Offshore wind turbine simulation: Multibody drive train. Back-to-back NPC (neutral point clamped) converters. Fractional-order control

Energy xxx (2014) 1e13

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Offshore wind turbine simulation: Multibody drive train. Back-to-back NPC (neutral point clamped) converters. Fractional-order control M. Seixas a, b, c, R. Melício a, b, *, V.M.F. Mendes a, c, * a

Department of Physics, Universidade de Évora, R. Romão Ramalho 59, 7002-554 Évora, Portugal IDMEC/Associated Laboratory for Energy, Transports and Aeronautics, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal c Department of Electrical Engineering and Automation, Instituto Superior de Engenharia de Lisboa, R. Conselheiro Emídio Navarro, 1950-062 Lisbon, Portugal b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 July 2013 Received in revised form 26 February 2014 Accepted 6 March 2014 Available online xxx

This paper is on offshore wind energy conversion systems installed on the deep water and equipped with back-to-back neutral point clamped full-power converter, permanent magnet synchronous generator with an AC link. The model for the drive train is a five-mass model which incorporates the dynamic of the structure and the tower in order to emulate the effect of the moving surface. A three-level converter and a four-level converter are the two options with a fractional-order control strategy considered to equip the conversion system. Simulation studies are carried out to assess the quality of the energy injected into the electric grid. Finally, conclusions are presented.
2014 Elsevier Ltd. All rights reserved.

Keywords: Offshore Wind energy Power converters Multibody drive train Fractional-order control

1. Introduction The anticipated exhaustion of fossil fuel energy and the climate change have led to deployment of renewable energy [1]. Renewable energy is an alternative non-scarce source of energy with a null anthropogenic greenhouse gases balance [2]. The integration of the renewable energy is viewed as an exploitation to meet both the needs for energy and for mitigating anthropogenic climate change [3]. The wind power industry and the construction of wind farms onshore and offshore are undergoing rapid development [4e10]. Offshore wind farms have a longer expected life time and have in favour [11] not only the favourable wind conditions on offshore as compared to sites on onshore [1,12e14], but also the scarcity of land for new onshore wind farms deployment. But, the favourable wind conditions over sea waters [15] and scarcity of land have to compensate for the additional cost of offshore deployment. Also,

* Corresponding authors. Department of Physics, Mechatronics Engineering, Universidade de Évora, R. Romão Ramalho 59, 7002-554 Évora, Portugal. Tel.: þ351 266 745372; fax: þ351 266 745394. E-mail addresses: [email protected], [email protected] (R. Melício), [email protected] (V.M.F. Mendes).

this deployment is viewed as a promising one in regard of the fact that ocean water covers about three-quarters of the earth surface [16]. Offshore allows increase in the rotor dimension due to smaller restrictions of scale which enables the use of larger turbines since there is less concern over visual impact [17,18] and noise. Offshore deployment is already in operation, for instances, in the United States, China, Japan, Norway and is based on experience with the deployment of structures for offshore oil and gas industry [19,20]. Europe has led offshore deployment, for instance, due to the limited land available in Denmark [6], the offshore development was an option and the first deployment fixed by foundation pile were installed at Vindeby in 1991, letting an estimated 20% higher output than with comparable land sites. In 2009, Statoil-Hydro and Siemens installed the first floating offshore wind turbine on the coast of Karmøy, Norway [13]. In 2011, Portugal had the first Southern Europe offshore floating wind turbine, operating at Aguçadoura Park installed in the north of the country. The type of power transmission technology in offshore depends on the distance from the floating platforms to onshore: for shorter distances, below 60 km HVAC (high-voltage ac) can be used, but for longer distances HVDC (high voltage dc) is required [21]. Most recent HVDC technologies use voltage source converters based on

http://dx.doi.org/10.1016/j.energy.2014.03.025 0360-5442/
2014 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Seixas M, et al., Offshore wind turbine simulation: Multibody drive train. Back-to-back NPC (neutral point clamped) converters. Fractional-order control, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.03.025

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M. Seixas et al. / Energy xxx (2014) 1e13

Nomenclature

Tst

u u0 n

ug

An

un Pt

r A cp

l q ut D Tt Ptt m anm gnm hn 4nm

ubi Jbi Tbi Dbi Tbih

uh Jh Thg Dh

wind speed value subject to the disturbance average wind speed kind of the mechanical eigenswing excited in the rotation wind turbine magnitude of the eigenswing n eigenswing frequency of the eigenswing n mechanical power of the wind turbine air density area covered by the rotor blades power coefficient tip speed ratio pitch angle of the rotor blades rotor angular speed at the wind turbine diameter of the area covered by the blades mechanical torque mechanical power of the wind turbine harmonic of the eigenswing normalized magnitude of gnm distribution between the harmonics in the eigenswing n modulation of the eigenswing n phase of the harmonic m in the eigenswing n rotor angular speed at the flexible blade section moment of the inertia at the flexible blade section mechanical torque at the flexible blade section resistant torque due to the damping at the flexible blade section resistant torque stiffness and damping between the flexible blade section and the hub rotor angular speed, at the rigid blades and the hub of the wind turbine moment of the inertia of the hub and the rigid blades torsional shaft torque (stiffness and damping) shaft resistant torque due to the damping

controlled turn-on and turn-off power semiconductors. HVDC VSC (high voltage dc voltage source converter) is suitable for interconnection between high-power plant production and low power electric grids, as in electric power floating platform [22e24]. The transmission might become too expensive [25] for deployment very far from onshore, therefore economic unviability is expected, i.e., although for far shore deployment higher outputs are expected [26], higher costs of hardware such as foundations and cables than for near shore plants are predictable. Also, as technology in the far shore is less proven risk is considered higher. Though, near shore is preferable for lesser cost solutions and lesser risk there are planning challenges and requirements of acceptance by local population to be addressed [26]. The increase in the rotor dimension and hence the size of the turbine leads to a need for a proper design of the drive train of the wind turbine beside higher requirements of turbine reliability [27]. Design of wind turbines adequately to their expected operation loading e especially for offshore applications with several MW e requires a very good knowledge of the dynamic loads on in situ operational conditions [28] and low maintenance strategies [29,30]. Wind farms are of two types: single-turbine-floater and multiple turbine floaters. Multiple turbine floaters are formed by several floating wind turbines laid nearby on a single pontoon floater, using the same infrastructures, as for instance, power transmission facilities to reduce costs. Floating platforms with wind

Jg Tg Dg ifk Ln ufk Rn uk

g Sik vdc Ck vCk

m <ðmÞ

G

f(t) e(t) Kp Ki S(eab,t) ε eab

sab X(k) x(n) XH XF

stiffness torque, acting on the structure and tower in the deep water responsible for the moving of the floating surface rotor angular speed at the generator moment of inertia of the rotating masses for the rotor of the generator electric torque resistant torque due to the damping at the generator currents injected into the electric grid inductance of the electric grid voltage at the filter resistance of the electric grid voltage at the electric grid switching variable of the IGBT state (conducting and blocking) of each IGBT total DC voltage on the capacitor banks capacitance of the capacitor banks voltage onCk fractional order of the derivative or the integral, (which can be a complex number) real part of m Euler’s gamma function output of the controller input error proportional constant integer constant sliding surface on sliding mode control error allowed for the sliding surface S(eab,t) error on the stator electric current in the ab plane voltage output of hysteresis comparators in the ab plane harmonic behaviour computed by DFT input signal, a complex number giving the amplitude and phase of the different sinusoidal components root mean square value of the signal root mean square value of the fundamental component

turbines mounted atop appear to be the most economical for the depths [12] on offshore. Offshore structures are influenced by marine waves in coastal waters. The generation of waves is due to the instability of the water surface layer in which the wind causes the wave. Two fluids with different speeds, i.e., water and air in this case, will generate instabilities at their interface if the densities and the speeds are different enough [31]. Due to the coupling effects of the wind and wave dynamics, the authors in Ref. [13] consider the effect of the floating support structure motion on the strength of the blades and shaft, and the inertia force induced by the combined rotational, translational and angular motion of the blades. These structures are design in order to have a fundamental frequency well below the ocean wave’s frequency lower bound [32] in order to avoid resonance. The model for a wind turbine system is highly nonlinear and the mechanical structure is very flexible due to the height and tends to oscillate [33]. The onshore conversion of wind energy into mechanical energy over the rotor of a wind turbine is influenced by various forces acting on the blades and on the tower of the wind turbine (e.g. centrifugal, gravity and varying aerodynamic forces acting on blades, gyroscopic forces acting on the tower), introducing mechanical effects influencing the energy conversion [34]. But in offshore the conversion of wind energy is further influenced by hydrodynamics depending on the foundation system and water

Please cite this article in press as: Seixas M, et al., Offshore wind turbine simulation: Multibody drive train. Back-to-back NPC (neutral point clamped) converters. Fractional-order control, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.03.025

M. Seixas et al. / Energy xxx (2014) 1e13

depth. Consequently, the experience of onshore wind turbines cannot be applied without further consideration to the development of offshore wind turbines. Offshore turbines might be more susceptible to instabilities because of the additional hydrodynamic interactions brought on by the ocean currents and surface waves [12]. Floating wind turbines have to be strengthened, because of the increased mechanic loads on turbine components as compared to the onshore ones, i.e., the platform motion induces fatigue loads on the turbine components that have to be taken into consideration [35]. Also, those loads disturb the conversion process. Hence, there is a need to access how important are the loads in disturbing the conversion in what regards electric energy quality. As wind energy is increasingly integrated into power systems, electric energy quality is becoming a concern of utmost importance, not only the stability of already existing power systems but also the THD (total harmonic distortion) coefficient must be assessed in order to ensure enough quality for the energy injected into the electric grid [34,36]. Variable speed generation is a convenient solution because of the characteristics to achieve maximum efficiency at all operational wind speeds, to improve energy capturing and to reduce the THD [37]. The wind turbine can be operated at the maximum power operating point for various wind speeds by adjusting the shaft speed. These characteristics are advantages of variable speed WECS (wind energy conversion systems) [37]. The variable speed turbine technology employed has been focused on DFIGs (doubly-fed induction generators) and synchronous generators equipped with full power converters. An advantage of using a PMSG (permanent magnet synchronous generator) in wind power applications as an alternative to conventional generators is the higher efficiency, as the copper losses in the rotor disappear [38]. Moreover, the employment of PMSG allows operating at low speed omitting the gearbox [39]. The elimination of the gearbox and introduction of the variable speed control further increases the efficiency of the system, reduces weight and dimensions of nacelle equipment, along with power losses in conversion and maintenance requirements [38,39]. A variable speed wind power generation system needs a full power converter, to convert variable frequency into constant frequency [37]. The PMSG, interfaced to the electric grid through a full power converter, is being increasingly adopted due to its greater power density, increased reliability and better controllability [38]. Thereby, power converters play a very important role as an enabling technology to operate at variable speeds, providing more effective power capture than the fixed speed ones [21]. The advantages of multi-level converters are known since the work presented in Ref. [40], proposing a back-to-back NPC converter topology. The increase on the number of voltage levels leads to high power quality waveforms [41] and decreases the THD. The multi-level converters are a good tradeoff solution between performance and cost in high-power systems [42]. A survey of topologies, controls, and applications for multi-level inverters is in Ref. [43]. However, multi-level converters are limited by the following drawbacks: voltage unbalances, high component count, and increased control complexity [44]. A critical issue in multi-level converters is the design of the DC-link capacitors in order to ensure a convenient operation, because, special attention in multilevel converters should be paid to unbalance voltage of the capacitors, which eventually is responsible for control malfunction. In this paper, an offshore variable-speed wind turbine installed on deep water is considered equipped with a PMSG using threelevel or four-level converter topologies, namely back-to-back NPC (neutral point clamped) converters topologies, converting the energy of a variable frequency source in injected energy into the electric grid with constant frequency through an AC link. The drive

3

train considered is described by a five-mass model with an input stiffness torque due to the need to take in consideration the effects of the moving floating surface. The moving floating surface is modelled by one mass describing the tower and the floating structure. Additionally, on the PMSG/full-power converter topology is considered a fractional-order control strategy. Simulation studies are carried out in order to adequately assess the THD of the energy injected into the electric grid. The rest of this paper is organized as follows. Section 2 presents the modelling for the offshore WECS. Section 3 presents the fractional-order control strategy. Section 4 presents the harmonic assessment. Section 5 presents the simulation results. Finally, concluding remarks are given in Section 6. 2. Modelling 2.1. Offshore wind energy conversion system The offshore wind energy conversion system is schematically shown in Fig. 1. The wind speed variation model reported in Ref. [45] is used in this paper. The wind speed carries one of the disturbances and the platform motion creates another one acting on the structure of the wind turbine. 2.2. Wind speed Although the wind speed has a stochastic nature and usually varies considerably, on the simulation approach for wind turbines of the type shown in this paper is normally admissible to model the wind speed as a sum of harmonics with the frequency range 0.1e 10 Hz [46], given by:

" u ¼ u0 1 þ

X

# An sinðun tÞ

(1)

n

2.3. Wind turbine The mechanical power Ptt of the wind turbine is given by:

Ptt ¼

1 rAu3 cp 2

(2)

The computation of the power coefficient cp requires the use of blade element theory and the knowledge of blade geometry. These complex issues are normally empirical considered. In this paper, the wind turbine in consideration is modelled as the one with the numerical approximation developed in Ref. [47], where the power coefficient is given by:

 cp ¼ 0:73

151

where li ¼

li

2:14

 0:58q  0:002q 1   0:003 3

1 ðl0:02qÞ

 18:4  13:2 e li

(3)

(4)

q þ1

the power coefficient is a function of the pitch angle q of rotor blades and of the tip speed ratio l, which is given by:

l ¼

ut D 2u

(5)

The mechanical power given by (2) captured from the wind is a function of cp given by (3), this function is a differentiable pseudo-

Please cite this article in press as: Seixas M, et al., Offshore wind turbine simulation: Multibody drive train. Back-to-back NPC (neutral point clamped) converters. Fractional-order control, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.03.025

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M. Seixas et al. / Energy xxx (2014) 1e13

u

Wave forces

Wave forces

Fig. 1. Offshore wind energy conversion system.

  cpmax lopt ð0Þ; 0 ¼ 0:4412; lopt ð0Þ ¼ 7:057

(6)

The mechanical torque Tt of the wind turbine is given by:

Pt wt

(7)

The mechanical power of the wind turbine disturbed by the mechanical eigenswings is given by: 2 X

#

anm gnm ðtÞ hn ðtÞ

(8)

m¼1

where gnm ¼ sin@

Zt

1 mun ðt Þdt þ fnm A 0

0

Tst Khs

Tb3h

u2 Tb2

2.4. Multibody drive train model One question arises with the increase in size of the wind turbines: Long flexible blades have an impact on the wind energy systems [48]? The answer to this question encompasses different studies as the ones proposed in this paper, concerning with the harmonic assessment due to the dynamic properties of the blades. One way to determine the dynamic properties of the blades is through the use of finite element methods, but this approach cannot be straightforwardly accommodated in the context of studies of power system analysis. Even though the model for the structure and the tower is a non-linear one described by finite element programming, for fatigue load simulations of the offshore wind turbine however the full complexity of the non-linear model is usually not required [49]. The use of an

Jb3

Db3

0

This mechanical power is the one responsible for the movement of the drive train.

u3

Tb3

(9)

Db2h

Jb2

Dh

Kgh

Jh

Tg

Jg

wg Dgh

Kb2h Tb2h

Db2

Tb1h

0

Db3h

An

n¼1

!

Kb1h

Pt ¼ Ptt 1 þ

3 X

Kb3h

"

Db1h

Tt ¼

approach not based on finite element theory implies the need to simplify the rotor dynamics as much as possible [34], i.e., escaping the use of the finite element theory implies the use of an approach of the rotor dynamics in a compromising way of accessing its dynamic and preserving desirable proprieties for convenient handling on power system analysis programs. One straightforward way to achieve this compromise, where the multibody drive train model is described by a torsional system [48] is schematically shown in Fig. 2. The following analysis discards the movement between the structure, the tower, the hub and the rigid section of the blades. Hence, the moment of the inertia of the structure, the tower, the hub and the rigid section of the blades are described by only one inertia mass; the flexible blade sections are described by three equal but independent inertia masses in order to simulate the three blades flexibility behaviour, but the rest of the rotating masses are discarded in the proposed model in order to assess the mechanical

Marine soil

concave one. The global maximum for the power coefficient is attained at a null pitch angle. Hence, a non-linear mathematical programming problem with the objective function (3) and subjected to (4) is used to assess at the null pitch angle the global maximum and the optimal tip speed ratio respectively given by:

Thg

Dg

Jb1 Db1 u1

Tb1 Fig. 2. Multibody drive train model.

Please cite this article in press as: Seixas M, et al., Offshore wind turbine simulation: Multibody drive train. Back-to-back NPC (neutral point clamped) converters. Fractional-order control, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.03.025

M. Seixas et al. / Energy xxx (2014) 1e13

5

Fig. 3. WECS equipped with three-level converter.

behaviour of the generator rotor. The equations modelling the multibody drive train are based on the torsional version of the second law of Newton [48]. These equations are given by:

dubi 1 ¼ ðT  Tbih  Dbi ubi Þ; Jbi bi dt

with i ¼ f1; 2; 3g

(10)

duh 1 T ¼ þ Tb2h þ Tb3h  Thg  Dh uh þ Tst Jh b1h dt

(11)

dug 1 Thg  Tg  Dg ug ¼ Jg dt

(12)

The switching variable gk is used to identify the state of the IGBT i in the leg k of the converter ascertain the switching function of each IGBT. The index i with i ˛ {1, 2, 3, 4} identifies the IGBT. The index k with k ˛ {1, 2, 3} identifies the leg for the rectifier and k ˛ {4, 5, 6} identifies the leg for inverter. The three valid conditions [54] for the switching variable of each leg k are given by:

8 < 1; ðS1k and S2k Þ ¼ 1 and ðS3k and S4k Þ ¼ 0 gk ¼ 0; ðS2k and S3k Þ ¼ 1 and ðS1k and S4k Þ ¼ 0 k˛f1; .; 6g : 1; ðS3k and S4k Þ ¼ 1 and ðS1k and S2k Þ ¼ 0 (14)

The model used for describing the moving surface is the random-phase/amplitude model, in which the surface oscillation is considered to be the sum of a large number of harmonic waves as in Ref. [31]. 2.5. Generator The generator considered in this paper is a PMSG. The equations for modelling a PMSG can be found in the literature [31,50]. In order to avoid demagnetization of permanent magnet in the PMSG, a null stator direct component current is imposed [51].

The switching variable gk and an auxiliary variable F1k is associated with the upper IGBT’s in each leg k. Also an auxiliary variable F2k is associated with lower IGBT’s. Each auxiliary variable depends on the conduction and blockade states of the IGBT’s. The voltage vdc is the sum of the voltages vC1 and vC2 in the capacity banks C1 and C2, modelled by the state equation [54] given by:

dvdc 1 ¼ C1 dt

3 X k¼1

F1k ik 

6 X k¼4

!

F1k ik þ

1 C2

3 X k¼ 1

F2k ik 

6 X

!

F2k ik

k¼4

(15) Hence, the three-level converter is specified by (14) and (15).

2.6. Electric grid

2.8. Four-level converter

A three-phase active symmetrical circuit in series models the electric grid [34]. The currents injected into the electric grid are modelled by the state equation given by:

The back-to-back NPC four-level converter is an AC-DC-AC converter, equipped with eighteen unidirectional commanded IGBT’s identified by Sik, implementing the rectifier, and with the same number of unidirectional commanded IGBT’s implementing the inverter. The rectifier is connected between the PMSG and a capacitor bank. The inverter is connected between this capacitor bank and a second order filter, which in turn is connected to an electric grid. The groups of six IGBT’s linked to the same phase constitute the leg k of the converter. The configuration of the simulated offshore WECS with four-level converter is shown in Fig. 4. The converter has p ¼ 4 voltage levels. The voltage level on each phase is associated with the switching variable nk which range from 0 to (p1) and is used to identify the state of the IGBT i in the leg k of the converter establishing the switching function of each IGBT. The index i with i ˛ {1, ..., 6} identifies the IGBT. The index k with k ˛ {1, 2, 3} identifies the leg for the rectifier and k ˛ {4, 5, 6} identifies the leg for the inverter. The configuration of the rectifier for the four-level converter is shown in Fig. 5.

difk 1 ufk  Rn ifk  uk ¼ Ln dt

k ¼ f4; 5; 6g

(13)

2.7. Three-level converter The back-to-back NPC three-level converter is an ACeDCeAC converter equipped with twelve unidirectional commanded IGBT (Insulated Gate Bipolar Transistors) identified by Sik, implementing the rectifier, and with the same number of unidirectional commanded IGBT’s implementing the inverter. The rectifier is connected between the PMSG and a capacitor bank. The inverter is connected between this capacitor bank and a second order filter, which in turn is connected to an electric grid. The groups of four IGBT’s linked to the same phase constitute the leg k of the converter. The configuration of the simulated offshore WECS with the tree-level converter [52,53] is shown in Fig. 3.

Please cite this article in press as: Seixas M, et al., Offshore wind turbine simulation: Multibody drive train. Back-to-back NPC (neutral point clamped) converters. Fractional-order control, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.03.025

6

M. Seixas et al. / Energy xxx (2014) 1e13

Fig. 4. WECS equipped with four-level converter.

The four valid conditions [54] for the switching voltage level variable of each leg k, at each level, are given by:

nk ¼

8 3; > > < 2; 1; > > : 0;

ðS1k ; ðS2k ; ðS  3k ; S4k;

S2k S3k S4k S5k

and and and and

S3k Þ S4k Þ S5kÞ S6k

¼ ¼ ¼ ¼

1 1 1 1

and and and and

ðS4k ; S5k ðS1k ; S5k ðS1k ; S2k ðS1k ; S2k

and and and and

S6k Þ S6k Þ S6k Þ S3k Þ

¼ ¼ ¼ ¼

The modelling for the rectifier considers the line voltage ucij as a function of the leg voltage usk or umk, respectively, given by:

ucij ¼ usi  usj ¼ umi  umj

i; j˛f1; 2; 3g

for isj

(17)

The difference between the line voltages as a function of usk or umk are given by:

uckj  uclk ¼ 2usk 

3 X

usk ¼ 2umk 

j¼1

j¼1

jsk

jsk

ksl

k˛f1; 2; 3g;

3 X

j˛f1; 2; 3g;

umk

The following disregards are assumed on: switch delays, onstate semiconductor voltage drops, and auxiliary grids. These dis-

0 0 0 0

k˛f1; .; 6g

regards are assumed on the regard of small dead times [55] and balanced capacitor voltage in steady state [56]. The leg voltage umk as a function of nk and vdc is given by:

umk ¼

k˛f1; 2; 3g

2 3 2n1 us1 1 4 us2 5 ¼ 4 n1 9 us3 n1 2

(22)

n2 2n2 n2

3 n3 n3 5vdc 2n3

(23)

(18) 3

ksl

l˛f1; 2; 3g

usk ¼ 0

Sik

Sik

i1

k¼1

From (18)e(19) the difference between the line voltages is given by:

k˛f1; 2; 3g

ic3

Sik

(19)

uckj  uclk ¼ 3usk

nk v 3 dc

From (21)e(22) the voltage usk, for the rectifier, as a function of nk is given by:

Assuming a balanced three phase electrical generator follows the condition given by: 3 X

(16)

i2

uc12

ic2 C2

uc31

Sik

j¼1

k˛f1; 2; 3g

vdc

(20)

usk

umk

vc2

1 ic1

Sik

3usk ¼ 2umk 

vc3

2

uc23 i3

From (18) and (20): 3 X

C3

um2

Sik

vc1

um1

(21) i=1, ,6

um3

C1

0

k=1,2,3

jsk ksl

Fig. 5. Four-level converter.

Please cite this article in press as: Seixas M, et al., Offshore wind turbine simulation: Multibody drive train. Back-to-back NPC (neutral point clamped) converters. Fractional-order control, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.03.025

M. Seixas et al. / Energy xxx (2014) 1e13

β

21

16

3

17;22

2 levels 4

1;14;27

18;23

19

Table 2 Three-level converter output voltage vectors selection for vC1 < vC2

3 levels

2;15

20

7

sb\sa

2

1

0

1

2

2 1 0 1 2

25 24 19 20 21

25 26 23 22 21

12 26; 11 1; 14; 27 22; 15 16

7 11 10 15 3

7 8 9 4 3

9

5;10

α

3. Control strategy 3.1. Fractional-order controllers

24

8

6;11 13;26

25

7

12

Fig. 6. Output voltage vectors for the three-level converter.

The modelling applied to the inverter is similar to the one of the rectifier. Consequently, the inverter output voltage is given by:

2 3 2n4 us4 1 4 us5 5 ¼ 4 n4 9 us6 n4 2

n5 2n5 n5

3 n6 n6 5vdc 2n6

(24)

The capacitors have an electric current which results from a relation of the combination each phase of the voltage converter. For instance, capacitor C3 is only charged by an electric current if at least one of the phases of the rectifier is at level 3, that is nk ¼ 3. When the level selected in nk ¼ 0 no capacitor is charged. The current on each capacitor bank icj is associated with the variable dnk, which defines if the generator output current has a null or a nonnull contribution, according to the phase voltage level. The variable dnk is given by:



dnk ¼

0 1

j > nk j  nk

A control strategy based on fractional-order PIm controllers is proposed for the variable-speed operation of wind turbines with PMSG/full-power converter topology. Fractional-order controllers are based on fractional calculus theory, which is a generalization of ordinary differentiation and integration to arbitrary not necessary integer order [58]. Applications of fractional calculus theory in practical control field have increased significantly [59]. Fractionalorder calculus used in mathematical models can improve the design, properties and controlling abilities in dynamical systems [60]. The fractional-order derivative or integral can be denoted by a m general operator a Dt [61], given by:

j˛f1; .; p  1g

n˛f3; 2; 1; 0g;

m

a Dt

¼

8 > > > > > > > > <

m

d ; dt m 1;

3 X

dnk ik 

k¼1

6 X

dnk ik j˛f1; .; p  1g

<ðmÞ ¼ 0

> > Zt > > > > ðdsÞm ; > > :

(28)

<ðmÞ < 0

a

The mathematical definition of fractional derivatives and integrals has been the subject of several descriptions. The most

(25) 29

13

Therefore, the current on each capacitor bank [57] is given by:

icj ¼

<ðmÞ > 0

(26)

14

β

45

25;46

9;30

4 levels

61

57

41;62

k¼4

The voltage vdc is the sum of the capacitor voltages vC1,vC2,vC3 in the capacity banksC1,C2,C3 modelled by the state equation given by:

5;26;47

j˛f1; .; p  1g

(27)

Hence, the four-level converter is specified by (16) to (27).

11;32

16

12

37;58 1;22;43;64

6;27;48

2

1

0

1

2

25 24 19 20 21

25 13 18 17 21

12 13; 6 1; 14; 27 17; 2 16

7 6 5 2 3

7 8 9 4 3

4

33;54

20

49

α

50

18;39;60

19;40

3;24

8

53

34;55 2;23;44

2 1 0 1 2

17;38;59

7;28

Table 1 Three-level converter output voltage vectors selection for vC1 > vC2

sb\sa

2 levels

21;42;63

10;31

15 p1 X dvdc 1 ¼ icj C dt j¼1 j

3 levels

35;56

36

51

52

Fig. 7. Output voltage vectors for the four-level converter.

Please cite this article in press as: Seixas M, et al., Offshore wind turbine simulation: Multibody drive train. Back-to-back NPC (neutral point clamped) converters. Fractional-order control, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.03.025

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Table 3 Four-level converter output voltage vectors selection for nk ¼ 0.

Table 5 Four-level converter output voltage vectors selection for nk ¼ 2

sb\sa

3

2

1

0

1

2

3

sb\sa

3

2

1

0

1

2

3

3 2 1 0 1 2 3

4 8 12 16 15 14 13

4 8 7 11 10 14 13

4 3 2 6 5 9 13

20 19 2 1 5 25 29

52 35 34 17 37 41 61

52 51 50 33 53 57 61

52 51 50 33 53 57 61

3 2 1 0 1 2 3

4 8 12 16 15 14 13

4 8 7 11 10 14 13

4 3 44 27 47 9 13

36 19 44 43 42 25 45

36 35 39 59 42 41 45

52 51 34 33 37 57 61

52 51 50 49 53 57 61

frequently encountered definition is called RiemanneLiouville definition, having the fractional-order integral given by: m a Dt f ðtÞ

1 ¼ GðmÞ

Zt

ðt  sÞm1 f ðsÞds

(29)

a

2 t 3 Z 1 dn 4 f ðsÞ ds 5 a Dt f ðtÞ ¼ Gðn  mÞ dt n ðt  sÞmnþ1

(30)

a

where the gamma function is normally given by:

GðxÞh

ZN

yx1 ey dy

f ðtÞ ¼ Kp eðtÞ þ Ki D t eðtÞ m

(34)

A classical PI controller is modelled by taking m ¼ 1 in (34). The transfer function of the fractional-order PIm, using the Laplace transform on (34) is given by:

and the definition of fractional-order derivatives is given by: m

order controller has a dynamical behaviour described by a fractional differential integral equation with a derivative or an integral having at least a non-integer order. The differential equation of the fractional-order PIm is given by:

(31)

GðsÞ ¼ Kp þ Ki sm

(35)

The fractional-order PIm controller is more flexible than the classical PI controller, because it has one more adjustable parameter, which can reflect the intensity of integration. The proposed control strategy for the fractional-order controllers is specified by (28) to (35).

0

In this paper m, is assumed as a real number that satisfies the restriction 0 < m < 1. The limit a of the integration is taken as a null m m value and the following convention is used: 0 D t hDt . The other approach is GrünwaldeLetnikov definition of fractional-order integral given by:

3.2. Converters control

An important property revealed by the RiemanneLiouville and GrünwaldeLetnikov definitions is that while integer-order operators imply finite series, the fractional-order counterparts are defined by infinite series [60,61]. This means that integer operators are local operators in opposition with the fractional operators that have implicitly more memory of the past events. The design of a fractional-order controller has the advantage of entailing more criterion than the classical one, augmenting the freedom for imposing an enhanced behaviour [62]. A fractional-

Power converters are variable structure systems, because of the on/off switching of their IGBT’s. As mentioned previously, the controllers proposed for the converters are fractional-order PIm controllers. PWM (Pulse width modulation) by SVM (space vector modulation) associated with SM (sliding mode) is used for controlling the converters. Sliding mode control presents special interest for systems with variable structure, such as switching power converters, guaranteeing the choice of the most appropriate space vectors. The aim is to let the system slide along a predefined sliding surface by changing the system structure. The sliding mode control presents attractive features such as robustness to parametric uncertainties of the wind turbine and the generator as well as to electrical grid disturbances [63,64]. The power semiconductors present physical limitations that have to be considered during not only design phase, but also on simulation studies. Particularly, they cannot switch at infinite frequency. Hence, an error on the electric currents will exist between the reference value and the control value for a finite value of the switching frequency, for instances, 2 kHz, 5 kHz or even 10 kHz are in use. Let eab be the error on the stator electric currents in the abplane, based on the Concordia (ab) transformation, in order to guarantee that the system slides along the sliding surface S(eab,t)

Table 4 Four-level converter output voltage vectors selection for nk ¼ 1

Table 6 Four-level converter output voltage vectors selection for nk ¼ 3

ta

 m a Dt f ðtÞ ¼ lim h

m

h/0

h X Gðm þ rÞ f ðt  rhÞ r!GðmÞ r¼0

(32)

and with the definition of fractional-order derivatives given by: ta

m a Dt f ðtÞ ¼ lim h

m

h/0

h X

ð  1Þr

r¼0

Gðm þ 1Þ f ðt  rhÞ r!Gðm  r þ 1Þ

(33)

sb\sa

3

2

1

0

1

2

3

sb\sa

3

2

1

0

1

2

3

3 2 1 0 1 2 3

4 8 12 16 15 14 13

4 8 7 32 10 14 13

20 24 23 6 26 30 29

20 19 18 22 21 25 29

52 56 18 38 21 62 61

52 51 34 54 37 57 61

52 51 50 49 53 57 61

3 2 1 0 1 2 3

4 8 12 16 15 14 13

4 8 12 32 15 14 13

4 24 44 48 47 30 13

36 40 60 64 63 46 45

52 56 60 59 63 62 61

52 51 50 54 53 57 61

52 51 50 49 53 57 61

Please cite this article in press as: Seixas M, et al., Offshore wind turbine simulation: Multibody drive train. Back-to-back NPC (neutral point clamped) converters. Fractional-order control, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.03.025

M. Seixas et al. / Energy xxx (2014) 1e13

9

Fig. 8. WECS equipped with three-level converter and using PIm controllers.

has been proven the necessity to ensure that the state trajectory near the surfaces is in accordance with the stability conditions [63,54] given by:



S eab ; t

 dS eab ; t dt

<0

(36)

In practice the sliding surface is chosen in way of letting a small error ε > 0 for S(eab,t) to be allowed, due to power semiconductors switching only at finite frequency. Consequently, the switching in practice strategy is given by:

 ε < S eab ; t < þε

(37)

A practical implementation of the switching strategy considered in (37) for the simulation is accomplished by using hysteresis comparators. The outputs of the hysteresis comparators are the integer variables sab ¼ (sa,sb) [63]. For the three-level converter, the output voltage vectors in the ab plane are shown in Fig. 6. The integer voltage variables sa and sb obey the following set relation given by:

sa ; sb ˛f  2; 1; 0; 1; 2g

(38)

redundant ones corresponding to combinations of IGBT states, different from each other, which in turn correspond to a different voltage level selection. Thus, the control strategy ensure stability for the four-level converter and mitigates the capacitors unbalancing voltages by selecting the most appropriate redundant vector, taking into account both external and internal hexagons formed by the output voltage vectors based on four tables, which define the vector to be used at each instant. Table 3 summarizes the vector selection for nk ¼ 0. Table 4 summarizes the vector selection for nk ¼ 1. Table 5 summarizes the vector selection for nk ¼ 2. Table 6 summarizes the vector selection for nk¼3. Hence, the proposed control strategy for the power converters is specified by (36) to (39). The control strategy for the WECS for the configuration with a three-level converter using PIm controllers has a block diagram as shown in Fig. 8. The design of PIm controllers follows the tuning rules in Ref. [66]. Power electronic converters are modelled as a pure delay [67] and the left-over dynamics are modelled by a second order equivalent transfer function, following the identification of a step response. The difference between the voltage vdc and the reference voltage v*dc is processed by the PIm controller in order to determine the reference stator currents. The difference between the stator current and the

This control strategy selects a new vector only when vC1 s vC2. The appropriate vector selection in order to ensure stability for the three-level converter [63,54] is shown in Table 1, for vC1 > vC2, and in Table 2, for vC1 < vC2. For the four-level converter, the output voltage vectors [65] in the ab plane are shown in Fig. 7. The integer voltage variables sa and sb obey the following set relation given by:

sa ; sb ˛f  3; 2; 1; 0; 1; 2; 3g

(39)

The integer voltage variables sa and sb allow choosing the most appropriate vector. On the inner output voltage vectors there are

Table 7 WECS data. Turbine moment of inertia Turbine rotor diameter Hub height Tip speed Rotor speed Generator rated power Generator moment of inertia

2500  103 kgm2 49 m 45 m 17.64e81.04 m/s 6.9e31.6 rpm 900 kW 100  103 kgm2

Fig. 9. Wind speed profile.

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M. Seixas et al. / Energy xxx (2014) 1e13

Fig. 10. Mechanical torque, the hub torque and the electric torque. Fig. 12. Three-level and four-level converter reference voltage and voltage vdc.

reference stator current is the error eab to be subjected to the output voltage vectors selection, after being processed by the hysteresis comparator specified by (38)e(39). The sliding mode control is a lower level of control as is normally envisaged with a PI controller.

Standards such as IEEE-519 [68] impose limits for different order harmonics and the THD. The limit for the THD is 5% for this standard and is used in this paper as a guideline for assessment of the viability on the quality of energy injected into the electric grid.

4. Harmonic assessment The harmonic content of the current injected into the electrical is expressed in percentage of the fundamental component via the THD given by:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2ffi XH THD ð%Þ ¼ 100

H¼2

(40)

XF

where the harmonics are computed by the DFT (Discreet Fourier Transform) given by:

XðkÞ ¼

N 1 X

5. Simulation results Both mathematical models for the offshore WECS with the three-level or with the four-level power converter topologies were implemented in Matlab/Simulink. The offshore WECS simulated in this case study has a rated electric power of 900 kW and is assumed that m ¼ 0.5 following the range discussed in Ref. [66]. Table 7 summarizes the offshore WECS data. The wind speed variation model is given by:

" u ¼ 13:5 1 þ

ej2pkn=N xðnÞ

for k ¼ 0; .; N  1

(41)

n¼0

Fig. 11. Rotor speed at the turbine, rotor speed at the hub and rotor speed at the generator.

X

# An sinðun Þ

0  t  10;

(42)

n

The wind speed profile is shown in Fig. 9. The mechanical torque acting in the rotor of the wind turbine disturbed by the mechanical eigenswings, the hub torque disturbed

Fig. 13. Three-level converter current injected into the electric grid.

Please cite this article in press as: Seixas M, et al., Offshore wind turbine simulation: Multibody drive train. Back-to-back NPC (neutral point clamped) converters. Fractional-order control, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.03.025

M. Seixas et al. / Energy xxx (2014) 1e13

11

Fig. 14. Four-level converter current injected into the electric grid. Fig. 16. Four-level converter THD of the current injected into the electric grid.

by the moving surface oscillation and the electric torque of the generator are shown in Fig. 10. The mechanical torque, blue line, over the rotor of the wind turbine shows a significant disturbance behaviour, while the hub torque, green line, shows an attenuation of that disturbance and finally the electric torque, red line, of the generator almost shows a total discard of the disturbance. The rotor speed at the turbine, green line (in web version), the rotor speed at the hub, red line, and the rotor speed at the generator, blue line are shown in Fig. 11. The reference voltage, the voltage vdc and the capacity voltages for the three-level converter and the four-level converter are shown in Fig. 12. The fractional-order controller on the three-level converter allows providing the instantaneous current injected into the electric grid shown in Fig. 13. The fractional-order controller on the four-level converter allows providing the instantaneous current injected into the electric grid shown in Fig. 14. The instantaneous current injected into the electric grid is almost a three phase sinusoidal current. But some disturbance

occurs as revealed by the THD. The THD of the current injected into the electric grid for the WECS with the three-level converter is shown in Fig. 15. The THD of the current injected into the electric grid for the offshore WECS with the four-level converter is shown in Fig. 16. The proposed five-mass drive train model incorporating the resistant stiffness torque, structure and tower of the WECS due to the moving surface oscillation gives an average value for the THD of the 0.78% with the three-level converter and of the 0.52% with the four-level converter. Accordingly, for the three-level and for the four-level converter topology using the drive train described by a five-mass model and considering the resistant stiffness torque of the offshore WECS in the deep water, the THD is lower than the 5% limit imposed by IEEE519 standard [68]. This THD is the result in part due to the moving surface oscillation influence on the offshore WECS energy conversion and the simulations are in favour of a convenient attenuation according to the imposed standard. Although IEEE-519 standard might not be necessarily applicable in such situation is used as a guideline for comparison purposes [34] in the case study. 6. Conclusions This paper assesses both the evaluation of offshore variablespeed wind turbines with PMSG with a back-to-back NPC three-level and with four-level converters topologies in what regards the THD on the current injected into the electric grid. The evaluation is made using a five-mass model with the consideration of stiffness torque, structure and tower in order to capture the behaviour of the moving surface oscillation. The simulation study revealed a good performance of the proposed offshore WECS with the three-level or with the four-level converter. Although, there are effects on the current output of the converters, those effects revealed by the THD of the current injected into the electric grid show a THD lower than 5% limit imposed by IEEE-519 standard. Acknowledgement

Fig. 15. Three-level converter THD of the current injected into the electric grid.

This work was partially supported by Fundação para a Ciência e a Tecnologia, through IDMEC under LAETA, Instituto Superior Técnico, Universidade de Lisboa, Portugal.

Please cite this article in press as: Seixas M, et al., Offshore wind turbine simulation: Multibody drive train. Back-to-back NPC (neutral point clamped) converters. Fractional-order control, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.03.025

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M. Seixas et al. / Energy xxx (2014) 1e13

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