Simulation and analysis of switched reluctance generator for renewable energy applications

The purpose of the paper is to investigate the application of a Switched Reluctance Generator (SRG) for future wave energy converters. This machine presents some favourable features as compared to others. The machine optimal operation can be obtained after a detailed assessment of the control strategy. For this purpose, a SRG model is implemented in Matlab® Simulink® environment to find out the values of the switching angles which maximize the converted power and to set up an appropriate control of the power exchange with the grid to which the machine is connected. The simulation results provide evidence that a proper advancing of the turn on angle enables an optimized power extraction, fulfilling at the same time the control references on DC link voltage and grid reactive power.

The difference between an intermittent and a non-intermittent energy source of equal rated power.

Abstract
The objectives of the investigation is to study a SRG (Switched Reluctance Generator) for future wave energy applications. In particular is intended to design and simulate a SRG model with MATLAB® SIMULINK® discovering the characteristic of the machine, the profile of current, torque ecc , figure out the best strategy in order to produce the large amount of power.
The machine is later connected to a grid converter in order to evaluate how much energy is possible to extract with this type of machine. A control on the grid side is also implemented to maintain the current in phase with the grid voltage. This work started from a demo of a Switched Reluctance Motor already implemented in the library of Simulink in SimPowerSystem that has been modify to convert the SRM in SRG. The grid side and control of reactive power started from a previous model already developed in the frame of the HVDC project. The model then is adapted to the requirements of the machine operations.
In the introduction is presented the general characteristics of a SR machine (Switched Reluctance machine), including the different possible configurations and the possible applications in the wave energy describing the technology and its advantages and disadvantages. In the second chapter advantages and disadvantages of the machine are presented and consideration on the torque production and characteristics are discussed with the machine operating as a motor in order to allow the reader to understand the changes apllied to the machine for the function as a generator, which is discuss in the chapter 3. So after a better explanation of the principle of the machine in the second chapter, the 3 rd chapter discuss about SR machine as a generator, so the differences are described and a general concept of circuit is explained. The 4 th chapter shows the model of the generator done with Simulink describing every part of the electric circuit. At the end results and graphics are described and explained figuring out the conclusions and the future work that will be done.
The purpose of this thesis is investigate and analyze the SRmachine and its applicability in the generation of electric energy from wave energy that is not usual for the limitation that it presents and that will be discussed later. As explained in the 1 st chapter this machine has a lot of advantages compare to other type of electric machine and this is why is interesting study the possibility of its implementation instead of other type like PM machine (Permanent Magnet) etc. In wave energy applications.

Switched reluctance technology
The SRM is an electric motor in which torque is produced by the tendency of its moveable part to move to a position where the inductance of the excited winding is maximized (or the reluctance is minimized).
SRM is a type of synchronous machine that has wound field coils of a DC motor for its stator windings and has no coils or magnets on the rotor. Both the stator and rotor have salient poles so the machine is a doubly salient, singly excited machine.
Stator windings on diametrically opposite poles are connected in series or parallel to form one phase of the motor. There are several combinations of stator and rotor poles , such as 6/4 (6 stator poles and 4 rotor poles) , 8/6 , 10/8 etc. and the configuration of higher number of stator/rotor pole combinations has generally less torque ripple.

Configuration
First of all an initial classification is made on the nature of the motion that can be rotating or linear.
The linear SRMs (LSRMs) have found application in the marketplace by catering to machine tool servos.
The rotary machine-based SRM is differentiated to radial field SRM and axial field SRM by the nature of the magnetic field path as to its direction with respect to the axial length of the machine.

.2 Rotary SRM
In the radial field SRM the magnetic field path is perpendicular to the shaft or along the radius of the cylindrical stator and rotor as we can see in figure 1.3, while in the axial field SRM the magnetic field path is along the axial direction as shown in figure 1.4. hjjj

LSRM (Linear Switched Reluctance Motor)
In this type of machine the motion of the motor is linear. A LSRM may have windings either on the stator or translator (the moving part). Fixed part is called track while the moving part is called translator.Compared to its single-sided counterpart, a double-sided LSRM (DLSRM) has a more propulsion force to mass ratio and normal force can be counteracted from serially connected excitation windings and a symmetric machine structure.

Features of Switched Reluctance Machine
In this chapter is explained the advantages that will obtain with the use of a Switched reluctance machine taken from [2].

Advantages
• Less materials is needed cause windings are on the stator only, without windings or magnets on the rotor so in term of cost is also more convenient; • For the fact that windings are concentric around a pole and not distributed like ac or dc machines means a greater manufacturing economy; • The inactive part of the materials are minimize for the fact that concentric windings reduce the endturn buildup and this mean a lower resistance and copper losses compared to the distributed winding of other machines.
• The rotor is the smallest of any machine and has the lowest moment of inertia, thus giving a large acceleration rate for the motor; • It is a brushless machine and therefore is superior from a maintenance point of view compared to dc machines; •It is mechanically robust and therefore naturally suited for high-speed operation because the rotor doesn't have windings or magnets • The higher heat is produced on the stator so cooling results easier for the accessibility of the stator compared to the rotor. The rotor losses are much smaller compared to the stator, unlike the case for dc and induction machines. Permanent magnet synchronous and brushless dc machines would be comparable in this respect.
• Skewing is not required to decrease cogging torque or crawling torque, as it is for induction and permanent magnet synchronous and brushless dc machines. This machine does not produce cogging or crawling torques.
•The power density is quite similar and sometimes also higher than for induction machines but lower than permanent magnet synchronous and brushless dc machines. However this thing is true only for low speeds (i.e., below 20,000 rpm); for higher speeds, the switched reluctance motor offers equivalent or higher power density. This has to remember when will speak about the possible applications in wave energy of the machine; •Because of the electrical separation between the windings from each other and the mutual coupling induction is negligible, generally the fault of one phase has no effect on other phases. This advantage is present only in Switched Reluctance Machine. For some permanent magnet brushless dc machines, such a concentric winding with one slot pitch is being introduced and may bestow some of the benefits of the SRM.
• The absence of current in the winding produce no induced emf in the SRM, because this one is a function of the current and a phase winding fault cannot be sustained if the input current is cut off. Such is not the case for induction or permanent magnet synchronous and brushless dc machines. This leads to higher reliability in an SRM compared to any other electrical machine.
• The freedom to choose any number of phases is inherent in the SRM and lends itself to high reliability if one or more phases fails during operation. Has to remember that all the phases are electrically independent.
• The machine is an inherent position transducer, as its inductance is uniquely dependent on rotor position and excitation current. During the inactive period of each phase winding, the rotor position can be extracted by measuring the inductance. Such a feature is difficult to exploit with induction and permanent magnet synchronous machines as there is no inactive period for the windings. The rotor position information is extracted in these machines in other ways, but all of the methods are fraught with complexities in implementation and signal processing. Extraction of discrete rotor position information is possible with a permanent magnet brushless dc machine because there is an inactive period for each winding in this machine, but a direct relationship between rotor position and a measurable quantity such as inductance is not available with the brushless dc machine as it is in the case of the SRM.

Disadvantages
• Torque ripple is high but can be reduced by controlling the overlapping phase currents.
• Acoustic noise is high, but its causes are being studied and some recommendations have resulted in considerable noise reduction compared to the first-generation machines. This will be interesting when it will speak about applications under sea.
• Because of the salient rotor , friction and wind losses are high at high speeds. They can be reduced by smoothing the rotor surface by filling in the rotor interpolar space with inert material.
• The SRM requires an electronic power converter to run and doesn't have line-start capability; therefore, it is difficult to compete in an application requiring this aspect, where an induction motor is an asset. Permanent magnet synchronous and brushless dc machines have the same disadvantages of the SRM in this regard.
•The information about the position is necessary for the SRM control, but also for permanent magnet synchronous and brushless dc machines; the only exceptions are for induction and dc machines. To compete for applications requiring no position sensor and absolutely low cost, the SRM must incorporate sensorless position control. Induction and dc machines are superior in this regard, at least for low-performance applications.
• Radial forces are high at aligned positions and minimum at unaligned positions; variation over half the rotor pole pitch can contribute to faster wear and tear of the bearings if there are rotor eccentricities and uneven air gap, which is the major source of noise in SRMs. Such a phenomenon is not present in other machines.

Advantages
• The SRM requires only unidirectional current for its four-quadrant operation and there are many power converter topologies with less than two switches per phase that can be used to operate the SRM. This is also translate in less cost for the converter cause is simpler and less power switching loss for the smaller number of switches.
• Because the power switches are always in series with the phase winding and together they are in parallel to the dc source voltage, at no time could a shoot-through fault occur, as in the case of the sixswitch H-bridge inverters. This assures high reliability for this converter compared to other converters.
• Failure of one switch in a converter phase need not interrupt operation of the remaining phase legs of the converter and hence the SRM drive system operation. That is not the case with the ac motor drives under the same circumstance.
• Because the number of power switches can be reduced in the SRM drive, a considerable saving can accrue from a reduction in the number of logic power supplies and gate drivers and reduced heatsink area and volume, resulting in great reduction in the converter packaging and enclosure size. In highvolume applications, this could lead to considerable savings of transportation costs and overall material and manufacturing labor costs. Such a distinct advantage is generally not available for other ac drives.

Disadvantages
• A separate freewheeling diode for each switch is necessary in all SRM converter topologies, which cannot use anti-parallel diodes in the switches, as in the case of the H-bridge inverters. This may increase the cost if two switches per phase converter are employed for an SRM in comparison to other inverter-driven machines. This may be mitigated by reducing the number of switches and also by a special converter module developed for high-volume applications so that the overall converter cost can be reduced.

Low speed wave energy application
There are not many literature about low and medium speed operation for a SRG. The current chopping mode control method is normally used at low speed for the wind energy conversion system (WECS) application [14] . The WECS application ensures an optimum operation of the wind turbine for a particular wind speed by relying on the conversion coefficient of the turbine, but this method requires measurement of wind speed . A power control strategy was used to maintain the required load demand. Another method to control the output power is a controller to regulate the turn on angle whilst the turn off angle is kept constant [15]. The resources on the application of the machine for the control strategies in wind energy are still limited. To keep constant the output power the machine has to follow a set of reference value power and this fact limit the potential of the machine to convert energy. Based on this the application of this machine to the wave energy will be interesting, also cause often a wind turbine is required. Ocean waves are a huge, largely untapped energy resource, and the potential for extracting energy from waves is considerable, but is relatively immature compared to other renewable energy technologies [6]. This renewable energy is characterized by the fact that , like wind energy and solar energy, is an intermittent source of energy. This mean that it is not able to product continue and constant energy but is characterized by irregular and unpredictable profile. In figure 5.1 is possible to see the difference between intermittent and not energy. The advantages of non-intermittent energy is that they are able to product every time energy at the rated power. This is good for the economy of the plant cause is exploited always at the maximum or a high power. For the intermittent energy source is different; the cost of the plant is determined by the rated power (the maximum power that can be extract) so is not connected to the effectiveness energy product. So the nominal power shouldn't fixed too high cause the cost will be high and the production of the power will never be like that cause the natural resource are limited and intermittent.

Benefits of wave energy
Using waves as a source of renewable energy has more advantages compared to the other type of generating energy.

Wave energy conversion
The devices for the energy conversion from the wave energy are called wave energy converters and they can be classified in different way: from their position, type and from their modes of operation.

Classification from the position
The energy carried from the waves can be influenced from different factors like the distance from the shore; one first classifications of WECs (Wave Energy Converters) is: c) offshore devices.

Shoreline devices
The advantages of shoreline is that they are close to the utility network so they can be moored on the seabed where the water is not deep or fixed to the rocks on the shore. They are easy to maintain , the access to them is every time possible and this mean a reduction of installation costs. As waves are attenuated as they travel through shallow water they have a reduced likelihood of being damaged in extreme conditions. Other advantages are that don't need deep moorings and neither long cables for the transport of the energy. However the biggest disadvantage is the quantity of energy from the wave that generally is lower near the shore.; this can be partially compensated by natural energy concentrated locations, called hot spots.

Nearshore devices
Nearshore devices are defined as devices that are in relatively shallow water (~10÷20 meters) and not so far from the shore (~100 ÷ ). Devices in this location are often moored to the seabed, which gives a suitable stationary base against which an oscillating body can work. This typology is a compromise between the others 2 configurations cause it tries to avoid the disadvantage of the shoreline being in more deep water where waves have more quantity of energy, and doesn't need too long moorings. On the other hand the installation cost results higher and devices have to hold up the higher load from the waves.

Offshore devices
Offshore devices are generally in deep water (usually more than 50 meters) so they can harvest the largest amounts energy because of the higher energy content in deep water. However, devices are more difficult to construct and maintain, and need to be designed to resist against more extreme conditions adding cost to construction because of the greater wave height and energy content in the waves. Other problems are found in the high loss for the length of cables to transfer the energy, and the interference that can create to navigation. Despite this, with more powerful waves floating devices in deep water offer greater structural economy .
A useful thing to know is that the higher part of wave energy contained in the movement of the water is closed the surface of the sea: up to 95 per cent of the energy in a wave is located between the water surface and one-quarter of a wavelength below it .

Type classification
Wave energy converters can be classified in three predominant types :

Attenuator
This type of converters lie parallel to the main wave directions so they ride the waves. An example of this type is the Pelamis, developed by Ocean Power Delivery Ltd. Later will be explained this technology more specifically.

Point absorber
The wave directions for this kind of devices is not important because of their small size adapted to the incident wavelength . They can be submerged under the sea surface exploiting the differential pressure, or they can float on the surface going up and down on the water.

Terminator
These devices lie perpendicular to the predominant wave direction so physically they intercept waves.

Modes of operation
Another classification can be done on the modes of operation on what the energy conversion is based:

Oscillating water columns
These converters consist in a chamber with an opening to the sea below the waterline where the level of the water goes up and down following the see waves profile. This variation of water level is translate in a differential pressure inside the chamber that generally activate the turbine.

Hinged contour devices
This type of device follows the motion of the waves; it creates power using the motion at the joints. It is commonly moored slackly to hold it in place.

Buoyant moored devices
This type of device floats on the surface of the water or below it. The energy is extracted during the oscillations from the bottom to the top or the opposite. It is moored to the seabed by either a taught or loose mooring system.

Overtopping devices
An Overtopping device captures sea water of incident waves in a reservoir above the sea level, then releases the water back to sea through turbines [10].
1.5 Existing technology applicable to SRG

Point absorbers
The point absorber is a type of wave energy device that could potentially provide a large amount of power in a relatively small device, compared to other technologies. While there are several different designs and strategies for deploying these types of devices, they all work in essentially the same manner. Point absorbers are relatively small compared to wave length, and may be bottom mounted or floating structures. The vertical motion of the buoy is utilized to alternate the compression of a gas or liquid in some form of container, converted into rotational movement of the power generator, or converted in other similar ways.

Single sensor
In these type of systems a buoy oscillate for the wave motion respect of a fixed reference system. One type can be a buoy connected with a cable, in tension by a spring, to a structure fixed on the bottom of the sea.
The relative motion between the buoy and the bottom of the sea supply a machine for the conversion and this one can be an LSRG (Linear Switched Reluctance Generator)

Double sensor
The single body structure has the limit of the distance between the surface and the bottom of the sea, moreover this distance is not constant but can vary by the tides. A different way was thought to use the relative motion between two structures oscillating in different manner.

Archimede Wave Swing (AWS)
According to Archimedes principle, when an object is emerged in a fluid, a force acts on the object which forces it upwards. This force is known as the buoyant force and is equal to the weight of the fluid displaced by the object. The weight of the displaced fluid can be found by the formula W = mg, where 'm' is the mass and the 'g' the gravitational force.
It is to note that the buoyant force does not depend on the weight or shape of the object, but on the mass of the fluid displaced. Also, when the density of the object is equal to the density of the fluid, the object will neither float or sink.
The Archimedes wave swing is a remarkably simple and elegant solution to tidal power generation. This wave energy converter is a cylindrical shaped buoy which is submerged and tethered to the ocean floor. Moored to the seabed, this generation unit has got only one moving part, the floater unit. The floater unit is an air-filled device which is connected to a lower fixed cylinder.
The Archimede Wave Swing is a system based on the undulating movement completely submerged developed in Holland. It consist in an oscillating top part and a fixed base. The buoy is pushed to the bottom by the water wave weight when it pass on the structure during its peak. When the peak is passed the buoy moves to the surface. The motion supply an LSRG (for example) while the air contained in it react as a spring pushing the buoy to the surface.

Oscillating water column (OWCs)
This is the relevant devices in once that use a turbine for the energy conversion, with the largest number of installations and life expect. The idea was born in 70's. As already said and shown in figure 1.10 this device consist in a chamber moored to the seabed and partially submerged with an opening to the sea below the waterline. Waves produce inside the chamber a variation of water level that is reflected in a variation of air pressure inside the chamber. This variation of pressure activate a turbine both when the water level is going up and down. The bidirectional flow of air involve that the turbine has to be realized in order to turn in the same direction, independently from the wind flows. For this reason usually Wells turbine are utilize in OWCs. The dimension of the chamber is important and has to be done based on the location where the device will be install, so in relation of the wavelength and typical period of waves that will find. OWCs can be installed onshore preferably on rocky shores; nearshore in up to 10m of water; or offshore in 40-80m deep water. They can be moored on the seabed or connected with a feeble fixing that consent a reply to the variation of water medium level. The moored devices contained the largest amounts of energy cause the resistance to the waves from the device is higher. However if there will be a high variation of the average sea level some problems with this devices can be presented, while feeble fixing consent a major flexibility. Also the cost are different cause, as easy to understand, the mobile devices are cheaper than fixed, cause they don't need strong foundations that need a fixed one. The OWCs category contain different devices inside itself but everyone is based on the principle of Oscillating water column. There are two big OWC in Europe:   When they are aligned , the stator current in the first phase is turned off and the corresponding situation is shown in the figure above. Now the second phase of the stator is excited pulling the rotor poles toward it in a opposite clockwise direction. This because as told before , the moving part of the motor tends to move in a position of minimum reluctance that corresponds to the aligned situation.

Torque production
The torque production in SRM can be explained using the elementary principle of electro-mechanical energy conversion [1].
During the conduction of a phase the torque tends to move the rotor in the direction that minimize the reluctance , causing at the same time an increment of the phase inductance as already said. So a positive torque can be obtained supply each phase when the inductance value is increasing. This process is explain by the principle of virtual works and it's possible to find a relationship between the electromagnetic torque and the variation of magnetic energy. In particular Where is the electromagnetic torque , ′ is the magnetic co-energy that represents the area under the magnetization curve at the position * from = 0 to = ( fig. 1.12) Where is the flux linkage.
In the same way is define the energy storage in the magnetic field as The inductance profile shown in figure 1.11 has a value during the unaligned position that seems zero but will be a different value, less than the aligned position but not zero.
So not all the energy is converted in mechanical energy but there is some energy that is stored in the field.

Energy Ratio
The motor is better exploitation if the difference of flux from the unaligned position and the aligned position is higher. This is obtained if the ≫ and if there is a high saturation of the magnetic circuit for high values of currents. If we consider one complete rotation of the machine, the work done is encloses inside the curve in the plane − .
is the not converted energy stored. The energy ratio is defined as = +

If
is high the exploitation of the machine is higher.

Figure 2.8:
W is the energy converted in one complete rotation ; R is the energy that will not be converted

Linear condition
In linear condition the curves − become straight lines [1]. In this condition we have And the relation of the torque becomes  Some considerations on the torque can be therefore outlined: a) It's proportional to the square of the current and hence, the current can be unipolar to produce unidirectional torque.
b) Since the torque is proportional to the square of the current, it has a good starting torque.
c) Because the stator inductance is nonlinear, a simple equivalent circuit development for SRM is not possible.
d) The torque characteristics of SRM are dependent on the relationship between flux linkages and rotor position as a function of current. Region1: is the region below the base speed ω b , which is the lowest possible speed for the motor to operate at its rated power. For the small back-emf in this region the current can be set at any desired level by means of regulators such as hysteresis controller or voltage PWM controller.

Torque-speed characteristic
Region2: is the region where the controller maintains the torque inversely proportional to the speed. In this region, the phase excitation time falls off inversely with speed and so does the current. Because torque is roughly proportional to the square of the current, the rapid fall in torque with speed can be countered by adjusting the conduction angle q dwell . By advancing the turn-on angle to increase the conduction angle until it reaches its upper limit at speed ω p , the phase current can be increased effectively to maintain the torque production at a high level.
Region 3: In this region, the q dwell upper limit is reached when it occupies half the electrical cycle. The torque in this region is governed by natural characteristics, falling off as 1/ 2 .

Equivalent circuit
Neglecting the mutual inductance between the phases is possible to find an elementary equivalent circuit for the SRM as well explained by Krishnan [2]. The applied voltage to a phase is equal to the sum of the resistive voltage drop and the rate of the flux linkages: where is the resistance per phase, and is the flux linkage per phase : where L is the inductance dependent on the rotor position and phase current. The last term may be write in a different way to be physically interpretable: The air gap power is the product of the electromagnetic torque and rotor speed: = from which is possible to obtained the torque (in linear condition) by equating these two equations:

The converter
The torque in SRM drives is independent of the excitation current polarity; for this reason the SRM drives require only one power switch per phase winding. In the following only one type of converter is considered, the same one which is used in the simulations [2].

Asymmetric bridge converter
The figure that we consider shows only one phase of the SRM because the rest of the phases are similar connected.

Figure 2.12: Asymmetric bridge converter
Turning on transistors T1 and T2 enable the supply of the phase A by the DC voltage Vdc. When the current rises above the commanded value, T1 and T2 are turned off. After this the energy stored in the motor winding of phase A will keep the current in the same direction until it is depleted. Diodes D1 and D2 will become forward biased leading to the source and a simultaneous current decrease reaching the value below the reference one. If a current Ip is desired during the positive inductance slope for motoring action, the A-phase current command is generated with a linear inductance profile. Phase advancing both at the beginning and during commutation are neglected. The current command is controlled with a feedback loop in which the current reference value is compared with the real phase current. The current error goes through a hysteresis controller with a current range of Δi. When the current error exceeds −Δi, the switches T1 and T2 are turned off simultaneously so diodes D1 and D2 take over the current and complete the path through the dc source. The voltage of phase A is then negative and will equal the source voltage, Vdc. During this interval, the energy stored in the machine inductance is sent to the source, thus exchanging energy between the load and source repeatedly in one cycle of a phase current. However this control strategy (strategy I) causes more ripples into the dc link capacitor that means that its life is reduced and also the switching losses are increased due to frequent switching. These can be ameliorated with an alternate switching strategy. This second strategy can obtain (strategy II) Turning off T2 only: the current will continue to flow through T1, phase A, and D1 and the voltage across the winding becomes zero. The current will pass from Ip+Δi to Ip−ΔI in a time greater than using the previous strategy and also the switching frequency is reduced and hence the switching losses. When the current command goes to zero, T1 and T2 are turned off simultaneously. During this interval , the voltage across the winding becomes -Vdc as long as D1 and D2 conduct and thereafter the winding voltage is zero. 3The Switched Reluctance Generator

Principle
The switched reluctance machine can operate as a generator by changing the firing angles [3].
In generating operation the angles are chosen in order to have the flowing of the current when < 0.
The circuit equation for one phase become: is the back-emf and = is the rotor speed as already said in the first chapter. Is easy to understand that the sign of is determined by ⁄ . When ⁄ < 0 the back-emf is negative and it tends to increase the current and convert the mechanical power into electrical power. The figure 2.1 shows this concept of the different supply during the inductance profile.     Between and the current is flowing through the phase so the transistors T1 and T2 are closed and the diodes are off while after the current is turned off so T1 and T2 are opened and diodes D1 and D2 are on cause the current close its circuit through them..
The peak of current is reached at or 1 .In the case (a) the current increases after turning off the switches at , when the back-emf in the coil is greater than the d.c.-bus voltage .
In the second case (b) the back-emf and have the same value so the current is constant until the pole overlap ends in 1 .In the third case (c) the back-emf is smaller than so the current decreases after .
From to the d. c. power source supplies the excitation power through the converter to the machine, and it's stored in the airgap as magnetic energy. After the switches are turned off regenerating current flows through the freewheeling diodes returning the generated power into the d. c. power supply until the current vanishes at .
If the generated power is greater than the excitation power from the d. c. bus-voltage, the system has net generated power by converting mechanical power into electrical power. The power generated will flow to the grid so the system needs an inverter to convert the power from DC to AC.
In many application, like the model that will be explained in the next chapter, all or part of the energy is stored inside batteries.
The waveform of the first case (a) has the smallest and (c) the largest. If the net generated current is the same for all three cases , the waveform of (a) is preferred cause the smallest reduces losses.
In the figure 2.6 the loci of the current on the − plane is shown , the current peak value being the same. The energy converted is proportional to the area enclosed by the loci. When the peak value is limited , (b) is expected to generate the largest energy.   When = at the rated current , is called the base speed (Miller, 1993) and in this case the current is constant during the period from to 1 provided > .
The current waveform of SRG may be controlled as follows:  Like the case (a) when the generating current is small and the peak current is less than its maximum value, in order to minimize the losses. is maintained at the nominal value.
 Like the case (b) after the peak current reaches its maximum value in order to utilize the maximum energy available for a given maximum current.

Inductance profile for different models machine configurations
Is now explained a description of inductance profile to enable a proper definition of the firing angles for different type of machine that will be used in the simulation model.

Switched Reluctance machine 6/4 (6 stator poles, 4 rotor poles)
Let consider a switched reluctance machine 6/4 (6 stator poles, 4 rotor poles) with 3 phase windings on the stator, one every 2 poles. Now to describe the inductance profile we have to consider that at the initial condition the rotor poles are aligned with the stator poles of the first phase like in the figure 2.9. So the initial conditions for each phase will be the angle between the stator and the corresponding rotor poles. The stator pole arc is 2 ⁄ while the rotor pole arc is 2 ⁄ so they will be respectively 60° 90°.This mean that the initial condition for the first phase will be 0°, for the second -30 ° and for the third -60°.
The inductance reproduces the rotor reluctance variation. This means that, being = 90°, between the maximum and the minimum value of inductance (aligned and unaligned position) there are 45° (90 2 ⁄ ).
The approximate inductance profile in linear condition for the first phase is reported in fig. 2.12 The profile for the second and the third phase will be exactly the same shifted by 30 and 60 degrees.
As in generating operation the firing angles are chosen so that the current flows when ⁄ < 0, so in this case the firing angles will be chosen between 0° 45°.
Then the conducting intervals ( − ) can be larger or smaller [4] [5]. A large conducting interval results in high peak flux-linkage value and consequently increased iron loss. A reduce conducting interval means that iron loss is reduced, however peak and rms current are increased and consequently copper loss is increased. Optimal efficiency operation is attained at a best balance between iron and copper loss.
Being the step angle And calling , the rotor polar arc and the stator polar arc, the conditions to impose in order to have eligible value are : The first one is necessary to avoid torque holes in a complete rotation, that are intervals where the machine doesn't produce torque.
The second one is necessary to avoid that one stator pole could overlap with two rotor poles at the same time because this situation would provide magnetic path that don't produce torque.
The third one is necessary for construction needs and also, with the second condition, provide the maximum ratio between the aligned and unaligned inductance that maximize the torque production.
After this definitions is possible to understand that the inductance will have the aligned inductance value for all the aligned period that is proportional to − .

Switched Reluctance machine 8/6 (8 stator poles, 6 rotor poles)
Such configuration consists of 4 phases with two stator poles simultaneously excite to enable the alignment of two rotor poles.  The others phases will have the same profile of inductance translate of −15°, −30°, −45° .

Switched reluctance machine 10/8 (10 stator poles, 8 rotor poles)
Such configuration consists of 5 phases with two stator poles simultaneously excite to enable the alignment of two rotor poles.  The other phases will have the same profile of inductance shifted by −9°, −18°, −27°, −36° .

SRG system modelling 4.1 Implementation
In Figure 3.1 is shown the general layout of the SRG without the connection to the grid side that is implemented inside the converter block for space reasons. In this paragraph is explained all the parts of the model, later also the simulations results are shown and the grid side and control is described. As is possible to see in figure 3.1 the machine is a 6/4 poles. The model can be divided in 4 main parts: the external voltage source, the converter, the machine, the position sensor; is now described each one of these part.
The simulations and analysis are done assuming a constant power, torque and speed as in steady state conditions. The model will be improved later in order to conduct the analysis with different torque load starting from an initial speed of zero but results are not discuss in this thesis cause will be a future work where will be possible to apply the waves torque profile.
The mechanical torque input TL provide an angular speed to the shaft, being: = So from the mechanical characteristic of the machine is possible to study it at different speed. At the beginning the simulations are done with constant value of load torque so with a constant angular speed. Another group of researcher is studying the wave profile in order to know which angular speed provide to the shaft. The other input is the current that is necessary through the phases to create the magnetic field; this is provided at the beginning from an external DC source voltage that will be disconnected when the capacitor is completely charged and can so becomes itself the DC bus voltage .

POSITION SENSOR
As described in the 2 nd chapter the machine has 3 phases on the stator poles and the current is provided in each one during the negative slope of ⁄ . So fixing the firing angle between 0 and 45 degrees the position sensor block provides the pulse for each phase. The angles 0 and 45 are the value for the first phase but summing these with the initial conditions as explained later also the value for the second and the third phase will figure out.  The angular speed [rad/s] is multiplied to a constant value, 180/π, to find [grad/s] that is integrated to find the angle of each phase between the phase and the closer rotor pole. This means that the output will be 3 values, the measures of the rotor poles position respect each phase. To find these the integrator needs a number of initial conditions equal to the number of phases. The machine used for the simulations and analysis is a 6/4 model and it has three phases, so considering the rotor pole alignment with the stator pole of the first phase, as already explained at 2.2, the initial conditions will be 0°, −30°, −60°. The next figure shows the positions of the rotor poles respect each phase: From the figure is possible to see that the rotor pole respect of the first phase is aligned at the beginning of the simulation, while the one respect the second and the third phases are out of phase respectively of -30 and -60 degrees. The angles can change from 0 to 90: when they reach this value the mod block (modulus after divided) divide them per 90 (the rotor angle) and the value restart from zero in order to use the same value of theta on and theta off, otherwise we should have to change the theta on and theta off angle values every time. Now the angles become the input signals in the Relational Operator block that compares two inputs using the Relational operator parameter that you specify. The first input corresponds to the top input port and the second input to the bottom input port.
<= TRUE if the first input is less than or equal to the second input >= TRUE if the first input is greater than or equal to the second input So if the angles are greater or equal than alfa and lower or equal than beta the output sig is 1 otherwise it will be 0 ; this is because the logical operator AND is TRUE (1) if all inputs are TRUE. Later the logical operator will be change in OR to be able to anticipate the theta on angle. The  The output signal of the sensor position goes inside a product with a constant value that is the reference current to be imposed in the windings. This value is compared with the real value of current that is feed-back from the output of the machine in order to maintain the current at a desired value. The error is now the input of a hysteresis control that output the specified 'on' or 'off' value by comparing the input to the specified thresholds. This signal command the switch on/off of the transistor inside the bridge converter in order to increase or decrease the current value. When the difference between the currents become +10 (the real current is lower than the reference value) it switch on and the current starts to increase. When it becomes -10 (real current higher than reference value) it switch off and the current start to decrease. The output when on is 1, when off 0 and it will control the gates of transistors of the converter. The current is provided from an external dc bus voltage how can we see in the general layout and as already said is disconnected when the capacitor across the bridge is charge so after the machine is autosupport.  The converter has three arms connected to the phases of the machine and in parallel to a capacitor where the energy is stored during the generating operation. As already said, when the capacitor charge is complete, the external dc bus voltage is disconnected and the capacitor becomes the source; this operation is implemented with two IGBT commanded by a timer as shown in figure 3.8.

CONVERTER
The figure 3.9 shows one of the converter arm: The gate of transistors is commanded by the output of the relay. When the output is 1,the transistor are turned on and the current flows inside the correspondent phase, C1 and C2. This create a back-emf during the negative slope of ⁄ . They are switched on and off to maintain the current at the reference value and when the angle reaches the value, the transistor are switched off and the back-emf charges the capacitor that is connected in parallel, between D1UP and D1DOWN. Figure  3.10 shows one cycle of one phase of the machine. The first profile is the position of the rotor pole respect to the first phase called theta angle; the second is the output of the sensor position that is one during the firing angle period that at the beginning was fixed at 0-45 degrees; the third is the current that flows in the first phase. This one is maintained at 250 A switching the transistors commanded with the output of the relay for the first phase (the fourth profile). As said the analysis is conduct in steady state conditions assuming a power of 60 [kW] (the machine is a 60kW SRG) and using a constant torque load. Later will be described the model with the integration of the mechanical block.
The input, , is scaled to the deg/s unit inside the position sensor block ; afterwards it is integrated in order to find the angle.
The circuit equation for a phase is

= +
The voltage drop is subtracted to in order to find ⁄ . Now with the angular speed is possible to calculate the current with the ITBL. At the end the current is used to calculate the electromagnetic torque and multiplying it per is possible to find the output power without losses.

Determination of optimum switching angles
The theta-on and theta-off angles play the major role in deciding whether a SRG develops positive or negative, high or less electromagnetic torque [5]. These angles can be considered function of speed. As explained in the previous chapter the electromagnetic torque is so apparently the angular speed has no effect on , cause the current is maintained constant with the switching at the reference value and the variation of the inductance respect of theta, ( , )⁄ , has every time the same profile, but = so it means that there is a relation between them. This can be explained with a sample reasoning. If is higher it means that the changing of positive and negative slope of is faster so the current flows through the phase for less time. When the transistors gates switch on, the current starts to flow through the phase , but its initial value is zero and need time to reach the reference value (250 A in this case) ; but this time is every time the same cause is the duration of the transitory. So if is high it means that angle has to be anticipate if the peak of current has to be reach when the negative slope starts. Moreover the current has a transitory time when is reached to come back to zero value when the negative slope is finished. So with high value of the theta off angle has to be anticipate (before the end of the negative slope) . This is necessary because if the angle is maintained like in ideal conditions so like if the current pass from 0A to 250A instantaneously , there is still current flowing through the phase during the positive slope and the electromagnetic torque pass from a positive value to a negative value in few time and every cycle. This is translate in less production of power, double side power flow unwanted and magnetic efforts that can damage the machine. After this discussion is now possible to understand how the firing angles can be evaluated. As shown in figure 3.14 and 3.15 there is a reference range of theta-on and theta-off:  The effect of changing firing angles is evaluated in terms of different performance indices but depends on the type of control that is wanted. For example if a speed control is the objective the performance indices, as well explained in [5], a potential strategy to be applied consists of the following specifications :  Time required by the machine to reach within 1% of rated speed;  Speed overshoot over and above 101% of reference value;  Speed undershoot under and below 99% of reference value;  Time required to reach and stay within 1% of rated speed;  Steady state speed ripple (peak to peak) on full-load;  Steady state torque ripple (peak to peak) on full load.
Moreover this thesis is initially studied with simple parameters so with constant angular speed and this approach is not applicable. A simple evaluation is done, looking the test results with a constant angular speed and choosing the angles that give the maximum output power but without inversion of torque sign. After several simulation the best angles is found in = -30 and = 40. For the mod block inside the sensor position that as explained permit to have the same value of theta angles, the turn on angle becomes = 60 cause is not possible to have negative value cause the measurement of theta goes from 0 to 90 degrees. To have this combination of angle is necessary to change the logical port in the sensor position from AND to OR: Now the output of the logical port is TRUE if at least one input is TRUE. It means that when theta is higher or equal than 60 one input is true, the one of alpha, when theta is from zero to 40 the second input becomes true. This block needs to be modify if a control speed is the objective cause every time that alpha condition change from zero to a value lower than beta the AND port is necessary. So the OR port is used when the angle needs to be anticipate from 0 while the AND port is the correct one if the angle value choose is zero or retarded from zero. The next figures show the improvements of the machine trying to find the correct firing angles from the initial situation :   It must remember that the output power shows in these pictures is not the output power that is flowing to the grid but is the electromagnetic output power storage in the air gap before the electric losses.
Simulations start from the initial condition of a firing angle of = 0 ÷ 45° where, considering the same period for every simulation, is not in steady state condition yet and it takes a lot of time cause the angles are of an ideal condition that neglect the mechanical closing time of the switches and the time to reach the maximum value. Finding the correct solution is possible to see that the output power increase but also the ripple because of the torque ripple but decrease the delta cause the power is changing at the beginning from 15kW to -15kW, after has a delta of 20kW and with the correct solution delta is 10 kW as shown in figures.
The next table shows the power that can be delivered to the grid by a parametric analysis involving the value of and in order to find the optimum angle for this machine at the constant rated value of rotor speed . Here are reported only the mains combinations that were more close to the solution but several simulation were conducted. So it's possible to see that anticipating the turn-on angle and the turn-off angle is translate in an increase of output power but anticipating too much decrease again the output power and increase also the ripple.

Grid side
In this paragraph is explained the grid side configuration referenced to figure 3.23:

Figure 4.24: Grid side configuration
The capacitor is connected to an inverter that convert the power from DC to AC for the power delivery to the grid. A line is simulate with an inductance L that is connected to the grid block that represents a three-phase voltage source with an inductance and a resistor in series as source impedance cause the grid can always be seen as an inductive load, seeing most of consumers are resistive-inductive. The parameters of the grid are chosen seeing that publication ratings on grid side are similar to ours. Also the power flow to be in the correct direction needs the grid voltage similar but a bit lower than capacitor voltage. The parameters are choose as To evaluate the amount of power that is possible to transfer from the inverter to the grid is necessary to elaborate the equation representing the situation that is shown in figure 3.24: The Grid1 represent the voltage source out of the inverter, the three-phase RLC branch represent the line and the Grid2 the real network. As known: Where: i. ∠ is the current flowing from the inverter to the grid; ii. ∠ is the inverter voltage; iii.
∠0 is the grid voltage; iv.
∠90 is the line impedance.

GRID CONTROL
As already said the objective is to maintained the voltage on the capacitor at a constant value of 240 [V] because it becomes the source voltage that have to supply the machine. This mean that when the voltage exceed the reference value of 240[V] is possible to extract more current and so power, when the voltage is lower than the reference value it means that is necessary to extract less current so less power. The control configuration adopted in this thesis is based on a balanced sinusoidal waveforms generator in which modulation index and phase shift are driven through two PI controllers as shown in Figure 3.25 increases it means more power should be extracted from the capacitor so the amplitude of grid current should increases too. A way to let grows is to increase voltage drop on grid inductance , and it can be done increasing through an enlarge value of . On the other hand, for the Power Factor correction, must be controlled in a way it brings parallel to that has zero phase so it means that ⟶ 0°. This can be done moving the voltage drop perpendicular to ⃗⃗⃗ through an equivalence shift of .
At the beginning the situation is what figure 3.26 shows:  As shown in figure 3.27 an increase of Vdc means that the drop voltage is not perpendicular to anymore(that is constant) and so the current has an angle ≠ 0 that cause reactive power ≠ 0 (< 0). To extract more power is necessary to bring again the current in phase with the grid voltage and to do it an increase of m is necessary.  Because now is greater for the increase of and also the angle so the sine of the angle.
Following the same way is easy to understand the opposite situation: if the voltage on the capacitor is lower than the reference value of 240 [V] means that less power has to be extract and that the angle delta becomes lower so the new configuration becomes: .30 shows that a decrease of Vdc means that the drop voltage is not perpendicular to anymore(that is constant) and so the current has an angle ≠ 0 that cause reactive power ≠ 0 (> 0). To extract more power is necessary to bring again the current in phase with the grid voltage and to do it a decrease of m is necessary in this case:  The power will be:

= sin < 41 [ ]
Because now is lower for the decrease of and also the angle so the sine of the angle. Figure 3.33 shows the control part of delta angle , while the all control is shown in figure 3.25: The first input is the three phase voltage grid measure; the second one is shown later in the control of modulation index and is the measure of three phase current to the grid; the third input is the measure of the voltage on the capacitor.

DELTA CONTROL
The measure of the voltage goes inside a sum block with the reference value of the voltage that wants to maintain constant. The error, that can be positive or negative , is going inside a proportional integrator PI and the output is sum with a constant value fixed at 30 degrees in order to have always a positive value of delta. The switch is at the beginning turn on a constant reference value of delta of 28.42 degrees and after 0.3 seconds is turned off and connected to the control. This is choose because until 0.2 sec the capacitor is charged by the external voltage source and becomes the source after but requires few time to arrive in steady state condition. So the delta output flows inside a three different blocks and after inside gains that transform degrees in radiant. In this way a three phase sinusoidal equilibrate voltage is created and goes inside the final block that is the product with the modulation index. The first input, the three phase voltage grid measurement, flows inside a discrete 3-phase PLL that provide the value of that will be sum with delta. In figure 3.34 are shown the profile of delta, the PI output and the error.   The second input as already said is the three phase current to the grid measurement. This is demux in order to extract the signal of one phase; this flows inside the discrete Fourier block that provide the output that is the phase of the current . The signal flows inside another PI that gives an output value that, sum with a constant value fixed at 0.5, gives the value of modulation index . As shown in figure 3.35 also here at the beginning there is a switch on a constant value of 0.735 (ideal condition) until when the external voltage source is disconnected and the capacitor reaches the steady state conditions, at 0.65 sec. Figure 3.36 shows the profile of the modulation index , the PI output and the phase of the current (with opposite sign):

Test results
In this chapter all the simulation results obtained with the SRG model with MATLAB® SIMULINK® are shown and discuss. All these results are referring to a 6/4 machine which parameters are now presented. In this example a DC supply voltage of 240 V is used. The converter turn-on and turn-off angles are kept constant at -30 (60degrees) deg and 40 deg, respectively [3.3]. The reference current is 250 A and the hysteresis band is chosen as ∓10 A.

Current-controlled mode
From stand still up to about 3000 rpm, the motor's emf is low and the current can be regulated to the reference value. In this operation mode, the average value of the developed torque is approximately proportional to the current reference. In addition to the torque ripple due to phase transitions, we note also the torque ripple created by the switching of the hysteresis regulator. This operation mode is also called constant torque operation

Voltage-fed mode
For speeds above 3000 rpm, the motor's emf is high and the phase currents cannot attain the reference value imposed by the current regulators. The converter operation changes naturally to voltage-fed mode in which there is no modulation of the power switches. They remain closed during their active periods and the constant DC supply voltage is continuously applied to the phase windings. This results in linear varying flux waveforms as shown on the scope. In voltage-fed mode, the SRM develops its 'natural' characteristic in which the average value of the developed torque is inversely proportional to the motor speed. Since the hysteresis regulator is inactive in this case, only torque ripple due to phase transitions is present in the torque waveforms.
The analysis is done using the current-controlled mode with a speed constant of 3000 rpm as already said. The simulation graphics show the results with a run time of 1.5 sec that is enough to reach the steady state conditions, considering the detachment of the external source and the connection of the capacitor as new voltage supply.
The next figure shows the complete profile of the main machine parameters: the flux-linkage , the current through the phases , the electromagnetic torque and the rotor speed that as already said is maintained at a constant value of 314 [rad/sec] that is 3000[rpm] , the machine rated speed. The first one at 0.2 sec as better shown in figure 5.2 ; there is an increase of the flux linkage and a decrease of the electromagnetic torque for a short time before to reach the same value of the beginning. This fact is caused by the disconnection of the external voltage and the connection of the capacitor like voltage supply; when the capacitor is charged (so when there is the connection) has a voltage value greater than the initial supply of 240 [V] and need time to reach that value with the control voltage. It's possible to see that at the moment of the connection the overlapping of the currents becomes greater for a short time and so the ripple of the electromagnetic torque as well, but in 0.01sec the steady state initial condition are re-established . The second instability, as shown in figure 4.3, starts at 0.3 sec. At this time the grid control is activate so the reference voltage value of the PWM change from an initial ideal value to the real one for the capacitor control voltage. This instability is caused by the fact that the power that is flowing to the grid is at a constant value with 0 reactive power but with an initial open loop so when the control is closed, the cycle need a short time to reach the correct value that the machine had before.  The capacitor is initially charged with the external voltage source until 0.2 sec when is reached its maximum value; at this point the external voltage source is disconnected and the capacitor becomes the supply voltage of the machine so the voltage on it decrease until 240 [V]. The exceed energy is delivered to the grid by the control on the capacitor voltage. At 0.3sec there is the second instability because of the connection of the grid control as already said. After less than one sec the correct value is re-established. Zooming on the profile is possible to see the continue charge and discharge of the capacitor around the reference value of 240 [V].          is the electromagnetic torque ;  is the load torque ;  is the inertial torque with J inertial momentum ;  = is the friction torque.
In the generating operation the load torque must be negative ,with the correct firing angles in order to extract power. Another improvement will be the insertion of mechanical characteristic and magnetization curve of a real machine in order to find results for an existing model , where saturation must be included and so conducting analysis in non-linear conditions. Simulations and results are not discussed in this thesis but could be the beginning of the next step.

Final considerations and future work
Created the model of the switched reluctance generator SRG that can be adopt different control strategy, it was possible to find an optimal condition for a constant speed in order to provide the maximum amount of power delivered to the grid. The conditions figured out is that for a 60 The analysis of the Switched Reluctance Machine done in this thesis is only the beginning of the final objective that is the study of the applicable of it in wave energy. This involve the study of the load torque profile extractable from wave energy and as a variable load causes a variable speed, a digital control on the firing angle must be implemented in order to find for every speed the optimum angle to extract the maximum power in every condition. This could be find basing on the principle of MPPT (Maximum Power Point Track) already used in other applications like photovoltaic. A possible way could be a program that try for every speed all the possible combinations finding a power function (if the maximum power extractable is the objective) that will increase until the reaching of a peak, after that it will starts to decrease so it means that the optimum angles are found and are the combination that provide the peak of the function.
Other characteristics of the machine must be discovered like the extractable power with phase failure for its capacity of working also in this condition for the independent of phases each other , and the study of the same machine with a different number of rotor and stator poles, based on this model. Some doubts about the applicability of SR machine has to be solved about the strong torque ripple that has to be demonstrated being admissible.
The study of this machine and the applicable of it in wave energy should be an interesting and clean way to produce energy in a different way, moreover convenient as already said at the beginning in the 1 st chapter compared to the utilization of other electrical machines more expensive.
Grazie a tutti i compagni di Padova, di corso e non , senza i quali non ricorderei questa esperienza con nostalgia ed io non sarei la persona che sono diventato.