevel converter.

The differential equation of the fractional-order PIcontroller is given by

)()()( teDKteKtu tip (8)

where pK is the proportional constant and iK is the integration constant. Taking 1 , a classical PI controller is obtained. In this paper, it is assumed that 5.0 . Using the Laplace transform of fractional calculus, the transfer function of the fractional-order PI controller is obtained, given by:

5.0)( sKKsG ip (9)

3.2 Converters Control

Power converters are variable structure systems, because of the on/off switching of their IGBTs. Pulse width modulation (PWM) by space vector modulation (SVM) associated with sliding mode is used for controlling the converters. The sliding mode control strategy presents attractive features such as robustness to parametric uncertainties of the wind turbine and the generator as well as to grid disturbances [13]. Sliding mode control guarantees the choice of the most appropriate space vectors.

The power semiconductors present physical limitations, since they cannot switch at infinite frequency. Also, for a finite value of the switching frequency, an error e will exist between the reference value and the control value. In order to guarantee that the system slides along the sliding surface

),( teS , it has been proven that it is necessary to ensure that the state trajectory near the surfaces verifies the stability conditions [12] given by:

0),(

),(dt

tedSteS (10)

4 Power Quality Evaluation The harmonic performance computed by the DFT is given by:

1

0

2 )()(N

n

Nnkj nxekX for 1,...,0 Nk (11)

where )(nx is the input signal and )(kX is a complex giving the amplitude and phase of the different sinusoidal components of )(nx .

The harmonic performance computed by the THD is given by:

F

HH

X

X 250

2100(%)THD (12)

where HX is the root mean square (RMS) value of the individual harmonic H components of the signal, and FX is the RMS value of the fundamental component. Standards such as IEEE-519 [10] impose limits for different order harmonics and the THD. The limit is 5% for THD.

5 Simulation Results The mathematical model for the WECS with multilevel power converter topology was implemented in Matlab/Simulink. The WECS considered in this case study has a rated electrical power of 900 kW. The wind speed variation is modelled as a sum of harmonics with the frequency range 0.1–10 Hz. The operational region of the WECS was considered for wind speed range from 5-25 m/s. The switching frequency used in the results is 5 kHz. The grid was modelled as a three-phase active symmetrical circuit in series, with 850 V at 50 Hz and 15% of fifth harmonic component.

The mechanical power of the wind turbine, the electrical power of the generator, and the difference between these two powers, i.e., the accelerating power, are shown in Figure 2.

0 0.5 1 1.5 2 2.5 3 3.5 4−200

0

200

400

600

800

1000

1200

Time (s)

Pow

er (

kW)

↑ Pg

↓ Pt

↓ P − Pt g

Figure 2: Mechanical, electrical and accelerating power.

The time-domain simulation with a grid RMS voltage decreasing 25% during 100 ms is shown in Figure 3 for the classical PI controller and in Figure 4 for the fractional-order controller, in order to see the comparative behaviour of both control strategies on the voltage at the capacitor bank.

0.4 0.45 0.5 0.55 0.6 0.65 0.72485

2490

2495

Time (s)

Vol

tage

(V

)

↑With fifth harmonic

↓Without fifth harmonic

Figure 3: Voltage at the capacitor bank during the grid RMS voltage decrease with classical PI controller.

0.4 0.45 0.5 0.55 0.6 0.65 0.72485

2490

2495

Time (s)

Vol

tage

(V

)

↑With fifth harmonic

↓Without fifth harmonic

Figure 4: Voltage at the capacitor bank during the grid RMS voltage decrease with fractional-order controller.

The current injected in the grid is shown in Figure 5.

3.4 3.42 3.44−1000

−500

0

500

1000

Time (s)

Cur

rent

(A

)

Figure 5: Current injected in the grid.

The harmonic behaviour computed by the DFT for the current injected in the grid with the multilevel converter is shown in Figure 6.

0 50 100 150 200 250 300 350 400 450 5000

20

40

60

80

100

Frequency (Hz)

Cur

rent

(%

)

Figure 6: DFT of the current injected in the grid.

The THD of the output current is shown in Figure 7.

0 1 2 3 43

4

5

6

Time (s)

TH

D (

%)

← Grid Voltage Decrease

Figure 7: THD of the output current.

Hence, it has been shown that non-ideal sinusoidal voltage waveforms on the grid affect the current from the converters.

Still, it should be noted that the average THD of the output current is lower than the 5% limit imposed by IEEE-519 standard [10], being equal to 4.75% for the classical PI controller and being equal to 4.21% for the fractional-order controller.

6 Conclusions This paper considers a PMSG-based WECS with multilevel power converter and a control strategy based on fractional-order controllers. Time-domain simulations are provided by means of a grid RMS voltage decrease. The simulation results show that the non-ideal sinusoidal voltage waveforms affect the current injected into the grid. Also, the fractional-order control strategy provides better results in what regards the THD of the output current, comparatively to a classical integer-order control strategy.

Acknowledgements R. Melício thanks FCT for a post-doctoral grant (SFRH/BPD/68585/2010).

References [1] V. Akhmatov, H. Knudsen, A.H. Nielsen, “Advanced

simulation of windmills in the electric power supply,” International Journal of Electrical Power Energy Systems, volume 22, pp. 421-434, (2000).

[2] R.G. Almeida, J.A. Peças Lopes, “Participation of doubly fed induction wind generators in system frequency regulation”, IEEE Trans. Power Syst.,volume 22, pp. 944-950, (2007).

[3] J.A. Baroudi, V. Dinavahi, A.M. Knight, “A review of power converter topologies for wind generators”, Renewable Energy, volume 32, pp. 2369-2385, (2007).

[4] A.J. Calderón, B.M. Vinagre, V. Feliu, “Fractional order control strategies for power electronic buck converters,” Signal Processing, volume 86, pp. 2803-2819, (2006).

[5] J.M. Carrasco, L.G. Franquelo, J.T. Bialasiewicz, E. Galvan, R.C.P. Guisado, A.M. Prats, J.I. Leon, N. Moreno-Alfonso, “Power-electronic systems for the grid integration of renewable energy sources: A survey”, IEEE Transactions on Industrial Electronics,volume 53, pp. 1002-1016, (2006).

[6] Z. Chen, F. Blaabjerg, “Wind farm—A power source in future power systems”, Renew. Sust. Energ. Rev.,volume 13, pp. 1288-1300, (2009).

[7] Z. Chen, J.M. Guerrero, F. Blaabjerg, “A review of the state of the art of power electronics for wind turbines”, IEEE Trans. Power Electron., volume 24, pp. 1859-1875, (2007).

[8] I. Erlich, J. Kretschmann, J. Fortmann, S. Mueller-Engelhardt, H. Wrede, “Modeling of wind turbines based on doubly-fed induction generators for power system stability studies”, IEEE Trans. Power Syst.,volume 22, pp. 909-919, (2007).

[9] A. Estanqueiro, R. Castro, P. Flores, J. Ricardo, M. Pinto, R. Rodrigues, J. Peças Lopes, “How to prepare a power system for 15% wind energy penetration: the Portuguese case study”, Wind Energy, volume 11, pp. 75-84, (2008).

[10] IEEE Standard 519-1992, IEEE Guide for Harmonic Control and Reactive Compensation of Static Power Converters.

[11] W. Li, Y. Hori, “Vibration suppression using single neuron-based PI fuzzy controller and fractional-order disturbance observer,” IEEE Transactions on Industrial Electronics, volume 54, pp. 117-126, (2007).

[12] R. Melício, V.M.F. Mendes, J.P.S. Catalão, “Wind energy systems with power-electronic converters and fractional-order controllers”, in: Proc. of the 5th IET International Conference on Power Electronics, Machines and Drives—PEMD 2010, Brighton, UK, pp. 1-6, (2010).

[13] R. Melício, V.M.F. Mendes, J.P.S. Catalão, Windturbines with permanent magnet synchronous generator and full-power converters: modelling, control and simulation. Wind Turbines, Ibrahim Al-Bahadly (Editor), IN-TECH. Vienna, Austria, pp. 465-494 (2011).

[14] C.-M. Ong, Dynamic Simulation of Electric Machinery: Using Matlab/Simulink. NJ: Prentice-Hall, pp. 259-350 (1998).

[15] I. Podlubny, “Fractional-order systems and PI-lambda-D-mu-controllers,” IEEE Transactions on Automatic Control, volume 44, pp. 208-214, (1999).

[16] R. C. Portillo, M. M. Prats, J. I. Leon, J. A. Sanchez, J. M. Carrasco, E. Galvan, L. G. Franquelo, “Modeling strategy for back-to-back three-level converters applied to high-power wind turbines”, IEEE Trans. Ind. Electron., volume 53, pp. 1483-1491, (2006).

[17] T. Senjyu, S. Tamaki, N. Urasaki, K. Uezato, “Wind velocity and position sensorless operation for PMSG wind generator,” in: Proc. of the 5th Int. Conf. on Power Electronics and Drive Systems, Singapore, pp. 787-792, (2003).

[18] E. Vittal, M. O'Malley, and A. Keane, “A steady-state voltage stability analysis of power systems with high penetrations of wind”, IEEE Trans. Power Syst.,volume 25, pp. 433-442, (2010).

[19] Z.X. Xing, Q.L. Zheng, X.J. Yao, Y.J. Jing, “Integration of large doubly-fed wind power generator system into grid,” in: Proc. of the 8th Int. Conf. Electrical Machines and Systems, Nanjing, China, pp. 1000-1004, (2005).