Thesis- Cont sys for wind turbines (Sept 2015 revision)

MODELING the CONTROLS ADVANCED RESEARCH TURBINE with MATLAB and SIMULINK

Research for ECET 7504Southern Polytechnic State University

By Larry Branscomb

In presenting this thesis in partial fulfillment of the requirements for a Master’s degree at Southern Polytechnic State University, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of this thesis is allowable only for scholarly purposes, consistent with “fair use” as prescribed in the U.S. Copyright Law. Any other reproduction for any purposes or by any means shall not be allowed without my written permission.

Signature

Date

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Abstract

Horizontal-axis wind turbine systems, such as the one pictured on the previous page, convert wind power into electrical power. The mathematics governing the aerodynamic, mechanical and electrical relationships of the system can be modeled with Matlab software. A system “virtual testbed” can be constructed to test the feasibility of system control techniques. Complex model-making should necessarily include a validation process that guarantees the accuracy of the model. This paper describes the building of five models of increasing complexity, culminating in an AC generator system with controls shown to maximize power production under variable wind conditions.

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TABLE OF CONTENTS

Introduction and organization of this paper: ..................................................5

Connection to the power grid .........................................................................7

Aerodynamics of wind turbines .....................................................................8

System #1: Matlab simulation of 600 kwatt DC machine, gearbox, damping, inertial components.....................................................11

System #2: Simulink model of complete plant ............................................21

System #3: Simulink model of plant with speed control..............................26

System #4: Simulink model of plant with torque control Indirect speed control method (ISC)..........................................32 Simulation of ISC with 2 wind speeds.......................................35 Simulation of ISC with realistic wind profile............................38 System #5: Replacing the DC generator with an AC generator ...................41 Volts-per-hertz operation...........................................................46 Simulation with Indirect Speed Control and realistic wind Profile……………………………………….............................50

Conclusion.....................................................................................................54

Appendix A: Derivation of System #1 State-Space to Transfer Function....54

Appendix B: Matlab program for System #1 State-Space Simulation..........57

Appendix C: Embedded AC Machine Used in Simulink Models.................59

References:....................................................................................................61

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Introduction and organization of this paper:

This paper presents the results of Matlab and Simulink modeling of a wind turbine power generation system. The wind turbine modeled is the controls advanced research turbine at the National Renewable Energy Laboratory in Golden, Colorado (NREL). This turbine is a test bed used to study wind turbine controls technology. Figure #1 contains a photograph of this system.

The purpose of modeling this system is to demonstrate the feasibility of various control system techniques that maximize wind turbine power capture. In the simulations to follow, the electrical and mechanical components are carefully modeled to closely match the actual components used at the NREL site.

Parameters for the wind turbine were taken from research papers published by the American Society of Mechanical Engineers. These parameters include wind turbine blade diameter, gearbox ratio, generator system electrical power, turbine moment of inertia, and a turbine power efficiency table derived from NREL measurements. AC and DC 600-kilowatt generator data were taken from catalogs of Siemens Energy and Automation. These parameters appear in tables #1 and #2.

The first topic is a description of system components and the connection to the power grid.

A brief analysis of the aerodynamics of wind energy follows. The mathematical relationships governing wind speed, torque, air density, turbine dimensions and turbine efficiency will form necessary building blocks in the models.

A series of simulations with Matlab and Simulink will then be performed and evaluated. Modeling will be done systematically in much the same way as an actual system would be built: Components of the system will first be modeled and tested; then the complete system model which will be tested and evaluated.

The first model, labeled System #1, will consist of a 600 kwatt DC motor/generator, a gearbox, and the wind turbine. Essentially, all hardware is modeled in this system, but the complex aerodynamics of wind torque is not yet included. Inputs to the system are torque and DC voltage. This multiple-input, multiple-output system (MIMO) will be simulated with Matlab using state-space analysis. Transient response and steady-state response of turbine rotor speed and armature current will be analyzed and validated.

System #2 will consist of the above system with the added complexity of wind torque as a disturbance input. This will be a complete model of the plant without

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control techniques to maximize power generation. For this model and all to follow, Simulink with block diagram format will be used.

System #3 adds a speed control technique to maximize efficiency of generated power. The pros and cons of the control method are evaluated.

System #4 replaces the speed control technique with a standard torque control method used by the wind energy industry. The method, known as Indirect Speed Control (ISC), is simulated first with steady wind conditions, then with a more realistic variable wind-speed profile. The control method is evaluated and discussed.

System #5 is the final system, which replaces the DC generator with an AC generator. Equations that represent the AC machine are developed and incorporated into the model. This system will then be tested and evaluated.

Throughout this paper I will validate each step of system simulation by comparing simulation results with those that can be predicted by calculations. It is important to ensure, for instance, that torque and current directions have been assigned correctly and that the signs of voltage and power are valid. Validation tests will also be done to ensure that modeled steady-state values agree with theoretical computations. Equations to ensure conservation of energy and power as well as torque balance will serve in the validation process as well. Checks are essential to guarantee confidence in the models as they grow increasingly complex.

Figure #1: One of the Controls Advanced Research Turbines.

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Connection to the Power Grid.

Figure # 2: A system to extract power from wind

Figure #2 shows the connection of a wind turbine to a power grid. The turbine is connected through a gearbox to a generator. The generator is connected to the power grid through a power converter and 3-phase transformer. The modeling in this paper will include only the components in the left half of the figure: the motor/generator, the gear-train, the wind turbine and all inertial components. Matlab, Simulink and the Simulink Controls System Toolbox will be used to create and test models of these components.

The components on the right side of figure #2 -- including the power converter, power transformer and supply grid -- will not be modeled here. It will be assumed that controllable DC or AC voltage and frequency will be furnished by the above-mentioned power conversion system. The basis for this assumption is that modern power conversion systems consisting of Insulated Gate Bipolar Transistors and pulse-width-modulation techniques can, with high efficiency, control DC voltage to a DC motor/generator. Modern power inverters can also efficiently control 3-phase AC voltage and frequency to AC machines in either motoring or generating mode. Though these modern power conversion components may add to costs, the added benefits of allowing generator voltage and frequency to be independent from grid voltage and frequency outweigh the costs.

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Aerodynamics of wind turbines.

Horizontal axis wind turbines convert the kinetic energy of wind into the mechanical energy of a rotating shaft called a rotor. Wind passing through aerodynamically designed blades creates torque at the rotor. Power captured by a wind turbine rotor is given by equation #1 [1,4,5,6]:

Equation #1: P = ½ R2Cp()v3

Where:P is power available in the wind in watts is density of air in Kg/m3

R is radius of turbine blade in metersv is wind speed in meters/secondCp is an efficiency function specific to each turbine.

The above relation shows that power available in wind is proportional to air density, the area swept out by the turbine blades, a turbine efficiency function, and the third power of wind speed. Equation #1 makes the following assumptions regarding the wind velocity, turbulence and direction[5]:

Constant wind speed that is uniform across the blades Wind perpendicular to the plane of rotation No turbulence or rotation of air induced by rotor blades

Most turbines can be turned (yawed) to face into the wind, thereby eliminating the need to account for oblique wind directions in the equation. Turbulence and non-uniformity of wind speed are not considered in equation #1. Still the relationship is considered a valid approximation of power available from the wind by wind generating industry and provides a commonly used aerodynamic model.

Cp, the wind turbine power coefficient and The tip-speed ratio.

The wind turbine power-coefficient function, Cp, is specific to each wind turbine design. Cp is a measure of mechanical power extracted by the turbine divided by available power from the wind:

Cp =

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Cp is a function of both tip-speed ratio () and blade pitch angle. Tip-speed ratio () is defined as the speed at the tip of the rotor divided by the average speed of the wind. Equation #2 states this relationship [1, 4, 5, 6]:

Equation #2 Where r = rotor angular velocity in radians/second

R = radius of turbine rotor (meters)v = wind speed (meters/sec)

Blade pitch angle, is another parameter affecting efficiency. Blade pitch will have an effect on the torque applied to the rotor at a given speed. The Cp and Cq functions for the Controls Advanced Research Turbine used in this paper came from modeling software developed by wind turbine researchers [1,4]. Plots of Cq() and Cp() are a 3-dimensional functions of blade pitch () and tip-speed ratio(). Blade pitch is often fixed during operation of a wind turbine, yielding two planar functions Cq() and Cp().

The resulting slices of the 3-D functions Cp and Cq for = -1O yield the planar plots of Cp() and Cq()[1]. These plots appear in Figure #3 and Figure #4. Data from researchers[1] were used to create these graphs. The graphs are piecewise-linear approximations which accounts for their not being smooth. The real functions are smooth and differentiable. Values used in simulations will come from these power efficiency and torque efficiency graphs.

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Figure # 3: turbine power efficiency (Cp) vs tip-speed ratio

Figure # 4: turbine torque coefficient (Cq) vs tip-speed ratio

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0 2 4 6 8 10 12 14 160

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45POWER EFFICIENCY vs TIP-SPEED RATIO

TIP-SPEED RATIO

PO

WE

R E

FFIC

IEN

CY

0 2 4 6 8 10 12 14 160

0.01

0.02

0.03

0.04

0.05

0.06

0.07TORQUE EFFICIENCY vs TIP-SPEED RATIO

TIP-SPEED RATIO

TOR

QU

E E

FFIC

IEN

CY

Air density constant, :

Air density at 0o C and 1 atmosphere of pressure is approximately 1.29 Kg/m3. Throughout this paper the air density constant, , will be set to a nominal value of 1 Kg/m3. This value is within the range of normal air densities for wind turbines; therefore, it will not affect the validity of the simulations. One is also a very convenient value to use in calculations.

Regions of operation of wind turbines:

The National Renewable Energy Laboratory defines three regions for wind turbine operation governed by wind speed [2, 4]. Region 1 is start-up. This is the lowest wind speed region where no power generation is possible. Wind speeds in Region 2 are high enough to allow power generation. Control algorithms in this region attempt to maximize power generation. Region 3 has the highest wind speeds. Control techniques in this region limit power generation to ensure safe operation of the turbine.

Simulations in this paper will be in region #2. Control techniques will therefore be evaluated according to how much power they can capture.

System #1: DC generator, geartrain with voltage and torque input. The first simulation is of a 600 kwatt DC generator connected to the NREL gearbox and turbine with inputs wind torque and DC voltage. Parameter values appear in Table #1

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Figure #5 Electromechanical diagram of wind turbine and components.

Figure #5 shows the electromechanical model of the system with relevant variables identified. The model considers all components to be rigid bodies, but frictional damping losses are considered. Polarities and directions of the model are such that wind velocity from right to left into the turbine causes clockwise torque. The DC generator equations are written so that a positive generator torque will produce positive power. The mechanical equations governing the motion of the turbine rotor, the turbine shaft, the motor shaft and gears are:

Equation #3:

Equation #4:

Equation #5: Gear ratio =

Variable definitions:

Ta = aerodynamic torque of the wind (clockwise looking into rotor)

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TLS = Torque applied to low-speed shaft of gearbox (CCW looking into rotor)THS = Torque applied to high-speed shaft of gearbox (CCW looking into rotor)Tg = Torque applied by generator (CW)JT = polar moment of inertia of the Turbine shaft and low-speed gearJg = polar moment of inertia of the Generator rotor and high-speed gearr = angular speed of wind turbine rotor (radians/sec, CW looking into rotor)g = angular speed of electric generator rotor (rad/sec, CCW looking into rotor)DT = frictional damping coefficient for Turbine side of gearbox DG = frictional damping coefficient for electric generator side of gearbox

The electro-mechanical equations relating DC generator voltage, armature current, back emf and generator torque are:

Equation #6:

Equation #7:

Equation #8: Vbemf = (Kb)(g)

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Variable definitions:

Ea = DC voltage applied to DC generatorIa = armature current (direction positive into applied voltage)Ra = armature resistanceL = self inductance of the armature circuitVbemf = back emf voltage to generator Kt = torque constant relating generator torque to armature amps (N-M/amp)Kb = constant relating back emf to generator rotor speed (Volt-sec/rad)

Generator constants are taken from a Siemens 600 kilowatt DC motor/generator.

We seek equations relating the inputs wind torque (Ta) and electrical voltage (Ea) to outputs rotor speed (r) and armature current (Ia). r and Ia are the state variables in this system.

Combining Eqn #3, Eqn #4, Eqn #5 yields:

Equation #9: Ta - JT - DTr = TLS = NgTHS = Ng[Tg + Jg + Dgg]

Replacing g with Ngr yields :

Equation #10: Ta - JT - DTr = TLS = NgTHS = Ng[Tg + JgNg + DgNgr]

Solving for Tg:Equation #11: Tg = 1/Ng[Ta - (JT + Ng2Jg) - (DT + Ng2Dg) r ]

Letting "J" = (JT + Ng2Jg) = polar moment of inertia reflected to turbine shaft, andLetting "D" = (DT + Ng2Dg) = frictional damping constants reflected to turbine shaft yields:

Equation #12: Tg = 1/Ng[Ta - J - D r ]

Replacing Tg with KtIa (Equation #7) and solving for yields the state equation:

State Equation #1: = - (NgKt/J)Ia - (D/J) r + Ta/J

Equations #6, and #8 involving the DC generator can be combined into the second state equation as follows:

State Equation #2: = - (Ra/L)Ia + (KbNg/L) r - Ea/L

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The resulting state space matrices are as follows:

= (-Ra/L) (KbNg/L) X Ia + -1/L 0 X Ea

(-NgKt/J)Ia (- D/J) r

0 1/J Ta

The A, B, C, D state space matrices for Matlab appear below.

A = (-Ra/L) (KbNg/L) (-NgKt/J)Ia (-D/J)

B = -1/L 0 0 1/J

C = [ 1 1 ]

D = 0

System #1 Simulation Results and Validation:

Thus far, the system represents a general DC generating system with a torque applied through a gearbox. To validate the mathematical representation of the system, parameter values must be filled in from Table #1 below and system will be simulated with Matlab. Table #1 contains a list of variables to be assigned. Generator variables are taken from the 600 kilowatt DC motor/generator discussed earlier in this paper. National Renewable Energy Laboratory wind turbine variables are taken from references [1,2]. The parameter J is the polar moment of inertia of the turbine (Jg) added to reflected polar moment of inertia of the DC generator(Ng2JT). Since the frictional damping coefficient, D, was not available for the NREL system, a value for friction was chosen that presented a torque component of 1% of the maximum allowable turbine torque(162 k-Newton) when operating at a speed of 3.24 rad/sec.

It can be seen that this is a two input system with voltage Ea and wind torque Ta as inputs. If we choose to observe the states Ia and r as outputs, the system is multiple-

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input, multiple-output system (MIMO). It is also a linear time-invariant system (LTI). This MIMO system can be represented by the 4 transfer functions derived in Appendix A:

Because the system is linear and time-invariant, the principles of linearity and superposition will allow the time domain response of the outputs due to each input to be plotted separately and the added to give the total response. Therefore Ia(t) due to generator voltage Ea can be added to Ia(t) due to wind torque Ta. Also, r(t) due to generator voltage Ea can be added to r(t) due to wind torque Ta. It should be pointed out that the large step inputs in this simulation are performed simply to view and validate transient and steady-state responses. As a practical matter, such values would damage the equipment.

The state-space representation is simulated with Matlab. Table #1 contains parameters used to simulate the DC motor/generator and wind turbine. Torque constant Kt and back EMF motor constant, Kb are derived as follows:

Torque constant: Kt = = 3.007 N-M/Amp

Back EMF constant: Kb = =

3.062 V/rad/sec

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As part of validity checking, the following conservation of power equations will be verified:

Equation #13 (motoring): |EaIa| = |Torque x r | + Ia2Ra + frictional losses

Equation #14 (generating): |Torque x r | = |EaIa| + Ia2Ra + frictional losses

Calculations of Kt and Kb above show two constants close to being equal (N-M/A and V/Rad/S are the same units). The slightly lower value for Kt allows for the frictional losses component in motoring mode only. In generator mode, a lower Kt causes negative losses, or efficiency > 100%.

The problem is solved and conservation of power is maintained in both motoring and generating if Kt = Kb = 3.062.

PARAMETER DESCRIPTION SYMBOL VALUE

DC motor/gen model # n/a 1GG5 402-5NK70-1VV5DC motor/gen Power rating n/a 600 KwRated Voltage n/a 420 VRated Armature Current IaRated 1490 ARated Speed n/a 1280 RPMField Voltage and Wattage n/a 360 volt , 3.9 WattRated Torque n/a 4480 N-MArmature Resistance @120oC Ra 6.4 mArmature Inductance L .11 mHDC motor/gen polar moment of inertia Jg 21 Kg-m2

Wind Turbine polar moment of inertia JT 388500 Kg-m2

Wind Turbine Rotor Radius Rad 21.65 mGearbox Ratio (low-speed = turbine) Ng 43.165Damping constant (referred to turbine) D 500 N-M/rad/secTorque constant (calculated) Kt 3.062 N-M/AmpBack EMF constant Kb 3.062 V/rad/secDC machine efficiency n/a 95%

Table #1: DC motor/generator, Wind Turbine Parameters

The Matlab program for this simulation is in Appendix B. In this simulation a step torque of 162 kilonewton-meters is applied (input Ta) and a step voltage input to the motor/generator of 250 volts (input Ea). Figure #6 shows four waveform step responses

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of output states Ia(t) and (r)t. The responses due to the separate inputs are then added together and appear in Figure #7.

System Poles calculated from Matlab program: Matlab is employed to find the system poles. Instructions are in Appendix B and results are below. We would expect with 2 states and no control compensators that the system be second order. Matlab results show the poles:

"dominant" pole = -7.1473 sec-1 2nd pole = -51.0357 sec-1

The -51.0357 sec-1 pole is due largely to the small L/R time constant (.11m Henry/6.4m ohms = .0172 seconds). The dominant pole at -7.1473 sec-1 is due to the large combined moment of inertia, "J". The response of the states Ia(t) and (r) appear in Figures #6 and #7.

FIGURE #6: Separate step responses of Ia and r due to inputs Ta and Ea

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0 0.5 1 1.5 20

0.5

1

1.5

2radial vel vs time w/ Ea = 250V step, Ta in = 0

time

radi

ans

per s

ec

0 0.5 1 1.5 20

0.01

0.02

0.03

0.04

0.05

0.06radial vel vs time w/ Ea = 0; Ta = 162 kN-M step

time

radi

ans

per s

ec

0 0.5 1 1.5 2-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0x 10

4 Ia vs time w/ Ea = 250V step; Ta = 0

time

Am

ps

0 0.5 1 1.5 20

200

400

600

800

1000

1200

1400Ia vs time w/ Ea = 0; Ta = 162 kN-M step

time

Am

ps

FIGURE #7: Combined step responses of Ia andr due to inputs Ta and Ea

Transient Response analysis of Matlab simulation :

Interpretation of machine response: it can be seen that there is an initial surge of about -32000 amps into the armature of the motor/generator. The negative sign indicates the DC machine is motoring due to the large step input voltage. It is nearly in locked rotor condition owing to the large polar moment of inertia of the wind turbine. Rotor speed increases until back EMF exceeds input voltage Ea and the DC machine is in generating mode. As mentioned earlier, these conditions would not be placed on actual equipment.

The step responses are what is expected of a second order system with one dominant pole at -7.1473 sec-1.

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0 0.5 1 1.5 20

0.5

1

1.5

2Radial velocity vs time step wind torque 162 kN-M & step 250 to DC gen

time

radi

ans/

sec

X: 1.938Y: 1.95

0 0.5 1 1.5 2-4

-3

-2

-1

0

1x 10

4 Ia vs time w/ Ta=162 kN-M step & Ea= 250 step

time

Am

ps

X: 1.956Y: 1218

0 0.5 1 1.5 2249

249.5

250

250.5

251DC motor voltage input

time

Vol

ts

0 0.5 1 1.5 21.62

1.62

1.62

1.62

1.62x 10

5 Wind torque input

time

Torq

ue N

-m

Steady State analysis of Matlab simulation :

The cursors on Figure #7 indicate approximate steady state values for state variables, armature current (Ia) and turbine rotor speed (r). They appear below:

Steady state r = 1.95 radians/sec

Steady state Ia = 1218 amps

Under steady state conditions, there should be no rotor acceleration component of torque. The resulting torque balance equation is then:

Generator torque = Wind torque - friction.

or: NgKtIa = Ta - D r

Substituting values above yields:

(43.165)(3.062)(1218) = 162000 - 500(1.95)

160,985 N-M = 161,024 N-M

This equality is accurate to 4 figures, the accuracy level of the simulation. Also under steady state conditions, we would expect conservation of power so that:

wind turbine power = generated power + I2R losses + frictional power losses.

(Torque)x(r) = (Ea)x(Ia) + Ia2R + (D)(r)2

(162000)x(1.95) = (250) x (1218) + (1218)2(.0064) + (500)x(1.95)2

315,900 watt = 304,500 + 9495 + 1901

315,900 watt = 315,896 watt

Again, this equality is accurate to 4 figures. We therefore conclude that the MIMO system and derived transfer functions produce a valid model of the simple system consisting of the DC generator and gear train with torque and voltage inputs.

Conversion from State-Space to Block Diagram :

Next, the aerodynamics of the wind turbine will be added to the model to make a complete model of the plant without any control system additions. Simulink will be used in future simulations. The state space MIMO system will first be converted and reduced

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through block diagram reduction methods to a signal flow graph in phase variable format. Figure #8 shows the resulting signal flow graph.

Figure #8: Signal flow graph of System #1

The signal flow is expressed in block diagram form in Figure #9. Block diagrams will be used for all further Simulink models.

Figure #9: Simulink model of System #1

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Table of Values:_____________________________R = .0064 ohm; armature resistanceL = 110 microhenry; Armature inductanceKt = 3.062; torque constant( N-m/amp)Kb = 3.062; emf constant (V/rad/sec)Kf = 500; N-meter-seconds (damping constant)J = 427,627; kg-meter^2Ng = 43.165; Gear ratio Rad = 21.65 meter (Rotor radius)p = 1kg/m^3 air density

Wind TorqueN-Meters

1J.s+Kf

Turbine rotor inertia,Gears, drivetrain

-Kt

Torque constant

1L.s+R

Mot/GenTransfer Function

Kb*Ng

Generator back emf constant

-Ng

GainDC Generator

Armature Voltage

Wr (rotor speed)Iarm

Vbemf

Tgen

1/s 1/s-1/L -NgKt/J

Ea

aI

Ia rr

NgKb/L

Ta1/J

System #2: Plant with DC generator, geartrain and wind turbine input. Simulation #2 will add the aerodynamic torque contributions of wind turbine to the 600 kwatt DC generator and NREL gearbox of simulation #1. System #2 is a complete model of the wind turbine plant with no extra control apparatus. For this and all future non-linear simulations, I chose Simulink to model the systems. In order to model the disturbance input -- wind torque -- the aerodynamics of the CART turbine must be represented with Simulink and added to the DC generator and NREL gearbox of simulation #1.

Equation #1 gives a starting point for calculating torque from the wind.

Given that: P = ½ R2Cp()v3 (Eqn#1)

and mechanical torque = Power/ rotational velocity

Ta= P/r

Then:

Equation #14:

Since blade pitch angle () is fixed at –1o, (Cp) is a function of only tip-speed ratio (). Another wind industry standard function called the wind turbine torque coefficient is defined as:

Equation #14 will be rearranged and new function Cq substituted to yield:

Equation #15:

In order to insert wind torque as an input to the system, the Cq function from figure #4 will be embedded in a Simulink table. The remaining components of wind torque will require use of the Simulink multiplication block. Feedback from the rotor will also be a component of wind torque, Ta. Figure #10 is the Simulink model of the complete plant thus far.

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Figure #10: System #2, Simulink model of complete turbine system.

The DC generator with gear train system of state equations #1 and #2 are represented in Figure #2 by the outer border of blocks and signal flow lines. The name of the frictional damping constant is "Kf" in these Simulink models (changed from "D" in the Matlab model).These include the upper row of blocks and the feedback loop with gain constant "Generator back emf constant". Wind torque is a disturbance input represented by all the blocks inside. These blocks perform the calculation of equation #15:

Note that blade pitch () is fixed at -1o and will not be a variable of Cq.


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tip-speed ratio outputfrom Windspeed inputand rotor speed input

Wind SpeedMeters/sec

1J.s+Kf


-Kt

Torque constant

Ta (Wind torque) output from Cq inputand intermediate product

-C-

Rotor radius(meters)

1L.s+R

Mot/GenTransfer Function

Kb*Ng

Generator back emf constant

-Ng

GainDC Generator

Armature Voltage

Cq,Lookup Table

f(u)

5XpiXRad 3̂XW 2̂ output from .Windspeed input

Wr (rotor speed)Iarm

Vbemf

Tgen

The purpose of the simulations for system #2 is to demonstrate and measure power generation for two cases: In the first simulation, DC generator voltage and wind speed inputs are applied to yield less than maximum power efficiency at steady state. The second simulation uses a combination of wind speed and DC generator voltage to give maximum efficiency of 43% at steady state. The two cases illustrate that efficiency could be maximized through controlling DC voltage. These simulations demonstrate the need for a control system that will maximize steady-state efficiency and generated power.

The Efficiency (Cp) waveform, as it appears in this and all remaining simulations, is simply the operating point on the Cp curve of Figure #3. The turbine efficiency Cp is a function of tip-speed ratio and is the instantaneous measure of wind power captured by the turbine compared to maximum available wind power: Cp = Pturbine/Pwind..

The first simulation is a step wind speed input of 9 meters per second and a step voltage input of 250 volts to the DC generator. Though these large step changes in voltage and wind speed are unrealistic, they provide the form of the step response for validation and the steady state response can be seen. The start time for both steps is delayed 1/10 second in order to give detail near the Y-axis. The results appear in Figures #11 and #12

Figure #11: Waveforms for System #2, Complete Plant

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Figure #12: System #2, Complete Plant Steady-State Response

Transient results-- Simulink model, System #2 case 1 :

Figure #11 shows that initially, the DC machine acts as a motor with rotor nearly locked by the high inertia load. This accounts for the spike in reflected generator torque up to a high positive value (4.3 x106 N-M) before settling to a final value of about -57 kiloN-M. It should be remembered that positive torque aids wind torque and negative torque opposes wind torque. Generator Power also spikes to over -8 megawatt. Negative power indicates motoring. The wind turbine torque builds slowly with rotor speed as is predicted by the shape of the Cq curve in Figure #4. Wind torque aids motor torque to

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4.601

tip-speed ratio

1.913

rotor speed Wr

-5.767e+004

generator torquereflected to turbine

generated PowerVoltsXIarm

Wrotorrad/sec

Windspeed9 Meters/sec

Wind torque

Wind Power

Voltage Input250 volts

1

J.s+Kf


5.864e+004

Torque from wind

Kt

Torque constant (N-M/amp)

-C-

Rotor radius

Product1

Product

1.122e+005

Mechanical Powerfrom wind

Lookup Table

436.3

Iarm-1

L.s+R

GenTransfer Function

GenPowerwatts

Gen torquereflected

-Ng

Gain

Efficiency (Cp)

1.091e+005

DC generated power (Pos=generating)

0.209

Cp

Kb*Ng

Back emf constant

f(u)

5XpiXRad^3XW^2 output from Windspeed input

Iarmature Generator Torque

Back emf

Relected gen torque(N-Meters)

increase rotor speed. After about 3/4 second, back emf to the DC machine builds until it exceeds the input voltage and the machine changes from motoring to generating. From 3/4 seconds until the end of the simulation, generator torque opposes wind torque and the DC machine is generating power. The plot of efficiency, Cp, rises until a maximum of .21 or 21%. As was the case for the MIMO simulation with Matlab, a large initial power is put into the system before power can be generated.

Steady State Results -- System #2 case 1: DC voltage = 250v Figure #12 contains the Simulink block diagram with displays that are fixed at their values after 3 seconds (long enough to be considered steady-state). For ease of interpretation, all displays are on the right side of the diagram. To validate the model, steady state values are inserted into torque balance and power balance equations and the results are checked below:

Torque balance eqn: |wind torque| = |generator torque| + frictional loss torque

|5.864x104| = |-5.767 x104| + (D)(r)

58,640 = 57,670 + 500x(1.913)

58,640 N-M = 58,626 N-M

Power balance eqn: Turbine power = generated power + I2R loss + frictional losses.

1.122x105 = 1.091 x105 + (436.3)2(.0064) + (D)(r)2

1.122x105 = 1.091 x105 + 1218 + 1830

112,200 watts = 112,148 watts

It is evident that for this simulink model, steady-state torques and steady-state powers balance to nearly 4 figures. It can be concluded that transient and steady-state responses for this second simulation prove the validity of this model.

Simulation Results, Steady State -- System #2 case#2: DC voltage = 410v :

A simulation of system #2 is again performed, but this time the DC input voltage is changed from 250 volts to 410 volts and wind speed is kept at 9 m/sec. The dynamic results are similar to the case 1simulation and need not be discussed. The steady state results of the simulation show power efficiency is at .4275, or nearly maximum. Figure #13 shows the steady-state results for system #2, case 2. This simulation demonstrates that for steady wind conditions, controlling voltage applied to the DC generator can maximize power efficiency. As with case #1, the model can be validated by inserting

26

steady state values into torque balance and power balance equations. This exercise is left for the reader.

Figure #13: System #2 Case #2: Maximum Power efficiency

System #2 is a complete model of the plant consisting of wind turbine rotor, gear train, DC machine and DC voltage supply line. The next systems to follow are examinations of control systems that will maximize the power efficiency of the plant.

System #3: Speed Control of Plant with DC generator .

It is apparent from the results of previous simulations that to maximize power efficiency, the DC voltage applied to the generator must change with wind speed. From figure #3, it can be seen that the optimum power efficiency is 43%. This maximum occurs at a tip-speed ratio () of 7.5. This means for a given wind speed, one wind turbine rotor speed would yield the maximum efficiency. Equation #16 gives this rotor speed.

27



7.525

tip-speed ratio

3.128

rotor speed Wr

-7.176e+004


generated PowerVoltsXIarm

Wrotorrad/sec


Wind torque

Wind Power

Voltage Input410 volts

1

J.s+Kf


7.333e+004Torque from wind

Kt


-C-

Rotor radius

Product1

Product

2.294e+005


Lookup Table

542.9

Iarm-1

L.s+R


GenPowerwatts

Gen torquereflected

-Ng

Gear ratio

Efficiency (Cp)

2.226e+005


0.4274

Cp

Kb*Ng

Back emf constant

f(u)


Iarmature Generator Torque

Back emf


Equation #16: =

Where r = rotor angular velocity in radians/secondR = radius of turbine rotor (meters)v = wind speed (meters/sec)

In System #3, a proportional controller is added before the DC voltage input that will force rotor speed to the optimum value given by equation #16. Figure #14 is a Simulink block diagram of System #3.

Figure #14: System #3:Plant with Speed control: Because of the non-linearities introduced the wind turbine, system #2 or system #3 are no longer linear time-invariant (LTI) systems. The principle of superposition and frequency domain analysis will not generally be valid. However, the steady-state error for system #1 can assist in assigning a value for proportional gain (K1) to minimize steady-state error. The gain, K1, can be used in the speed control loop of system #3.

28

System #1 has two inputs: a speed reference (ref) and a torque disturbance due to the wind. Steady-state error can be separately calculated for a step wind torque input and and a step speed reference. The steady-state error can be calculated by then adding the two components of steady-state error.

Steady-State Error contribution due to ref :

Effects of ref on steady-state error will be considered first by removing (or zeroing) input Ta and reducing the system to a unity feedback system through techniques of block diagram reduction. System #1 can be reduced to a unity feedback system of figure #15 with G(s) given by equation #17

Equation #17: G(s)=

Figure #15: SS error due to ref: System #3 reduced to unity feedback.

The steady-state error in rotor due to a step ref is:

Equation #18: SSErr(ref) = lim

s-->0

Equation #18 simplifies to:

Equation #19: SSErr(ref) = =

This steady-state error must be added to the SS error contribution of Ta calculated next.

Steady-State Error contribution due to wind torque Ta :

29

G(s)K1

_+

ERRORrefrotor

Techniques of block reduction can be used to find the steady-state error in rotor due to wind torque input Ta. Referring to Figure #14 (System #3), a new block diagram relating Ta to "error" must be made. For the contribution due only to Ta, the input ref must be zeroed. Note that the blocks contributing to Ta are eliminated and replaced with a step input for this steady-state error calculation. This block diagram is shown in figure #16.

Figure #16: SS error due to Ta: System #3

Equation #20: SSErr(Ta) = lim

s-->0

Equation #21: SSErr(Ta) = =

Adding the two error components together yields:

Equation #22: SSErr (rotor)= -

The steady state error equation shows that proportional gains (K1) under 100 would yield relatively large percentage errors.


Simulation Results, Steady State -- System #3 :

To validate these steady-state calculations, a simulation will be done with the following parameters:

Wind speed = 9 meter/second step input

30

-1 . Js + Kf

-(Ng2KbKt + K1KtNg) Ls + Ra

ERRORTa

_

+

Proportional gain K1 = 80 ref = .346 x windspeed (from Equation #16)

The Simulink model appears in Figure #17 which shows that system #3 has a single input, wind speed, which governs wind torque input (Ta), and DC generator voltage (Ea). Steady-state values are displayed at the right of the figure.

Figure #17: System #3, Simulink model for speed control

The setpoint commanded was .346 x 9 m/s wind speed or 3.114 rad/sec. This optimum rotor speed would have yielded an optimum tip speed ratio of 7.5 which in turn would have produced an efficiency of 43 %. But the displays show the final turbine rotor speed is 1.175 radians/sec. Due to the large error, the efficiency is only .8% (see display labeled Cp1 in Figure #10). The low value for K1 is the reason for the poor efficiency.

These results will be compared with results predicted by steady-state error equation #22

SSErr (rotor)= -

31



2.826

tip-speed ratio

1.175rotor speed Wr

-3425


generated PowerVoltsXIarm Wrotor

rad/sec


Wind torque

155.3

Vbemf

1J.s+Kf


4014Torque from wind

Kt


-C-

Rotor radius

.346

ReferenceGain

80

ProportionalGain

Product1

Product

4716


Lookup Table

-1L.s+R


GenPowerwatts

Gen torquereflected

-Ng

Gain

Efficiency Cp)

DCVoltage4020


0.008787Cp1

Kb*Ng

Back emf constant

f(u)


Iarm Gen Torque

Back emf

Torque from wind (N-Meters)


Turbine Rotor Speed

=1.9400 - .000916 rad/sec

Combined SSErr = 1.939 rad/sec

As a check on the validity of this error, the final rotor speed should be predicted as:

final rotor = ref - error = 3.114 - 1.939 = 1.175 rad/sec.

The predicted value for rotor agrees to 4 figures with the value shown for rotor in Figure #17.

The large error is due to system#3 being a type 0 system (no pure integrators in the forward transfer function). By nature there must exist a steady state error.If steady-state conditions were all that needed considering, the proportional gain could be increased to reduce SS error.

The above calculations show frequency domain methods can be used to design proportional gain values for a given error. Additionally, an integral component in parallel to the proportional gain would reduce the steady-state error to zero. Before continuing the design of this speed control method, however, the transient response of the simulation should be examined.

Figure #18: Generated Power System #3 (Speed Control)

32

Simulation Results, Transient response :

A plot of Generated Power in Figure #18 shows an enormous negative spike of -7 Megawatt in generator power in the first ,2 second of the simulation. As with previous simulations, the large initial motoring is due to a large initial step change to the DC voltage. Not only does motoring represent power loss, but the huge armature current would damage the DC machine and could upset the power grid. It can be concluded that while speed control offers promise for maximizing efficiency under steady state conditions, the transient responses are not acceptable. In the following section, a torque control scheme will be simulated.

System #4: Torque Control of Plant with DC generator .

A limitation that might be placed on a wind generating system is that the generator be prevented from entering a motoring mode of operation. This would prevent the large power losses with speed control while accelerating or decelerating rotor speed to an optimum tip-speed ratio. Such a restriction is possible if the sign of the torque supplied by the generator is consistent with generating power instead of motoring power. A torque control scheme could guarantee this.

In the wind energy industry, a standard region 2 torque control method is frequently used [1,4]. The method is called indirect speed control (ISC). With this scheme, power generation under steady-state conditions is maximized by setting generator torque equal to optimum aerodynamic torque (Ta) for a given wind speed. Optimum Ta is the value that will produce the most power at a given wind speed.

An expression for wind torque can be derived from equation #1 as follows:

Equation #1: P = ½ Rad 2Cp()v3

Since blade pitch () is fixed, an expression for maximum power is given by:

Equation #23: Pmax = ½ Rad2Cp(optimal)v3

Power is aerodynamic torque (Ta) times rotor speed (rotor), so:

Equation #24:

Substituting equation #2 into equation #24 yields:

Equation #25:

33

Since maximum efficiency, Cp, is .43 at a tip-speed ratio of 7.5, then optimum = 7.5. Lumping constants in equation #25 together yields [4]:

Equation #26: Ta = Krotor2

where constant K = = 7615

And since generator torque (Tg) is gear-transformed and negative to oppose aerodynamic torque:

Equation #27: Tg = - rotor2 = -176rotor

2

(For the above derivation, the torque component due to friction is neglected.)

Torque control method under transient conditions:By design, this torque setpoint is the value of generator torque to yield maximum power under steady state conditions for a given wind speed and rotor speed. It remains to be shown that this control scheme will give satisfactory results during transient conditions. It can be shown that the turbine rotor speed is be pushed toward optimum rotor speed whether the turbine is rotating faster or slower than optimum. This requires that:1). zero torque exists at optimum efficiency2) an accelerating torque results if turbine rotor speed is less than optimum efficiency3). a decelerating torque results if turbine rotor speed is greater than optimum efficiency. Rotor acceleration ( ) can be expressed as [1]:

Equation #28: = (Ta - TgNg)

Combining equations #1, #26, #27 and #28 yields:

Equation #29: =

Referring to the Cp efficiency curve of Figure #3, it can be seen that Cp(optimal) = .43 and optimal = 7.5 . Since Cp() < Cp(optimal), Equation #29 demonstrates that, for tip-speed ratios above optimal, the bracketed term will be negative and rotor acceleration will be negative. For < optimal, it must be shown that Cp()/3 > Cp(optimal)/(opt)3 in order for positive acceleration to result. Transposing the equation and inserting values for opt and Cp(opt) yields: Cp() > .0010193. When this cubic curve .0010193 is plotted against the Cp() curve of Figure #3, It can be seen that Cp() > .0010193over the range of 3.3 < < optimal. This comparison demonstrates that acceleration will be positive for below optimal..

It is in this manner that implementing the torque control method known as Indirect Speed Control (ISC)[4] maximizes power by maintaining optimum tip-speed ratio (). It

34

should be noted that for < 3.3, rotor acceleration will be negative and generator torque will slow the turbine instead of allowing it to accelerate to maximum efficiency. This fact points out the need for ISC control method to maintain above 3.3. This is not a flaw in ISC control method, as ISC is only one of the control methods governing control in region 2 wind speeds; The control method for region 1 (start-up) would likely be introduced if drops below 3.3 or if wind speed is insufficient.

System #4 implements this torque control method and appears in Figure #19. The changes from system #3 to system #4 are the replacement of speed reference with the torque reference given by equation #27 and the addition of generator torque feedback. It should be noted that generator torque feedback appears to be positive instead of negative. This is because the pickoff point for generator torque is before the -Ng gain, whose sign corrects the torque sign but also reflects torque through gearbox.

Figure #19: System #4 Indirect Speed Control.


35



2 squared


1J.s+Kf


Kt


-C-

Rotor radius

-176Optimum gain (.5pAR^3Cp)/NgtipspdOpt^3

Lookup Table

-1L.s+R


-Ng

Gain

-K-

Back emf constant

f(u)


uv

IarmGen Torque

Back emf



Torque setpoint

Wr^2

Turbine rotor speed

As with system #3 (speed control), system #4 is a single input system in which wind speed governs the wind torque input (Ta), and DC generator voltage input (Ea). The difference is that a torque control method governs the setpoint for DC generator voltage instead of a speed control method.

The first simulation demonstrates the indirect speed control method in both accelerating and decelerating operation: The turbine is accelerated when tip-speed is lower than optimum and decelerating the turbine when tip-speed is higher than optimum. The input for this simulation is wind speed of 11 meter/sec for 50 seconds followed by 7 meter/sec for 50 seconds.

For this simulation The torque efficiency curve (Cq) was modified slightly for this simulation. When blade pitch is set to -1O torque is negligible at zero rotor speed. Under these conditions, no wind torque would be generated and the system would remain in a stall condition. A liberty was taken of modifying the table so that a small starting torque would result. Practically, this could be achieved by changing blade pitch to an angle that could "catch the wind". Alternatively, the turbine could be motored to a minimal speed.

36

Figure #20: System #4 Waveforms, Indirect Speed Control

Transient results-- Simulink model, System #4 :

Figure #20 contains transient response waveforms for the system with zero initial rotor speed and a step wind input of 11 meters /second then dropping to 7 meters /second. The two wind speed in the simulation allow the ISC method to be tested when the rotor is both accelerating and decelerating.

Owing to the small initial wind and generator torques, this simulation takes longer to reach steady-state values. In the first 80 seconds, the turbine accelerates until leveling off at about 3.5 rad/sec. The “Torque from Wind” waveform clearly shows the non-linearity of the torque calculation table. It can be seen that wind torque slowly builds as turbine rotor speed builds and that generator torque is always negative and also slowly building. The power efficiency (Cp) climbs irregularly before leveling off at just over .40 (nearly maximum efficiency). Significantly, this is the first simulation in which generator power is always positive. This means the control system has prevented the DC machine from motoring and dissipating power.

37

In the last 70 seconds of the simulation, the wind speed has gone from 11 meter/sec to 7 meters/sec. Generator torque now exceeds wind torque and the turbine decelerates to a steady-state speed of just over 2 radians/sec. Power efficiency drops initially, but climbs back to nearly maximum efficiency over time. This proves the ISC method optimizes efficiency by either accelerating or decelerating to optimum tip-speed ratio.

Figure #21: System #4 Indirect Speed Control Simulation: Steady-State Results

38


6.625

tip-speed ratio

tip-speed ratio

2 squared

2.142

rotor speed Wr

-4.701e+004


generated PowerVoltsXIarm Wrotor

rad/sec


Windspeed

Wind torque

Wind power

283.1

Vbemf

1J.s+Kf


4.803e+004

Torque from wind

Kt


-807.6

Torq Reference

1089

Torq Feedback

SignalGenerator

-C-

Rotor radius

Product1

Product


1.029e+005


1/s

Integrator1

2

Initial Region 2rotor speed

-1L.s+R


GenPowerwatts

Gen torquereflected

-Ng

Gain

Efficiency Cp)

2.137e+007

Display2DCVoltage

1.002e+005


CqLookup Table

0.4075

Cp1

-K-

Back emf constant

f(u)


uv

IarmGen Torque

Back emf

Cq



Torq setpoint

Wr^2

Steady-state results-- Simulink model, System #4 :

Figure #21 shows that an efficiency of nearly 41% is achieved when the wind energy standard torque control is used (Cp = .40751). This is nearly the peak value of 43% indicating little steady-state error. As with the speed control system (system #3), this type 0 system requires a P-I controller or an I controller to reduce steady state error to zero.

But of greater importance than achieving optimal steady-state response is the goal of maximizing power under realistic conditions of variable and random wind speeds. The next simulation uses a realistic wind-speed profile as input.

Simulations of Indirect Speed Control with Realistic Wind Profiles.

Having found a control method that maximizes generated power for steady state wind conditions, the method must be simulated with realistic wind profiles. By its nature, wind speed is stochastic. A realistic wind-speed profile must be variable, within speed boundaries and within limits of frequency. The varying wind speed will prevent the turbine from maintaining optimal rotor speed and thereby reduce its efficiency. The generated energy over time ( ) can be compared with the theoretical maximum energy generated by the wind given by the time integration of equation #23 (

). This comparison will be a meaningful measure of efficiency.

For the simulation to follow, a mean wind speed of 7 meters/second is added to a random number generator with magnitude limited to +/– 3 m/sec with a period of 5 seconds. This wind profile was chosen to replicate typical wind conditions as measured by the NREL. The simulation is given an initial condition of r = 2 rad/sec in order to start the system in region 2 operation. A realistic wind profile input is added to System #4 and the resulting block diagram is Figure #22. Figure #23 has the resulting waveforms.

39

Figure #22: System #4 ISC with Realistic Wind Profiles

40

Figure #23: System #4 waveforms for ISC with Realistic Wind Profiles

41

System #5: Simulations Introducing an AC generator into the Model

AC generators are more common in energy generating applications owing to their lower cost and simpler maintenance than DC generators. Advances in power electronic devices have led to greater control of voltage, frequency and phase of the applied AC waveform to AC motors and generators.

Figure 24 is an industry-recognized equivalent circuit of a squirrel cage or wound rotor AC machine [7,9]. This model represents a per-phase equivalent circuit. All rotor circuit parameters have been referred to the stator. The three branches of the circuit represent referred rotor current Ir, magnetizing current Im, and windage and friction loss current Iwf. Constants Rs and Rr represent stator resistance and referred rotor resistance, respectively. L is combined stator inductance and referred rotor inductance. Lm represents magnetizing inductance and Rm is the resistance that causes windage and friction loss. Eg is the applied AC voltage. An AC inverter is assumed to supply a pulse-width-modulated waveform whose magnitude and frequency can be controlled.

The values chosen for the circuit model are those of a 600 kwatt, 4-pole, 3-phase AC squirrel-cage induction machine. The rated line voltage of the machine is 460 volt; the synchronous speed is 1800 RPM; rated speed is 1790 RPM; rated torque at 1790 RPM at 1790 RPM. The following equations are used to develop an embedded Matlab function to represent the AC generator. Definitions of parameters appear in Table #2.

42

PARAMETER DESCRIPTION SYMBOL VALUE AC Generator parametersAC motor/gen Power rating n/a 600 KwattRated lineVoltage n/a 460 VRated phase-neutral Current IaRated 895 ARated Speed n/a 1790 RPMSynchronous speed n/a 1800 RPMRated Torque n/a 3183 N-MRotor resistance referred to stator Rr 1.7 mStator resistance Rs 7.2 mStator + referred rotor inductance L .15093 mHWindage +friction resistance Rm 24.5 Magnetizing inductance Lm 2.281 mHAC motor/gen polar moment of inertia Jg 21 Kg-m2

AC machine efficiency n/a 95%

System ParametersApplied AC voltage Es variableFrequency of Es in(radians/second) e variableGenerator rotor speed (radians/second) g variableGenerator synchronous speed (rad/sec) syn variableGenerator torque at generator shaft Tg variableSlip s variable

Table #2: Parameters for AC Generator

Figure #24:

43

1 2Lstator + referred rotor

150.93microH

1

2 Lm

2.281mh

Rstator

.0072ohm

Rr

.0017ohm

Rm

24.5

Istator--->

Ilosses-->

Imagnetizing->

Ireferred rotor-->

Es

Per-Phase circuit model for 600 kwatt, 3-phase AC machine

The following relationships describe the operation of the AC machine in steady state operation where frequency is fixed:

Equation #30: syn =e /2

Synchronous speed of generator rotor is ½ of electrical speed because the AC machine has 4 poles (two pole pairs)

Equation #31: slip = s = =

From the equivalent circuit, referred rotor current is:

Equation #32: Ir =

Definitions of power components:

Active power to rotor =

Power loss in rotor =

(Note that factor of three is summation of the per-phase quantities)

PowerMechanical = Pm = Tgxg

By conservation of energy, in motoring mode:

Active power to rotor = Power loss in rotor + PowerMechanical

This same equation holds in generating mode if it is recognized that slip and torque are negative. In either case:

+ Tgg

44

Substituting equation # 31 into above yields:

Equation #33:

Equations #32 and 33 can be combined to give torque as a function of electrical frequency (e), slip (s) and voltage (Es).

Equation #34:

A plot of torque versus rotor speed for the rated frequency of 60 Hz appears in Figure #25. The sharp sensitivity of torque to rotor speed changes near synchronous speed should be noted. This is the “low-slip” region in which the generator will be operating.

45

Figure #25: Torque vs rotor speed for 600 kwatt AC Machine at 460 volts

Figure #26 shows a family of torque-speed curves for frequencies 45 hertz to 65 hertz illustrates that torque can be increased or decreased sharply by changing applied frequency. The figure also shows that peak torque rises with decreasing applied frequency. This effect is due to reduce reactance of the magnitizing branch of the circuit model (exLm) thereby increasing the Im component of torque.

46

0 200 400 600 800 1000 1200 1400 1600 1800 2000-1.5

-1

-0.5

0

0.5

1x 10

4 Torque vs rotor speed for a 600 kwatt 3-phase, 4-pole AC machine

rotor speed in rpm

Torq

ue in

N-M

Figure #25: Family of Torque vs rotor speed curves for AC Machine at 460 volts

Volts-per-Hertz Operation

The region near synchronous speed known as the “low-slip” region. The phenomena of increasing voltage with decreasing frequency in the low-slip region is eliminated if applied voltage Es is reduced proportionately to applied frequency. In this “volts-per-hertz”operation, Es/e is held constant. Torque is very nearly proportional to applied frequency or voltage. An analytical benefit is that the relationship of torque to applied voltage or frequency is linear. An AC machine in volts-per-hertz operation is mathematically similar to a DC machine with constant applied field. Volts-per-hertz is a common mode of operation for AC machines controlled by sophisticated motor drives.

A new family of torque speed curves in Figure #26 illustrates volts-per-hertz operation of an AC motor/generator.

47

0 200 400 600 800 1000 1200 1400 1600 1800 2000-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5x 10

4 Family of Torque vs rotor speed Curves showing rise in peak torque with decreasing frequency

rotor speed in rpm

Torq

ue in

N-M

45Hz 50Hz

55Hz 60Hz 65Hz

Figure #26: Family of Torque vs speed curves for AC Machine in Volts per Hertz operation

For large horsepower machines operating near their synchronous speed in the low-slip region (s < .01):

Rr/s >> Rs

Rr/s >>exL

These relationships lead to the following approximations for Equation #35.

Equation #35:

Substituting into equation #33 yields:

48

0 200 400 600 800 1000 1200 1400 1600 1800 2000-1.5

-1

-0.5

0

0.5

1x 10

4

rotor speed in rpm

Torq

ue in

N-M

Family of Torque vs rotor speed Curves in volts-per-hertz operation

45Hz50Hz 55Hz

60Hz 65Hz

Substituting “volts-per-hertz” constant: yields:

Equation #36:

For the 600 kwatt AC machine:

m =.7056 volt/rad/sec

Substituting into equation #35 and supplying rotor resistance Rr = .0017 yields:

Equation #37: Tg = 1757(e- 2g)

The above equation shows that generator torque is proportional to applied electrical frequency minus twice the generator rotor speed feedback. (The factor of two due to the 4-pole motor halving the rotor speed of the machine).

To incorporate the AC generator into the simulink two changes need to be made:

1). torque equation #37 can replace the simulink DC machine block, -Kt/(Ls + Rr)

2). In place of back emf voltage feedback inherent in the DC machine, the AC machine must have generator rotor speed feedback.

49

Only the torque relationship need go in the simulink control loop. An embedded Matlab block is used to calculate generated power, stator current, and power factor. Figure #27 shows the new model of the wind turbine with AC generator incorporated into it. The control strategy used is Indirect Speed Control (ISC).

Figure #27:System #5, Indirect Speed Control with an AC machine

50

AC Generator Table of Values:_____________________________Rr = .0017 ohm; Rotor resistanceRs = .0072 ohm; Stator resistanceL = .15093 mh; stator + referred Rotor ind.Kf = 500; N-meter-seconds (damping constant)J = 427,627; kg-meter^2Ng = 43.165; Gear ratio Rad = 21.65 meter (Rotor radius)p = 1kg/m^3 air density

1

torq-to-freq conversion gain

7.113

tip-speed ratio

tip-speed ratio

2 squared

Es

We

Wr

T

s

MagIs

theta

PF

P

SiemensACGenerator

rotor speed=inputtorque=out

3.614

rotor speed Wr

-112615.87


generated power

Ng

gear ratio

-3.005

angle Istator

Wrotorrad/sec


Windspeed

Wind torque

Wind power

We

.7055

V/HzFactor

1J.s+Kf


114422.76

Torque from wind

Slip

-C-

Rotor radius

Random Signal:5 sec period,

+/-3 amplitude

Product

-0.9908

PowerFactor


4.135e+005


614.5

Istator

-2669

Generator Torq

Gen torquereflected

-2609

Gen Torq

Ng

Gear Ratio

Efficiency Cp)

CqLookup Table

0.422

Cp1

-4.001e+005

AC generated power (Negative=generating)

1757

AC Generator

f(u)


.

2

uv Cq


Wr^2

Wr (turbine rotor speed)

Torq setpoint

WgeneratorRotor

Gen Torq We(electricalfrequency rad/sec)

Simulation Results and Validation of AC generator system using ISC:

A simulation is performed with the following conditions:

AC generator replacing DC generator.

Control method: Indirect Speed Control (ISC)

Wind speed: 11 meters/second step input

Simulation time: 150 seconds

Figure #28: System #5 waveforms, constant wind

51

Transient results-- Simulink model, AC generator, Indirect Speed Control, constant wind :

The transient results in Figure #28 are similar to those in Figure #20 – System 4 indirect speed control with the DC generator. Generator torque is negative indicating a torque opposite wind torque. This assures the AC machine is generating and not motoring. Generator torque ramps to match the rising torque exerted by the wind. Turbine rotor speed rises until nearly maximum efficiency and stays there. Generator power is shown as negative power as calculated in the embedded Matlab block for the AC machine. The slip of the AC generator can be seen to ramp negatively to -.005 and stay there. This is validation that the AC machine stays within the low-slip region and is acting as a generator.

Steady-state results-- Simulink model, AC generator, Ind. Speed Control, constant wind :

Figure #27 has values at the end of the 150 second simulation. The tip-speed ratio of 7.118 is very nearly at optimum efficiency . The embedded AC generator calculates an angle of stator current of –3.006 radians (172o) indicating current flow into the voltage source. The power factor is -.99, which is favorable.

Simulations of AC generator using ISC with Realistic Wind Profiles. The final simulation is under identical conditions as System #4 with realistic wind profile but with the AC generator. the following conditions apply:

AC generator.

Control method: Indirect Speed Control

Wind speed: 11 meter/sec step +5 second period random number generator w/ mag limit +/– 3 m/sec.

Simulation time: 150 seconds

Transient results-- Simulink model, AC generator, Indirect Speed Control, Realistic wind :

Results in Figures #29 and 30 are qualitatively the same as the results of realistic wind profiles with the DC machine. Of note is the efficiency curve (Cp) which can be seen to move toward the optimum .43 value even in varying wind conditions. Slip is again between 0 and -.005, thereby justifying the AC machine approximations.

These results are validation of the model for the indirect speed control method when used with an AC generator.

52

Figure 29: System #5, AC Generator with realistic wind profile

53

AC Generator Table of Values:_____________________________Rr = .0017 ohm; Rotor resistanceRs = .0072 ohm; Stator resistanceL = .15093 mh; stator + referred Rotor ind.Kf = 500; N-meter-seconds (damping constant)J = 427,627; kg-meter^2Ng = 43.165; Gear ratio Rad = 21.65 meter (Rotor radius)p = 1kg/m^3 air density

1

torq-to-freq conversion gain

5.966

tip-speed ratio

tip-speed ratio

2 squared

Es

We

Wr

T

s

MagIs

theta

PF

P

SiemensACGenerator

rotor speed=inputtorque=out

2.659

rotor speed Wr

-63583.00


generated power

Ng

gear ratio

-3.064

angle Istator

Wrotorrad/sec


Windspeed

Wind torque

Wind power

We

.7055

V/HzFactor

1J.s+Kf


91616.52

Torque from wind

Slip

-C-

Rotor radius

Random Signal:5 sec period,

+/-3 amplitude

Product

-0.997

PowerFactor


2.436e+005


345.9

Istator

-1511

Generator Torq

Gen torquereflected

-1473

Gen Torq

Ng

Gear Ratio

Efficiency Cp)

CqLookup Table

0.3682

Cp1

-1.669e+005

AC generated power (Negative=generating)

1757

AC Generator

f(u)


.

2

uv Cq


Wr^2

Wr (turbine rotor speed)

Torq setpoint

WgeneratorRotor

Gen Torq We(electricalfrequency rad/sec)

.

.

Figure 30: System #5 waveforms with realistic wind profile

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Conclusion: Systems #1 through #5 are a progression of Matlab and Simulink models culminating in a 3-phase AC generating system employing a torque control method proven to maximize region #2 power capture. The final model could serve as a platform for further modeling. The per-phase, steady-state model of the AC generator could be replaced by a more exact dynamic d-q model (direct and quadrature) of an AC machine. A d-q AC generator model would be useful in evaluating sudden torque load that would yank the generator out of the low-slip region. Blade pitch could be added as another input to the system, if the mathematics behind blade pitch were furnished.

APPENDIX A: Derivation of System #1 State-Space to Transfer Function

Given:

= (-Ra/L) (KbNg/L) X Ia + -1/L 0 X Ea

(-NgKt/J)Ia (- D/J) r

0 1/J Ta

then:

The A, B, C, D state space matrices for Matlab appear below.

A = (-Ra/L) (KbNg/L)

(-NgKt/J)Ia (-D/J)

B = -1/L 0

0 1/J

C = [ 1 1 ]

D = 0

Definitions of state vector and input vector:

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X =

U =

In general:

X(s) = (sI-A)-1 BU(s)

Y(s) = C(sI-A)-1 BU(s) (in the case where D = 0)

T(s) = = C(sI-A)-1 B

For system #1,

(sI-A) =

and

determinant (sI-A) = |sI-A| =

so

(sI-A)-1 =

where

and so:

(sI-A)-1 B =

and

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T(s) =

The four transfer functions represented here are:

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APPENDIX B: Matlab program for System #1 State-Space Simulation% DCStatesIarmSpeedDocumentedOct10 % State-space simulation of multiple-input, multiple-output system (MIMO)% % Equipment simulated:% 1). Siemens 600 kwatt DC motor/generator % (Siemens Model# 1GG5 402-5NK..-1VV5).% 2). Wind turbine rotor and geartrain from test site of% the National Renewable Energy Laboratory (Golden Colorado)%%% Inputs: % 1). A Step torque of 162 kilonewtons is applied at low-speed side of geartrain% 2). A Step voltage of 250 volts is applied to the armature of the DC machine.%% States:% 1). Armature current (Ia)% 2). Turbine rotor speed (Wr)% % Outputs:% 1). the states above are the outputs in this simulation. R = .0064; % Armature resistance L = .00011; % Armature inductance = .11 milliHenry TRated = 4480; % torque at rated currentIRated = 1490; % Current at rated torqueVRated = 420; % Rated voltage at rated speedSRated = 1280; % Rated speed in RPMKb = 3.007; % Back EMF constant (V/rad/sec)Kf = 500; % N-meter-seconds (damping constant reflected to turbine low speed shaft.)Kt = 3.062 % Torque constant Trated / IratedJrotor = 388500; % kg-meter^2 (rotor polar moment of inertia (J)Jmotor = 21; % kg-meter^2 (motor/generator polar moment of inertia (J)Ng = 43.165; % Gear ratio = 43.165Rad = 21.65; % Rotor radius for CART turbinep = 1.00; % Air density in kg/m^3J = 427627; % total polar moment J ( Wind Turbine Rotor + Reflected generator rotor)

time = 0:.002:1.998; % establish time vectorVin=250*ones(1,1000); % step voltage to DC motor/generatorwindin=162000*ones(1,1000); % step torque to low-speed shaft (Maximum torque at NREL is 162000) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Inputs are 1): DC motor/Gen voltage; and 2): Ta torque from wind% States are armature current and shaft radial velocity.% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PLOT #1 radial velocity vs time% INPUTS: Vin = 250 V step ;torque = 0% INITIAL CONDITIONS: Rad Vel = 126 rad/sec%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A = [-R/L Kb*Ng/L;(-Ng*Kt/J) -Kf/J];B = [-1/L ; 0]; % B vector for Voltage inputC = [0 1]; % [0 1] C vector for radial velocity state variable as output; D = [0]; x0 =[0;0]; % initial condition: radial vel = 2.9 rad/sec; Iarm = 0 ampssys1= ss(A,B,C,D);w_Volts = lsim(sys1,Vin,time,x0);

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subplot(2,2,1);plot (time,w_Volts);title('radial vel vs time w/ Ea = 250V step, Ta in = 0')xlabel('time')ylabel('radians per sec')grid on [num1,den1] = ss2tf (A,B,C,D,1) %(A,B,C,D,1) selects transfer function for input #1 (DC mot/Gen voltage)[zz,pp,kk]= tf2zp(num1,den1) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PLOT #2 radial velocity vs time% INPUTS: torque step of 162 K-Newton-Met ;Vin = 0% INITIAL CONDITIONS: Rad Vel = 126 rad/sec%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B = [0 ; 1/(J)]; % B vector for Torque inputx0 =[0;0]; % initial condition: radial vel = 2.9 rad/sec; Iarm = 0 ampssys2= ss(A,B,C,D);w_Wind = lsim(sys2,windin,time,x0);subplot(2,2,2);plot (time,w_Wind);title('radial vel vs time w/ Ea = 0; Ta = 162 kN-M step')xlabel('time')ylabel('radians per sec')grid on %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PLOT #3 Armature current vs time% INPUTS: Vin = 250 V step ;torque = 0% INITIAL CONDITIONS: Rad Vel = 126 rad/sec%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A = [-R/L Kb*Ng/L;(-Ng*Kt/J) -Kf/J];B = [-1/L ; 0]; % B vector for Voltage inputC = [1 0]; % [1 0] C vector for Armature current state variable as output; D = [0]; x0 =[0;0]; % initial condition: radial vel = 2.9 rad/sec; Iarm = 0 ampssys3= ss(A,B,C,D);Ia_Volts = lsim(sys3,Vin,time,x0); subplot(2,2,3);plot (time,Ia_Volts);title('Ia vs time w/ Ea = 250V step; Ta = 0')xlabel('time')ylabel('Amps')grid on %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PLOT #4 Armature current vs time% INPUTS: torque step of 162 K-Newton-Met ;Vin = 0% INITIAL CONDITIONS: Rad Vel = 2.9 rad/sec%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B = [0 ; 1/(J)]; % B vector for Torque inputx0 =[0;0]; % initial condition: radial vel = 2.9 rad/sec; Iarm = 0 ampssys4= ss(A,B,C,D);Ia_Wind = lsim(sys4,windin,time,x0);subplot(2,2,4);plot (time,Ia_Wind);title('Ia vs time w/ Ea = 0; Ta = 162 kN-M step')xlabel('time')ylabel('Amps')grid on

pause%

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% Now add effects of both DC 250 volt input and wind speed torque input%w = w_Wind + w_Volts;Ia =Ia_Wind + Ia_Volts;subplot(2,2,1);plot (time,w);title('Radial velocity vs time step wind torque 162 kN-M & step 250 to DC gen ')xlabel('time')ylabel('radians/sec')grid on subplot(2,2,3);plot (time,Vin);title('DC motor voltage input')xlabel('time')ylabel('Volts')grid on subplot(2,2,2);plot (time,Ia);title('Ia vs time w/ Ta=162 kN-M step & Ea= 250 step')xlabel('time')ylabel('Amps')grid on subplot(2,2,4);plot (time,windin);title('Wind torque input')xlabel('time')ylabel('Torque N-m')grid on

APPENDIX C: Embedded AC Machine Used in Simulink Models

function [T,Is,Ir,Im,PF,P] = SiemensIndMotor(Es,We,Wr)% This block supports an embeddable subset of the MATLAB language.% See the help menu for details. % % This program is a Matlab embedded AC motor/generator model used in simulink.%%% The following AC machine ratings apply:%% 4-pole, three-phase cage induction motor/generator.% 600 kilowatt rated power % 460 volt rated line-to-line input voltage % Rated speed 1790 RPM % Rated torque 3183 N-meter @ 1790 RPM%% % Variables List:% % Es = Rms line-to-neutral voltage (Per-phase) % Rstator = stator winding resistance% Rrotor = rotor winding resistance (referred to stator)% L = combined leakage inductance in henries (Lstator = L rotor Referred)% Lm = magnetizing inductance% Rm = stator loss resistances% Ir = rotor current (referred to stator, Per-phase)% Is = Stator current (Per-phase)% Im = magnetizing component of current (Per-phase)% Wsyn = synchronous speed of generator shaft% We = electrical frequency in radians/sec of Es.% s = slip (Wsyn-Wrotor)/Wsyn% T = torque% P = power % % ================================================================%

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% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% assign motor constants %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Rstator = .0072; % Rrotor = .0017;L = .00015093; % leakage inductance Lstator + Lrotor reflectedLm = .002281; % magnetizing inductanceRm = 24.5; % resistance of magnetizing branch Wsyn = We/2; % Wsyn = We/(# of pole pairs)s = (Wsyn-Wr)/Wsyn; % slip Ir = Es/((Rstator + Rrotor/s) + 1j*We*L); % rotor current Im = Es/(Rm + 1j*We*Lm); % magnitizing current Is = Ir + Im; MagIr = abs(Ir); T = (6*(MagIr^2)*Rrotor)/(We*s); theta = angle(Is);PF=cos(theta); P = 3*Es*abs(Is)*PF;

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REFERENCES: 1. Kathryn E. Johnson, Lee J.Fingersh, Mark J. Balas and

Lucy Y. Pao, “Methods for Increasing Region 2 Power Capture on a Variable-Speed Wind Turbine,” in Transactions of the ASME Journal of Solar Energy Engineering, Vol. 126, Nov, 2004.

2. Alan Wright, Lee J.Fingersh, Mark J. Balas, “Testing State-Space Controls for the Controls Advanced Research Turbine,” in Transactions of the ASME Journal of Solar Energy Engineering, Vol. 126, Nov, 2004.

3. Alan Wright and Mark J. Balas,“Design of State-Space Control Algorithms for Wind Turbine Speed Regulation,” in Transactions of the ASME Journal of Solar Energy Engineering, Vol. 125, Nov, 2003.

4. B. Boukhezzar, H. Siguerdidjane and M. Maureen Hand,“Nonlinear Control of Variable-Speed Wind Turbines for Generating Torque Limiting and Power Optimization,” in Transactions of the ASME Journal of Solar Energy Engineering, Vol. 128, Nov, 2006.

5. David M. Eggleston and Forrest S Stoddard, “Wind Turbine Engineering Design,” New York: Van Nostrand Reinhold, 1987.

6. David A. Spera, “Wind Turbine Technology: Fundamental Concepts of Wind Turbine Engineering,” New York: ASME Press, 1994.

7. Bimal K. Bose, “Modern Power Electronics and AC Drives,” Upper Saddle River, NJ: Prentice Hall, 2002

8. I. Boldea and S.A. Nasar, “Vector Control of AC Drives,” Boca Raton: CRC Press, 1992.

9. Theodore Wildi, “Electrical Machines, Drives, and Power Systems,” 4th ed., Upper Saddle River, NJ: Prentice Hall, 2000.

10. Norman S. Nise, “Controls Systems Engineering,” 4th ed.,John Wiley and Sons(Asia) Pte. Ltd., 2004.

11. B.P. Lathi, “Signals, Systems, and Controls,” New York: Intext Educational Publishers, 1974.

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Documents

Thesis- Cont sys for wind turbines (Sept 2015 revision)