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Simulated generator for wave energy extraction in deep water
Ocean Engineering 32 (2005) 1664–1678 www.elsevier.com/locate/oceaneng
Simulated generator for wave energy extraction in deep water ˚ gren, M. Leijon I.A. Ivanova*, H. Bernhoff, O. A The A˚ngstro¨m Laboratory, Division of Electricity and Lightning Research, Department of Engineering Science, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden Received 4 August 2004; accepted 1 February 2005 Available online 28 April 2005
Abstract A directly driven permanent magnet linear generator for ocean wave energy conversion has been simulated for a specific site with large waves. Wave characteristics influence the generator power output, the electromagnetic processes, geometrical sizes, and efficiency. The buoy and the generator dimensions are important parameters to get high efficiency of the converter. Electromagnetic and mechanical simulations of the generator with realistic wave conditions are presented. q 2005 Elsevier Ltd. All rights reserved. Keywords: Renewable energy; Wave power; Linear generator; Energy conversion
1. Introduction Wave energy conversion possibility is not a newborn idea. Investigations on the subject have been started many years ago. Ocean waves have high energy density compared to other new types of renewable sources, see Wave Net (2000–2003). Waves at open ocean sites are situated far away from coastline, have high amplitude, and large wavelength. There is no friction of the waves with sea bottom and a lot of energy is available. Concern is to extract this energy efficiently and convert it to the consumer grid with minimal losses. During the last years, a large number of technical principles of wave energy conversion have been developed, presented in publications, constructed and tested. Several principles rely on the use of conventional rotation generators, see Washio et al. (2000). * Corresponding author. Fax: C46 18 471 5810. E-mail addresses: [email protected] (I.A. Ivanova), [email protected] (I.A. Ivanova).
0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.02.006
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Linear generators have previously been used for wave application; see Mueller et al. (2000) and Mueller and Baker (2002). Waves at different places have certain character and energy density. Wave energy converters (WEC) should be designed for the particular specific location, taking into account the available wave energy resources there. Challenges in WEC and random wave conditions that lead to difficulties to achieve survivability of the system, unstable voltage level and variable power output are irregular. WEC’s should be provided with a good control system. Start up and shut down procedures, abnormal modes of operation needs to be controlled in any electrical devices. At normal operational modes it is necessary to control fluctuations in power output. In the present work, octagonal linear permanent magnet generator with axial flux has been simulated for selected place near the coast of Ireland, with large waves. A scheme of the generator is drawn in Fig. 1a. Principle of operation of the converter has been considered by Ivanova et al. (2003). A buoy on the sea surface moves vertically driven by the waves. The buoy is connected to a linear electric generator by a rope, which is connected to the permanent magnet (PM) generator piston (rotor). The piston moves inside the stator. The stator is made of electric steel and has slots, where the winding is located. The buoy provides piston movement relative the stator that induces electromotive force (EMF) in the winding. A large number of magnet poles placed on the piston increases the rate of magnetic flux change in the generator and as a consequence, increase the EMF. The buoy’s dimensions, mass and shape are important for effective wave energy extraction. The buoy should be large enough to absorb the maximum energy, but on the other hand, a substantially large buoy could destroy the generator in stormy weather. Spring serves to retract the generator piston on the ocean floor and increase the speed of the piston downward motion (Leijon et al., accepted for publication). In case of reasonable spring, energy extracted from the waves when the piston moves up is approximately equal to energy extracted when piston moves down. In the absence of the spring, the piston downward motion would be slow since only the gravity acts on the piston in such a case (Eriksson et al., 2004). The end stops serve to protect the generator at high waves and prevents the piston to go out of the stator. In the present work, the main emphasis is put on the generator design to be able to extract energy efficiently. The paper is organized as follows. Section 1 includes introduction of the subject and a brief description of the converter. Some generator theory is presented in Section 2. The simulation method is described in Section 3. Results are presented in Section 4. Sections 5 and 6 complete the paper. 2. Theory 2.1. Wave climate and motion equation Most of the energy stored in ocean waves is associated with wave periods within the interval from 5 to 15 s. In deep water, that corresponds to wavelengths from 40 to 350 m (Falnes, 2002).
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Fig. 1. (a) Scheme of WEC with linear generator, (b) and above view.
In these studies, harmonic sinusoidal wave propagation is considered with 5 m wave height and 11 s wave period. According to the European Wave Energy Atlas, place Shannon at Atlantic Ocean near south–west coast of Republic Ireland has been chosen due to the good perspective to extract energy there. The occurrence of waves with certain
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Table 1 Occurrence of significant wave height H and wave period T per year at Shannon H (m)
T (s) 0
2
3
4
5
6
7
8
9
10
11
12
13
14
15
18
2
3
4
5
6
7
8
9
10
11
12
13
14
15
18
20
0.0–0.5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.5–1.0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
1
4
6
1
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
11 6 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
32 40 23 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
55 51 38 30 14 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0
26 44 36 32 27 18 8 3 1 0 0 0 0 0 0 0 0 0 0 0
3 25 30 36 22 19 17 12 8 4 1 0 0 0 0 0 0 0 0 0
0 5 19 20 19 16 15 15 11 9 6 3 1 0 0 0 0 0 0 0
0 0 3 7 11 13 12 10 8 6 6 4 4 1 1 0 0 0 0 0
0 0 0 1 2 7 7 7 6 5 5 4 3 2 2 2 1 1 0 0
0 0 0 0 0 1 2 3 2 2 3 3 2 1 1 1 1 1 1 0
0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 0 1 1 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1.0–1.5 1.5–2.0 2.0–2.5 2.5–3.0 3.0–3.5 3.5–4.0 4.0–4.5 4.5–5.0 5.0–5.5 5.5–6.0 6.0–6.5 6.5–7.0 7.0–7.5 7.5–8.0 8.0–8.5 8.5–9.0 9.0–9.5 9.5–10 10–11 11–12
parameters and energy flux J transported by waves at Shannon are presented in Tables 1 and 2. The annual mean wave energy available is 68 kW/m. This site has been chosen due to the large energy potential. The wavelength at deep water can be found from the following formula (Falnes, 2002) lZ
g 2 T 2p w
(1)
where Tw is wave period. For deep water, the following condition is valid: DO Lwave =2
(2)
The wavelength at the Shannon is equal to LwaveZ188 m. The longer the wave, the more energy is transported. The simulated waves in the present studies are sinusoidal. The energy flux transported by purely progressive sinusoidal wave is given by Falnes (2002) Jsin Z kTw Hw2
(3)
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Table 2 Energy table for significant wave height H and period T per year at Shannon (energy unit is [kW h/m year]) H (m)
4
5
6
7
8
9
10
11
12
13
14
15
18
5
6
7
8
9
10
11
12
13
14
15
18
20
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 13.5 414 443 122 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 64 1433 3488 3315 1292 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 111 2823 5131 6320 7453 4845 1386 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 20.9 1512 5017 6785 9010 10,618 9424 5380 2520 1026 0 0 0 0 0 0 0 0 0 0 0
0 0 195 3186 6320 8182 9669 11,118 12,777 11,266 9175 5503 1625 0 0 0 0 0 0 0 0 0
0 0 0 704 4424 6956 9230 10,348 12,640 15,565 13,944 13,685 10,779 6286 2417 0 0 0 0 0 0 0
0 0 0 0 765 2666 5852 9208 10,918 11,365 11,107 9992 11,805 9180 10,590 3025 3428 0 0 0 0 0
0 0 0 0 0 414 1157 5389 6922 8647 9054 9051 10,693 9978 8633 6577 7453 8384 4685 5205 0 0
0 0 0 0 0 0 0 832 2136 4002 3260 3910 6929 8082 6216 3551 3995 4527 5059 5621 6519 0
0 0 0 0 0 0 0 0 1147 0 1750 2100 2481 2894 3338 3815 4323 0 5434 6037 0 0
0 0 0 0 0 0 0 0 0 0 0 0 2823 0 0 4341 0 0 0 0 7968 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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0–0.5 0.5–1.0 1.0–1.5 1.5–2.0 2.0–2.5 2.5–3.0 3.0–3.5 3.5–4.0 4.0–4.5 4.5–5.0 5.0–5.5 5.5–6.0 6.0–6.5 6.5–7.0 7.0–7.5 7.5–8.0 8.0–8.5 8.5–9.0 9.0–9.5 9.5–10 10–11 11–12
T (s)
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where kZ976 W/s m3, Hw is the height of the sinusoidal wave. For the chosen wave parameters, JsinZ275 kW/m. For more realistic ocean waves, coefficient k is 500 W/s m3 as stated by Duckers (2000). The energy flux transported by realistic nonsinusoidal waves is J Z kTw Hs2
(4)
where Hs is significant wave high found from wave energy spectra. The energy transported by real waves is approximately half of the flux transported by sinusoidal waves and equals to JZ137 kW/m in the cases under investigation. Practical utilization of wave energy shows that in the range of 20% of the energy J can be absorbed by WEC in reality (Kim Nielsen, 2000). Assumption of 20% absorbed energy, gives JZ30 kW/m. The power delivered by the buoy with sinusoidal waves is equal to Pb Z Jsin dbuoy
(5)
where dbuoy is the buoy diameter. For a 6 m buoy diameter, chosen in present studies, power delivered by the buoy is Pb Z 20% Jsin dbuoy Z 20%ð6 !275Þ Z 330 kW: Absorption ability of the oscillating body is limited. Although the energy stored and transported in the sea is large, only a fraction of it can be extracted by the oscillating body. Most of the available energy cannot be extracted. The ratio of the produced energy P to the volume of the point absorber Vbuoy plays a significant role to the amount of extracted energy. For a point absorber, i.e. the body with linear extension much smaller than the wavelength, inequality stated by Falnes (2002) is as follows P p Hw ! gr Vbuoy 4 Tw water
(6)
where rwaterZ1020 kg/m is the sea water density and g is gravity acceleration. As stated by Falnes (2002), the theoretical maximum of the absorbed by point absorber energy does not exceed 50%. For the chosen wave parameters, the energy is P 1 p Hw ! gr Z 3:6 kW=m3 Vbuoy 2 4 Tw water The vertical piston motion is driven by the wave motion and acting spring. The motion induces an electromagnetic force Fem between the stator and the piston. Spring accelerates the piston at the second phase of wave; speed of piston motion from the wave top to the trough is increased by spring force Fspring. End stop force Fstop, is not active at normal operation conditions and acts only when the piston stroke inside the stator becomes out of limits due to the too high wave heights. The gravity force also acts in the system and buoy force Fbuoy is lift force due to the Archimedes force.
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Equation of motion below includes all mentioned forces m
d2 x Z Fbuoy C Fem C Fspring C Fstop C Fg dt2
(7)
where m is equal to sum of piston and buoy mass, and only piston mass if there is a slack in the rope. 2.2. Electromagnetic model In this study, an octagonal shape of the generator have been chosen, see Fig. 1b. Edge effects can be avoided with circular shape of machine, but drawback is more complicated construction and increased cost of the machine. Permanent magnets piston moves vertically, and thus induce electromagnetic field in the generator, which is described by Maxwell’s equations: V !H Z jf V !E Z K
(8) vB vt
(9)
The simulations are done in two dimensions and vector magnetic potential A has only one component: B Z VA !z
(10)
Permanent magnets are modeled by jump conditions at the surface: Jm$z Z fn !Mg Vector magnet potential is derived from Ohm’s and Ampere’s laws and is determined by V2 A K sm0 mr
vA Vm$VA vU Z C sm0 mr vt m vz
(11)
where s is the conductivity, m is the permeability and U is the applied voltage. 2.3. Simulation method The simulation of the generator is done in two dimensions. The saturation effects and the three-dimensional coil-end effect are taken into account by separate three-dimensional calculations. The problem has complicate dependence of many parameters. The main emphasis in the present studies has been placed on finding suitable compromises between geometry, efficiency, a smooth power output and price. The system of equations stated in Section 2 is completed by the stator circuit equations and heat equation and it is solved by finite element method (FEM) code.
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The region of interest is a sector of the two-dimensional cross-section of a generator. The simulation code divides the cross-section of the generator into triangle elements and simulates magnetic field distribution. From the field other parameters, e.g. length, efficiency, load angle, currents, voltages, losses, forces, power output, are defined. The multi-physics aspects of the generator are modeled and results of the simulations are fully described generator.
3. Simulations In the present study, waves are large, the wave height is 5 m and the wave period is 11 s. This kind of waves take place during 131.4 h per year at Shannon and this wave corresponds to the central part of the table of occurrence (Table 1). These waves have large energy as can be seen from the table of energy (Table 2). The generator and the buoy are sized according to the wave parameters. Spring is sized to have about equal energy at the upward and the downward piston motion. The piston is chosen 500 mm longer than the stator to achieve effective electricity production. More specific parameters of the investigated system are presented in Table 3. The studies have been done for the generator nominal for wave conditions stated above and then estimations have been made under different conditions situated at two opposite points of the wave occurrence table, for lower and higher waves. The wave characteristics are presented in Table 4. The energy flux Jsin is transported by waves with stated parameters Hw and Tw. The average power output of the generator is obtained from the results of simulations for that wave conditions. Table 3 A selection of design parameters for the generator Nominal power (kV A) Stator voltage (kV) Piston length (mm) Stator length (mm) Added piston (rotor) length (mm) Static spring force (kN) Spring constant (kN) Buoy diameter (m) Buoy height (m) Buoy weight (kg) Load angle (degrees) Efficiency (%)
88 2.2 4970 4470 500 250 40 6 2 1719 8.6 88.7
Table 4 Wave parameters Hs (m) Tw (s) Jsin (kW/m) Save (kV A)
5 11 275 88.5
2.5 8 50 51.5
7 13 637 102.3
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4. Results The forces acting in the system are presented in Fig. 2. End stop force is zero according to Fig. 2, since the simulation is for normal operation conditions. The end stops become active at too high waves. The buoy force is counteracted by the electromagnetic force. The fluctuations in the electromagnetic force is very low and could be estimated from Fig. 3 where the cogging, e.g. the electromagnetic force fluctuations, for one piston stroke is presented. The cogging is equal to 2.7 kN. Power output of the generator under nominal wave conditions is presented in Fig. 4. The average power is equal 88.5 kV A. As obtained from the picture, the power at both the upward and the downward motion is nearly equal. This means that the spring is chosen suitable and retract the piston with necessary speed. The fluctuation level equals to 5 kV A and is rather low that characterize a good machine design. Intervals between rising and failing wave phases when power is zero correspond to the upper and lower piston positions, where the piston speed is zero, as seen in Fig. 6. In case of high waves, zero power intervals are longer than in the two other cases, as can be seen in Fig. 8. This is due to the end stops acting in the system. It takes place since the piston stroke become longer and end stops act to keep the piston inside the generator (Fig. 5). Phase and line voltages and harmonics during one piston stroke are presented in Fig. 4 and Table 5, respectively. The third harmonic is quite high in cases under investigations and is equal to 10.6% of the fundamental, but it is absent in the line voltage due to symmetry. The line voltage is slightly distorted from a sinusoidal shape.
Forces 600 Buoy,kN
Spring, kN
500
EM force, kN
End stop, kN
400
Gravity, kN
Force, kN
300 200 100 0
-100 -200 -300 -400
time, s
0
2
4
6
Fig. 2. Forces acting in the generator.
8
10
12
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Cogging 157
Cogging force, kN
156
155
154
time, s 153 0.00
0.02
0.03
0.05
0.07
0.08
0.10
Fig. 3. Cogging force between piston and stator.
The speed and position of the wave and piston are presented in Fig. 6. At the turning points the piston is situated at the center of the stator. The zero wave position corresponds to the middle of the wave. At first second wave speed decreases, but piston speed increases and opposite situation takes place as fifth Power
200
Absorbed power, kVA
180
Average output power, kVA
Absorbed power, kVA
160 140 120 100 80 60 40 20
time, s
0 0
1
3
4
5
6
8
Fig. 4. Power output of the generator at nominal wave conditions.
9
10
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Stator voltages 4000 Uab, V Ub, V
3000
Ua. V Uc. V
Voltage, V
2000 1000 time, s
0 0.00 -1000
0.02
0.04
0.06
0.08
0.10
0.12
-2000 -3000 -4000 Fig. 5. Stator line and phase voltages.
second wave speed starts to increase, but the piston speed still decreases as seen in Fig. 6. This shift between the wave and the piston speed curves is due to the piston inertia. The shift between wave and piston position depends on the spring configuration and the generator design. The piston follows wave motion due to the acceptable spring, otherwise the piston downward motion could be too slow and energy would be lost. The piston would not go down to the lower position and would not use the stator properly. In Fig. 7, the output power for lower than nominal waves conditions is presented. The power has the same character as at nominal conditions, but average power is equal to 51 kV A due to the less energy stored in lower waves compare to nominal case. Fig. 8 shows the power at higher than nominal waves. The power-decreasing interval interrupts the second period when piston moves from wave top to wave trough thereby increasing the power. It corresponds to the spring, which retracts the generator with high force. Under the simulation of the generator in the present case, the end stop constant has been chosen to be 990 kN/m, compared to low and nominal case when this constant has been equal to 800 kN/m. This has been done to reach reliability at high waves, when piston stroke is longer and it could lead to abnormal operation of the generator. Table 5 Voltage harmonics Harmonic
Phase V (%)
Line V (%)
1 3 5 7 9
100 10.6 0.5 0.4 0.7
100 0 0.5 0.4 0
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Speed and position 3
Wave speed, m/s Piston speed, m/s Wave position, m Piston position, m
speed [m/s] and position [m]
2
1
time, s
0 0
2
4
6
8
10
12
-1
-2
-3 Fig. 6. Wave and piston speed and position.
Power
100
Absorbed power, kVA Average output power, kVA
Absorbed power, kVA
90 80 70 60 50 40 30 20 10
time, s
0 0
1
2
3
4
5
5
Fig. 7. Power output of the generator at low waves.
6
7
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Power 250 Average output power, kVA
Absorbed power, kVA
200
150
100
50 time, s
0 0
1
3
4
5
7
8
10
11
12
Fig. 8. Power output of the generator at high waves.
During the simulations the generator under high waves, first, the end stop spring constant has been kept equal to 800 kN/m similar to the other two considered cases. The results have shown a bad design of a machine, since the end stops chosen are too weak and cannot withstand high waves and keep the piston safely inside the stator. Therefore, the end stops have been strengthened to reach necessary safety level. The copper and iron losses in the generator are listed in Table 6. The major part of copper losses is present in the stator core in the winding due to a long cable, which is 10.6 km long. Iron losses are mainly due to the hysteresis effect in the stator steel. Eddy current losses are rather low due to the lamination of the stator steel.
5. Discussions The results show difficulties to achieve survivability of the system at high waves. Proper design of the generator, spring and end stops have to be chosen. With a weak Table 6 Losses in the generator Cupper losses Stator core Coil end Iron losses Hysteresis Eddy currents Total
8.9 kW 1.3 kW 6.8 kW 0.9 kW 17.9 kW
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spring, the piston downward motion can be slow which leads to an output power reduction. With a too strong spring, the piston will be retracted to its lowest position to early, during the falling phase of the wave. This results in that the piston does not reach wave top and again the power output will be reduced. In the present work, the efficiency is for the generator, not to the whole converter. Additional losses can be expected in AC–DC and back DC–AC voltage conversion, and transmission of the power to the consumer grid via underwater cable (Thorburn et al., accepted for publication). A step up transformer can also be required and it would increase the price for the whole plant. The results show the effectiveness of the proposed WEC at special site of location with specific waves, but more realistic irregular waves should be taken into account in further studies.
6. Conclusion The simulations results for the octagonal linear generator for site Shannon near the south–west coast of Ireland have been presented and analyzed. This site has large waves with 5 m height and 11 s period, with a high energy equal to 68 kW/m, potentially available. The electromagnetic characteristics, properties, efficiency reaches quite a good values. The results are promising and show economic perspective and technological possibility to construct and exploit the converter. At low and nominal wave conditions, the power output has similar behavior. At higher waves, the end stops act to prevent the piston to have too long stroke. Power is interrupted by decreasing intervals at high wave conditions.
Acknowledgements Dr Karl Erik Karlsson and Dr Arne Wolfbrandt are thanked for developing the software for the simulations. Swedish Institute is acknowledged for financial support of the project.
References Duckers, L., 1996. Wave Energy. Renewable Energy ed. Boyle G. Oxford University Press. Ch. 8, ISBN 0-19856451-1. Eriksson M., Karin Thorburn, Bernhoff, H., Leijon, M., 2004. Dynamics of a linear generator for wave energy conversion. 23rd International Conference on Offshore Mechanics and Arctic Engineering, Canada, OMAE2004-51205. Falnes, J., 2002. Ocean Waves and Oscillating Systems, Linear Interaction including Wave-Energy Extraction. Cambridge University Press, Cambridge. ISBN 0 521 78211 2, pp. 75–83. Ivanova, I., Agren, O., Bernhoff, H., Leijon, M., 2003. Simulation of a 100 kW permanent magnet octagonal linear generator for ocean wave conversion, Fourth European Wave Energy Conference, Ireland. Kim Nielsen, 2000. Carsten plum point absorber—numerical and experimental results, Fourth European Wave Energy Conference, Denmark. ISBN 87-90074-09-2.
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˚ gren O., Isberg J., Sundberg J., Berg M., Karlsson K.-E., Wolfbrandt A., 2004. MultiLeijon M., Bernhoff H., A physics simulations of wave energy to electric energy conversion by permanent magnet linear generator. IEEE Journal of Energy Conversion 19. Mueller M.A, Baker N.J., 2002. A low speed reciprocating permanent magnet generators for direct drive wave energy converters. ‘Power Electronics, Machines and Drives’. Conference Publication No. 487. Mueller, M.A., Baker, N.J., Spooner, E., 2000. Electrical aspects of direct drive wave energy converters, Fifth European Wave Conference, Taborg, Denmark. Thorburn K., Bernhoff H., Leijon M., 2004. Wave energy transmission system concepts for linear generators arrays. Ocean Engineering Journal 31, 1339–1349. Washio, Y., Osawa, H., Nagata, Y., Fujii, F., Furuyama, H., Fujita, T., 2000. The offshore type wave power device ‘Mighty Whale’: open sea tests, Proceedings of 10th International Offshore and Polar Engineering Conference, Seattle, USA. Wave Net, 2000–2003. Results from the work of the European Thematic Network on wave energy. European Community, ERK5-CT-1999-20001.
Simulated generator for wave energy extraction in deep water ˚ gren, M. Leijon I.A. Ivanova*, H. Bernhoff, O. A The A˚ngstro¨m Laboratory, Division of Electricity and Lightning Research, Department of Engineering Science, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden Received 4 August 2004; accepted 1 February 2005 Available online 28 April 2005
Abstract A directly driven permanent magnet linear generator for ocean wave energy conversion has been simulated for a specific site with large waves. Wave characteristics influence the generator power output, the electromagnetic processes, geometrical sizes, and efficiency. The buoy and the generator dimensions are important parameters to get high efficiency of the converter. Electromagnetic and mechanical simulations of the generator with realistic wave conditions are presented. q 2005 Elsevier Ltd. All rights reserved. Keywords: Renewable energy; Wave power; Linear generator; Energy conversion
1. Introduction Wave energy conversion possibility is not a newborn idea. Investigations on the subject have been started many years ago. Ocean waves have high energy density compared to other new types of renewable sources, see Wave Net (2000–2003). Waves at open ocean sites are situated far away from coastline, have high amplitude, and large wavelength. There is no friction of the waves with sea bottom and a lot of energy is available. Concern is to extract this energy efficiently and convert it to the consumer grid with minimal losses. During the last years, a large number of technical principles of wave energy conversion have been developed, presented in publications, constructed and tested. Several principles rely on the use of conventional rotation generators, see Washio et al. (2000). * Corresponding author. Fax: C46 18 471 5810. E-mail addresses: [email protected] (I.A. Ivanova), [email protected] (I.A. Ivanova).
0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.02.006
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Linear generators have previously been used for wave application; see Mueller et al. (2000) and Mueller and Baker (2002). Waves at different places have certain character and energy density. Wave energy converters (WEC) should be designed for the particular specific location, taking into account the available wave energy resources there. Challenges in WEC and random wave conditions that lead to difficulties to achieve survivability of the system, unstable voltage level and variable power output are irregular. WEC’s should be provided with a good control system. Start up and shut down procedures, abnormal modes of operation needs to be controlled in any electrical devices. At normal operational modes it is necessary to control fluctuations in power output. In the present work, octagonal linear permanent magnet generator with axial flux has been simulated for selected place near the coast of Ireland, with large waves. A scheme of the generator is drawn in Fig. 1a. Principle of operation of the converter has been considered by Ivanova et al. (2003). A buoy on the sea surface moves vertically driven by the waves. The buoy is connected to a linear electric generator by a rope, which is connected to the permanent magnet (PM) generator piston (rotor). The piston moves inside the stator. The stator is made of electric steel and has slots, where the winding is located. The buoy provides piston movement relative the stator that induces electromotive force (EMF) in the winding. A large number of magnet poles placed on the piston increases the rate of magnetic flux change in the generator and as a consequence, increase the EMF. The buoy’s dimensions, mass and shape are important for effective wave energy extraction. The buoy should be large enough to absorb the maximum energy, but on the other hand, a substantially large buoy could destroy the generator in stormy weather. Spring serves to retract the generator piston on the ocean floor and increase the speed of the piston downward motion (Leijon et al., accepted for publication). In case of reasonable spring, energy extracted from the waves when the piston moves up is approximately equal to energy extracted when piston moves down. In the absence of the spring, the piston downward motion would be slow since only the gravity acts on the piston in such a case (Eriksson et al., 2004). The end stops serve to protect the generator at high waves and prevents the piston to go out of the stator. In the present work, the main emphasis is put on the generator design to be able to extract energy efficiently. The paper is organized as follows. Section 1 includes introduction of the subject and a brief description of the converter. Some generator theory is presented in Section 2. The simulation method is described in Section 3. Results are presented in Section 4. Sections 5 and 6 complete the paper. 2. Theory 2.1. Wave climate and motion equation Most of the energy stored in ocean waves is associated with wave periods within the interval from 5 to 15 s. In deep water, that corresponds to wavelengths from 40 to 350 m (Falnes, 2002).
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Fig. 1. (a) Scheme of WEC with linear generator, (b) and above view.
In these studies, harmonic sinusoidal wave propagation is considered with 5 m wave height and 11 s wave period. According to the European Wave Energy Atlas, place Shannon at Atlantic Ocean near south–west coast of Republic Ireland has been chosen due to the good perspective to extract energy there. The occurrence of waves with certain
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Table 1 Occurrence of significant wave height H and wave period T per year at Shannon H (m)
T (s) 0
2
3
4
5
6
7
8
9
10
11
12
13
14
15
18
2
3
4
5
6
7
8
9
10
11
12
13
14
15
18
20
0.0–0.5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.5–1.0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
1
4
6
1
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
11 6 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
32 40 23 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
55 51 38 30 14 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0
26 44 36 32 27 18 8 3 1 0 0 0 0 0 0 0 0 0 0 0
3 25 30 36 22 19 17 12 8 4 1 0 0 0 0 0 0 0 0 0
0 5 19 20 19 16 15 15 11 9 6 3 1 0 0 0 0 0 0 0
0 0 3 7 11 13 12 10 8 6 6 4 4 1 1 0 0 0 0 0
0 0 0 1 2 7 7 7 6 5 5 4 3 2 2 2 1 1 0 0
0 0 0 0 0 1 2 3 2 2 3 3 2 1 1 1 1 1 1 0
0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 0 1 1 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1.0–1.5 1.5–2.0 2.0–2.5 2.5–3.0 3.0–3.5 3.5–4.0 4.0–4.5 4.5–5.0 5.0–5.5 5.5–6.0 6.0–6.5 6.5–7.0 7.0–7.5 7.5–8.0 8.0–8.5 8.5–9.0 9.0–9.5 9.5–10 10–11 11–12
parameters and energy flux J transported by waves at Shannon are presented in Tables 1 and 2. The annual mean wave energy available is 68 kW/m. This site has been chosen due to the large energy potential. The wavelength at deep water can be found from the following formula (Falnes, 2002) lZ
g 2 T 2p w
(1)
where Tw is wave period. For deep water, the following condition is valid: DO Lwave =2
(2)
The wavelength at the Shannon is equal to LwaveZ188 m. The longer the wave, the more energy is transported. The simulated waves in the present studies are sinusoidal. The energy flux transported by purely progressive sinusoidal wave is given by Falnes (2002) Jsin Z kTw Hw2
(3)
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Table 2 Energy table for significant wave height H and period T per year at Shannon (energy unit is [kW h/m year]) H (m)
4
5
6
7
8
9
10
11
12
13
14
15
18
5
6
7
8
9
10
11
12
13
14
15
18
20
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 13.5 414 443 122 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 64 1433 3488 3315 1292 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 111 2823 5131 6320 7453 4845 1386 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 20.9 1512 5017 6785 9010 10,618 9424 5380 2520 1026 0 0 0 0 0 0 0 0 0 0 0
0 0 195 3186 6320 8182 9669 11,118 12,777 11,266 9175 5503 1625 0 0 0 0 0 0 0 0 0
0 0 0 704 4424 6956 9230 10,348 12,640 15,565 13,944 13,685 10,779 6286 2417 0 0 0 0 0 0 0
0 0 0 0 765 2666 5852 9208 10,918 11,365 11,107 9992 11,805 9180 10,590 3025 3428 0 0 0 0 0
0 0 0 0 0 414 1157 5389 6922 8647 9054 9051 10,693 9978 8633 6577 7453 8384 4685 5205 0 0
0 0 0 0 0 0 0 832 2136 4002 3260 3910 6929 8082 6216 3551 3995 4527 5059 5621 6519 0
0 0 0 0 0 0 0 0 1147 0 1750 2100 2481 2894 3338 3815 4323 0 5434 6037 0 0
0 0 0 0 0 0 0 0 0 0 0 0 2823 0 0 4341 0 0 0 0 7968 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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0–0.5 0.5–1.0 1.0–1.5 1.5–2.0 2.0–2.5 2.5–3.0 3.0–3.5 3.5–4.0 4.0–4.5 4.5–5.0 5.0–5.5 5.5–6.0 6.0–6.5 6.5–7.0 7.0–7.5 7.5–8.0 8.0–8.5 8.5–9.0 9.0–9.5 9.5–10 10–11 11–12
T (s)
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where kZ976 W/s m3, Hw is the height of the sinusoidal wave. For the chosen wave parameters, JsinZ275 kW/m. For more realistic ocean waves, coefficient k is 500 W/s m3 as stated by Duckers (2000). The energy flux transported by realistic nonsinusoidal waves is J Z kTw Hs2
(4)
where Hs is significant wave high found from wave energy spectra. The energy transported by real waves is approximately half of the flux transported by sinusoidal waves and equals to JZ137 kW/m in the cases under investigation. Practical utilization of wave energy shows that in the range of 20% of the energy J can be absorbed by WEC in reality (Kim Nielsen, 2000). Assumption of 20% absorbed energy, gives JZ30 kW/m. The power delivered by the buoy with sinusoidal waves is equal to Pb Z Jsin dbuoy
(5)
where dbuoy is the buoy diameter. For a 6 m buoy diameter, chosen in present studies, power delivered by the buoy is Pb Z 20% Jsin dbuoy Z 20%ð6 !275Þ Z 330 kW: Absorption ability of the oscillating body is limited. Although the energy stored and transported in the sea is large, only a fraction of it can be extracted by the oscillating body. Most of the available energy cannot be extracted. The ratio of the produced energy P to the volume of the point absorber Vbuoy plays a significant role to the amount of extracted energy. For a point absorber, i.e. the body with linear extension much smaller than the wavelength, inequality stated by Falnes (2002) is as follows P p Hw ! gr Vbuoy 4 Tw water
(6)
where rwaterZ1020 kg/m is the sea water density and g is gravity acceleration. As stated by Falnes (2002), the theoretical maximum of the absorbed by point absorber energy does not exceed 50%. For the chosen wave parameters, the energy is P 1 p Hw ! gr Z 3:6 kW=m3 Vbuoy 2 4 Tw water The vertical piston motion is driven by the wave motion and acting spring. The motion induces an electromagnetic force Fem between the stator and the piston. Spring accelerates the piston at the second phase of wave; speed of piston motion from the wave top to the trough is increased by spring force Fspring. End stop force Fstop, is not active at normal operation conditions and acts only when the piston stroke inside the stator becomes out of limits due to the too high wave heights. The gravity force also acts in the system and buoy force Fbuoy is lift force due to the Archimedes force.
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Equation of motion below includes all mentioned forces m
d2 x Z Fbuoy C Fem C Fspring C Fstop C Fg dt2
(7)
where m is equal to sum of piston and buoy mass, and only piston mass if there is a slack in the rope. 2.2. Electromagnetic model In this study, an octagonal shape of the generator have been chosen, see Fig. 1b. Edge effects can be avoided with circular shape of machine, but drawback is more complicated construction and increased cost of the machine. Permanent magnets piston moves vertically, and thus induce electromagnetic field in the generator, which is described by Maxwell’s equations: V !H Z jf V !E Z K
(8) vB vt
(9)
The simulations are done in two dimensions and vector magnetic potential A has only one component: B Z VA !z
(10)
Permanent magnets are modeled by jump conditions at the surface: Jm$z Z fn !Mg Vector magnet potential is derived from Ohm’s and Ampere’s laws and is determined by V2 A K sm0 mr
vA Vm$VA vU Z C sm0 mr vt m vz
(11)
where s is the conductivity, m is the permeability and U is the applied voltage. 2.3. Simulation method The simulation of the generator is done in two dimensions. The saturation effects and the three-dimensional coil-end effect are taken into account by separate three-dimensional calculations. The problem has complicate dependence of many parameters. The main emphasis in the present studies has been placed on finding suitable compromises between geometry, efficiency, a smooth power output and price. The system of equations stated in Section 2 is completed by the stator circuit equations and heat equation and it is solved by finite element method (FEM) code.
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The region of interest is a sector of the two-dimensional cross-section of a generator. The simulation code divides the cross-section of the generator into triangle elements and simulates magnetic field distribution. From the field other parameters, e.g. length, efficiency, load angle, currents, voltages, losses, forces, power output, are defined. The multi-physics aspects of the generator are modeled and results of the simulations are fully described generator.
3. Simulations In the present study, waves are large, the wave height is 5 m and the wave period is 11 s. This kind of waves take place during 131.4 h per year at Shannon and this wave corresponds to the central part of the table of occurrence (Table 1). These waves have large energy as can be seen from the table of energy (Table 2). The generator and the buoy are sized according to the wave parameters. Spring is sized to have about equal energy at the upward and the downward piston motion. The piston is chosen 500 mm longer than the stator to achieve effective electricity production. More specific parameters of the investigated system are presented in Table 3. The studies have been done for the generator nominal for wave conditions stated above and then estimations have been made under different conditions situated at two opposite points of the wave occurrence table, for lower and higher waves. The wave characteristics are presented in Table 4. The energy flux Jsin is transported by waves with stated parameters Hw and Tw. The average power output of the generator is obtained from the results of simulations for that wave conditions. Table 3 A selection of design parameters for the generator Nominal power (kV A) Stator voltage (kV) Piston length (mm) Stator length (mm) Added piston (rotor) length (mm) Static spring force (kN) Spring constant (kN) Buoy diameter (m) Buoy height (m) Buoy weight (kg) Load angle (degrees) Efficiency (%)
88 2.2 4970 4470 500 250 40 6 2 1719 8.6 88.7
Table 4 Wave parameters Hs (m) Tw (s) Jsin (kW/m) Save (kV A)
5 11 275 88.5
2.5 8 50 51.5
7 13 637 102.3
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4. Results The forces acting in the system are presented in Fig. 2. End stop force is zero according to Fig. 2, since the simulation is for normal operation conditions. The end stops become active at too high waves. The buoy force is counteracted by the electromagnetic force. The fluctuations in the electromagnetic force is very low and could be estimated from Fig. 3 where the cogging, e.g. the electromagnetic force fluctuations, for one piston stroke is presented. The cogging is equal to 2.7 kN. Power output of the generator under nominal wave conditions is presented in Fig. 4. The average power is equal 88.5 kV A. As obtained from the picture, the power at both the upward and the downward motion is nearly equal. This means that the spring is chosen suitable and retract the piston with necessary speed. The fluctuation level equals to 5 kV A and is rather low that characterize a good machine design. Intervals between rising and failing wave phases when power is zero correspond to the upper and lower piston positions, where the piston speed is zero, as seen in Fig. 6. In case of high waves, zero power intervals are longer than in the two other cases, as can be seen in Fig. 8. This is due to the end stops acting in the system. It takes place since the piston stroke become longer and end stops act to keep the piston inside the generator (Fig. 5). Phase and line voltages and harmonics during one piston stroke are presented in Fig. 4 and Table 5, respectively. The third harmonic is quite high in cases under investigations and is equal to 10.6% of the fundamental, but it is absent in the line voltage due to symmetry. The line voltage is slightly distorted from a sinusoidal shape.
Forces 600 Buoy,kN
Spring, kN
500
EM force, kN
End stop, kN
400
Gravity, kN
Force, kN
300 200 100 0
-100 -200 -300 -400
time, s
0
2
4
6
Fig. 2. Forces acting in the generator.
8
10
12
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Cogging 157
Cogging force, kN
156
155
154
time, s 153 0.00
0.02
0.03
0.05
0.07
0.08
0.10
Fig. 3. Cogging force between piston and stator.
The speed and position of the wave and piston are presented in Fig. 6. At the turning points the piston is situated at the center of the stator. The zero wave position corresponds to the middle of the wave. At first second wave speed decreases, but piston speed increases and opposite situation takes place as fifth Power
200
Absorbed power, kVA
180
Average output power, kVA
Absorbed power, kVA
160 140 120 100 80 60 40 20
time, s
0 0
1
3
4
5
6
8
Fig. 4. Power output of the generator at nominal wave conditions.
9
10
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Stator voltages 4000 Uab, V Ub, V
3000
Ua. V Uc. V
Voltage, V
2000 1000 time, s
0 0.00 -1000
0.02
0.04
0.06
0.08
0.10
0.12
-2000 -3000 -4000 Fig. 5. Stator line and phase voltages.
second wave speed starts to increase, but the piston speed still decreases as seen in Fig. 6. This shift between the wave and the piston speed curves is due to the piston inertia. The shift between wave and piston position depends on the spring configuration and the generator design. The piston follows wave motion due to the acceptable spring, otherwise the piston downward motion could be too slow and energy would be lost. The piston would not go down to the lower position and would not use the stator properly. In Fig. 7, the output power for lower than nominal waves conditions is presented. The power has the same character as at nominal conditions, but average power is equal to 51 kV A due to the less energy stored in lower waves compare to nominal case. Fig. 8 shows the power at higher than nominal waves. The power-decreasing interval interrupts the second period when piston moves from wave top to wave trough thereby increasing the power. It corresponds to the spring, which retracts the generator with high force. Under the simulation of the generator in the present case, the end stop constant has been chosen to be 990 kN/m, compared to low and nominal case when this constant has been equal to 800 kN/m. This has been done to reach reliability at high waves, when piston stroke is longer and it could lead to abnormal operation of the generator. Table 5 Voltage harmonics Harmonic
Phase V (%)
Line V (%)
1 3 5 7 9
100 10.6 0.5 0.4 0.7
100 0 0.5 0.4 0
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Speed and position 3
Wave speed, m/s Piston speed, m/s Wave position, m Piston position, m
speed [m/s] and position [m]
2
1
time, s
0 0
2
4
6
8
10
12
-1
-2
-3 Fig. 6. Wave and piston speed and position.
Power
100
Absorbed power, kVA Average output power, kVA
Absorbed power, kVA
90 80 70 60 50 40 30 20 10
time, s
0 0
1
2
3
4
5
5
Fig. 7. Power output of the generator at low waves.
6
7
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Power 250 Average output power, kVA
Absorbed power, kVA
200
150
100
50 time, s
0 0
1
3
4
5
7
8
10
11
12
Fig. 8. Power output of the generator at high waves.
During the simulations the generator under high waves, first, the end stop spring constant has been kept equal to 800 kN/m similar to the other two considered cases. The results have shown a bad design of a machine, since the end stops chosen are too weak and cannot withstand high waves and keep the piston safely inside the stator. Therefore, the end stops have been strengthened to reach necessary safety level. The copper and iron losses in the generator are listed in Table 6. The major part of copper losses is present in the stator core in the winding due to a long cable, which is 10.6 km long. Iron losses are mainly due to the hysteresis effect in the stator steel. Eddy current losses are rather low due to the lamination of the stator steel.
5. Discussions The results show difficulties to achieve survivability of the system at high waves. Proper design of the generator, spring and end stops have to be chosen. With a weak Table 6 Losses in the generator Cupper losses Stator core Coil end Iron losses Hysteresis Eddy currents Total
8.9 kW 1.3 kW 6.8 kW 0.9 kW 17.9 kW
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spring, the piston downward motion can be slow which leads to an output power reduction. With a too strong spring, the piston will be retracted to its lowest position to early, during the falling phase of the wave. This results in that the piston does not reach wave top and again the power output will be reduced. In the present work, the efficiency is for the generator, not to the whole converter. Additional losses can be expected in AC–DC and back DC–AC voltage conversion, and transmission of the power to the consumer grid via underwater cable (Thorburn et al., accepted for publication). A step up transformer can also be required and it would increase the price for the whole plant. The results show the effectiveness of the proposed WEC at special site of location with specific waves, but more realistic irregular waves should be taken into account in further studies.
6. Conclusion The simulations results for the octagonal linear generator for site Shannon near the south–west coast of Ireland have been presented and analyzed. This site has large waves with 5 m height and 11 s period, with a high energy equal to 68 kW/m, potentially available. The electromagnetic characteristics, properties, efficiency reaches quite a good values. The results are promising and show economic perspective and technological possibility to construct and exploit the converter. At low and nominal wave conditions, the power output has similar behavior. At higher waves, the end stops act to prevent the piston to have too long stroke. Power is interrupted by decreasing intervals at high wave conditions.
Acknowledgements Dr Karl Erik Karlsson and Dr Arne Wolfbrandt are thanked for developing the software for the simulations. Swedish Institute is acknowledged for financial support of the project.
References Duckers, L., 1996. Wave Energy. Renewable Energy ed. Boyle G. Oxford University Press. Ch. 8, ISBN 0-19856451-1. Eriksson M., Karin Thorburn, Bernhoff, H., Leijon, M., 2004. Dynamics of a linear generator for wave energy conversion. 23rd International Conference on Offshore Mechanics and Arctic Engineering, Canada, OMAE2004-51205. Falnes, J., 2002. Ocean Waves and Oscillating Systems, Linear Interaction including Wave-Energy Extraction. Cambridge University Press, Cambridge. ISBN 0 521 78211 2, pp. 75–83. Ivanova, I., Agren, O., Bernhoff, H., Leijon, M., 2003. Simulation of a 100 kW permanent magnet octagonal linear generator for ocean wave conversion, Fourth European Wave Energy Conference, Ireland. Kim Nielsen, 2000. Carsten plum point absorber—numerical and experimental results, Fourth European Wave Energy Conference, Denmark. ISBN 87-90074-09-2.
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˚ gren O., Isberg J., Sundberg J., Berg M., Karlsson K.-E., Wolfbrandt A., 2004. MultiLeijon M., Bernhoff H., A physics simulations of wave energy to electric energy conversion by permanent magnet linear generator. IEEE Journal of Energy Conversion 19. Mueller M.A, Baker N.J., 2002. A low speed reciprocating permanent magnet generators for direct drive wave energy converters. ‘Power Electronics, Machines and Drives’. Conference Publication No. 487. Mueller, M.A., Baker, N.J., Spooner, E., 2000. Electrical aspects of direct drive wave energy converters, Fifth European Wave Conference, Taborg, Denmark. Thorburn K., Bernhoff H., Leijon M., 2004. Wave energy transmission system concepts for linear generators arrays. Ocean Engineering Journal 31, 1339–1349. Washio, Y., Osawa, H., Nagata, Y., Fujii, F., Furuyama, H., Fujita, T., 2000. The offshore type wave power device ‘Mighty Whale’: open sea tests, Proceedings of 10th International Offshore and Polar Engineering Conference, Seattle, USA. Wave Net, 2000–2003. Results from the work of the European Thematic Network on wave energy. European Community, ERK5-CT-1999-20001.