A methodology for the transient behavior of horizontal axis hydrokinetic turbines

Energy Conversion and Management 87 (2014) 1261–1268

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A methodology for the transient behavior of horizontal axis hydrokinetic turbines André Luiz Amarante Mesquita a, Alexandre Luiz Amarante Mesquita a,⇑, Felipe Coutinho Palheta a, Jerson Rogério Pinheiro Vaz a, Marcus Vinicius Girão de Morais b, Carmo Gonçalves c a b c

Federal University of Pará, Faculty of Mechanical Engineering, Av. Augusto Correa, s/n, Belém, PA 66075-900, Brazil University of Brasília, Department of Mechanical Engineering, Av. L3 Norte, Asa Norte, Brasília, DF Cep. 70.910-900, Brazil Centrais Elétricas do Norte do Brasil S.A., ELETRONORTE, Brasília, DF, Brazil

a r t i c l e

i n f o

Article history: Available online 30 June 2014 Keywords: Hydrokinetic turbine Blade Element Method Dynamic modeling

a b s t r a c t In recent years, increasing attention is being given to the study of hydrokinetic turbines for power generation due to the use of clean energy by using renewable sources. This paper aims to present a general methodology for the efficient design of horizontal axis hydrokinetic turbines with variable rotation. The approach uses the Blade Element Method (BEM) for determining the power coefficient of the turbine. The modeling of the hydrokinetic rotor is coupled with the model of the drive line of the system, including the multiplier and the electric generator. Therefore, the modeling of the whole system comprises the hydrodynamic information of the rotor and the characteristics of the inertia of whole system, frictional losses and electromagnetic torque of the generator. The results of numerical simulation are obtained for the rotational speed of the rotor as well as the results of the torque, mechanical and electrical power. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Hydrokinetic turbines are used in free water flow environment, like as marine, tidal, and river currents, without the need to use dams for power generation. Bahaj [1] presents a current status of wave and marine current energy conversion technologies, providing a large review of the literature, including the analysis of the interactions and impacts on the marine environment and economic assessment. Güney and Kaygusuz [2] provide a review on the kinds of hydrokinetic turbines employed for river and tidal current, analyzing also the generator, control technologies and environmental impacts. Khan et al. [3] present a detailed assessment of various turbine systems (horizontal and vertical axis), along with their classification and qualitative comparison. Lago et al. [4] focuses on innovative concepts and trends in hydrokinetic system development, looking for the future scenario of this technology. Ramos and Iglesias [5] develop a procedure to compare the performance of different tidal stream turbines at a given site, using an index which combines the flow and water depth information and present a case study for an estuary. These turbines are receiving very attention due to the use of clean energy by using renewable sources and also ⇑ Corresponding author. E-mail address: [email protected] (A.L. Amarante Mesquita). http://dx.doi.org/10.1016/j.enconman.2014.06.018 0196-8904/Ó 2014 Elsevier Ltd. All rights reserved.

because this kind of power generation represent a good alternative for both isolated locations and modern grid-connected turbine farm applications. Kusakana and Vermaak [6] analysis a case of power supply project for a rural and isolated load and compare the hydrokinetic system from other power supply options such as standalone PV, wind, diesel generator and grid extension line. In order to improve the accuracy in estimating the energy cost of a tidal current turbine farm, Li et al. [7] propose a model which uses a scenario-based cost-effectiveness analysis to identify the minimum energy cost, including hydrodynamic effects and O&M cost for the different scenarios. Therefore, it is necessary to develop methodologies capable of implementing efficient hydrokinetic systems, considering the influence of all components of the machine. The procedures for obtaining a mathematical model of a hydrokinetic turbine are similar to a wind turbine. In this work, the Blade Element Method (BEM) is used for determining the power coefficient of the turbine. The BEM model is well recognized method to design and analysis of wind turbine [8] and is also widely used for predicting the performance of hydrokinetic turbines. For example, Goundar and Ahmed [9] employed the BEM theory to develop a hydrokinetic turbine with rotor diameter of 10 m designed for the rated tidal current velocity of 2 m/s and maximum theoretical power of 150 kW. This work present a methodology to analysis the dynamic behavior of a hydrokinetic turbine. The modeling of the hydrokinetic rotor is coupled with the model of the drive line of

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Nomenclature Arabic symbols a, a0 axial and tangential induction factors Cn; Ct coefficients of the normal and tangential forces CP power coefficient dM the bearing pitch diameter (m) ft radial load factor F Prandtl tip-loss factor F n and F t normal and tangential forces (N) Fr radial load G geometry factor base on the internal dimensions of the bearing J total total inertia of the system ðkg m2 Þ Jf moment of inertia from the added mass of the fluid around the turbine blades ðkg m2 Þ J blade moment of inertia of the blade ðkg m2 Þ J hub moment of inertia of the hub ðkg m2 Þ JL moment of inertia from generator ðkg m2 Þ J GT moment of inertia from gearbox ðkg m2 Þ Jp moment of inertia of the planet gear ðkg m2 Þ Jc moment of inertia of the carrier ðkg m2 Þ Js moment of inertia of the sun gear ðkg m2 Þ L and D lift and drag forces (N) mp planet gear mass (kg) n rotational speed (rpm) nblade number of blades Np number of the planet gears PT output power of the hydrokinetic turbine (W)

the system, including the planetary multiplier gearbox and the electric generator. The mathematical model of the turbine includes both steady state and the transient operation, where the minimal rotational speed of the rotor for electrical generation and the over speed of the rotor (loss of load) are evaluated. The model assumes that the shafts are infinitely rigid since the vibration modes of the system are located in a frequency range much higher than the operational frequency range. The mathematical model also takes into account the inertia of the fluid and the gearbox inertia, the frictional losses of the bearings, and the electromagnetic torque. The results of numerical simulation are obtained for the rotational speed of the rotor and the electrical generator as a function of time, as well as the results of the torque, mechanical and electrical power.

speed ratio torque of the turbine rotor (N m) torque of the generator (N m) equivalent load torque (N m) total frictional torque (N m) windage frictional torque loss (N m) frictional torque into a loading dependent component (N m) frictional torque influenced by the viscous property of lubricant (N m) velocity of the center of mass from planet gears (m/s) undisturbed freestream (m/s) relative velocity on the elementary blade section (m/s) local-speed ratio (LSR) tip-speed ratio (TSR)

T0

vp V W x X

Greek symbols a angle of attack b twist angle g gearbox efficiency m0 lubricant kinematic viscosity ðm2 =sÞ xc angular speed of the carrier (rad/s) xL angular speed of the generator (rad/s) xM angular speed of the rotor (rad/s) xp angular speed of the planet gears (rad/s)

analysis. T LM is the equivalent load torque of the generator. Its expression is given by:

T LM ¼

1

g

rT L

ð2Þ

where T L is the torque of the generator, g is the transmission efficiency and r is the speed ratio r ¼ xL =xM . The total inertia of the system (Jtotal ) is the sum of following moments of inertia: the inertia of the turbine (JT ); the inertia of the added mass of the fluid around the turbine blades (J f ), the inertia of the multiplier planetary with respect to the input shaft of the multiplier (J GT ), and the inertia of the generator (JL ), i.e:

J total ¼ J T þ J f þ J GT þ J L

ð3Þ

In next subsections the other terms in Eqs. (1) and (3) are discussed in details.

2. Dynamical model A hydrokinetic turbine system consists of a turbine rotor with mass moment of inertia J T connected to a generator (load) with mass moment of inertia J L , through a multiplication system with speed ratio r and efficiency g, as shown in Fig. 1. The assumption is made that the shafts and the gears are infinitely rigid. Such consideration is valid since the vibration modes of the system are assumed to be in frequency range higher than the operational frequency range. The dynamic equation governing power transmission system shown in Fig. 1 is given by:

T M ¼ T T  T D  T LM ¼ J total

r TT TL T LM TD Tv T1

dxM dt

2.1. Rotor torque The hydrokinetic rotor torque is obtained from the hydrodynamic forces acting on the rotor blades. In this work such

ð1Þ

The driving torque ðT M Þ is equal to the torque of the turbine (T T ) minus the dissipative torque (T D ) (friction torques of bearings), and minus the equivalent load torque T LM . xM is the angular speed of the rotor. The torque of the turbine is the result of hydrodynamic

Fig. 1. Illustration of the complete system of a hydrokinetic turbine.

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interaction is determined by the classical BEM method, which is the model most frequently used by scientific communities for design and analysis of hydrokinetics and wind rotors [10]. This method is essentially an integral method, with semi-empirical information from hydrodynamic forces in hydrofoil sections issued from two dimensional hydrofoil flow model or experimental data, as given by Sheldahl and Klimas [11] in a wind tunnel test series for four symmetrical airfoil sections, and by Abbott and Doenhoff [12], where is presented an important data bank on the aerodynamics of airfoil at subcritical speeds, very useful for the turbomachinery design. Hydrofoils can be designed by studying the airfoil characteristics as described by Goundar et al. [13]. The forces on a blade element can be obtained though the velocity diagram shown in Fig. 2, where a and a0 are the axial and tangential induction factors, respectively, V is the undisturbed freestream velocity, W is the relative velocity, a is the angle of attack, b is the twist angle, / is the angle of flow, rM is the radial position, L and D are the lift and drag forces, respectively, F n and F t are the normal and tangential forces, respectively. The torque of the rotor can be expressed as a function of the induction factors a and a0 , which are given by [14].

a rC n ¼ 1  a 4F sin2 /

ð4Þ

a0 rC t ¼ 1 þ a0 4F sin / cos /

ð5Þ

where r is the solidity of the turbine, F is the Pradntl tip-loss factor [10]. C n and C t are the coefficients of normal and tangential forces. In this case, the power coefficient, C P , for the hydrokinetic rotor is [15]:

CP ¼

8 X

2

Z

X

a0 Fð1  aFÞx3 dx

ð6Þ

0

where X ¼ xM R=V is the tip-speed ratio (TSR) and x ¼ xM rM =V is the local-speed ratio (LSR). Finally, the rotor torque T T can be expressed as [16]:

TT ¼

PT

xM

2

¼

3

1 qpR V CP 2 xM

ð7Þ

where P T is the output power of the hydrokinetic turbine, q is the water density, R is the radius of the turbine.

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2.2. Generator torque Hong et al. [16] show that the electricity generation systems are attracting great attention, since that can be operated with constant speed or variable speed operations using power electronic converters. Among them, the variable-speed generation system is more attractive than the fixed-speed system because of the improvement in energy production. Variable-speed power generation enables operation of the turbine at its maximum power coefficient over a wide range of speeds. Hasanien [17] presents a control system for permanent magnet synchronous motor (PMSM), which is used for torque ripple minimization of this type of motors. The dynamic response of the PMSM with the controller is studied during the starting process under the full load torque and under load disturbance. Therefore, the permanent magnet synchronous generator is a good option for the high performance energy generation in variable-speed operation. In this case, the torque of a permanent magnet synchronous generator can be given through following algebraic equation:

TL ¼

3 pWisq 2

ð8Þ

where p is the pole pair number, W is the magnet flux and isq is the electric current one of the synchronous phases. However, it is emphasized here, that the one of the objectives is only to considering the effect of the electromagnetic torque on the dynamic behavior of the system. Detailing on the control system, connection to the electrical grid, and the operation with constant speed or variable speed using power electronic converters are detailed in Ref. [18]. Thus, in order to reduce the complexity of the electromagnetic torque equation is assumed that the relation between synchronous generator torque and its angular speed is given by an approximate linear equation [19]. In this case, a first order linear function that describe the electromagnetic torque as a function of the generator angular speed is used, which is given by:

T L ¼ K e xL þ K e0

ð9Þ

where K e and K e0 are obtained by a linear fit with experimental data. This methodology is reasonable, since that the generator electromagnetic torque is always available from the manufacturer. 2.3. Friction torques According to the loads acting on the bearings and the bearing types used throughout the system, is possible to estimate the overall friction torque in the system. The bearings are located on outlet of rotor turbine, at planetary gearbox, and at generator. In this work, the transmission losses are taken into account in the efficiency of the gearbox. There are two types of bearings in the turbine system: tapered roller bearings and rolling bearings. The formulas for estimating the friction torque used in this study follow the methods of Witte [20] for tapered roller bearings and Palmgren [21] for rolling bearings. Witte [20] empirically studied the running friction torque, which resulted in Eq. (10):

 1 1 Fr 3 T D;tapered ¼ 3:35  1011 Gðnm0 Þ2 ft K

Fig. 2. Velocity diagram for a rotor blade section.

ð10Þ

where G is a geometry factor based on the internal dimensions of the bearing, n is the rotational speed, m0 is the lubricant kinematic viscosity, ft is a radial load factor and F r is the radial load. Palmgren [21] separated the friction torque into a loading dependent component (T 1 ) and a load independent component (T 0 ) which is influenced by the viscous property of lubricant type,

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the amount of the lubricant employed and bearing speed. Thus, the total friction torque (in N m) is given by Eq. (11):

J blade ¼

N N X X mi r 2i þ J root ¼ mi r 2i þ mroot r2root i¼1

T D;rolling ¼ T 0 þ T 1

ð11Þ

where

T 1 ¼ 103  f1  F b  dM 10

T 0 ¼ 10

 f0 ðnm0 Þ

2 3

ð12Þ ð13Þ

where f1 is a factor depending on the bearing design and relative bearing load, F b depends on the magnitude and directions of the applied load, dM is the bearing pitch diameter and f0 is a factor depending on the type of bearing and the method of lubrication. More detail about those formulas can be found in Harris and Kotzalas [22]. According to Carbo and Malanoski [23] there is another friction torque that must be considered is the windage frictional torque loss that occurs due to viscous fluid drag on immersed impellers. Windage friction generates damping, unlike coulomb friction, since the frictional force generally increases with speed. This damping is usually negligible for impellers surrounded by air such as compressor and turbine rotors. However, pump impellers immersed in a highly viscous liquid can generate damping torques that are not insignificant. Ker Wilson [24] provides the following equation for the torque due to friction acting on all surfaces of a hollow disk:

h

Tv ¼

plxM R30 ðR0 þ 2LÞ  R4i

i

2 mhub r 2hub 3

ð17Þ

In the mass of the cube should be deducted the masses corresponding to the holes that exist in the hub. In the expression of the total inertia of the system should also be taken into account the added mass of the fluid around the blades. The model described by Maniaci and Li [25] the added mass for a blade was assumed to be equal to the mass of a cylinder (with length L equal to the length of the blade R) whose diameter is equal to the chord length C. Thus, the expression of added mass is given by:

1 2 pC qL 4

ma ¼ pR2cil qL ¼

where T v is the frictional torque (in.lbf), l is the fluid dynamic 2 viscosity ðlbf  s=in ), xM is the disk angular velocity (rad/s), R0 is the disk outside radius (in), Ri is the disk inside radius (in), h is the radial clearance or axial clearance (assumed equal) (in) and L is the disk length (in). 2.4. Inertia calculations 2.4.1. Turbine rotor and added mass The moment of inertia of the turbine rotor is given by:

ð15Þ

where nblade is the number of blades, Jblade is the moment of inertia of a blade and Jhub is the moment of inertia of the hub. The moment of inertia of a blade is calculated as follows: the blade is divided into several small volumes along their profile, and in each volume is determined the center of mass of the volume, the mass and the distance between the center of mass to the center of rotation of the blade, as shown in Fig. 3a. It is seen that in Fig. 3a there is also the root of the blade, which can be approximate as a thin cylindrical shell. Therefore, the equation that provides the moment of inertia of the blade is given by:

ð18Þ

where Rcil is the radius of cylinder. In order to calculate the inertia of the fluid, it is proposed here in inserting the added mass calculated in Eq. (18) into Eq. (16) as follows:

J blade;fluid ¼

N  X

mi þ

ma  2 r þ J root N i

mi þ

ma  2 r þ mroot r 2root N i

i¼1

ð14Þ

h

J T ¼ nblade J blade þ J hub

The moment of inertia of the hub has a geometry shown in Fig. 3, which can be approximated by a hollow hemisphere. Thus, the expression of its moment of inertia is given by:

J hub ¼

3 dM

ð16Þ

i¼1

¼

N  X i¼1

ð19Þ

Therefore, the moment of inertia of the turbine rotor and the added mass is given by:

J T þ J f ¼ nblade J blade;fluid þ J hub

ð20Þ

2.4.2. Gear train The gear train multiplier used in the hydrokinetic turbine consists of a planetary gearbox with two stages, as illustrated in Fig. 4. The inertia of this planetary multiplier (J GT ) with respect to the input shaft is given by [26]:

" J GT ¼ J c1 þ Np1 J p1 þ J c2 þ J s2



2



2

xc2 xc1

xs2 xc1



xp1 xc1

#

2 þ "

þ Np2 J p2

2 mp1 d1



xp2 xc1

2

þ J s1



þ mp2

xs1 xc1



2

v p2

2 #

xc1 ð21Þ

In Eq. (21), xs ; xc ; xp and Js ; Jc ; Jp are the rotational speeds and moments of inertia of the sun gear, carrier and planet gear,

Fig. 3. (a) Blade of the hydrokinetic turbine. (b) Hub of the hydrokinetic turbine.

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A.L. Amarante Mesquita et al. / Energy Conversion and Management 87 (2014) 1261–1268 Table 1 Parameter values of the rotor. R¼5m JT ¼ 89; 310 kg m2 r ¼ 34 V 0 ¼ 2:5 m/s

Fig. 4. Illustration of the planetary gearbox used as a multiplier in the turbine system.

respectively, and NP is the number of planet gears. The subscripts 1 and 2 refer to stages of the multiplier. The mass and velocity of the center of mass of each planet gear are given by mp and v p , respectively. Note that xc1 ¼ xM . 2.4.3. Generator inertia The equivalent inertia of the generator (J LM ) is given by:

J LM ¼

1

g

r2 JL

ð22Þ

where JL is the inertia of the generator, g is the transmission efficiency and r is the speed ratio r ¼ xL =xM . For the inertia of the generator rotor it was considered a cylinder of mass mgen , radius Rgen and length Lgen , i.e:

JL ¼

1 1 mgen R2gen ¼ qgen pR4gen Lgen 2 2

r hub ¼ 0:75 (m) Jf ¼ 282; 590 kg m2

q ¼ 997 kg=m3 n0 ¼ 14:66 rev/min

is obtained from classical BEM method [15]. It is important to note that the BEM method takes into account the rotor hydrodynamic shape in a more detailed manner, different of the functional form as described in several references available in the literature [29]. In this work, at each time step (see Eq. (1)) the hydrodymamic torque (T T ) is calculated. The BEM model considers the turbine rotation as a constant for each time step, allowing calculation of the power coefficient, where R is the radius of the rotor, V 0 is the velocity of the river, q is the water density and n0 is the rotation of the rotor. Tables 2 and 3 refer to gearbox multiplier inertias e speed of the rotational elements. It is considered an gearbox with 98% efficiency. The data in Tables 2 and 3 were obtained from a manufacturer of commercial multipliers. For the generator, it is considered a commercial permanent synchronous generator with 93.7% efficiency. The electromagnetic torque is shown in Fig. 6. A linear fit shows that K e ¼ 182:7 N ms and K e0 ¼ 12:83 N m. Thus, the electromagnetic torque is given by:

T L ¼ 182:7xL  12:83 ðN mÞ

ð24Þ

The friction torque calculation was performed considering all axial and radial reactions on the bearings and the size and type of the bearings. Therefore, using Eqs. (10), (11) and (14) the total expression for bearing friction torque as function of rotational speed of the turbine is given by: 1

ð23Þ

T D ¼ 12:5712 þ 132:106xM þ 0:0000525x2M 2

þ 3:55534x3M ðN mÞ

ð25Þ

3. Results and discussions 3.1. Rotor, gearbox and generator data

Table 2 Angular speed of the gear train as a function of angular speed of the rotor. Equation

Fig. 5 shows the complete hydrokinetic system used in the simulation, which has a 3-bladed rotor with 10 m of diameter and 1.5 m hub diameter (other rotor data are shown in Table 1). The blade was designed using the classical Glauert’s optimization [27], and NACA 653 – 618 hydrofoil. With respect to the powertrain, it is considered 1:34 gearbox transmission and a permanent magnet synchronous generator under rated conditions: 500 kW (output power), 500 rpm (rotation), 95.6 kN m (shaft torque) and 93.7% of efficiency. The power coefficient of the hydrokinetic rotor

Variables of the first stage

xp1 xs1 Variables of the second stage xc2 ¼ xs1

xp2

v p2 xs2

2:68xc1 4:9411xc1 4:9411xc1 11:7647xc1 xc2 d2 34xc1

Table 3 Parameter values of the gear train. Values Variables of the first stage Jc1 (kg m2 ) N P1 Jp1 (kg m2 ) mp1 (kg) d1 (mm) Variables of the second stage N P2 Jp2 (kg m2 ) mp2 (kg) d2 (mm) Js2 a (kg m2 ) Fig. 5. Illustrations of the gearbox and generator of the hydrokinetic turbine system.

146.4086 4 5.7649 20.6198 338 3 3.2232 11.0754 294 (0.1765 + 0.1121)

a In the variable Js2 are included the mass moment of inertia of the sun gear in the second stage and the output shaft of the multiplier.

A.L. Amarante Mesquita et al. / Energy Conversion and Management 87 (2014) 1261–1268

Electromagnetic torque, TL − (kNm)

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determination of the water velocity required for starting the turbine is not easy. However, it is possible to determine the speed needed to start generating electricity. In this case, using Eq. (24), the generation occurs, when T L > 0, this implies that the starting electrical energy generation occurs for value higher than 60 K e0 =ð2prK e Þ ¼ 0:02 rpm. For river velocity of 0.4 m/s the turbine has zero electric generation in the range of 0–0.02 (see Fig. 7a). At low speeds the rotor acceleration is very small, as shown in Fig. 7b. In this condition the system stabilizes after 60 s at a rotation of 0.05 rpm. Another feature is that at low speeds the start of electrical generation is retarded, as seen in Fig. 7b. The water speed of electricity generation is 0.2 m/s, with starting over 20 s.

10

8

6

4

2

Experimental data Linear fit 0 0

10

20

30

40

50

60

Angular speed of the generator, ω − (rad/s) L

3.3. Transient and steady-state

Fig. 6. Experimental and linear fit for the electromagnetic torque.

3.2. Start up Wood [29] developed a detailed study on the starting behavior of small wind turbines, which the initial ’’idling period’’ is characterized by small acceleration, where it is usually much longer than the subsequent period of rapid acceleration to reach operational rotor speed. In this case, the idling reduces the power generation potential of any turbine. Eq. (1) shows that the turbine acceleration occurs when the rotor torque (T T ) is greater than the resistive torque. In the present work, the resistive torque is given by the sum of the dissipative torque (T D ) with equivalent torque load (T LM ). The

Fig. 8 shows the transient behavior of the turbine from the solution of the dynamic equation of the power train, Eq. (1). The results shown were obtained for a constant water speed of 2.5 m/s. The electric output power in this case stabilizes at 496.83 kW and rotation of 14.65 rpm. In both results the effect of the added mass can be noted. The inertia of the system is increased, and the machine takes longer to reach the steady-state. More details about the influence of the added mass can be found in Maniaci and Li [25]. 3.4. Over speed An important effect, and should be considered in the hydrokinetic turbines design corresponds to the behavior of the system 0.1

4

Electric power Mechanical power

3.5

0.08

Rotation − (rpm)

Power − (W)

3 2.5 2 1.5 1

0.06 0.04 0.02

0.5 0

V = 0.2 m/s V = 0.4 m/s V = 0.6 m/s

Starting generating electricity

0

0.01

0.02

0.03

0.04

0.05

0 0

0.06

20

40

60

Rotation − (rpm)

Time − (s)

(a)

(b)

80

100

Fig. 7. (a) Power as a function of rotor rotation. (b) Shaft rotation as a function of time.

500

16

450

Electric power − (kW)

Rotation − (rpm)

14 12 10 8 6 With added mass Without Added mass

4 20

40

60

80

100

400 350 300 250 200 150 100 50 20

With added mass Without Added mass 40

60

Time − (s)

Time − (s)

(a)

(b)

Fig. 8. (a) Rotational speed as a function of time. (b) Electric power as a function of time.

80

100

A.L. Amarante Mesquita et al. / Energy Conversion and Management 87 (2014) 1261–1268

60

Rotation − (rpm)

50 40

Stability at around 50 rpm

30 20

Point where the electrical charge is removed

10 0

0

200

400

600

800

1267

The dynamic model presented is very useful for control analysis strategies in all operating range of the wind turbine, where electricity is generated. Finally, it was verified that the influence of added mass on the dynamics of the system is not significant. The proposed methodology presents still some limitations which need to be investigated through comparisons with experimental data for a detailed analysis of the proposed model behavior. It is noteworthy that experimental data are scarce in the literature for hydrokinetic turbines. A detailed experimental analysis of the start-up stage is necessary, in order to evaluate the behavior of the turbine at low flow velocity. Despite of theses aspects, the proposed model is a good tool for the dynamic analysis of hydrokinetic turbines and can be employed for design purposes.

Time − (s) Fig. 9. Over speed of the hydrokinetic turbine.

Acknowledgments The authors would like to thank CNPq, INCT – EREEA, and ELETRONORTE for financial support.

when there is no electric charge. The rotational speed of the turbine increases abruptly due to the lack of electrical resistive torque. Thus, the elements that constitute the machine must be designed considering the effect of rapid acceleration. In Fig. 9 it is shown that when the electrical charge is removed, the rotational speed is increased from 14.65 rpm to almost 50 rpm in 150 s or so, therefore a safety system must be properly designed.

4. Conclusions The proposed methodology represents an alternative approach for the efficient design of the horizontal-axis hydrokinetic turbines, considering the inertial effects and energy loss in the complete power system. This method solves the dynamic equation of the power train, using a scheme with lower computational cost and advantages in the implementation of the design procedure. An important aspect of this methodology is the use of the classical BEM method, which is applied in each time step in order to calculate the hydrodynamic torque of the turbine. This processing is not current in the literature, being an important contribution of this work. Since in general, functional mathematical formulas are used for calculating the power coefficient, as given by Bao and Ye [19], which use an approximated expression regarded as the polynomial of the tip-speed ratio and pitch angle. In the same manner, Slootweg [28] uses a numerical approximation developed to calculate the power coefficient for given values of tip-speed ratio and pitch angle. For start-up analysis, it is necessary to consider the following facts: (1) if the hydrodynamic torque (T T ) is higher than the combined resistive torque of the power train and electrical generator (T R ¼ T D þ T LM ) the system accelerates (T T > T R ); (2) if T T ¼ T R the system presents a stationary behavior; (3) if T T < T R the system decelerates. In all the cases the flow velocity is an input data in the proposed methodology, in which the hydrodynamic influence needs to be taken into account at any operating condition of the turbine. This aspect shows that the determination of the flow velocity for the rotor start up is strongly dependent of the resistive torque modeling from both generator and power train. In practice the generator torque characteristics is available from the manufacturer, and therefore, the efforts must be addressed to the friction torque model at low rotational speed. In the case of the over speed determination, very important for structural resistance of the blades at extreme load condition, the available friction torque models for higher rotational speed are more precise. Furthermore, the coupling with the hydrodynamic torque calculated from the BEM model is more suitable for the dynamic analysis, and for the quasi-static procedure, the model gives good results as well.

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