Aleksander M. Derdowski. Wind Turbine optimized for VSC-DC transmission

Transcription

1 Aleksander M. Derdowski Wind Turbine optimized for VSC-DC transmission Master Thesis, June 2008

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4 Preface This is the master thesis for Aleksander M. Derdowski and is the final step in obtaining the Masters of Science (M.Sc.) degree at the Technical University of Denmark. The thesis has been done under supervision of Professor Zhao Xu, Tonny Wederberg Rasmussen, Arne Hejde Nielsen and industrial supervisor German Cláudio Tarnowski. The project was a great opportunity to expand my knowledge within the field of wind energy, which will be most valuable in my future work in Norway. It is also an honour to being part of a sustainable future, by contributing to the fast and innovative development of renewable energy resources. Finally I want to thank all people around me, for support and attention, both socially and academically. Technical University of Denmark June 2008 Aleksander M. Derdowski 1

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6 Wind Turbine optimized for VSC-DC transmission This report was prepared by: Aleksander M. Derdowski Supervisor(s): Germán Claudio Tarnowski Zhao Xu Tonny Wedeberg Rasmussen Arne Hejde Nielsen Department of Electrical Engineering Centre for Electric Technology (CET) Technical University of Denmark Elektrovej Building Kgs. Lyngby Denmark Tel: (+45) Fax: (+45) Release date: Category: Edition: Comments: 1(Public) 1st edition This report is part of the requirements to achieve the master of science MSc. Degree in Wind Energy - Electrical at the Technical University of Denmark. This report represents 35 ECTS points. Rights: Aleksander M. Derdowski,

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8 Abstract This thesis investigates the possibility of increasing power production connecting a Rotor Current Control (RCC) generator wind turbine to a Voltage Source Converter (VSC) HVDC transmission system. The generator is optimized in order to produce maximum power at every wind speed until nominal power is reached. The project contains an optimization algorithm to provide the reader with the optimum power production possible when being able to vary stator frequency, stator voltage and rotor resistance. The power production is presented and compared with AC connected wind turbines. The comparison of power production is performed in both steady state and in dynamic simulations. In steady state the VSC HVDC connected generator is compared with both a variable resistance and constant resistance AC connected wind turbine, in the dynamic simulation the comparison of the VSC HVDC generator is only done with the constant rotor resistance AC connected generator, as this is believed and demonstrated to be sufficient for showing the benefits of the new topology. The project proposes a control system for optimized power production using wind speed to generate reference signals for stator voltage and frequency. The new control system is submitted to a tripping of the receiving end converter (grid-side VSC) and the reaction of the new topology is shown for this type of faults. The project shows that there is a possibility of increasing the power production of wind turbines by allowing the generator to work with variable stator voltage and frequency and the operational range of the generator is presented. In Further research losses in the PWM converters have to be investigated to show if there is increased power submitted to the Grid. Other ideas for improvements on this field of study are also presented for future research. Key Words: Wind power, wind turbines, induction generator, variable speed, control, power electronics, HVDC, power systems. 5

9 Acknowledgements This project has been carried out at the Centre for Electric Technology at the Technical University of Denmark. I would like to thank my supervisors German Cláudio Tarnowski, Tonny Wedeberg Rasmussen and Zhao Xu, where especially the help and constructive discussions with my industrial supervisor German Cláudio Tarnowski were of great inspiration and help for this project. Further I want to thank two fellow master degree students János Hethey for interesting point of views especially on simulation technical issues and Xavier Le Mestre for improving my time schedule by driving me to school during this whole period. 6

10 PREFACE 1 ABSTRACT 5 ACKNOWLEDGEMENTS 6 1. INTRODUCTION Review of related research Purpose and contribution THE MODEL REPRESENTATION Mechanical Representation Electrical Representation Transformer representation Tuning of the Steady state model STEADY STATE OPERATION, MECHANICAL POWER AND LOSSES Torque curve Currents and slip Mechanical Power Losses Generated Power Chapter Evaluation STEADY STATE OPTIMIZATION AND OPERATIONAL POINTS The Optimization Algorithm Operational Points AC connected Constant Rotor resistance Wind Turbine Operational Points Variable Rotor Resistance AC Connected Turbine Operational Points VSC HVDC Connected Wind Turbine Necessary operational range Summary of Power Production and operational points 36 7

11 5. STEADY STATE COMPARISON OF POWER PRODUCTION Comparison Strategy Power Production Evaluation DYNAMIC MODELING Wind Turbine Model Turbine model Pitch Controller Model Induction Machine Model PWM Model HVDC Model Combined Controller Model COMPARISON BETWEEN THE STEADY STATE AND THE DYNAMIC MODEL DYNAMIC COMPARISON BETWEEN THE AC AND DC CONNECTED WIND TURBINE The Dynamic Comparison Strategy Evaluation DYNAMIC RESPONSE TO SYSTEM FAULT DISCUSSION CONCLUSION Future Research 65 APPENDIX 68 A NOMENCLATURE 68 B CALCULATIONS 69 C MATLAB OPTIMIZATION SCRIPT 72 8

12 1. Introduction Wind energy installations are faced with demands from the grid operators regarding frequency and voltage variations. Increased demands for controllability have pushed the technologies both within the wind turbine and in the power transmission system. To meet the desired controllability, either synchronous generators with full-scale frequency converters or doubly-fed induction generators with slip-recovery frequency converters are favoured in the wind turbines. Meanwhile, the AC transmission systems are reinforced with static VAr compensation and other FACTS devices. Such topologies will remain suited for connection of wind parks to relatively strong grids, low power and short transmission distances. However, for particular wind parks and transmission systems, it may be advantageous to use DC rather than AC for power transmission from a wind park to a transmission grid. In this case, HVDC transmission using voltage-source converters (VSC) is suitable. The HVDC-VSC is just as controllable as the wind turbine itself, and it decouples the AC transmission system from the wind park AC grid. Hence, all the uncontrolled frequency & voltage variations normally present at the turbine terminals disappear. As a consequence, a less complex wind turbine may be used, for example the rotor current controlled (RCC) induction generator. One operation mode that may be envisaged is that the High Voltage Direct Current Voltage Source Converter (HVDC-VSC) controls voltage and frequency to match the mean wind conditions, while each RCC turbine thus only requires a limited speed range. The research presented will contain the RCC induction generator turbine control and performance with variable stator frequency and voltage, necessary RCC wind turbine speed range when connected to VSC-HVDC, Energy production comparison between DC and AC transmission solutions and the Wind turbine response to faults in back bone transmission grid. (Vestas) 1.1 Review of related research Literature considering power optimization of RCC generator connected to VSC HVCD transmission was during this project not possible to find. Although a paper considering DFIG generators connected to VSC HVDC transmission is found in (Erlich, 2007), where a proposed control system for coordinated control is presented. Their main approach is to reduce the size of the grid-side converter of the DFIG while maintaining the same operational range, this is done by using the average wind farm slip to generate reference signal for the stator frequency and stator voltage, where the voltage is direct assignment of the frequency (no saturation in generator). The frequency controller is implemented without using a feed back control system. It is purposed for future research to keep the grid-side converter constant and increase the operational range of the DFIG. Considering loss minimization more literature was possible to find, in (Ahmed G. Abo-Khalil, 2004) a control algorithm for minimizing losses is proposed taking into account stator and rotor iron and copper losses. The paper shows power increase when using variable stator frequency. The project was validated using experimental results, other research considering loss minimization of variable 9

13 speed connected induction machines can be found in (Robert Leidhold, 2002) and (G. O. Garcia, 1994). Considering fault ride trough of wind turbines research has been found in (Watson, 2005). The paper is based on the modification of the Irish grid code in 2004 where new stricter requirements for wind turbines where included. The paper presents the comparison of fault ride through of Fixed Speed active stall regulated wind turbines based on induction generators connected to VSC HVDC and HVAC. The study shows that the fault clearing time is significantly expanded using the VSC HVDC connection. Paper regarding modelling of VSC HVDC systems can be found in for instance (Florin Iov, 2006). 1.2 Purpose and contribution The main purpose of this project is to show the possible power production gain of connecting a RCC generator to VSC HVDC transmission system for both steady state and in dynamical simulation. Secondly propose a control system to reach this power production and check the control systems liability when the transmission is submitted to a fault. In order to present the power production a representative model had to be made and is therefore modelling has a significant part in this project. The details of the contribution to research are: Chapter 4 presents an optimization algorithm for finding best possible operational points for the RCC generator considering rotor and stator losses in the generator. The optimization allows variable rotor resistance stator voltage and stator frequency. The results are presented for steady state operation and the new generator operational span is compared with already known AC transmission solutions. Based on the operational points in Chapter 4, Chapter 5 shows a steady state comparison of the power production from the new VSC HVDC generator topology with AC connected variable and constant rotor resistance generators. The steady state results show positive effect on the power production of the new topology. Chapter 6 presents a dynamic control system for power optimization based on the operational points found in Chapter 4. The control system uses the wind to generate reference signal for stator frequency and stator voltage, consisting also a feedback loop in the control system in order to enable rate of change limiting of frequency and voltage. Chapter 8 shows the gain of using the new control system in dynamic simulations and comparing the power production to the same wind turbine connected to an AC transmission. The dynamic simulation results supports the steady state results and shows that there is possible gain in power production when using the presented control system Finally in Chapter 9 the proposed control system is exposed to a tripping of the receiving end converter and its response is monitored and discussed. The control system shows relatively good coping with the problem and a proposal for improvement is presented. 10

14 2. The Model Representation This chapter will in detail describe how the mechanical and electrical model was represented and implemented. The chosen models are presented and substantiated, also numeric modifications of former models has been performed to provide better correlation with the RCC Vestas V80 wind turbine which is used as a base reference throughout this project. 2.1 Mechanical Representation This section will describe the mechanical equations and explain why this representation is chosen. The mechanical power is the power in the wind that is possible to convert to the rotor. Therefore the static characteristics of the wind can be described by the relation between the total power in the wind and the mechanical power of the wind turbine (Ackermann, 2005). It can be shown that an approximation of the total power in the wind can be expressed as: (2.1) Where ρ is the air density kg/m 3, A is the rotor radius and v w the wind speed. However it is not possible to extract all the kinetic power in the wind since this would result in a stand-still of the wind on the back side of wind turbine blades, therefore the maximum theoretical utilization of the power in the wind is given by Betz law to be 59.3%,(Betz, 1966). This utilization of the wind is denoted C p and will here be described as a function of tip-speed ratio, λ and blade pitch angle, θ. At modern wind turbines this efficiency will lay somewhere in the area of 70-80% of the maximum given by Betz law (Ahmed, 2006) and can be expressed as (Bose, 1983): (2.2) The choice of this representation of the Cp curve is based on its simplicity together with good base for correlation with the power curve provided in the Vestas V80 product description (Vestas, 2008). The constants a= 0.76 and b=2.64, are self defined and chosen iteratively to give a better correlation to the actual data. This correlation will be presented later in the project. θ is the blade pitch angle in deg. and the tip speed ratio is [2]: (2.3) Where R is the rotor diameter, ω r is the rotor speed and v w is the wind speed. The mechanical power of the wind turbine will therefore be expressed as (Ackermann, 2005): (2.4) 11

15 Figure 1: Cp curves at different wind speeds as a function of Rotational Speed When allowing variable rotational speed the optimum efficiency point can be regulated to fit a given wind-speed as seen in Figure 1 and the mechanical power in Figure 2 can be obtained. Figure 2: Mechanical Power at different Wind Speeds as a function of Rotational Speed, with the maximum mechanical power for each wind-speed marked with a cross 12

16 2.2 Electrical Representation This section will describe the electrical representation of the Rotor Current Control generator. A RCC is a single fed, wound rotor generator with the possibility to vary the rotor resistance. A highly simplified sketch of this generator might be seen in Figure 3. The requirements for the electrical model in this project is that it is able to react to changes in rotor and stator frequency, change in external rotor resistance and change in stator voltage. The chosen model can be found in (Paul C. Krause, 2002) and is presented in Figure 4. Al values are from now on assumed to be in p.u. if not otherwise specified. Figure 3: Rotor Current Control Generator Figure 4: Arbitrary reference-frame equivalent circuits for 3-phase symmetrical induction machine 13

17 Since in this project it is chosen to investigate the generator losses in the desired operational points, the necessary equations have to be deduced. The voltages can be described as: ψ (2.5) ψ (2.6) ψ (2.7) Where the flux linkage is described as: ψ (2.8) ψ (2.9) ψ (2.10) ψ (2.11) ψ (2.12) By working in the arbitrary reference frame the number of unknowns can be reduced to improve calculation time. Therefore the following values are set: (2.13) This is verified since fulfilling: (2.14) (2.15) Further the rotor voltages can be set to zero: (2.16) This is true when defining the rotor voltages over the short circuited rotor side as shown in Figure 4 and additional rotor resistance (R add ) is included in R r. Further by combining equation(2.16), (2.5), (2.8), (2.10) and (2.12) i dr can be written as: 14 (2.17)

18 Since the losses are highly dependent on the currents, the currents are of great interest to single out. This is chosen to be done with a Newton Raphson algorithm because of its history of solving this type of initialization problems. In order to use the Newton-Raphson method to solve three implicit unknowns, three equations has to be set to zero (William H. Press, 1992). This is done by combining all equations from equation(2.5) until equation(2.17) obtaining: (2.18) (2.19) (2.20) The deductions can be found in the Apendix. The steady state electrical torque of the machine can be written as: Where: (2.21) (2.22) (2.23) (2.24) Further the stator voltage, stator frequency and air gap flux is co-dependent as shown in (Matsch, 1977) and can be presented as: (2.25) (2.26) 15

19 Where f s is the stator frequency, N s is the number of turns in the stator a 1 is the number of the current paths, k is the pitch factor and ψ a is the air-gap flux, the subscript nom indicates nominal operation. To be able to create an optimization algorithm within a reasonable time-span a simplification had to be made, saying that the air-gap flux is constant for all stator frequencies: (2.27) Assuming this is representative and that the factors k, N and a 1 remain constant the voltage dependence on frequency will be written as: (2.28) This is valid until V s = V s,nom where the air-gap flux has to be reduced in order to keep the voltage constant (Ion Boldea, 2001) a typical voltage frequency characteristic for a 220 V 50 Hz induction machine is shown in Figure 5 Figure 5: Voltage-Frequency Characteristic for a 220 V 50Hz induction machine (Ion Boldea, 2001) The rotor copper losses are expressed as: (2.29) The stator copper losses are expressed as: (2.30) Therefore the generated power of the generator is: 16

20 (2.31) 2.3 Transformer representation This section will describe the two winding transformer connecting the wind turbine to the sending end converter. The reason for including the transformer is to investigate the impact of the frequency variations on the losses in the transformer. This will show if the transformer losses should be taken into account when considering the combined control system. Figure 6: Equivalent diagram for transformation An equivalent diagram of a two winding transformer is seen in Figure 6 and can be found in (Matsch, 1977). Here the assumption is that the current generated by the generator is fed in to the primary side of the transformer hence: (2.32) The losses in the transformer can then be calculated: Where the magnetization current can be approximated: (2.33) (2.34) And i m,rm is the real part of i m further: 17

21 (2.35) Where i s and i m are presented as vectors. Typical values for a transformer was provided by my industrial supervisor and presented in Table 1 Table 1: Suitable Transformer values Lpri Rpri Lsec Rsec Lm Rm p.u p.u p.u p.u. 175 p.u. 300 p.u. 2.4 Tuning of the Steady state model This section will present how the model was tuned in order to become representative for its purpose. In order to get reliable results from the further calculations it is necessary for the model to represent a real life turbine power production as good as possible. And since some information about the power production of the Vestas V80 is provided in (Vestas, 2008), together with electrical parameters from a 2 MW double fed induction generator found in (Petterson, 2005) the steady state power production could be closely represented. The model tuning was performed on the fixed stator frequency model (AC transmission connected), and tuned to fulfill the criteria known to the author during this project. Figure 7: Power curve of the Vestas V kw Wind Turbine 18

22 From the product description (Vestas, 2008) it is known that the nominal power of the wind turbine is reached at wind speed equal to 15 m/s. It is also known to the author that turbine works with a fixed optimum slip, and the slip range is from 1-10% slip (Vestas, 2008). The rotor speed of the generator is based on a combined control between the pitch angle of the blades and the control of the additional rotor resistance (Vestas, 2008). Although for this project the optimization considering change in pitch angle will not be considered, consequently the optimum slip was not able to follow for all wind speeds. Therefore the chosen constrain was to reach maximum power production for all wind turbine topologies at a slip higher than -10%. This was done by adapting the self defined a) and b) coefficients in equation(2.2) together with choosing a step size of the additional rotor resistance, which would provide a satisfactorily power output ( P Generated =P Mechanical -P loss ). This was performed manually and is therefore no guaranty that this provides the optimum correlation between Figure 7 and Figure 8, although for the scope of this project this is considered by the author to give a good enough base for the power comparison. The original power curve is shown in Figure 7 and the power curve obtained using equation(2.31) for all wind speeds is shown in Figure 8. The data needed to obtain the power curve in Figure 8 is presented in Table 2 and Table 3. Figure 8: The calculated Power Curve for the AC connected variable rotor resistance wind turbine Table 2 Mechanical Parameters: Rotor Diameter Cut in wind Speed Cut out wind Speed Nominal wind Speed 80 m 4 m/s 25 m/s 15 m/s Table 3 Electrical Parameters: Rotor Resistance p.u. * Rotor Inductance 0.07 p.u. Stator Resistance 0.01 p.u. Stator Inductance 0.18 p.u. 19

23 Magnetizing Inductance Nominal Power Additional Rotor Resistance 4.4 p.u kw 10 steps of 0.02 p.u. (self defined) *In (Petterson, 2005) the value of the Rotor resistance was p.u. This project however is using p.u. as minimum value of the rotor resistance and happened because of a mathematical overlook in the optimization algorithm. This change in rotor resistance has no effect on the main idea of the project and it was therefore considered unnecessary to recalculate all the presented results. 20

24 3. Steady State operation, Mechanical Power and Losses The following chapter will provide the base for the further optimization of the VSC HVDC RCC generator. In order to find the optimum operational points the models response to changes in the stator voltage, stator frequency and additional rotor resistance has to be examined. This will give a thorough examination of how torque curve, generator losses, rotational speed and internal currents will behave when controlling the generator. The results are all based on the steady state model derived in Chapter 2 and will now be presented. The purpose of this chapter is to show the high unlinearity of power and losses based on operational points. This is therefore meant to substantiate the choice of optimization algorithm. 3.1 Torque curve The torque curve is defined by the electrical parameters and provides the base for the operational point. The torque curve expression can be found in equation(2.21) and will now be presented for different stator voltages/frequencies and rotor resistance. Figure 9: Shows the Electrical torque as a function of rotational speed, for different stator frequencies 21

25 Figure 10: Shows the torque curve as a function of rotational speed for different rotor resistances As can be seen in Figure 9 and Figure 10 the torque characteristic is highly dependent on both the rotor resistance and stator frequency, and therefore might be used in the tracking of the optimal operation point of the generator. 3.2 Currents and slip This section will show how the currents and slip will behave in the generator when adjusting stator frequency and rotor resistance. The wind is chosen to be the constant factor. Figure 11: Slip as a function of stator frequency and rotor resistance at wind speed = 9 m/s 22

26 Figure 12: Rotor Current as a Function of rotor resistance and stator frequency at wind speed = 9 m/s As can be seen from the Figure 11 and Figure 12 the slip and currents are highly un-linear when changing the rotor resistance and stator frequency. One thing of interest is the impact of the stator frequency on the currents in the generator. It can be seen that the Frequency variations have a much greater impact on the currents than the change in rotor resistance when using the step sizes and operational span of this project. 3.3 Mechanical Power Figure 13: Mechanical Power as a function of stator frequency and rotor resistance 23

27 In Figure 13 one can see the mechanical power possible to track by adjusting the stator frequency and rotor resistance. As can be seen the optimization based on the rotor resistance is modest for this wind speed compared with the optimization based on the power tracking capability enabled by varying the stator frequency. Although by looking on the raw data, the tracking done by the rotor resistance can be shown to be in the order of 1-6 % of the mechanical power at the given stator frequency. Table 4: Showing 4 operational points, the impact of the stator frequency and rotor resistance on the mechanical power StatorFrequency/RotorResistance p.u p.u p.u. Pm = p.u. Pm = p.u p.u. Pm = p.u. Pm = p.u. What can be noticed in Table 4 is that the optimum mechanical power is not given explicitly by the rotor resistance or the stator frequency. This means if the stator frequency [p.u.] is above the rotational speed [p.u.] that provides the optimum mechanical power, a low resistance will provide a better operational point for mechanical power. Again, if the stator frequency is lower than the rotational speed that provides the optimum mechanical power a high resistance will provide a better operational point for mechanical power. This might be easier to visualize by studying Figure 9 together with Figure 10 and imagine how this will provide operational points in Figure Losses Figure 14: Generator power loss as a function of stator frequency and Rotor resistance at constant wind speed = 9 m/s 24

28 The loss calculations are based on stator and rotor losses showed in equation(2.29) and equation(2.30). And will vary dependant on stator frequency and rotor resistance. In Figure 14 it can be seen how the losses are distributed for a constant wind speed when varying the additional rotor resistance and stator frequency. As can be seen when imagining the stator frequency constant the losses increase with increased rotor resistance. The reader can also keep in mind that the operational point changes with the wind speed, consequently the losses will change with the wind speed. In Figure 15 the losses in the transformer calculated according to equation(2.33) are presented. Figure 15: Transformer power loss as a function of stator frequency and Rotor resistance at constant wind speed = 9 m/s As can be seen when comparing the Figure 14 and Figure 15 it can be seen that the losses in the transformer are modest compared to the losses in the generator. 25

29 3.5 Generated Power Figure 16: Generated Power as a function of Rotor Resistance and Stator Frequency Finally the generated power is presented in Figure 16. This figure is a result of subtracting Figure 13 from Figure 14. The optimum operational point at this wind speed is found at Stator frequency = 0.97 p.u. and with the minimum value of rotor resistance. 3.6 Chapter Evaluation This thorough deduction was made during the progress of this project when the operational points of the generator jet were not known. The aim was to see and understand how all possible operational conditions to provide an optimization algorithm that was sufficient to show the possible gain of combined controlling stator frequency, stator voltage and additional rotor resistance. But what will be shown later is that when being able to change the stator frequency in such a wide interval, the tracking of mechanical power can be done by the stator frequency alone, therefore the rotor resistance can be kept at a minimum and reduce the losses in the generator, and the losses in the transformer are not of a size which effects the operational points considering the step sizes used in the optimization algorithm of this project. This would maybe not be the case if this project had to consider stricter limits on the stator frequency variation, or used a higher resolution of step-size for rotor resistance and frequencies. This was not obvious for the author until the later part of the project. Since the losses in the transformer can be shown by the author to not impact the power production and the operational points of the generator in a substantial way, this is from now on neglected in the project. On the other hand, the rotor resistance has a positive effect on the AC connected model, where there is no possibility of power tracking by using variable stator frequency. Therefore the rotor resistance is used in the power optimization of the AC connected model. 26

30 4. Steady state optimization and Operational Points This chapter will describe the optimization algorithm and how it was implemented in MatLab. As mentioned earlier the objective is to find the maximum power output for the generator, as shown in Figure 17 within the operational boundaries shown in later presented in Table 8. The variables in the algorithm are the rotational speed, the stator frequency, stator voltage and rotor resistance. Figure 17: Energy yield for the RCC generator model When examining the steady state model presented in Chapter 2.2 and the mechanical equations in Chapter 2.1 it is seen that we are operating with highly un-linear co-dependent equations. The main thought for the algorithm was for it to be robust and thorough, and to be able to check a wide range of operational conditions to show the picture of how the power output of the generators where responding according to these changes. The algorithm then stores the necessary data for validation, power production, losses, currents and optimum operational points for each wind speed. The simplified diagram of the algorithm is presented in Figure 19. The further presentation might require some knowledge about MatLab, but is tried to be presented in a more general way. 4.1 The Optimization Algorithm The algorithm is based on for loops and is starting by setting wind speed equal to 4 m/s and proceeds to the setting of the stator frequency equal to 0.5 p.u. (30 Hz) where the stator voltage is calculated according to equation(2.28). The rotor resistance is then set to p.u.. Based on this information the rotor speed can be calculated by using equation(2.4) and equation(2.21) resulting in: (4.1) 27

31 This is solved by plotting the two graphs on a common axis of rotational speed (step size of 1/33 p.u.) where the operational points in between the steps is approximated using a spline interpolation. The intersection point is found using the Matlab function fsolve, which is a solver for implicit unknowns similar to a Newton-Raphson algorithm. The benefit of using this function instead of a self defined algorithm is its possibility of providing an exitflag which tells something about the accuracy of the solution. This exitflag is therefore used as the first constrain to ensure that the intersection of the two graphs is of desired mathematical accuracy (converging to a real root). Other constrains at this point in the algorithm is that the stabile operational point marked with a circle in Figure 18 and that the difference in between the two curves illustrated by the arrows in Figure 18 is sufficient. And since the generator is not supposed to work at slips lower than -10% and the rotor speed should be less than 1.5 p.u. these are included as constrains. Figure 18: Mechanical Power and Electrical Power Torque If all constrains are not fulfilled a bypass is initiated, setting rotational speed, currents, powers and slip equal to zero before storing the data with label. Than the rotor resistance is increased and the same calculations and verification is initialized. When all constrains are fulfilled the rotor speed is used for current calculations and mechanical power calculations. The currents are calculated using a Newton-Raphson method on equation(2.18) to equation(2.20) and finally finding the last current by solving equation(2.17) with the currents already obtained. The choice of using the Newton-Raphson method was based on its history of solving this type of mathematical problems. Since all operational points here already where filtered earlier, the problem with the algorithm converging towards unrealistic solutions is not an issue at this point in the algorithm. More information about Newton- Raphson can be found in (William H. Press, 1992).The mechanical power is calculated according to equation(2.4) Based on the currents the losses in the stator and rotor are calculated according to equation(2.29) and equation(2.30). The generated power is calculated according to equation(2.31). The data is 28

32 stored in a 3D matrix where the data is stored with its respective label of [ω s,r r, V w ]. ω s is the stator frequency, R r is the rotor resistance and V w is the wind speed. When R r reaches R r,max, ω s is increased with one step, and the same procedure is initiated. When ω s reaches ω s,max, V w is increased by one step until V w,max is reached. This ensures that all operational conditions are checked, verified and stored. 29

33 Figure 19: Simplified diagram of the optimization algorithm 30

34 When all data is stored a second algorithm is initiated finding the maximum power production and at which operational conditions these are obtained, these operational conditions together with their power production are saved and used as the base for the rest of the project. This same algorithm is used for all three turbine topologies, only difference is the amount of variables chosen to be constant. The VSC HVDC topology is the one described above, the AC connected variable resistance is keeping the stator frequency constant, and the AC connected constant rotor resistance topology additionally keeps the rotor resistance constant. A summary is seen in Table 5. Table 5: Operational range and step size for the different generator topologies Limits/topology VSC HVDC AC Variable Resistance Rotor resistance Resistance step size 1/50 p.u. 1/200 p.u. - Stator frequency p.u. 1 p.u. 1 p.u. Frequency step size 1/33 p.u. - - AC Constant Resistance The values chosen in Table 5 where changed and modified continuously during this project, the final solution however ended with the presented values. This because: The VSC HVDC topology prefers to find operational points by changing stator frequency and keeping the rotor resistance minimum, in order to minimize losses. Therefore the resistance could in general be set to constant, but for illustrative reasons it is changing to verify this statement. Since the AC variable resistance is more dependent on the rotor resistance a higher resolution on the step size was needed in order to provide smooth operational points. The AC connected topology with constant resistance, has rotor resistance = since this is the value the Variable Resistance topology uses at wind speed = 15 m/s and reaches the power production of 1 p.u. (can be seen in Figure 21) If the reader should be interested in reading the MatLab script appended the constrains are presented more orderly in Table 6. Table 6: List of Constrains MatLab Denotation Mathematical Denotation 1 st Constrain Choosing the correct intersection 2 nd Constrain Solution has to be a real root 3 rd Constrain Sufficient difference between electrical power torque and mechanical power 4 th Constrain Slip > -10% 5 th Constrain Rotational speed

35 4.2 Operational Points AC connected Constant Rotor resistance Wind Turbine For comparison of the three types of wind turbine generator topologies it was chosen to show how the slip changes of the AC connected constant speed generator. This generator will have no possibility of performing any kind of power tracking, and the rotational speed will only be affected by the change in wind speed. The slip characteristic is shown in Figure 20 Figure 20: Slip of generator with Constant Rotor Resistance 32

36 4.3 Operational Points Variable Rotor Resistance AC Connected Turbine Since the Vestas V80 OptiSlip has a very advanced optimization system the author had to simplify this in order to be able to provide a model within the time scope of this project. In reality the OptiSlip is combined with an OptiTip system, which provides coordinated control of the slip of the generator and the slip will further be used as the active power reference. This system has many advantages like increased power quality, less flicker, and less harmonic disturbance together with higher power output(vestas, 2008). Since this project mainly focus on the investigation of power production, the author has simplified the control system and will use only the rotor resistance to provide a maximum power production tracking. This simplification distance the model from the real V80 since an optimum slip function is not possible to obtain with this control system. On the other hand, the VSC HVDC model is also not considering the OptiTip function, so a slight justification of the comparison of the models can be drawn. Although this has to be kept in mind when the final results of this project is presented. If this report should be basis for a new control strategy of wind-turbines it is highly recommended to include possible gain of using the OptiTip function. Figure 21: Optimum Rotor Resistance for AC connected Wind Turbine As can be seen in Figure 21 the rotor resistance is kept low when the wind is low to minimize losses and a good mechanical power tracking is performed. When the wind speed increase additional rotor resistance is provided so the rotational speed increase to perform a better tracking of the mechanical power, while maintaining low losses. When the wind reaches 13 m/s the mechanical power is high so the most profitable solution is to reduce the losses by reducing the rotor resistance. How these operational points effect the slip is shown in Figure

37 Figure 22: Slip of the AC connected Wind Turbine 4.4 Operational Points VSC HVDC Connected Wind Turbine In order to create the dynamic control system the optimum operational points of the new topology has to be singled out. This section will provide the reader with the operational points and the approximation used in between operational points for use in the dynamical simulation program DIgSILENT. Figure 23: Optimum Stator Frequency as a function of Wind Speed 34

38 Figure 24: Optimum Rotational Speed as a Function of Wind Speed Figure 25: Optimum Slip as a Function of Wind Speed In Figure 23, Figure 24 and Figure 25 the optimum operational points are published. Al figures are presented until wind speed equal to 13 m/s since here the nominal power production is reached. In Figure 23 and Figure 24 the spline approximation of the operational points is also presented. This line shows how the operational points later implemented in the dynamic simulation in DIgSILENT Power Factory. In the VSC HVDC connected generator the optimum operational points always resulted in the minimum value of the rotor resistance, this since all the power tracking is performed by the converter regulating the stator frequency. 35

39 4.4.1 Necessary operational range From the operational points presented the necessary operational range can be derived. To be able to follow the tracking method presented it will be necessary for the rotor to rotate in the interval showed in Table 7. Table 7: Operational range needed for Induction machine Min rotational speed Max Rotational Speed 0.5 p.u p.u. 30 Hz 83.9 Hz 4.5 Summary of Power Production and operational points As is shown, the effect of controllability of the generators has a significant impact on the power production performed by the generators. The more controllable the generator topology, the more power can be withdrawn from the wind, while limiting the losses so the generated power will maximize. To present a final steady state overview, the author has chosen to present the operational points of the three wind turbine topologies on common mechanical power curves. This is meant to illustrate and describe for the reader, the essence of the Power Optimization Theory. Figure 26: Operational Points of the three generators on the power curve provided by a wind of 6 m/s 36

40 Figure 27: Operational Points of the three generators on the power curve generated by a wind of 13 m/s As is seen in Figure 26 and Figure 27 the operational points of the three generators are different for the different wind speeds. The controllability of the VSC HVDC generator provides it with the benefit of keeping the rotor resistance low, and use the stator frequency to decide the rotational speed. This is an optimal solution since this provides minimum losses and maximum mechanical power for all wind speeds. The generator with variable rotor resistance has the ability to consider if the extra rotational speed obtained with the increase of extra rotor resistance will provides a favourable relation between losses and mechanical power. When comparing Figure 21 and Figure 27 one can see that the impact of decreasing the rotor resistance is favourable for this wind speed. This because the mechanical power curve here is relatively flat, so the gain in mechanical power of adding extra rotor resistance will be lost with interest when considering the losses. The normal AC connected generator has no chance of optimization and will therefore changing rotational speed based on the wind only. 37

41 5. Steady State Comparison of power production This chapter will use the models derived earlier to show the difference of the energy production from the AC transmission connected RCC generator and the VSC HVDC connected RCC generator. The chosen way of performing this analysis is described and how the results can be interpreted 5.1 Comparison Strategy In order to provide reasonable results in the power production comparison between the generators connected to the VSC HVDC and the generator connected to regular AC transmission, the general details has to be defined. By checking the product description (Vestas, 2008), the operational interval of the generator is spanning from 1-10 % slip. Therefore it is assumed that this same constrain applies to the new wind turbine design with variable stator frequency. The three turbines investigated in steady-state are the same turbines presented in Chapter 4, and the power curves derived from the presented operational points. As mentioned, for simplicity the pitch angle is not used in the optimization algorithm. The pitch angle is firstly used when the generators reach nominal power, and is then used to reduce the mechanical power. When the nominal power is reached, the optimization is terminated and both generators keep producing nominal power until the cut out wind speed of 25 m/s is reached. Both turbines are assumed to have the same cut in wind speed, 4 m/s. The yearly energy production will be based on weibull distributed wind speeds multiplied with the respective power curves, assuming that both generators run without stop for the whole year. A more tidy presentation of the physical constrains is listed in Table 8. Table 8: Constrains for the two generators Generator connected to AC Generator connected to VSC Transmission HVDC Transmission Nominal Power 1800 kw 1800 kw Nominal Wind Speed 15 m/s Unknown (13 m/s) Operational stator frequency 50 Hz 25 to 75 Hz Operational rotor frequency 50 to 55 Hz 25 to 75 Hz Maximum Slip 10% 10% Cut in wind speed 4 m/s 4 m/s Cut out wind speed 25 m/s 25 m/s 38

42 5.2 Power Production This section will present the results of the two different wind turbine connections. Using the optimization algorithm the power curves for the different wind turbines was obtained. To make the power calculations valid the chosen wind speeds where taken from Sklinna Fyr in Norway (Norwegian Metrological Institue, 2008). Based on 1 hour s values from May 2007 until May 2008 a weibull distribution was generated in MatLab. The yearly energy production can be described as: (5.1) Where V w is the wind speed and Wblpdf represents the Weibull distribution, the reader can refer to (Johnson, 1994) for the weibull calculation technique is the number of hours in a year considering no operational problems for the wind turbine. For Illustrative purposes also the energy production at given wind speeds is presented and can be calculated like: (5.2) The power curves together with the weibull distribution for the wind is presented in Figure 28 and the power production generated at different wind speeds is shown in Figure 29. Figure 28: Power Curves for the two wind turbines together with the weibull distribution of the wind 39

43 Figure 29: Power Production distributed at different wind Speeds Figure 30: The Power Curves of the three wind-turbine topologies Based on the yearly energy production equation it is now possible to compare the production of the generators and the results are presented in Table 9. Table 9 Turbine Type AC Connected Constant Resistance AC Connected Variable Resistance VSC HVDC Connected Power Production MWh MWh MWh 40

44 Increase compared to: Power Production Increase of the VSC HVDC Connected Wind Turbine AC Connected Constant Resistance 7.25 % AC Connected Variable Resistance 5.49 % Further a dynamic computation of energy will be performed with high resolution wind data in Chapter 8. The wind data used is described more in detail in chapter 8.1. Therefore firstly a steady state computation of the energy gain will be presented, for later comparison with the dynamic model. In order to save some time in the dynamic modelling, the choice was to compare the VSC- HVDC topology with the AC connected constant rotor resistance topology. This was considered by the author to be sufficient to illustrate the possible gain of the new topology. This assumption was made by analysing Figure 30, seeing that the difference between the constant and variable resistance AC connected wind turbine is very small compared with the difference up to the VSC HVDC topology. This simplification saved the author potential time used for implementing and tuning a rotor resistance control system. Figure 31: Power Production distributed on different wind speeds Figure 31 shows the weibull distribution of the high resolution wind together with the energy production of the two wind turbine topologies calculated according to equation(5.2). The Power Production calculated according to equation (5.1) is presented in Table 10. Table 10 Turbine Type AC Connected Constant Resistance MWh VSC HVDC Connected MWh Increased Power Production 3.56% 41

45 5.2.1 Evaluation It can be seen from Figure 28 that if the shape of the wind will have a higher density in the area above 10 m/s the difference will be more favorable for the DC connected wind turbine. On the other hand if the density of the wind is compressed more in the below 10 m/s region the difference in power production might decrease. The favorability of this new concept using VSC HVDC transmission will therefore be strongly dependant on the wind speed. The author also find it reasonable to assume that the shape of the wind will have a strong impact on the favorability since the power production of the VSC HVDC connected and the variable rotor resistance wind turbine has a very comparable power production around the average wind area of 7-10 m/s. Therefore the favorability of the new VSC HVDC concept has to be determined sight specific in order to see the possible gain when looking at power production. What has not been considered in these calculations and will most likely determine if there will be any benefit, are the losses in the PWM converters compared to normal HVAC transmission systems. A general loss description of the most common transmission systems was found in (Watson, 2005) and is presented in Table 11. Table 11: Losses in typical transmission systems, losses expressed in percent of nominal power Alternative Convention Losses Line Losses Total Losses AC 0 % 1.2 % 1.2 % Conventional HVDC 1.4 % 0.5 % 1.9 % VSC HVDC 5 % 1.5 % 6.5 % As seen from Table 11 the total losses for VSC HVDC transmission is 5.3 % higher at nominal power in the VSC HVDC transmission compared with the AC transmission. More detailed data considering the two topologies is provided in (Watson, 2005). The details of how the state of the art PWM converter losses are compared to the ones presented here, and how the losses are represented in all operational conditions was beyond the scope of this project. Including these losses is highly recommended by the author in future research. Conventional HVDC is tyristor/diode based HVDC solutions and will not be discussed in detail in this project. If the reader wants more information on this topology a detailed description can be found in (Kundur, 1993). 42

46 6. Dynamic Modeling For further investigation of the stability of the new topology, a dynamic model has to be developed. The chosen program is DIgSILENT Power Factory because of its history of being used for representing wind turbine models. The model has to contain all components important for dynamic investigations and to show dynamic power optimization. Further the VSC HVDC transmission receiving end station control system need to be represented realistic, and a control system for the sending end converter has to be implemented. The generator and PWM converters are chosen from the Power Factory component library and modified to fit better with the wind turbine investigated. 6.1 Wind Turbine Model The wind turbine model implemented in Power Factory was based on the model from the steady state analysis. This was done to obtain correlation and therefore a dynamic validation of the steady state results. Since the steady state analysis was performed without the considering the effect of the shaft, this was also neglected in the dynamic model. Therefore the whole inertia was moved to the generator. The wind turbine generator is therefore represented by a pitch controller, turbine model and the wound rotor single cage induction generator Turbine model The turbine model is implemented using the equations derived in Chapter 2.1. The C p curve is provided by a spline approximation between values of and θ from equation(2.2). The lambda and mechanical power is computed as in equation(2.3) and equation(2.4) Pitch Controller Model The pitch angle controller used is a slightly modified version of the pitch controller used in the Power Factory DFIG example. This is implemented by comparing the rotational speed of the generator to a given reference. This signal is fed to a PI controller with limits of the maximum and minimum pitch angle. This generates the reference signal for the pitch angle, which is compared with the actual pitch angle and providing the desired pitch angle. In order to smoothen the pitching a rate limiter on the rate of change of the pitch angle is implemented. If the reader wants to use a similar approach to reconstruct this model, remember that the reference speed has to be changed in order to cope with the variations desired in the rotational speed of this project. For more detailed information about the pitch controller the reader can referee to (Anca D. Hansen, 2003). 43

47 6.1.3 Induction Machine Model The induction machine used in the dynamic model is the Type ASM Asynchronous Machine. The electrical equations of the machine are derived in the same way as the ones presented in chapter 2.2. The voltage equations are described as: (6.1) (6.2) Removing the derivatives from equation (6.1) and (6.2) then comparing with equation(2.5) to equation(2.8) the equation are identical. DIgSILENT state in the technical reference that stator transients are neglected in this model, so ω ref is fixed to global or local reference.the author has not found which reference is used in the presented model, but based on the results presented in Chapter 7 it is clearly shown that the change in stator frequency affects the generator model. One theory is that the local reference frame chosen in Power Factory is linked to the output frequency of the PWM converter of the sending end of the HVDC, and therefore identical with the statorfrequency. The very good correlation between the steady state model and the dynamic model presented in Chapter 7 was only obtainable when considering change in generator reactance based on stator frequency in the steady state model! All other approaches ended in larger deviation for the two models (for instance setting the ω s = ω n in the steady state model and running the dynamic model in an EMT simulation (DIgSILENT2, 2007)). A more detailed description of the induction machine can be found in the DIgSILENT technical reference for induction machines (DIgSILENT2, 2007). 6.2 PWM Model The PWM Model chosen for this project is also one of the standard models from the DIgSILENT Power Factory component library. The model represents a self commutated voltage source AC/DC converter. The model is based on the fundamental frequency approach and supports sinusoidal and rectangular modulation. Further the model supports a number of different control conditions (DIgSILENT, 2007): V ac -phi: Specifies magnitude and phase of AC-terminal. Typical control modes for motor-side converters in variable speed drive applications. V dc -Phi: Specifies the DC-voltage and the AC-voltage phase. No typical application. PWM-phi: Load-flow setup without control. The pulse-width modulation factor is directly set in magnitude and phace. V dc -Q: Specifies DC-voltage and reactive power. Typical applications: STATCOM, shuntconverter of UPFC, grid side converte of doubly fed induction machine and VSC-HVDC applications 44

48 V dc -P: Specifies AC-voltage magnitude and active power. This is equivalent to a PV characteristic of conventional synchronous machines. Typical applications: Grid side converter of converter driven synchronous machines, VSC HVDC. P-Q: Specifies P and Q at the AC-side. This control mode is equivalent to a PQ - characteristic of synchronous machine. Typical applications: Same as Vac-P For more detailed information the reader can refer to the DIgSILENT technical reference (DIgSILENT, 2007). 6.3 HVDC Model This section will determine the choice of VSC HVDC model and present a stepwise modeling solution for use in Power Factory. During the history of VSC based HVDC models, many control strategies for the converter has been proposed. This project will not consider them all but concentrate at some solutions fitted for this project. Firstly, even though the two converters topologies are equal, the main functions of the two converters are different, for the receiving end converter the important property is to feed the active power transmitted by the sending end converter and maintaining the DC voltage at a desirable level and controlling the reactive power (Erlich, 2007),(Florin Iov, 2006). The sending end converter will transmit the power from the wind farm and control/maintain the frequency and voltage on the wind turbine AC side (Erlich, 2007). Based on this information the implemented control system for the receiving end converter was made to control the reactive power and DC link voltage, the system is chosen to operate in the d-q reference frame and is highly inspired by the control system presented in (Florin Iov, 2006). The sketch of the control system is shown in Figure 32 and the supported control condition chosen from Chapter 6.2 is the V dc Q control. The DC link voltage is controlled using a PI controller where the measured voltage from the DC-Link is compared with a fixed reference. The fixed reference is set to be the nominal voltage of the DClink, 150 kv (1 p.u. of connected bus-bar). The output from the DC-Voltage controller is used as the reference for the current in the d-axis. Referring to (Florin Iov, 2006) the set point of the reactive power can be set to zero, it should be noticed that the system operator can demand a different set point. Although for simplicity this is used in this project and therefore the output of the reference output (current reference in q axis) of the Q-controller is set to zero. 45

49 Figure 32: Control system for the receiving end Converter Now when the reference parameters are determined the measured currents have to be transformed into the d-q axis. This is done by implying a Phase-Locked-Loop to calculate the currents from real and imaginary to the d-q axis as shown in equation (6.3) and (6.4) (6.3) (6.4) Where i d is the d-axis current, i q is the q-axis current, i r is the real part of grid current and i i is the imaginary part of the grid current. The Phase-Locked-Loop measures sin(ϕ) and cos(ϕ). These currents are now fed into a first order filter before compared to the reference signals. The offset is than fed to the current controller and generate the control signals P md and P mq of the PWM converter. 46

50 The relation between the AC and DC voltage is then described as(digsilent, 2007): (6.5) (6.6) Where U AC,d is the d-axis grid voltage, U AC,q is the q axis grid voltage, U DC is the DC voltage and K 0 is based on the type of modulation, in this project a sinusoidal modulation is chosen and therefore K 0 is defined as(digsilent, 2007) : (6.7) 6.4 Combined Controller Model The combined controller model is designed to increase the maximum power production of the wind turbine generator. By using the sending end converter the voltage and frequency of the wind turbine stator side can easily be controlled. Of the supported control conditions in Power Factory described in Chapter 6.2 the sending end controller will utilize the V AC phi system. This system easily provides the possibility of quick and accurate settings of voltage and frequency. The sketch of the complete control system can be seen in Figure 33 47

51 Figure 33: Control system for the sending end controller The input signals of the converter in this model will be P m_in and f 0. And since the receiving end converter now controls the DC Link voltage, and this can be assumed to be constant at 150 kv in normal operation, the voltage of the AC side in kv is: (6.8) The input signal f 0 is directly proportional to the AC frequency in p.u. The generator stator frequency will therefore be described as: (6.9) For simplicity the grid voltage was considered proportional to P m_in. This is true for normal operation, but as shown in Chapter 9 is not representative when the system is submitted to a fault, although it is considered by the author to be a sufficient approximation in this project. 48

52 In (Anca D. Hansen, 2003) it is stated that the use of a moving average is a realistic way of filtering wind data in power optimization. The method is based on a fixed window, moving in time, averaging the values within, and can be expressed mathematically as: (6.10) Where t is the instant time, and t 1 is the time with one step delay (e.g. one second ago). x is the value at the given time. The averaging time is determined experimentally by trying with different numbers until an averaging time providing favorable power production was obtained. The averaging time is chosen to be 1 sec averages. To impose a control system that changes the stator voltage and frequency it is very important to take the inertia of the generator into account. If the stator frequency is increased to fast the rotational speed of the generator might not be able to follow, and the generator can be forced into under magnetization and in worst case resulting in a change in operational mode. If the stator frequency exceeds the rotational frequency, the generator turns to motor and starts consuming power from the grid. And if the stator frequency is lowered to fast, the slip might increase, providing unfavorable high currents (Kundur, 1993). This problem has been solved by implementing a rate limiter on the output of the frequency and voltage controller. One way of implementing a rate limiter in Power Factory is shown in Figure 34 49

53 Figure 34: PI controller with rate limiter in Power Factory As can be seen the error is multiplied with a constant to represent the actual frequency. Than the rate limits can be set by setting a maximum and a minimum on the raw error signal. When integrating between these boundaries the actual frequency is obtained, with the desired rate of change limits. The rate of change limits where set manually at the rate that provided smooth variations of the stator frequency and power production. The same rate of change was imposed on the voltage controller to ensure correspondence between the voltage and frequency. The reference signals of the frequency and voltage is generated as a function of wind speed, based on the operational points presented in Figure 23 where the voltage is limited on 1 p.u. This is implemented in Power Factory using a lookup table where a spline interpolation provides the values in-between the operational points. 50

54 7. Comparison between the Steady State and the Dynamic model This chapter will provide a systematic comparison between the steady state model and the dynamic model presented earlier. The purpose of this comparison is to increase the reliability of the results presented, and provide an estimate of how the results should be interpreted. The method used in this project was to force the dynamic model to work in steady state To force the dynamic model into steady state is done by fixing a number of parameters that result in a steady output. In this project the author chose to fix the rotor resistance, stator frequency, stator voltage and wind speed. The outputs chosen for comparison was P-mechanical, P-generated and rotational speed, ω r. The reason for the choices is as followed: P-mechanical: The mechanical Power is used to see how accurate the mechanical representation of the model in Power Factory was achieved. This contains the accuracy of the C p and λ representation although it has to be noticed that this is affected by the operational point ω r. One of the important issues of this comparison was to find a suitable resolution of the C p -curve. The C p curve in Power Factory was implemented as a spline approximation matrix, where the inputs where the tip speed ratio (λ) and pitch angle (θ). This was made on a try and fail basis until a sufficient correspondence was obtained. Rotational Speed: The rotational speed was a good measure to see the influence of the rotor resistance on the operational point. This shows the intersection between the electrical and mechanical torque curves. This also increased the demands on the number of operational points in the optimization algorithm. It was shown that to provide a sufficient accuracy there was needed a step size of 1/33 p.u. of the stator frequency and rotor frequency. Also it showed that the suitable approximation in the optimization algorithm was to use a linear approximation between the operational points, when this gave the lowest error between the steady-state and dynamic model. P-generated: The generated power is used to see the accuracy of the operational point. Since the optimization algorithm is considering losses as an optimization factor it is important to see if the dynamic model has comparable losses at the same operational points. The chosen operational points for comparison are the optimum operational points from the optimization algorithm. This is because these values are already calculated and presented and is of interest for the project. The correspondence between the steady-state and dynamic model is shown in Table 12 51

55 Table 12: Comparison of the Steady-state and Dynamic model at optimum operational points Wind Speed Operational Conditions Steady State Model Dynamic Model fs Vs Rr ω r P mechanical P Generated ω r P mechanical P Generated [p.u.] [p.u.] [p.u.] As can be seen from the table the two models correspond with a relative good accuracy. The error in rotational speed can be shown by the author to be removed if the step-size of the rotor and stator frequency in the optimization algorithm is decreased, tough this is too time consuming to perform for all wind speeds. The less obvious error is the P generated which seems to be consequently deviating from the steady-state model. Although this error is small in p.u. it can be seen in Table 13 that this have an effect in percent, especially at the lower wind speeds. The author was not able to find the reason for this difference during this project, but consider the results to be corresponding adequately to perform further analysis. Table 13: Comparing the steady-state and dynamic model losses Wind Speed Steady-State model Dynamic Model Error [p.u.] Losses in p.u. Losses in % of Pm Losses in p.u. Losses in % of Pm DM-SSM

56 8. Dynamic Comparison between the AC and DC Connected Wind Turbine This chapter is meant to show a more realistic power production where the wind is fluctuating and the inertia of the generator is taken into account. Since rotational speed cannot be changed instantaneous because of the inertia of the generator the optimum tracking of the generator is provided with a delay. This is why the proposed control system is of great interest to analyse in dynamic simulation. The full system implemented in Power Factory is shown in Figure 35 Figure 35: Full system for Dynamic Simulation, implemented in Power Factory The system shown is by the author divided in to three areas, shown in Figure 35 to be area A, B and C. The idea of this is to keep are A in both simulations, The HVDC connected simulation and the AC connected simulation. The External grids in B and C are identical. The HVDC model contains 2 x 150 kv 75.2 µf (Du, 2007) capacitors to decrease fluctuations of the DC-Link. The cable is modeled as a 150 kv DC cable with resistance of Ω/km and reactance of Ω/km the DC voltage is 150 kv and the external grid is connected at 30 kv. In area A the generator with nominal voltage of 690 V is connected to a transformer with the ratio of 0.69/30 kv (Bonus Energy a/s, 2001). The inertia of the generator is determined by acceleration time constant and is set to 2 s. When simulating the AC connected system area B is disconnected and put out of operation, so only system C is connected to system A. When the HVDC connected system is simulated, system C is disconnected and put out of operation. Here only system A and system B is in operation. 53

57 8.1 The Dynamic Comparison Strategy Since the aim of this project was to create a control system for optimum operation of the generator designed to increase the yearly production, a reasonable comparison strategy was needed. The high resolution wind data available for the author during this project was based on a wind generator algorithm provided by the industrial supervisor. This data is divided in a step-size of 0.05 s and have duration of 10 min. Further details of the wind data are presented in Table 14 and its weibull distribution was earlier presented in Figure 31. Table 14: Characteristic of the Wind data used in the dynamic model Mean Wind Speed 9.65 m/s Min Wind Speed Max Wind Speed 7.11 m/s m/s The two system combinations presented in Figure 35 are now exposed to the same wind data, and the power production data is stored and exported to MatLab. Figure 36: The wind used for dynamic comparison 54

58 Figure 37: AC connected generator Power Production presented in p.u. Figure 38: DC Connected generator Power Production presented in p.u. 55

59 Figure 37 and Figure 38 shows the energy production of the two generator topologies based on the wind in Figure 36. The reason for the negative scale of the y-axis of the power production graphs is because the machine is in generator mode. The data stored in these graphs could easily be exported to MatLab using the Power Factory export to windows clipboard feature, and the results where integrated and divided by time. (6.11) Where P pu is the power in p.u. stored in the graphs of Figure 37 and Figure 38, and 600s is the simulation time. The answer obtained is then the average Power Production in p.u. Assuming the wind data is representative for a year, the yearly energy production can be written as: (6.12) Therefore the increase of power production based on the dynamic model can be written as: (6.13) The results of the dynamic simulation are shown in Table 15 Table 15: Average Power Production [p.u.] for the two wind turbine topologies p.u. AC connected wind turbine VSC HVDC connected wind turbine p.u. Increased Power Production = 2.62% 56

60 8.1.1 Evaluation The results so far show the possibility of increasing the power production of a wind turbine both in steady state and in dynamic models by varying the stator frequency and stator voltage. Although it has to be remembered that in order to compare this strategy with a state of the art wind turbine like the Vestas V80, a more accurate model has to be developed. Also the possibility of regulating the slip by actively using the pitch angle should be included in the optimization algorithm and this control might be needed to implement in the dynamic simulation. Lastly the power submitted to the grid will include losses in the DC transmission and cables. This should be considered in future research. 57

61 9. Dynamic Response to system fault As the last part of this project it was desired to check how the model with control system reacts to tripping of the one of the converters. The base of the dynamic study is to see how the active and reactive power, stator voltage rotational speed and the receiving end controller react to the tripping of the converter. The system study investigated will be tripping of the receiving end controller. This can represent a malfunction of the converter, or a fault on the external grid which forces the converter to disconnect. The system is modelled as shown in Figure 35 and contains system A and B. The analysis of the tripping of the grid side converter is performed by opening the switch connected to T DC shown in Figure 35. The opening of the switch occurs at time = 10 s and the closing of the switch at time = 10.3 s. The parameters for investigation are DC Voltage, stator voltage, stator frequency, rotor speed, receiving end current controller, active power and reactive power. Figure 39: Active power during Converter tripping 58

62 Figure 40: Reactive Power during Converter Tripping 150 Figure 41: Rotational speed during Converter tripping 59

63 Figure 42: Stator side Frequency during Converter Tripping Figure 43: Receiving End Current controller during Converter Tripping 60

64 Figure 44: DC -Link Voltage during Converter tripping Figure 45: Generator Side AC Voltage during Converter Tripping 61

65 Figure 46: Rotor speed during converter tripping, with constant wind = 12 m/s As can be seen from the presented results the VSC HVDC system reacts properly to a situation where the receiving end converter needs to trip. The generator will continue energy production during the fault and charge the DC capacitors. This result in an increase of the DC Link voltage as seen in Figure 44 and the receiving end control system will try to retain the DC voltage, Figure 43. The DC voltage will then oscillate until the PI controller has managed to stabilize the voltage. Since the Sending end converter is implemented on the assumption of a constant DC Voltage, the oscillation of the DC voltage is directly proportional to the generator AC side voltage as seen in Figure 45, the reason the AC voltage stabilizes around 33 kv is because the stator frequency here is over 1 p.u., earlier showed in Figure 5. The increase of the rotor speed is not a sign of instability thus only a result of continues controlling of the stator frequency based on the wind- speed as seen when comparing Figure 41 with Figure 42, for illustrative purposes it was also decided to show the rotational speed for the same fault with wind speed constant equal to 12 m/s as shown in Figure 46. The measurement of the voltage of the sending end converter was for simplicity in this project assumed to be proportional with the P m_in signal to the converter. This is true when the system operates in normal operation. Although as was seen in the simulations presented this assumption provides direct proportionality between the voltage variations in the DC-Link to the generator side AC terminal. In order to reduce this influence of DC voltage variations it might be interesting to use an external voltage measurement on the station1/pcc in Figure 35 instead. This might give lower dependency between the AC and DC voltage, although this hypotheses was not tested by the author during this project. 62

66 10. Discussion This project has investigated the possible gain of using a VSC HVDC transmission system for increasing the power production. The results presented show a positive effect of the new control possibilities when connecting such a generator to HVDC transmission system. Although the reader should have in mind the assumptions and simplifications this project has taken to account. Firstly wind farms connected to a VSC HVDC system will most likely not be represented by a single wind turbine, but rather by a significant number, as an example Horns Rev is represented by 80 wind turbines. When working with these types of wind farms there is no guaranty that the wind speed is equal for all the turbines, therefore common optimal operational point might be hard to find and the efficiency and power production presented in this project hard to obtain. The shaft system is neglected during the whole project and will influence the dynamic analysis and also possible change the operational points, although for the power production comparison it is assumed by the author the comparison is representative since none of the turbines in the project has this model included. This project has also not considered the effect of saturation in the generator which might be found in (Maria Imecs, 2001). The assumption of neglecting the saturation was based on the simplicity obtained. If saturation should be included the whole optimization algorithm would have to be reconstructed and the difficulty level increased since the co-dependency of currents and flux no longer would allow the setting of rotational speed in the way presented earlier. The model presented is also based on picking parts (data) from different wind turbines where references have been found. In order to provide a product for use in real life these parameters have to be more exact, so the models can be customized in a way that produces the best possible correlation with real life. The losses in the transformer was initially included in the model, but removed during the project because of its modest impact considering losses and operational points. What should be included in future research is the losses in the PWM converter, as these are believed by the author to be determinant for the new topology feasibility considering increased power production compared to standard AC transmission. 63

67 11. Conclusion The project shows, based on its assumptions and simplifications, that there is a possible gain in power production by implementing coordinated control between VSC and the RCC generator, although it prevail uncertainty if this gain will compensate for the losses in the PWM converters. The power production increase is shown in both steady state and in dynamic simulations. Basing the conclusion on the wind data used in the dynamic simulation the steady state model shows an increase in power production of 3.56 % compared to the same generator working as a constant rotor resistance AC-transmission connected wind turbine. The dynamic control system is based on using wind speed to provide a reference signal for stator frequency and stator voltage, and a rate limiter implemented so the variations should fit better with the inertia of the generator. The gain of using the control system based on the same wind speed as the one used in the comparison in steady state is shown to be 2.62%. The difference from steady state to dynamic simulation is because of the inertia of the generator that withstands sudden changes in frequency. But if losses in the PWM are of the size shown in Table 11 the VSC-HVDC generator will provide a lower power output to the grid than a conventional AC transmission at this wind distribution. Therefore it is believed by the author that the new control system presented will have a future in the situations where PWM converter transmission is the only topology that fulfils the requirements set by the transmission system operators. In these cases the optimization performed in this project will be pure gain. The models presented in this project are believed by the author to be representative to show the idea of the new topology. Although to initiate product development of such a system more exact data is needed, and considerations of state of the art OptiPitch control and HV PWM converters should be investigated and possibly included. 64

68 11.2 Future Research The following topics are recommended as future research in this field and recommended to do in the following order: Including losses of state of the art PWM converters in order to determine power output to the grid compared with the alternative AC transmission Use wind data from existing wind parks to see possibilities for coordinated control of a whole wind park, alternatively see how many sections a wind park should be divided into in order to get sufficient controllability. Include saturation and iron losses in the generator model 65

69 [1] Ackermann, T. (2005). Wind Power in Power Systems. Wiley. [2] Ahmed G. Abo-Khalil, H.-G. K.-C.-K. (2004). Maximum Output Power Control of Wing Generation Systems Considering Loss Minimization of Machines. Busan: IEEE. [3] Ahmed, N. &. (Aug 2006). Stand-Alone Hybrid Generation System Combining Solar Photovoltaic and Wind Turbine with Simple Maximum Power Point Tracking Control. IEEE Power Electronics and Motion Control Conference, s [4] Anca D. Hansen, C. J. (2003). Dynamic wind turbine models in power system simulation tool DIgSILENT. Roskilde: Pitney Bowes Management Sevice Denemark. [5] Betz, A. (1966). Introduction to the Theory of Flow Machines. Oxford: Pergamon Press. [6] Bonus Energy a/s. (2001). Middelgrunden Offshore - The Project. [7] Bose, P. M. (Dec 1983). Stability Simulation of Wind Turbine Systems. IEEE Transactions on Power Apparatus and Systems, s [8] DIgSILENT. (2007). Technical Documentation - PWM Converter. Gomaringen, Germany. [9] DIgSILENT2. (2007). Technical Documentation Induction Machine. Gomaringen, Germany. [10] Du, C. (2007). VSC-HVDC for Industrial Power Systems. Göteborg: CHALMERS UNIVERSITY OF TECHNOLOGY. [11] Erlich, C. F. (2007). Variable Frequency Operation of DFIG based Wind Farms Connected to the Grid through VSC-HVDC Link. Duisburg: IEEE. [12] Florin Iov, P. S. (2006). Modeling and Control of VSC Based DC Connection for Active Stall Wind Farms to Grid. Aalborg/Risø: IEEJ Trans. [13] G. O. Garcia, J. C. (1994). AN Efficient COntroller for an Adjustable Speed Induction Motor Drive. IEEE. [14] Ion Boldea, S. A. (2001). The Induction Machine Handbook. Florida: CRC Press. [15] Johnson, R. A. (1994). Miller and Frund s Probability and Statistics for Engineers. Englewood Clifs: Prentice-Hall, Inc. [16] Kundur, P. (1993). Power System Stability and Control. New York: McGraw- Hill, Inc. [17] Maria Imecs, I. I. (2001). A SIMPLE APPROACH TO INDUCTION MACHINE. Workshop on ELECTRICAL MACHINES' PARAMETERS (s. 80). Technical University of Cluj-Napoca. [18] Matsch, L. W. (1977). Electromagnetic & Electromechanical Machines. New York: A Dun-Donnelley Publisher. [19] Norwegian Metrological Institue. (mai 2008). Hentet fra [20] Paul C. Krause, O. W. (2002). Analysis of Electric Machinery and Drive Systems. Wiley. [21] Petterson, A. (2005). Analysis, Modeling and Control of Doubly-Fed Induction Generators for Wind Turbines. Department of Energy and Environment, Division of Electric Power Engineering. Göteborg: CHALMERS UNIVERSITY OF TECHNOLOGY. 66

70 [22] Robert Leidhold, G. G. (2002). Field-Oriented Controlled Induction Generator With Loss Minimization. IEEE. [23] Vestas. (2008). [24] Vestas. (u.d.). Project Description. [25] Watson, A. R. (2005). Comparison of VSC based HVDC and HVAC Interconnections to a Large Offshore Wind Farm. IEEE. [26] William H. Press, B. P. (1992). Numerical Recipes in C: The Art of Scientific Computing. Cambridge: Cambridge University Press. 67

71 Appendix A Nomenclature Symbol A a 1 C p f f 0 ϕ ψ i k k 0 L λ N ρ air P c P gen P m P md P mq R s T e θ v V w Description Rotor Radius Number of Current Paths Power Coeficient Frequency Frequency modulation Angele between Current and Voltage Flux Current Pitch Factor Inductance Tip Speed Ratio Number of Turns Air Density Copper Loss Generated Power Mechanical Power Modulation d axis Modulation q axis Resistance Slip Electrical Torque Blade Pitch Angle Voltage Wind Speed Subscripts: a avg AC DC d q r s Description Air Gap Average Alternating Current Direct Current d axis q axis Rotor Stator 68

72 m nom max min pu Magnetizing Nominlal Maximum Minimum Per Unit B Calculations Matrix([[Rs,w*(Ls+Lm),0,w*Lm],[-w*(Ls+Lm),Rs,-w*Lm,0],[0,(wwr)*(Lm),Rr,(w-wr)*(Lm+Lr)],[-(w-wr)*Lm,0,-(w-wr*(Lm+Lr)),Rr]]); > A:=Vector([iqs,ids,iqr,idr]); > Matrix([[Rs,w*(Ls+Lm),0,w*Lm],[-w*(Ls+Lm),Rs,-w*Lm,0],[0,(wwr)*(Lm),Rr,(w-wr)*(Lm+Lr)],[-(w-wr)*Lm,0,-((w-wr)*(Lm+Lr)),Rr]]); >B:=Matrix([[Rs,w*(Ls+Lm),0,w*Lm],[-w*(Ls+Lm),Rs,-w*Lm,0],[0,(wwr)*(Lm),Rr,(w-wr)*(Lm+Lr)],[-(w-wr)*Lm,0,-((w-wr)*(Lm+Lr)),Rr]]); > é ë é ë é B := Rs w ( Ls + Lm) 0 w Lm -w ( Ls + Lm) Rs -w Lm 0 0 ( w - wr) Lm Rr ( w - wr) ( Lm + Lr) -( w - wr) Lm 0 -w + wr ( Lm + Lr) Rr Rs w ( Ls + Lm) 0 w Lm -w ( Ls + Lm) Rs -w Lm 0 0 ( w - wr) Lm Rr ( w - wr) ( Lm + Lr) -( w - wr) Lm 0 -( w - wr) ( Lm + Lr) Rr ë Rs é iqsù ids A := iqr ë idrû w ( Ls + Lm) 0 w Lm -w ( Ls + Lm) Rs -w Lm 0 0 ( w - wr) Lm Rr ( w - wr) ( Lm + Lr) -( w - wr) Lm 0 -( w - wr) ( Lm + Lr) Rr ù û ù û ù û 69

73 > B.A; é ë Rs iqs + w ( Ls + Lm) ids + w Lm idr -w ( Ls + Lm) iqs + Rs ids - w Lm iqr ( w - wr) Lm ids + Rr iqr + ( w - wr) Lm + Lr -( w - wr) Lm iqs - ( w - wr) Lm + Lr ù ( ) idr ( ) iqr + Rr idrû > > > idr:=solve(-(w-wr)*lm*iqs-((w-wr)*(lm+lr)*iqr)+(rr*idr),idr); > subs(idr=solve(-(w-wr)*lm*iqs-((wwr)*(lm+lr)*iqr)+(rr*idr),idr),b.a); > B:=Matrix([[Rs,w*(Ls+Lm),0,w*Lm],[-w*(Ls+Lm),Rs,-w*Lm,0],[0,(wwr)*(Lm),Rr,(w-wr)*(Lm+Lr)],[-(w-wr)*Lm,0,-((w-wr)*(Lm+Lr)),Rr]]); > B; é B := é ë ë idr := Rs Lm iqs w - Lm iqs wr + w Lm iqr + iqr w Lr - iqr wr Lm - iqr wr Lr Rr > A:=Vector([iqs,ids,iqr,idr]); B. A w ( Ls + Lm) 0 w Lm -w ( Ls + Lm) Rs -w Lm 0 0 ( w - wr) Lm Rr ( w - wr) ( Lm + Lr) -( w - wr) Lm 0 -( w - wr) ( Lm + Lr) Rr Rs w ( Ls + Lm) 0 w Lm -w ( Ls + Lm) Rs -w Lm 0 0 ( w - wr) Lm Rr ( w - wr) ( Lm + Lr) -( w - wr) Lm 0 -( w - wr) ( Lm + Lr) Rr é A := ë iqs ids iqr Lm iqs w - Lm iqs wr + w Lm iqr + iqr w Lr - iqr wr Lm - iqr wr Lr Rr ù û ù û ù û 70

74 > B.A; é ë é ë w Lm Lm iqs w - Lm iqs wr + w Lm iqr + iqr w Lr - iqr wr Lm - iqr wr Lr Rs iqs + w ( Ls + Lm) ids + ( ) Rr ù, û [-w ( Ls + Lm) iqs + Rs ids - w Lm iqr], é ( w - wr) Lm ids + Rr iqr ë ( + w - wr ) ( Lm + Lr) ( Lm iqs w - Lm iqs wr + w Lm iqr + iqr w Lr - iqr wr Lm - iqr wr Lr ) ù, [ Rr û -( w - wr) Lm iqs - ( w - wr) ( Lm + Lr) iqr + Lm iqs w - Lm iqs wr + w Lm iqr + iqr w Lr - iqr wr Lm - iqr wr Lr ] ù û 71

75 C Matlab Optimization Script % This M-File Contains the Newton Raphson initialization of Generator % Currents, rotor speed dependancy on rotor resistance, and optimization % regarding the maximum power output clear all close all clc syms iqs ids iqr Rr ws wr vs %Electrical Constants Ls=0.18; %Stator Inductance Lr=0.07; %Rotor Inductance Lm=4.4; %Manetizing Inductance Rs=0.01;%Stator Resistance Lss=Lm+Ls; Lrr=Lm+Lr; %Mechanical Constants teta=0; % Blade Pitch Angle beta=0; % Blade Pitc Angle rho=1.225; % Air Density Awt=40; %Radius of turbine Blade Wmin=4; %Minimum Wind Speed Wmax=4; %Maximum Wind Speed Omega_s=33; %Inverse stepsize factor for stator frequency Omega=33; %Inverse stepsize factor for rotor frequency %Base Values wb=1; % Base value for Frequencies, could be removed Pnom=1.8*10^6; %Nominal Power Of Wind Turbine %Self defined factors for computation R_factor=200; %Rotor resistance step size lfac=2.64; % Lambba factor prop=0.355; %Proporsionality Factor %Trafo Parameters Kundur 2-vinding transformer Rprim= ; %Primary Resistance Lprim=0.0216; %Primary Inductance Rsec= ; %Secondary Resistance Lsec=0.0216; %Secondary Inductance Lmt=175; %Magnetizing Inductance Rmt=300; %Magnetizing Resistance LV=690; %Low Voltage HV=33000; %High Voltage % Electrical Equations equal to Zero F1=(-vs)+(Rs*iqs)+(ws*(Ls+Lm)*ids)+(ws*Lm*((Lm*iqs*ws)- (Lm*iqs*wr)+(ws*Lm*iqr)+(iqr*ws*Lr)-(iqr*wr*Lm)-(iqr*wr*Lr)))/Rr; F2=(-ws*(Ls+Lm)*iqs)+(Rs*ids)-(ws*Lm*iqr); F3=((ws-wr)*Lm*ids)+(Rr*iqr)+((ws-wr)*(Lm+Lr)*((Lm*iqs*ws)- (Lm*iqs*wr)+(ws*Lm*iqr)+(iqr*ws*Lr)-(iqr*wr*Lm)-(iqr*wr*Lr)))/Rr; %Creating Jackobian Matrix 72

76 G1=[diff(F1,iqs) diff(f1,ids) diff(f1,iqr)]; G2=[diff(F2,iqs) diff(f2,ids) diff(f2,iqr)]; G3=[diff(F3,iqs) diff(f3,ids) diff(f3,iqr)]; %Defining Matrixes for Newton - Raphson J=[G1;G2;G3]; F=[F1;F2;F3]; %Unknown currents, Initial guess iqs(1)=0.005; ids(1)=0.005; iqr(1)=0.005; x=zeros(3,4); x(1,1)=subs(iqs); x(2,1)=subs(ids); x(3,1)=subs(iqr); for Vw =Wmin : Wmax; for Ws = 1 :Omega_s; ws=(0.5-(1/(omega_s-1)))+(ws/(omega_s-1)); %Calculation of stator frequency if ws <= 1 ; vs=ws %Calculation of stator voltage else vs=1; %Calculation of stator voltage end for rr=1:10 Rr=0.009+(rr/R_factor); %Calculation of Rotor resistance %Initiazing Step for Finding and veryfing opperational point for Ws_in = 1 :Omega_s; ws_in=(0.5-(1/(omega_s-1)))+(ws_in/(omega_s-1)); if ws_in<=1 vs_in=ws_in; else vs_in=1; end for Wr_in = 1:Omega wr_in=(0.5-(1/(omega-1)))+(wr_in/(omega-1)); %Equations seen from the Electrical Side s(wr_in,ws_in)=(ws_in-wr_in)/ws_in; topp(wr_in,ws_in)= (ws_in/wb)*(lm^2)*rr*s(wr_in,ws_in)*vs_in^2; bottom(wr_in,ws_in)=(rs*rr+s(wr_in,ws_in)*((ws_in/wb)^2)*(lm^2- Lss*Lrr))^2+((ws_in/wb)^2)*(Rr*Lss+s(Wr_in,Ws_in)*Rs*Lrr)^2; Te(Wr_in,Ws_in)=topp(Wr_in,Ws_in)/bottom(Wr_in,Ws_in); PmEl(Wr_in,Ws_in)=(-1)*wr_in*Te(Wr_in,Ws_in); %Equations seen from the mechanical side lambda_in(wr_in)=(awt*wr_in*lfac)/vw; Cp_in(Wr_in)=prop*(lambda_in(Wr_in)-0.022*(teta^2)- 5.6)*exp(-0.17*lambda_in(Wr_in)); 73

77 end Pm_in(Wr_in)=((rho/2)*pi*Awt^2*Cp_in(Wr_in)*Vw^3)/Pnom; %Finding the rotor rotational speed in the opperation Point wr_plot=linspace(0.5,1.5,omega)'; interp1(wr_plot,pm_in,a,'linear')- interp1(wr_plot,pmel(:,ws_in),a,'linear'); %First Constrain [wr_vector(ws_in,rr),feval(ws_in,rr),exitflag(ws_in,rr)]=fsolve(ff,(ws_in)) ; wr_vector1(ws_in,rr,vw)=(wr_vector(ws_in,rr)); % Calculating third constrain a=(pmel(:,ws)==max(pmel(:,ws))); aa=ind2sub(size(pmel),find(a==1)); % Initializing second, third and fourth constrain if exitflag(ws,rr) == 1 & PmEl(aa,Ws)>Pm_in(aa) & wr_vector(ws,rr)/ws <= 1.1 wr=wr_vector(ws,rr); slip(ws,rr,vw)=(ws-wr_vector(ws,rr))/ws; % Calculating Curents using Newton-Raphson for i=2:4 subs(j)\subs(f)); x(:,i)=subs([x(1,i-1);x(2,i-1);x(3,i-1)]- iqs=x(1,i); ids=x(2,i); iqr=x(3,i); idr=(((lm*iqs*ws)- (Lm*iqs*wr)+(ws*Lm*iqr)+(iqr*ws*Lr)-(iqr*wr*Lm)-(iqr*wr*Lr))/Rr); iqs1(ws,rr,vw)=x(1,i); ids1(ws,rr,vw)=x(2,i); iqr1(ws,rr,vw)=x(3,i); idr1(ws,rr,vw)=(((lm*iqs1(ws,rr,vw)*ws)- (Lm*iqs1(Ws,rr,Vw)*wr)+(ws*Lm*iqr1(Ws,rr,Vw))+(iqr1(Ws,rr,Vw)*ws*Lr)- (iqr1(ws,rr,vw)*wr*lm)-(iqr1(ws,rr,vw)*wr*lr))/rr); flux_qs(ws,rr,vw)=ls*iqs1(ws,rr,vw)+lm*(iqs1(ws,rr,vw)+iqr1(ws,rr,vw)); flux_ds(ws,rr,vw)=ls*ids1(ws,rr,vw)+lm*(ids1(ws,rr,vw)+idr1(ws,rr,vw)); flux_qr(ws,rr,vw)=lr*iqr1(ws,rr,vw)+lm*(iqs1(ws,rr,vw)+iqr1(ws,rr,vw)); flux_dr(ws,rr,vw)=lr*idr1(ws,rr,vw)+lm*(ids1(ws,rr,vw)+idr1(ws,rr,vw)); flux_s(ws,rr,vw)=sqrt(flux_qs(ws,rr,vw)^2+flux_ds(ws,rr,vw)^2); flux_r(ws,rr,vw)=sqrt(flux_qr(ws,rr,vw)^2+flux_dr(ws,rr,vw)^2); flux_m(ws,rr,vw)=sqrt(flux_r(ws,rr,vw)+(lr^2).*(sqrt(iqr1(ws,rr,vw)^2+idr1( Ws,rr,Vw)^2))); 74

78 ir(ws,rr,vw)=sqrt(iqr1(ws,rr,vw)^2+idr1(ws,rr,vw)^2); is(ws,rr,vw)=sqrt(iqs1(ws,rr,vw)^2+idr1(ws,rr,vw)^2); end else wr_vector(ws,rr)=0; wr=0; slip(ws,rr,vw)=0; ids=0; idr=0; iqr=0; iqs=0; ir(ws,rr,vw)=0; is(ws,rr,vw)=0; end end end end Vw end %Calculating losses in the transformator Zprim=sqrt((ws*Lprim)^2+Rprim^2); Zsec=sqrt((ws*Lsec)^2+Rsec^2); Zmt=((1/((ws*Lmt)^2))+(1/Rmt))^-1; PrimLoss=(iqs^2+ids^2)*Rprim; Imt=(iqs^2+ids^2)*(Zprim/(Zmt+Zprim)); MagLoss=Imt*((ws*Lmt)/(Rmt*(ws*Lmt)))*Rmt; SecLoss=((iqs^2+ids^2)-Imt)*Rsec*(LV/HV); Tloss(Ws,rr,Vw)=PrimLoss+MagLoss+SecLoss; lambda=(wr*awt*lfac)/(vw); Cp=prop*(lambda-0.022*(teta^2)-5.6)*exp(-0.17*lambda); Ploss(Ws,rr,Vw)=((((iqr^2+idr^2))*Rr)+((iqs^2+ids^2)*Rs)); Pm(Ws,rr,Vw)=((rho/2)*pi*Awt^2*Cp*Vw^3)/Pnom; Pgen(Ws,rr,Vw)=Pm(Ws,rr,Vw)-Ploss(Ws,rr,Vw); P(Ws,rr,Vw)=Pgen(Ws,rr,Vw)-Tloss(Ws,rr,Vw); Ps(Ws,rr,Vw)=vds*ids+vs*iqs; Qs(Ws,rr,Vw)=vs*ids+vds*iqs; %initializing 5th Constrain if ws >= wr wr > 1.5 Pm(Ws,rr,Vw)=0; Pgen(Ws,rr,Vw)=0; Ploss(Ws,rr,Vw)=0; P(Ws,rr,Vw)=0; Ps(Ws,rr,Vw)=0; Qs(Ws,rr,Vw)=0; slip(ws,rr,vw)=0; ir(ws,rr,vw)=0; is(ws,rr,vw)=0; end 75

79 % Finding Maximal Power Production and Corresponding operational points for Vw=Wmin:Wmax g=(p==max(max(p(:,:,vw)))); [I1(Vw),I2(Vw),I3(Vw)] = ind2sub(size(p(:,:,vw)),find(g==1)); PowerCurve(Vw)=P(I1(Vw),I2(Vw),Vw); RotorSpeed(Vw)=wr_vector1(I1(Vw),I2(Vw),I3(Vw)); end %Operational Points omega_s=((0.5-(1/(omega_s-1)))+(i1./(omega_s-1)))*50 Rotor_resistance=0.009+(I2./R_factor) 76

80 Department of Electrical Engineering Centre for Electric Technology (CET) Technical University of Denmark Elektrovej 325 DK-2800 Kgs. Lyngby Denmark Tel: (+45) Fax: (+45)