development of a variable electromotive-force

DEVELOPMENT OF A VARIABLE ELECTROMOTIVE-FORCE GENERATOR WITH AN ACTIVE CONTROL SYSTEM W. D. Zhu Mechanical Engineering Department University of Maryland, Baltimore County Baltimore, MD 21250 [email protected]

N. Goudarzi Mechanical Engineering Department University of Maryland, Baltimore County Baltimore, MD21250 [email protected]

X. F. Wang Mechanical Engineering Department University of Maryland, Baltimore County Baltimore, MD 21250 [email protected]

P. Kendrick Mechanical Engineering Department University of Maryland, Baltimore County Baltimore, MD 21250 [email protected]

ABSTRACT A variable electromotive-force generator (VEG), which is a modified generator with an adjustable overlap between the rotor and the stator, is proposed to improve the efficiency and/or expand the operational range of a conventional generator, with particular applications to wind turbines, hybrid vehicles, and so on. A mathematical model of the VEG is developed, and a novel prototype is designed and fabricated. The performance of the VEG with the active control system, which adjusts the overlap ratio based on the desired output power at different input speeds, is theoretically and experimentally studied. The results show that reducing the overlap between the rotor and the stator of the generator at low speeds results in a reduced torque loss of the generator and an increased rotational speed of the generator rotor. NOMENCLATURE A Rotor swept area - m2 B Magnetic flux density - Wb/m2 CP Power coefficient ds Change in the magnetic field length - m E electromotive force - V F Factor to condition the normalized output power I

Current - Amp 1

Currently works at Bell Helicopter Company

1

lf N P Pem Pg Ploss Pr/Ppm Q R T Tpm U V X Y Z δ ηmech θ ω Ф 



Width of the magnetic field - m Number of coils Normalized output power - W True power of a synchronous generator - W Output generator power - W Power loss - W Output rotor power - W Reactive power– VAR Resistance – Ohm Torque – Nm Prime mover torque – Nm Wind speed- m/s Terminal voltage – V Reactance – Ohm Pulse width modulation signal value Impedance – Ohm Load angle Mechanical efficiency Phase angle Rotational speed - rad/s Magnetic flux –Wb Air density - kg/m3 Tip speed ratio

INTRODUCTION Crude oil, natural gas, and coal have been the main energy sources for a long time. In recent decades, various types of renewable energies have been introduced to decrease the dependency on the oil industry. Among them wind power has been drawing more interest and improving its market share more rapidly due to advantages of the wind power and limitations of other types of renewable energies [1]. Wind turbines convert the kinetic energy in wind into mechanical power, and generators convert the mechanical power into electricity. Figure 1 shows three regions of a power curve for a typical variable speed pitch control wind turbine [2,3], which is obtained from the wind turbine output rotor power equation: Pr 

1 2

3

 AU C p mech

(1)

where Pr is the output rotor power,  is the air density, A is the rotor swept area, U is the horizontal wind speed, CP is the nondimensional power coefficient that represents the fraction of the wind power that is extracted by the rotor, and ηmech is the mechanical efficiency of the drivetrain. There is not any power generation in region one, where the wind speed is less than a minimum value called the cut-in speed. In region two, a wind turbine operates at its optimum CP values from the cut-in speed to the rated speed at which the rated power of a wind turbine generator is achieved. To maximize the wind power capture in region two, the blade pitch angle is kept constant (equal to the blade pitch angle value at the cut-in speed) and the generator torque controls the turbine rotor speed to maintain a constant tip speed ratio λ corresponding to the optimum Cp value. In region three, the generator output power is kept constant between the rated speed and a maximum speed called the cut-out speed, at which a high tip speed and undesirable noise emission occur.

Fig. 1 A typical power curve of a wind turbine

There is not any power generation above the cut-out speed to prevent structural damage [2,4]. US patents 7863789 and 6492753 describe a brushless permanent magnet electric machine with a fixed air gap that is operated to have a much higher speed than the normal speed by changing the overlap between the rotor and the stator to eliminate the induced magnetic torque loss in the generator [5,6]. Figure 2 shows the torque drag versus the generator rotor speed in a permanent magnet motor [5]. While at the full overlap between the rotor and the stator, the torque drag due to bearing friction and iron loss increases rapidly with the increase of the rotor speed, at the minimum overlap between the rotor and the stator, the torque drag has a very slow increase that is just due to the bearing friction. Thus, at a constant input power, by decreasing the induced magnetic torque loss of a generator, the generator rotor can rotate faster. The variable electromotive force feature can be employed in a hybrid vehicle with frequent stops and starts. As the vehicle decelerates, the overlap between the rotor and the stator can be increased via a higher engagement to initiate generating power. High-efficiency ultra capacitors can be used for rapid storage and discharge. When the car is accelerating, the motor boost provides a full torque at the zero speed. It provides a faster acceleration and increases the fuel efficiency [6]. This concept in a different way can be employed in a wind turbine to expand its operational range. A larger capacity generator can be used to capture more wind power in the entire operational range of a wind turbine, and its starting operational point can be reduced by means of a variable electromotive-force generator (VEG). While a larger capacity generator can generate more power, it has higher torque loss and lower efficiency, especially at low wind speeds, and need a higher initial rotational speed to start generating useful power [1-3].

Fig. 2 Torque drag versus the generator rotor speed in a permanent magnet motor in two different situations with the minimum and full overlaps between the rotor and the stator

In this work, a VEG design is presented to improve the efficiency and/or expand the operational range of a conventional generator, mainly by adjusting the overlap between the rotor and the stator at different low input powers. A mathematical model of a VEG is developed, a prototype of a VEG is fabricated, and an active control law for adjusting the overlap between the rotor and the stator based on the desired output power at different input speeds is developed. Employing the VEG feature for the wind turbine application is theoretically and experimentally studied to see how the electromotive force of the generator and the generator rotor speed change with the overlap between the rotor and the stator. The stability of the control system is studied and the control results are presented. Mathematical Modeling of the VEG The fundamental operation of a synchronous generator or other electric machines can be understood by applying the Faraday’s law of induction on a conductor moving in a static magnetic field [8]: Emf  NB

dA ds  NBl f  NBl f v dt dt

(2)

where Emf is the induced voltage or the electromotive force that is proportional to the time varying flux enhanced by the circuit; l f and ds are the width and the change in the length of the moving surface area, respectively; and v is the velocity of the moving conductors. In a constant magnetic field and a constant velocity of the moving conductors, the electromotive force can be changed by changing lf in the magnetic field. This approach is taken here by keeping the rotor position fixed, and adjusting the overlap between the rotor and the stator by moving the stator relative to the rotor. In effect, the difference in the overlap can be thought of as having different generators with a varying length and a similar width in series. Figure 3 shows the schematic of a VEG with a fixed rotor and a movable stator; the overlap between the rotor and the stator is adjusted, when the wind speed falls in the range between the reduced cut-in speed of the modified wind turbine and the cut-in speed of the current wind turbine, to smooth the decreasing rate of the output power of the modified generator compared with that of the current generator.

Fig. 3 Schematic of a VEG with a movable stator relative to the rotor

The full overlap between the rotor and the stator occurs when the wind speed is slightly higher than the current cut-in speed and lower than the cut-out speed to produce the maximum power in this range. The one-phase equivalent circuit of a synchronous generator of a wind turbine is shown in Fig. 4 [9]. At the steady state, the mechanical torque from a prime mover should balance the electromagnetic torque provided by the generator and the torque losses due to friction and wire winding: (3)

Tpm  T  Tloss

Multiplying Eq. (3) by the synchronous speed

sys yields

the power balance equation: (4)

Ppm  Pem  Ploss

where Ppm  T pm sys is the mechanical power supplied by the prime mover, Pem  T  sys is the electromagnetic power of the generator, and Ploss  Tloss  sys includes the mechanical and electrical power losses. Using the phasor analysis, the real power for a synchronous generator with a cylindrical rotor is defined by Pem 

EV Zs

cos( s   ) 

V

2

Zs

cos( s )

(5)

where E and  are the magnitude and the phase angle of the induced voltage, respectively, V is the terminal voltage, Z is the synchronous impedance, and  is the phase angle of the synchronous impedance. If the effective armature resistance is neglected ( R0 ), the magnitude of the synchronous impedance becomes that of the synchronous reactance ( Z s  X s ). The real power and the electromagnetic torque for three phases with   90 can be simplified to s

Pem  3EV sin   Pmax sin  Xs Tem 

Pem

 syn



3 EV

 syn X s

sin 

(6) (7)

Fig. 4 One-phase equivalent circuit of a synchronous generator

Note that Eq. (6) is derived under the assumption of a uniform air gap between the rotor and the stator. In the case of a non-uniform air gap, Eq. (7) becomes [9-11]

Tem 

Pem



 syn

3 EV

 syn X d

sin  

3V

2

(

1

2 syn X q



1 Xd

) sin 2

where Ai and Amin are the areas of the effective moving surfaces at any wind speed and the current minimum speed, respectively. Thus, the ratio of the overlap between the rotor and the stator at any wind speed to that at the current minimum speed can be obtained from the corresponding area ratio:

(8)

 XS  i  )  RL  X S   min  XS  XS i sin(arctan  min  )  RL    sin(arctan 

where X q and X d are the quadratic-axis and direct-axis synchronous reactance, respectively. With the assumption of a uniform air gap, X q  X d  X s . The load angle can be estimated using the phasor diagram [12]:

  arctan(

IX S P  IRQ VS  IX S Q  IRP

)

(9)

where Q is the reactance power and S is the apparent power. The rotor power at the cut-in speed of a wind turbine speed is

Pr

min

1 3  mech Cpmin  AU min min 2

(10)

where U min is the current cut-in speed of a wind turbine,

mech and Cpmin are the mechanical efficiency and the power min

coefficient at the current cut-in speed, respectively. The ratios of the rotor power and the generated power at any wind speed to those at the current minimum input speed (cut-in speed) are Pri Pr

min



 mechi CpiU i3

sin(arctan( Pg i Pg min



Ei Emin

(11)

3 mechmin CpminU min

sin(arctan(

I i X S P  I i Ri Q i

)) Vi S  I i X S Q  I i Ri P i I min X min P  I min Rmin Q

Vmin S  I min X S Q  I min Rmin P min

Xs

min

(12)

Xs )) i

respectively, where the subscript i in Pri , Cp i , U i , Pgi , Ei , and  i denotes any wind speed. Under the assumptions of Xs >> R and the unit power factor of the generator, making use of the power relations and Eq. (12), one has

Pg Pg

 XS  i )   XS R L  Ai  min  Amin  XS  XS min  i sin(arctan  )  RL    sin(arctan 

i

min

(13)

Ai Amin



Pri Pr

min

(14)

For a specific generator in a wind turbine, the overlap ratio will be obtained based on the generator specifications and the ratio of the input power at any wind speed to that at the current minimum (cut-in) speed. The changes in the generator parameters at different overlap ratios and input powers can be expressed as a polynomial of degree n:

Y n    Ai Pi P  ai ,1  ai , n      i kn  Amin Pmin  1  Pmin  

(15)

where the second and third ratios on the right-hand side, are expressed as a polynomial kn of degree n, in which n and the coefficients ai , j depend on the type of the generator and the test conditions, and Y depends on the generator specifications and input rotational speed of the generator. Note that the polynomial kn can be obtained through a set of tests for a specific range of the input power and a specific overlap ratio, and be modified for other input powers and overlap ratios. Design Procedure for the VEG Choosing a Generator and a Prime-Mover. A 12 V synchronous direct current (DC) generator is modified at low rotational speeds through the use of the VEG feature. The advantages of a DC generator in terms of having a higher efficiency, a lower cost, and a simpler structure compared with an asynchronous generator were among the main reasons of choosing this kind of generator for a laboratory sample test [13,14]. A 24 V DC motor with an output power of 700 W in a wide range of rotational speeds, with the maximum rotational speed of 19000 RPM is used to model the output rotor power of a wind turbine for the VEG test stand. Design and Fabrication of the VEG. Figure 5 shows the VEG test stand with its main components, including an electric motor, a modified generator with an adjustable overlap between the rotor and the stator, and an active controller [14]. The electric motor is mounted by calm-shell style mounting

brackets that constrain it in two locations. The output shaft of the electric motor and the generator shaft are attached by a love-joy shaft coupler as shown in Fig. 5. A movable stator carrier is designed to have an axial motion along the generator rotor axis, which adjusts the overlap between the rotor and the stator from 0% to 100% of the full overlap. It is supported with four stainless steel rods parallel to the generator rotor axis. Since the air gap between the rotor and the stator has a major impact on the generator output, careful attention to machining tolerances and adjustment of the moving stator carrier throughout the range of motion of the stator are required. Accomplishing this high precision task requires careful consideration of the alignment and balance of the system, with tolerance to1/1000th inch. To facilitate assembly, the stator is chilled in a refrigerator at a temperature of approximately 37˚F, and the aluminum carrier is warmed using a heat gun to a temperature of approximately 150˚F. This allows assembly to be a nearly sliding fit, with some room to ensure that the two parts remain in alignment while cooling to a snug interference fit. The stator carrier is mounted into the position in line with the center of mass of the stator. An important feature to consider when designing the carrier and its supporting rods is the mass of the stator itself. It is critical to analyze the rods so that bending does not occur. For this test sample with a relatively low power yield, bending of the rods may be inconsequential; however, for a large assembly, bending of the rods may be quite significant, especially as the rotor length, and consequently the stator movement length, increase. A rotor extension is necessary to allow the stator to move over the entire length of the rotor. This can be done in several ways. The ideal scenario would be a custom designed generator with an extended snout one-piece rotor. If one uses a consumer off-theshelf generator, it is necessary to fabricate an extension shaft. As the rotor size increases, the extension shaft and its interface with the rotor must be robust enough to prevent its bending and twisting while remaining stationary. The stator carrier, all the plates and fixtures, and the base-plate are made from 6061 aluminum here. There are various methods of achieving a linear motion of the stator carrier; a linear actuator, an ACME threaded rod, a timing pulley, and a hydraulic assembly can all be modified or designed to move the stator. However, some considerations should be taken into account in choosing a desired method. For instance, a hydraulic assembly may be ideal but its design should allow compensation for environmental factors. The ACME rods with extremely high precision compared with standard threaded rods are used to translate rotational motion to linear motion. A high precision low cost stepper motor, with an output torque of 465oz-in (3.284 Nm), fixed to a mounting plate and connected to an ACME threaded rod via a love-joy coupler, is used to control the position of the stator carrier at different rotational speeds (Fig. 5). It is designed to have 200 steps per revolution for the adjustment of the overlap between the rotor and the stator from 0% to 100% of the full overlap, which is equivalent to approximately 1.8 degrees of rotation per step. An IFI Robotic-Victor 885 speed controller that operates at the 24 V

Fig. 5 The VEG test stand

nominal voltage is wired in series between the power supply and the motor; it processes the output PWM signal of the Arduino Uno microcontroller. The PWM signal is used to control the transistors in the speed controller, allowing the voltage to flow through the controller. Active Control System of the VEG. In order to automatically control the VEG to generate a desired output power, the overlap between the rotor and the stator can be adjusted by an active control system that has two main parts: the data acquisition part (DAP), and the control (Ctrl) part, as shown in Fig. 6. The DAP is performed by using a precise proximity sensor to record the rotational speed, as shown in Fig. 5, and calculate the corresponding desired output power that will be generated by the VEG. The Ctrl part that includes a feedback control system is to make the VEG to generate the desired output power by adjusting the overlap between the rotor and the stator. The desired output power calculated from the DAP is the set point for the Ctrl part; the output power of the VEG is the feedback, and the difference between the desired and real output powers multiplied by a factor F, which will be discussed in the results, is the error between the real output power of the VEG and the desired one calculated in the DAP. The adjustment procedure will be applied based on either a fixed desired output voltage that will be given to the control system or an optimum output voltage that can be obtained for a specific generator.

Fig. 6 Active control system for the VEG with two main parts: the DAP and the Ctrl part

The control law for the Ctrl part is performed using national instruments LabVIEW system design platforms including the SCB-68 data acquisition interface to monitor the output voltage of the generator, and a stepper motor as the actuator of the control system that is controlled by the UMI-7772 motion control interface. A proportional-integral (PI) controller with a constant P  250 converts the error to a signal that is transmitted to the stepper motor control unit to change the output power by adjusting the overlap between the rotor and the stator. The more accurate the output power reading of the DAP, the more robust and reliable the control system to achieve a faster and more accurate overlap adjustment. Note that higher accuracy in reading data and designing the control system is required, especially at lower input speeds and for a smaller overlap between the rotor and the stator. RESULTS Impact of the VEG on the generator rotor speed At the steady state with constant input speeds, the normalized output voltage versus the normalized rotor speed is shown in Fig. 7 for different overlap ratios; the output voltage decreases with the overlap. The first goal here is to determine whether a modified generator with an adjustable overlap between the rotor and the stator can generate electricity at input powers lower than the one that a regular generator starts working. Having a modified generator operating under a lower input power than that of a regular one in a wind turbine means the modified generator can start working at a lower wind speed than the current cut-in speed of the regular one. A multi-variable 7th order polynomial is developed using the results in Fig. 7, which defines the output voltage of the generator as a function of the rotor speed and the overlap between the rotor and the stator: V   (op ) ( RPM )

(16) where and  are functions of the overlap (op) and the rotor RPM value, respectively: 3

variations at different input power levels due to decreasing the overlap between the rotor and the stator. The input powers are indicated in a range from 1 to 10 where 1 is the minimum input power that makes the generator rotor to start spinning; 6 and 7 are the current start-up power of the generator, and 8 to 10 are the powers that the generator works as a regular generator following its power curve. The input power level values are obtained by the input PWM signals from the Arduino Uno microcontroller and they increase by the input signal level. It is observed that at a constant input power, by decreasing the overlap between the rotor and the stator, the generator rotor speed is increased. Up to 12% increase in the generator rotor speed is achieved at low rotational speeds by decreasing the overlap between the rotor and the stator. In addition, three regions with specific speed ranges are observed in Fig. 8. The first region relates to the very low input speed values lower than 110 RPM; the second region is a transition between the low and high input speed values; and the third region relates to the input rotational speeds larger than 160 RPM. Figure 9 shows the changes in the normalized output power for different input levels of the Arduino Uno microcontroller. While there is up to 12% increase in the generator rotor speed when the overlap between the rotor and the stator is at 20%, there is a significant reduction of up to 65% in the generator output power. In addition, there is a jump in the generator rotor speed from 110 to 140 RPM when the input power level increases from the 6th to the 7th level. It is desired to control the overlap ratio variations in order to generate the maximum power, while keeping the generator rotor spinning at its maximum speed. While the theoretical analysis shows the possibility of increasing the rotational speed by decreasing the overlap between the rotor and stator to 20% or even lower and still making the generator working, the experimental results show that a more steady behavior and smoother changes of the generator output power are achieved at overlap ratios larger than 50%.

2

  A1op  A2 op  A3 op  A4 4

3

 ( RPM )  k1 RPM  k 2 RPM  k3 RPM  k 4 RPM  k5

(17) 2

(18)

in which the coefficients are

A1  0.8972 A2  1.7216 A3  1.1130 A4  0.0924

k1  0.0386

k2  1.0

k3  9.6408

k4  41.3116 k5  65.8427 The generator rotor speed and output power is measured at any input power. Figure 8 shows the generator rotor speed

Fig. 7 Normalized output voltage values at different rotational speeds and for different overlap ratios

Validation of the Mathematical Model The mathematical model is checked against experimental measurements obtained from the test stand developed. A series of tests for different overlap and input power ratios are performed and the output rotational speeds and output powers are measured. Three overlap ratios of 100%, 50%, and 20% at rotational speeds lower than 150 rpm are represented here. Looking first at the case where the overlap ratio is 100%, the kn coefficients for different input powers at rotational speeds lower than 150 rpm can be obtained from tests:

k100%  0.0021Y 3  0.0196Y 2  0.0767Y  1.0588 Fig. 8 Generator rotor speed variations for different input power levels and overlap ratios

There is a trade-off between having a VEG that works at smaller input powers by rotating faster and having a stable and smooth output power; the optimum condition is obtained when the overlap between the rotor and the stator is changing from 50% to 100% and the input rotational speed is less than 150 RPM, while the full overlap between the rotor and stator is preferred for rotational speeds above 150 RPM. Higher overlap ratios (more than 50%) between the rotor and the stator will result in higher output powers at lower rotational speeds, and lower overlap ratios (less than 50%) between the rotor and the stator will result in lower output powers at higher rotational speeds. Note that the three regions of low, transition, and high output powers in Fig. 9 are equivalent to those in Fig. 8 at different output rotor speeds.

Fig. 9 Normalized output power variations for different input power levels and overlap ratios

(19)

According to Eq. (15), the overlap ratio for any input power with the specific polynomial kn obtained here should have a value of around 1. Table 1 shows the comparison between the generator specifications ratios in Eq. (15) obtained from the experimental tests versus the mathematical model for different input power levels at rotational speeds lower than 150 rpm for the 100% overlap ratio. The same procedures for other overlap ratios can be performed. The experimental kn polynomials for the two cases of 50% and 20% overlap ratios are provided below: k50%  0.00115Y 2  0.0016Y  0.5

k20%  0.0029Y  0.2042

(20) (21)

In the cases where the overlap ratios are 50% and 20%, the mathematical and experimental results for the generator specifications ratios are compared in Table 1 as well. There is a very good compatibility between the mathematical model and experimental test results for different input powers. The maximum errors for the 100% ,50%, and 20% overlap ratios do not exceed 5%, 2%, and 2%, respectively. The modified generator will work as a regular generator as the input power reaches the start-up input power of the regular generator and follow the power curve of a regular generator. Active Control System Figure 10 shows the generator output power by changing the overlap between the rotor and the stator of a modified generator. The control is unstable at small overlaps. When the normalized desired output power is 0.5 and the overlap is about 75% of the full overlap, the normalized output power of the VEG is stable after a transient. The small oscillations after the transients are introduced by the VEG when the regular generator is modified to allow the overlap adjustment. When the normalized desired output power is less than 0.4 and the overlap is less than 20% of the full overlap, the normalized output power of the VEG is unstable.

Table 1 Experimental and the mathematical comparison of the generator specifications ratios at 100%, 50%, and 20 % overlap ratios

smaller at a small overlap, and the output of the VEG is stable at every overlap ratio, as shown in Fig. 12. Equation (8) can be used for different overlaps and rotational speeds. Hence the fixed or optimum output power of the VEG can be achieved based on the input power and rotational speed. The stability of the fixed or optimum output power can be improved by the correction factor.

Fig. 10 output power of the VEG which is stable when the normalized desired output power is larger than 0.5 and unstable when the normalized desired output power is less than 0.4

Before the differential node to obtain the error in Fig. 6, there are two identical blocks (F) to condition the desired and real output powers, and the conditioned values can be used to obtain the conditioned error. Since the relation between the output power and the overlap is nonlinear, as shown in Fig. 11 where the normalized output power is the output power divided by that at the full overlap between the rotor and the stator, the slope of the factor F changes with the desired or real output power. If the overlap gets smaller, the change in the output power with the overlap will be steeper which means that the output power at a small overlap is more sensitive to the overlap change than that at a large overlap. In order to make the control system stable at every overlap, the slope of the factor F should be changed with the overlap or the normalized desired output power. A variable factor F, which is a function of the normalized desired or real output power Pg, is fitted using the data in Fig. 11:

F  2.16Pg 2  0.6Pg  1.93

Fig. 11 Normalized output power of the VEG versus the overlap

(22)

Equation (22) shows that when the normalized desired or actual output power is smaller, the slope of the factor is smaller so that the sensitivity of the output power with respect to the overlap is

Fig. 12 Output power of the VEG with the variable factor in Eq. (22), which is stable at every overlap ratio

An active control system is developed to obtain either a fixed or an optimum VEG output power. While the fixed output power can be chosen arbitrarily, the optimum output power is obtained under two considerations: the VEG generates the maximum output power, and it works at the maximum rotational speeds at lower input speeds. The optimum control module is implemented for low rotational speeds lower than 150 RPM (i.e., the 7th input power level) before jumping to higher rotational speeds and output power values. The optimum generator output power can be obtained by a logarithmic relation between the input speed and the output power: Pg  0.7631ln( )  3.0204

(23)

Fig. 13 Generator rotor speed variations with time using the active control module

where Pg is the generator output power, and ω is the rotational speed in RPM. The equivalent voltage with respect to the generator power will be the input to the PI controller module to adjust the overlap between the rotor and the stator. Figure 13 shows the changes in the generator rotor speed as the control law is applied to the VEG. The speed fluctuations in the range from 130 to 150 RPM have been controlled based on the optimum output power having an approximately constant optimum value of 140.90 RPM in the transition region between the very low rotational speeds and low rotational speeds. The generator rotor speed at low input rotational speeds, lower than 120 RPM, is being controlled under two considerations of generating the maximum output power and having the generator working continuously at the maximum optimum rotational speeds. The generator rotor will spin approximately at an average speed of 96.25 RPM in this low speed range. Figure 14 shows the actual and desired output voltage values of the VEG at different input speeds using the active control Module together with the mathematical modeling. The actual output voltage values start from zero and the desired output voltage values, which are calculated from Eq. (23) with the input speeds start from 0.78. The actual output voltage is controlled to be close to the desired output value with low fluctuations by applying the control law. While the input speed from the electric motor (prime mover) is larger than 150 RPM, the VEG works as a regular generator with a full overlap between the rotor and the stator, which results in generating the maximum output voltage and power values, as shown in Fig. 14 in the time range from 0 to 450 seconds. After the high speed region, there is the transition region where the desired output power of the generator has a large decreasing rate due to the drop of the input speed. The actual output voltage is controlled by adjusting the overlap between the rotor and the stator to have the optimum value. The normalized output voltage of the VEG will have an average value of 0.75 with 5% fluctuations around the average rate. The last region in Fig. 14 represents the normalized actual and desired output voltage values of the VEG at very low input speeds. The actual output voltage in this range has an average of 0.47 with 3% fluctuations around the average rate.

Fig. 14 The actual and desired normalized output power variations with time using the active control module

CONCLUSION A novel VEG system, which is a modified generator with an adjustable overlap between the rotor and the stator, is developed. A mathematical model of the VEG is derived and a prototype, which includes a 12V synchronous DC generator with an adjustable overlap between the rotor and the stator, a 700 W electric motor as the prime mover, and a stepper motor as the actuator for an active control system of the VEG, is designed and fabricated. A reduced torque loss of the generator and an increased rotational speed of the generator rotor are achieved by recuing the overlap between the rotor and the stator. By using the derived mathematical model with a high compatibility with the experimental results, an active control system with a variable factor that guarantees the stability of the system is designed to adjust the overlap ratio and obtain an optimum generator output power based on the input power at different speeds. A VEG provides an expanded operational range and higher efficiency for a wind turbine by decreasing the current cut-in speed and the power loss. A VEG can also be used to increase the fuel efficiency of hybrid vehicles with frequent stops and starts.

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