Recent Developments of Electrical Drives - Part 6 doc

Coupled Model for PMSM 37 The dynamic equation expressed in the torque-slip plane is, 8: −1 p J · w2 0· s ds d δ = T sδ + T as − T cs 8 Where: J is the combination inertia of the motor a

Figure 13 Graphic representation of speed vs time for different inertia torque (M) with T l= 0 and

Sl= 1

The permanent magnets influence about the transient behavior of the PMSM is very important As higher are the equivalent currents of the magnets as lower are the slips of the transient response In Fig 18 we can observe that if the current of the magnet decrease below a certain value the motor is not capable of take up the overload and the motor lost the synchronism The optimum value of this current depends, amount other factors, of the magnets braking torque at the synchronous speed proximity If this value is overcome they will appear higher oscillations during the transient operation

The rotor geometry and in consequence the relationship between the d-axis and q-axis

reactances, also modify the PMSM behavior, as in the same way that for the equivalent

current we must obtain an optimum value for the relation Xd/Xq and also it is important to

obtain the most appropriate squirrel cage resistance value

Figure 14 Graphic representation of the torque of the squirrel cage vs time for a sudden decrease

of the load

I-3 Coupled Model for PMSM 35

Figure 15 Graphic representation of the torque of the squirrel cage vs time for a sudden increase of

the load

Really the number of variables that have influence about the starting and synchronization processes of a PMSM, take into account the motor and also the load, is very higher and then it

is very difficult know in advance a set of necessary conditions for the correct synchronization

of the PMSM Then for develop an analyze of this type it is necessary take into account the parametric variation of the main magnitudes that have influence about the synchronization process

In this particular case we have analyzed this process in term of his synchronization energy, specially we have considered the property “capacity of synchronization” of the motor, that

Figure 16 Graphic representation of the magnets and reluctance torques vs time during a sudden

increase of the load

Figure 17 Graphic representation of the magnets and reluctance torques vs time during a sudden

decrease of the load

we can defined it like a set of critical combinations of inertia and load torque in which the PMSM is capable to obtain the synchronization

For obtain the synchronization energy we use a set of simple expressions that permit determine this magnitude in the last stage of the synchronous operation of the motor just when the machine describe a limit circle

Figure 18 Graphic representation of speed vs time for different values of the equivalent current of

the magnets with Sl = 1 and Tl = 0.

I-3 Coupled Model for PMSM 37 The dynamic equation expressed in the torque-slip plane is, (8):

−1

p J · w2

0· s ds

d δ = T s(δ) + T a(s) − T c(s) (8) Where: J is the combination inertia of the motor and the load, Ts is the sum of all the synchronization torques, Ta include all the asynchronous average torques, and Tc is the sum

of the load, slip, and ventilation torques

The equation (8) describe the critical trajectories of the polar slips on the load angle-slip plane

Conclusions

We have developed along this paper a coupled model for accurate representation of the characteristics of permanent magnet synchronous motors and we have proposed the

de-termination of the direct axis reactance, “Xd,” and the quadrature axis reactance, “Xq,” by

calculus and texts with the permanent magnet synchronous machine under generator duty

We also have analyzed the starting and synchronization processes of the PMSM and the influence that on transient behavior of the motor produce different values of the main motor parameters

References

[1] J Salo, T Heikkil¨a, H.T Pyrh¨onen, “New Low-Speed High-Torque Permanent Magnet Syn-chronous Machine With Buried Magnets”, International Conference on Electrical Machines (ICEM 2000), Espoo, Finland, 2000, pp 1246–1250

[2] F Parasiliti, P Poffet, A model for saturation effects in high field permanent magnet synchronous motors, IEEE Trans Energy Convers., Vol 4, No 3, pp 487–494, 1989

[3] M.P Donsion, M.F Ferro, Motores sincronos de imanes permanentes, Research book published

by the University of Santiago de Compostela, Spain

[4] M.P Donsi´on, J.F Manzanedo, C Iglesias, “Coupled Model of the Interior Type Permanent Magnet Synchronous Motor Application to a Siemosyn Motor”, International Conference on Electrical Machines (ICEM’94), Par´ıs, France, 1994, pp 144–147

[5] M.F Ferro, M.P Donsion, “Torques Analysis in Permanent Magnet Synchronous Motors”, IASTED Power High Tech’89, Valencia, Spain, 1989, pp 271–275

[6] M.F Ferro, M.P Donsi´on “Transient Behavior of Permanent Magnet Synchronous Motors Under Sudden Change in Load”, IASTED Ninth International Symposium, Modelling, Identification and Control, Innsbruck, Austria, pp 406–410

[7] M.F Ferro, M.P Donsi´on, “Specific Characteristics of the Interior Type Permanent Magnet Synchronous Motors Aplication so a Siemosyn 1FU3134”, International AEGEAN Conference

on Electrical Machines and Power Electronics, Turkey, Vol 2, pp 378–382

Abstract A vernier hybrid machine has been developed for use as a linear generator in a wave energy

converter Accurate predictions for power capture require testing this machine in a nonsinusoidal manner A dynamic model capable of predicting the machine’s behavior for this kind of mechanical excitation is presented Simple equivalent circuit models have been found to be unsuitable for these machines and a flux-linkage map approach is instead used Experimental results are used to verify this approach and the functioning of a unity power factor controller

Introduction

The vernier hybrid permanent magnet machine (VHM) is a member of the family known as variable reluctance permanent magnet machines These machines are known to produce air gap shear stresses significantly higher than conventional machines Weh et al [1] measured

a shear stress in the transverse flux machine (TFM) of the order of 200 kN/m2 However, the TFM is a very complex machine to construct Mecrow and Jack [2] investigated the use of VRPM topology in a more conventional machine structure and demonstrated that improvements over conventional machines could be made Spooner and Haydock [3] de-veloped the VHM, which is easier to construct than the TFM, but also benefits from high shear stress A shear stress of 106 kN/m2has been measured for a prototype linear VHM [4] Combined with the high shear stress capability and the effect of magnetic gearing the VHM is a suitable machine for low speed high torque (or force) applications One such application is in direct drive wave energy converters as proposed by Mueller and Baker [5]

In this application a linear generator is directly coupled to the wave energy device, such as

a heaving buoy or the Archimedes Wave Swing [6] so that the generator experiences the same displacement as the device Ideal waves are monochromatic resulting in sinusoidal motion Under this condition the displacement of the generator is well known, but the induced voltage is variable in both frequency and magnitude In real sea conditions waves from different frequencies combine to give a very random motion Wave data has been collected

S Wiak, M Dems, K Kom˛eza (eds.), Recent Developments of Electrical Drives, 39–49.

2006 Springer.

40 Mueller et al.

Figure 1 Open circuit voltage in a linear VHM.

over many years using instrumentation buoys, and data from one particular buoy has been collated to demonstrate the nature of the actual displacement a direct drive generator will experience The data is stored as one dimensional wave data, a compact form summarizing the displacement characteristics of the sea surface [7] Fig 1 shows the recreation of typical displacement and velocity waveform obtained from the data of one such buoy The amplitude and period of the signal have been scaled down to preserve the shape of the signal whilst allowing for a manageable sized test rig [8]

It is quite clear that the machine is expected operate in a very dynamic environment Hence a dynamic model of the VHM has been developed to investigate its performance and how it interacts with the wave energy device The model can be used to investigate the use of the electrical generator itself to control and tune the actual characteristics of the device An indication of how this can be achieved is given in the paper In addition the model has been developed to enable the designer to investigate the conditions under which the machine has to operate to overcome the inherent high inductance and hence to extract maximum power Based upon the displacements in Fig 1 the power output is likely to be random and pulsating Output from the dynamic model can be used to investigate control strategies for energy storage to smooth the power

The model is described in the paper and results are presented to verify it and to illustrate its use with this application in mind It has been developed using MATLAB and incorporated into SIMULINK such that it can be represented as a block to enable it to be connected to models of different marine energy prime movers

Experimental machine and test rig

A prototype linear VHM has been designed and built, details of which are given in [9], and

a photo is shown in Fig 2 Table 1 summarizes the main geometrical data The translator

is driven via a crank mechanism connected to a variable speed drive via a 14:1 step down gearbox In this way the translator can be driven at frequencies typical of those expected in

Figure 2 Linear VHM prototype.

a direct drive wave energy converter The output of the generator is fed into a three-phase ac/ac converter, the details of which can be found in reference [4]

Model description

Fig 3 shows a block diagram of the complete generator model including a block to maximize the power output of the electrical generator and the prime mover The complete electrome-chanical model consists of three blocks, A, B, and C, corresponding to the prime mover, generator, and power conversion equipment, respectively The contents of each block will

be described in this section

Block A: Prime mover model

In this application the prime mover is a marine energy device, which could be a wave or tidal stream energy converter The inputs to the model are the force imparted by the waves

Table 1 Main dimensions of the prototype

42 Mueller et al.

F gen

desired

Î

i(t)

Generator circuit model

V(t)

Prime mover

map

UPF model

Force model

Figure 3 Model block diagram.

say and the generator reaction force, with displacement being the output A wave energy device is simply modeled as a mass-spring damper system according to equation (1)

where F wave is the force on the device from the incident wave, F genis the generator reaction

force, M is the device mass, K is the buoyancy force, and B is the mechanical resistance.

The buoyancy force is essentially a spring force The mechanical resistance is the sum of the radiation resistance due to waves being created by motion of the device and any mechanical and viscous losses

A test rig has been developed to emulate the mass-spring damper system in a wave energy device The structure of the rig is shown in Fig 4 and has a similar response to equation (1)

An induction motor stepped down through a gearbox drives the crank which drives the rotor (or translator) of the linear generator via the springs and steel cables Without the springs the amplitude of the rotor is 0.2 m, but with the additional energy from the springs the rotor can achieve much greater amplitudes, depending upon the choice of spring

Rotor

Springs

x

x

x a

a 2x

a

o

a 2r x

r o

o

o

K1

K2

rl

xl

x

B

a

r w

rR

D A

Rotor

Figure 4 Wave energy emulator.

0 -15 -10 -5

5 10

Current (A) Position (mm)

Figure 5 Flux-linkage map for the VHM prototype.

stiffness and the friction The displacement, x(t), is nearly sinusoidal, thus representing monochromatic waves In this case F wave is now the force imparted to the crank by the

drive motor, K is the spring stiffness, M is the mass of the translator, and B represents the

mechanical losses in the rig, which are principally in the pulleys

Block B: Generator flux-linkage model The flux-linkage-position map provides a complete electromagnetic description of the ma-chine Because of the small magnet pitch in the VHM it was found that a simple equivalent circuit approach used for PM synchronous machines was not accurate enough 2D finite element analysis was used to generate the flux-linkage map, which is essentially used as

a look-up table in the model Knowing the position and current the flux-linkage is then determined, which is used in the force and circuit models Fig 5 shows the flux-linkage map generated from a 2D finite element model [9]

Block B: Generator force model The flux-linkage map generated from 2D finite element analysis is used to determine the co-energy at a particular position and coil current Force is then calculated from the rate of change of co-energy with displacement according to equation (2):

Block B: Generator circuit model The generator circuit model is shown in Fig 6, in which the no-load induced emf and inductance are lumped into one represented by the rate of change in flux linkage The

44 Mueller et al.

R

V(t) dt

dΨ(i, x)

Figure 6 Phase equivalent circuit for the VHM.

terminal voltage is described in equation (3)

V (t)=d (x, i)

Block C: Unity power factor correction model

A three-phase active rectifier connected to the terminals of the VHM is controlled to ensure that the generator current is in phase with the induced emf in order to maximize the power generated at the terminals By controlling the machine in this way reactive power flows from the active rectifier to compensate for the high inductance in the machine This has been implemented and demonstrated on the prototype machine [4] A measure of the no-load induced emf is required so that PWM signals can be generated to control the switches

in the active rectifier and thus ensuring the current tracks the emf exactly Search coils are used on the rig to achieve this, but in the simulation the induced emf is obtained from the flux-linkage map at zero armature current (equation (4)) The desired armature current

is simply scaled from the no-load induced emf according to the ratio of the peak desired current and the peak no-load induced emf as shown in equation (5)

E(t)= d (x, i = 0)

i (t )= ˆdesired

ˆ

The generator current then feeds the force model and the flux-linkage model and hence used to calculate the new position, the next value of flux linkage, and so on

Verification of the generator models in block B

Experimental results were used to verify the model and algorithm developed Displacement measurements taken from the test rig were fed into the model to generate these results Fig 7 shows the no-load emf

Fig 8(a,b) shows experimental and computed results for the prototype machine, when operating in unity power factor mode