Real-time Process Simulator for Evaluation of Wind Turbine Control Systems

Transcription

1 ECN-E Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Modelling and Implementation E.L. van der Hooft; T.G. van Engelen, J.T.G. Pierik, P. Schaak June 2007

2 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Keywords Wind Energy, Wind Turbine, Control, Simulation, Modelling ii ECN-E

3 Abstract The development of a real-time simulator for a complete wind turbine system has been carried out for the evaluation of the overall control system. The real-time simulator software has been developed in Matlab/Simulink and supports automated real-time compilation (Real Time Workshop) to a real-time code for use at a hardware platform (dspace, xpc-target) The following program modules were developed and implemented: efficient integrated linear structural models for the rotor, drive-train and support structure in a working point range; an interpolation method between these models has been derived; non-linear aerodynamic (BEM) and hydrodynamic (Morison) conversion models; a blade effective windspeed model, which account for the rotational sampling of spatial turbulence, for tower shadow and wind shear, and for oblique inflow; a wave generation model (Airy) in order to cope with offshore situations; an electric system model in a rotating reference frame (Park) consisting of a doubly fed induction generator, converter, transformer and cabling. quasi-steady and easy to parametrise models for turbine specific peripheral devices like pumps, motors, valves, brakes, heat exchangers. models of peripheral devices which comprise discontinuous behaviour such as switching and Coulomb friction. generic models for the thermic behaviour of the heat generating systems like gearbox, brake and generator. These subsystem models were integrated in an overall Simulink scheme for time-domain simulation and compilation to real-time code. ECN-E iii

4 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Acknowledgement This project has been done on a partial grant of the Ministry of Economic Affairs of the Netherlands in the SenterNovem programme, the Dutch Agency for Energy and the Environment, under grants DEN and BSE The authors would like to thank the following people for their technical involvement: Mr. Johan Morren (Delft University of Technology), for his contribution to the electric model. Mr. Stoyan Kanev (ECN Wind Energy) for implementation and reviewing of the generation of blade effective wind speeds. Mr. Servaes Ramakers (ECN Wind Energy) for inventarisation of peripheral devices. Furthermore, we are gratefull to Mr. Ben Hendriks (group leader of ECN Wind turbine technology) for financial project concerns and to Mr. Dennis Wouters (ECN Wind Energy) for his practical experience and recommendations in using Matlab/Simulink and RealTimeWorkshop. Finally, we would like to thank many employees of different wind turbine manufacturers for the given insights in their turbine systems and having discussions to guarantee the industrial relevance of this project. iv ECN-E

5 CONTENTS SUMMARY ix 1 INTRODUCTION 3 2 TURBINE SYSTEMS Simulation blocks Modelling approach Peripheral device modelling principles Turbine dynamic system modelling principles External influence modelling principles PERIPHERAL DEVICES General systems Induction motor Centrifugal pump Coulomb friction Gearbox system (GBX) Description Definition Gearbox losses Gearbox oil pump drive Gearbox oil circulation Gearbox heat exchanger Generator system (GEN) Description Definition Generator losses Generator water pump drive Generator water circulation Generator radiator Yaw system (YAW) Description Definition Yaw brake oil pressure Yaw drive system Yaw sensors Brake system (BRK) Description Definition ECN-E v

6 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Brake oil pump drive Brake oil circulation Mechanical brake Pitch system (PIT) Electric blade pitching Working principles Effective pitch speed servo behaviour TURBINE DYNAMIC SYSTEMS Structural dynamic system (SDS) Rotor Drive Train Support structure Integrated structural dynamic model Aerodynamic conversion (ACS) Hydrodynamic conversion (HCS) Thermic conversion (TCS) Gearbox system Generator system Brake system Electric conversion (ECS) Modelling in a dq0-reference frame Doubly-Fed Induction Machine model Converter model Control of the DFIG converters Other electrical component models Implementation of the DFIG model in Simulink Response of the DFIG system to a setpoint change and a disturbance EXTERNAL INFLUENCES Wind Generation (WIN) Blade effective turbulence Periodic wind speed variations by tower shadow and wind shear Wind speed affection by oblique inflow Blade effective turbulence realisations for variable speed and oblique inflow Wave Generation (WAV) Realisation algorithm for water surface elevation Wave speed and acceleration from elevation realisation Grid (GRD) vi ECN-E

7 CONTENTS 5.4 Gravity (GRV) WIND TURBINE PROCESS SIMULATION Wind turbine simulation model Data input and processing Time domain simulation Blade effective wind speed realisations Blade effective wind speed realisation including oblique inflow Blade and tower loading Gearbox temperatures Oil pump behaviour Real time wind turbine simulation CONCLUSIONS 167 References 169 ECN-E vii

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9 SUMMARY The commissioning of a (prototype) windturbine comprises the testing of the control system. This concerns the feedback control of the wind turbine in operation as well as the on/off and status control of the auxiliary systems. It appears that commissioning can be cumbersome and expensive, especially for offshore sited wind turbines. Extensive testing of the control system in advance is expected to significantly accelerate the commissioning and increase the turbine reliability and safety. Therefore, the development of a real-time simulation of complete wind turbine systems has been carried out. The real-time simulator software has been developed in Matlab/Simulink. Automated compilation to a real-time code for use at a hardware platform (dspace, xpctarget) can then be done by using Real Time Workshop 1. The program modules were implemented in Simulink for time domain simulation. Simulink facilitates to a large degree in convenience and flexibility (model modification and exchange) for the user. The modular framework makes future extension and modifications tranparent. A large number of program modules has been developed for real-time process simulation. These modules can be tailored on demand in order to derive an implementation of any wind turbine. The developed Matlab/Simulink modules enable the real-time simulation of the wind turbine itself, the so called peripheral devices and the driving sources. Unfortunately, it was not possible to realize an implementation of a real wind turbine in the Process Simulator software, due to the withdrawal of the industrial partners from the project. The implementation has been realised in such a way that it allows for automated compilation to a real-time hardware target like dspace or xpc-target via the Real-Time Workshop toolbox. This is a proven procedure and guarantees compatibility to industrial standards. A specific application of TURBU Offshore has been developed for the generation of an integrated linear structural model for the rotor, drive-train and support structure in a working point. Structural models are derived in a number of working conditions and simultaneously used during real-time simulation. For this, an interpolation method between these models has been derived. The integrated model processes all mechanical interactions between the rotor, drive-train and support structure and interacts with the peripheral devices and the aero- and hydrodynamic conversion system; it provides kinematic responses to aero- and hydrodynamic loads and to loads from the peripheral devices. The relatively low order linear models can be configured in advance and yield time-efficient simulation. The aerodynamic and hydrodynamic conversion models are implemented as non-linear simulation modules because of their non-linear nature. The aerodynamic conversion model is based on BEM theory and converts the wind speed to external forces and torques on the blade elements; dynamic inflow is accounted for. The hydrodynamic conversion is based on Morison s equation and converts wave speed and acceleration to external forces on the tower elements. The blade effective windspeed model has proven to be an effective driving source for realtime simulation. The windspeeds on the blades account for the rotational sampling of spatial turbulence, for tower shadow and wind shear, and for oblique inflow. A wave generation model has been developed in order to cope with offshore situations. It is based on the water surface elevation spectrum in conjunction with Airy s linear wave theory that maps the surface elevation to wave speed and acceleration. As in reality, the directions of the waves and the water current may both differ from the average wind direction. The electric system is modelled by simulation modules for a doubly fed induction generator, 1 Matlab, Simulink, RealTimeWorkshop and xpc-target are products of The Mathworks Inc., dspace is a product of dspace GmbH ECN-E ix

10 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems converter (rotor), transformer (stator) and cabling. The simulation time for the fast dynamics, which are inherently associated with this kind of electric systems, is significantly reduced by a transformation to the rotating reference frame (Park s transformation). Although, the grid is assumed to be rigid, the electric system model is already prepared for dealing with grid transients and events. The models for turbine specific peripheral devices like pumps, motors, valves, brakes, heat exchangers are based on their quasi-steady behaviour. The involved parameters are conveniently accessible for engineers. The implemented models cope with discontinuous behaviour such as switching and Coulomb friction. Turbine specific modifications can be applied easily, because of the generic character of the models and the leveled implementation for fluid systems, losses and heat exchange. Generic models have been derived for the thermic behaviour of the gearbox, brake and generator. The models are driven by power losses. They provide temperatures on different (system) locations, based on first order dynamic behaviour. The parameters are derived from the involved thermic capacities and conductivities. Based on industrial contacts and presentations, it can be concluded that the interest in real-time process simulation still increases. A simple real-time simulation program has been transferred to a wind turbine manufacturer. Wind manufacturers are convinced about the value of real-time simulation, in particular for offshore turbines. However, they also experience problems with the inclusion of so much turbine behaviour for which this project will support them. Further developments are actually being carried out within the European Project UPWIND. Also, the simulation of the whole wind turbine, as modelled in this project, will be used in the EOS Project SUSCON (SenterNovem grant EOS-LT-02013) when dealing with extreme event handling and with turbine shut-down in case of emergency. Finally, ECN will also trigger bilateral cooperation with wind turbine manufacturers on the subject of process simulation. x ECN-E

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13 1 INTRODUCTION The commissioning of a (prototype) windturbine comprises the testing of the control system. This concerns the feedback control of the wind turbine in operation as well as the on/off and status control of the auxiliary systems. It appeared that commissioning can be cumbersome and expensive, especially for offshore sited wind turbines. The testing of the control system in advance is expected to significantly accelerate the commissioning. With this in mind, we drafted a proposal for the development of a software platform for real-time simulation of complete wind turbine systems, the process simulator. This two-phase proposal was approved by SenterNovem, the Dutch Agency for Energy and the Environment, under grants DEN and BSE The first phase resulted in the well-defined problem based on an exensive inventarisation. This phase was completed in 2002 [1]. It pointed out that it was worthwhile to carry on with the development. The second phase is reported here and involves the modelling and the implementation of the so called turbine systems, viz: rotor, main shaft, gearbox, support structure, generator system; wind, wave and grid; yaw, brake, pitch system. The turbine system models were implemented in a Simulink scheme. The Simulink scheme facilitates integrated real-time simulation. Simulink is a well-known graphically interfaced toolbox of the MATLAB software platform [2]. In Chapter 2 it is described how each turbine system is implemented as a (set of) simulation block(s). It also contains the applied modelling approaches to the turbine systems. A simulation block is a code block for simulation of elementary process behavior. The distinghuised types of simulation blocks are turbine dynamic systems, external influences and peripheral devices. In Chapter 3 simulation blocks for the peripheral devices are dealt with. These blocks realise control actions and power losses, governed from the control system and the turbine dynamic systems respectively. Their implementation heavily depends on the specific manufacturing concepts. Nevertheless, basic building blocks are supplied for pumps, motors and friction generating devices. The servo and power loss behaviour of the gearbox, generator, yaw, pitch, brake system is implemented in the actuation devices. In Chapter 4 simulation blocks for the turbine dynamic systems are dealt with. These blocks realise the principal conversion flow in a wind turbine system as well as the thermic behaviour. The involved models are highly generic. The aero- and hydrodynamic conversion behaviour, the structural dynamic behaviour of the rotor, main shaft, gearbox, generator, support structure, the thermic and electric conversion behaviour are implemented in the turbine dynamic systems. In Chapter 5 simulation blocks for the external influences are dealt with. These blocks realise the generation of excitation signals from the wind, waves, grid and gravity. The involved models are highly generic. In Chapter 6 the overall simulation scheme which comprises all previously simulation blocks and some simulation results generated by Simulink are shown and discussed. Finally, conclusions are listed and a preview is made on how to continue with real-time implementation in close cooperation with wind turbine industry. ECN-E

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15 2 TURBINE SYSTEMS In this chapter it is described how each turbine system is implemented as a (set of) simulation block(s). It also contains the applied modelling approaches to the turbine systems. A simulation block is a code block for simulation of elementary process behavior. The distinghuised types of simulation blocks are turbine dynamic systems, external influences and peripheral devices. The models of the turbine systems have been implemented as MATLAB code blocks in a Simulink scheme. The models for generic behavior and for specific manufacturing concepts are stored in different code blocks such as for aerodynamic conversion and for pitch servo actuation. These simulation blocks are dealt with in par The separation principle enables to tailor the basic process simulator software to a specific wind turbine in a transparant way. As a consequence, a simulation block may represent only a part of a turbine system like the thermic conversion of the gearbox. Such a block accommodates a clear-cut portion of process behavior like servo actuation, aerodynamic conversion or structural dynamics. The level of modelling detail in the simulation blocks was tuned to the foreseen actions from the control system. This is subject of par Simulation blocks The implemented turbine systems in the process simulator are: rotor, main shaft, gearbox, generator system, support structure; wind, wave, grid and gravity; yaw, brake and pitch system The real-time process simulator is built up from subsystems that each simulate a part of the behaviour of a turbine system, the simulation blocks. Only three types of simulation blocks exists. A simulation block type is linked to a specific kind of process behaviour, viz. the main and thermic conversion in a wind turbine (turbine dynamic systems); realisation of excitation signals (external influences); realisation of demanded actions or behaviour (peripheral devices). Tables 2.1 and 2.2 give a complete overview of the subdivision of the turbine systems in simulation blocks. Figure 2.1 shows the layout of the process simulator in simulation blocks. Peripheral devices are e.g. the torque servo in the generator system and the gearbox oil pump drive ((GEN), but also the generating mechanism for friction loss in the main bearing (GBX). The subsystems for the structural dynamic behaviour (SDS) and electric conversion (ECS), but also those for thermic conversion (TCS), belong to the turbine dynamic systems simulation block type. The external influences are the wind (WIN), waves (WAV) and the grid (GRD). The turbine system implementation in more simulation blocksenables to seperate the [continuous] dynamic behavior, which can be genericly modelled, from the turbine specific realisation of actuation concepts. ECN-E

16 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 2.1: Process simulator layout in simulation blocks 6 ECN-E

17 2 TURBINE SYSTEMS turbine system simulation block description type name rotor dynamic system ROT-SDS blade motion and deformation in rotor model ACS profile behaviour external influence WIN sample wind field for blade positions main shaft system actuation device SHAFT-FRIC generate loss by viscous and Coulomb main bearing friction dynamic system SHAFT-TCS temperature response on loss power SHAFT-SDS shaft motion and deformation in drive-train model gearbox system actuation device GBX-FRIC generate loss by viscous friction GBX-PUMP generate oil flow in gearbox GBX-OIL oil circulation along gears for lubrication GBX-HEAT exchange heat between oil and cooling system dynamic system GBX-TCS temperature response on loss power GBX-SDS transmission ratio generator system actuation device GEN-COPP generate loss power from windings GEN-PUMP generate water flow in generator GEN-WTR water circulation along windings for cooling GEN-HEAT exchange heat between water and radiator GEN-SERV realise desired generator torque dynamic system GEN-TCS temperature response on copper loss GEN-SDS generator inertia in drivetrain GEN-ECS electro/mechanic torque conversion support structure dynamic system SUPP-SDS foundation, tower and nacelle motion and deformation HCS drag and appparent mass behaviour Table 2.1: Subdivision of turbine systems in simulation blocks(1) ECN-E

18 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems turbine system simulation block description type name wind external influence WIN generate blade wind speed wave external influence WAV generate wave speed and acceleration grav external influence GRV periodic gravity effects grid external influence GRD grid voltage and frequency yaw system peripheral device YAW-COPP generate loss power from windings YAW-FRIC generate loss by Coulomb friction YAW-SERV realise desired yawing speed dynamic system YAW-TCS temperature response on copper loss brake system peripheral device BRAKE-FRIC generate loss by Coulomb friction dynamic system BRAKE-TCS temperature response on friction loss pitch system peripheral device PIT-COPP generate loss power from PIT-FRIC windings generate loss by Coulomb friction PIT-SERV realise desired pitching speed dynamic system PIT-TCS temperature response on copper and friction loss Table 2.2: Subdivision of turbine systems in simulation blocks(2) 8 ECN-E

19 2 TURBINE SYSTEMS 2.2 Modelling approach In this section it is described how the different types of process behaviour in the process simulator are modelled. This modelling approach is described in relation to the three types of simulation blocks in the next three subsections: peripheral devices; turbine dynamic systems; external influences Peripheral device modelling principles Process behavior of the peripheral devices is typed by five modelling principles: actuation and measurement, fluid flow, mechanic loss, electric loss, heat transfer. Actuation and measurement Based on measured signals and received detections, the turbine control system determines commands and calculates setpoints. Pitch setpoint realisation is performed by servo loops [3]. The servo loop models only includes the relevant, as simple as possible dynamic characteristics of the equipment. Oil pumps, water pumps are grid-connected via on/off contactors which are commanded by the turbine control system. The yaw motors are able to rotate clockwise and counterclockwise. The motor models are based on the stationary torque-speed and current-speed curve, which are voltage and frequency dependent. This enables to simulate the stationary behaviour as well as the start-up and overload behaviour. The parameters of the motor models are derived from common motor date (name plate and catalogue information). The power loss due to motor slip is modelled for thermic purposes. Valves are modelled as gradually switching devices. The models include hysteresis to avoid shuttling. The measured values (pressure, speed, current, voltage, wind etc.) are first order filtered before they enter the turbine control system. These filters cater for delays and limited bandwidths in practice. Also quantisation and sampling effects are taken into account. Fluid flow Fluid systems like oil and water circulation are based on non-linear stationary pressure-flow curves. The pressure dynamics are first order modelled as hydraulic capacitors (storages). The time constant is then related to the volume and physical properties of the fluid. System components like tubes, filters and valves are modelled as hydraulic resistances (or reciproce conductivities). Specific valves like unidirectional valves, control valves, overpressure valves and thermovalves are modelled as non linear (discontinuous) elements [4]. Mechanic loss Mechanic loss is due to friction which causes torque loss and heat power. The losses of mechanical devices (e.g. gearbox) are dependent of speed and torque. Speed dependent losses are caused by windage or churning (oil mist), while load dependent losses are due to sliding or rolling (gears). The loss models are quasi-steady [5] [6]. Furthermore, three types of frictions are considered: viscous friction (proportional with speed), Coulomb friction (constant and discontinuous at standstill) and stick friction (extra friction at ECN-E

20 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems standstill which should be overcome before motion starts). Viscous friction (bearings, windage) is linearly modelled and related to the efficiency. Coulomb and stick friction are modelled in a discontinuous way via a state machine approach. This type of friction is relevant for braking devices and for rotating devices around standstill (startup, speed reversals). Electric loss The electric losses of the generator are classified in copper losses (electric resistance) and iron losses (core saturation) of the stator and rotor. When saturation effects are left out of consideration, the copper losses are much larger than iron losses. Copper losses are proportional to the square current in the windings. The models for the electric losses of motor drives are quasi-steady. The actual slip has been taken as a measure for heating loss (difference between electric input power and mechanical shaft power)) Heat transfer The modelling of heat transfer between devices or from a device to the environment is based on a linear approach of conduction of heat power between two heat storages (thermic capacitors). Heat power is exchanged between storages when there is a temperature difference. The temperature of thermic capacitors increases when heat power (losses) is supplied and it decreases when heat power is extracted (cooling). The time constant of a thermic capacitor is proportional to its volume and the properties of the medium (oil, water, air) [7] Turbine dynamic system modelling principles There are five different types of process behaviour in the turbine dynamic systems: structural dynamics, aerodynamic conversion, hydrodynamic conversion, electric conversion, thermic conversion. Structural dynamics The structural model is set up by beam elements that consist of rigid bar with a lumped mass and with angular and linear springs and dampers between two elements. Sets of beam elements model the substructures of the wind turbine (blades and tower). The lumped mass of the elements is accompanied by moments of inertia. The inertia properties are matched to the mass distribution of the substructure-partition that is replaced by the element. The springs are matched to the stiffness proporties of the substructure-partition that surround the connection point of two elements; the dampers are matched to the structral damping rate of the first deformation modes of the blades and tower. Further, distinct degrees of freedom are included that allow for foundation compliance, rotor shaft deformation, blade hinges, variable speed operation, blade pitching, yawing and gearbox house compliance [8] [9]. The structural dynamic behaviour is modelled by linearised dynamic deformation around the average deformed state. The average deformed state is derived from linear stress-strain relationships in a non-linear geometry under non-linear aerodynamic equilibrium assessment. The non-linear equations of equilibrium are stage-wise simultaneously solved via nested iteration procedures. The non-linear geometry is established by a co-rotational formulation of the beam elements; the aerodynamic equilibrium is based on Blade Element Momentum theory. The linearised structural behaviour around an equilibrium is obtained by applying Newton s 2 nd and 3 rd law on rigid bodies in their local coordinate systems (F = ma, F react = F act ). Structural component models are derived for the rotor blades, drive-train and support structure. These are connected under elimination of interacting load and kinematic variables. The 10 ECN-E

21 2 TURBINE SYSTEMS model orders for the blades and tower are reduced to the desired number of deformation modes [10] [11]. Azimut-dependent model parameters are eliminated via a multi-blade transformation [12]. The last steps constraints the rotor layout to 3 blades or more. A concatenation of linear structural models is simulated in order to cover a wide range of operation while the relevant non-linear structural dynamic behavioral aspects are taken into account. Aerodynamic conversion The modelling of the aerodynamic conversion is based on Blade Element Momentum theory (BEM) extended with a dynamic term for the evoluation of the wake, as proposed by Snel and Schepers [13] in the ECN s Differential Equation model. The conservation of both axial and tangetial impulse is catered for in the rotor annuli. The rotor annuli correspond with the swept areas by the beam elements of the rotor blades. The axial impulse equation is extended to a first order differential equation in each annulus. Effects of the blade tip and blade root are taken into account via Prandtl s correction factor. The influence of oblique inflow on the induction behaviour is taken into account by a first order approximation of the well-know scheme proposed by Glauert and an emperically derived scheme by Schepers. This approximation effectively yields an azimut dependent addition to the axial wind speed that depends on the oblique inflow angle and average wind speed. The influence of individual blade deformation and local flow conditions on the induction speeds is not taken into account. Thus, the induction state is assumed annulus-uniform except as to the effect of oblique inflow which is effectively moved to the (undisturbed) axial wind speed. Hydrodynamic conversion The hydrodynamic conversion is modelled in accordance with Morison s hydrodynamic load model, which includes drag and mass loads. The water current, the horizontal wave speed and the tower speed on a distinct number of underwater levels is taken into account in the hydrodynamic drag load calculation. The mass loads are derived from the horizontal wave acceleration and tower acceleration on these underwater levels. The underwater levels correspond with the centre points of the underwater beam elements of the structural tower model. Different directions of waves and current are allowed for. The direction of neither the waves nor the current has to coincide with the direction of the wind speed. The reduction of mass loads because of diffraction is catered for via the McCamy-Fuchs correction of the inertia coefficient in Morison s equation. This has been carried via a frequencydependent scaling factor in the realisation algorithm for the wave acceleration. In this approach, the reduction of mass load due to tower acceleration is neglected [14]. Electric conversion The electric conversion is catered for by a model for a doubly fed asynchronous machine including power electronic converter connected to the rotor, transformer and cable. Balanced conditions are assumed. The electrical signals are transformed to a rotating frame of reference (Park-transformation, dq0-frame [15]). Because this frame rotates at the grid frequency, all electrical quantities are constant in steady state conditions. This enables easy controller design and increases the simulation speed of the fast electric dynamics significantly. The rotor converter sets the rotor current with respect to the position of the magnetic flux (vector control). The converter is modelled by a pair of controlled voltage sources: on the machine as well as on the grid side. ECN-E

22 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Thermic conversion For thermic modelling a device is broken down in different thermic storages. For each temperature location of interest a storage (thermic capacitor) is defined, which can be heated up by injected power losses (heat power source). Heat is exchanged by temperature differences between thermic storages. The thermic capacitors are related to the volume of the storage and the properties of the medium. The thermic heat exchange is modelled by linear conductivities. This results in sequences of first order thermic models [7] External influence modelling principles The behaviour of external influences is typed by three modelling principles: wind turbulence and shear, waves and current, electric grid rigidity. Wind turbulence and shear The concept of blade effective wind speed signals is applied in real-time process simulation instead of the simulation of a complete three-dimensional wind field [16]. The three wind speed signals are designed such that they cause realistic blade root loads; the loads are similar to those that arise when a rotating rotor blade samples a homogeneous turbulent wind field for longitudinal turbulence only, affected by wind shear and tower shadow. Also oblique inflow is accounted for, viz. (i) in the way the longitudinal turbulence is sampled and (ii) by the appearance of periodic tangential wind velocity components through the average sideway and vertical wind speed in the rotor plane; finally, the azimut dependent axial induction variation by oblique inflow, as mentioned in the paragraph on aerodynamic conversion, is added to the turbulence. The homogeneous turbulence is taken into account by time-domain realisations for the three rotor blades that are obtained by Fourier synthesis of a 3 3 spectral matrix. This spectral matrix is obtained from the auto power spectrum of the wind in a fixed location and the coherence function over the rotor plane (ESDU-power spectrum; Kaimal coherence function) [17]. The 3 3 matrix is obtained from radius-dependent weighted cross power spectra between wind speeds in the center points of all 3N blade elements. These turbulence realisations can be considered as helices from which the wind speed values are sampled by the rotating blades as a function of the azimut angle for a constant rotor speed. Each blade has its own helix. The difference between the helices consist of (i) the phase shift that corresponds with the azimut separation of the rotor blades and (ii) deviations in time because of the evoluation of the wind speed in the fixed locations of the rotor cylinder. Because of oblique inflow, and also because of rotor speed variation, the rotor blade is ahead of or lags behind its helix with wind speed values. This effect is taken into account by interpolating between helices. Tower shadow is accounted for in accordance with the behaviour of a semi-infinite dipole. Wind shear is modelled by exponential height dependence in accordance with the IEC norm [18]. All blade effective wind speed variations, except those related to the homogeneous turbulence, pertain to the 2 -radius location of the blades. 3 Waves and current The hydrodynamic load computation depends on the horizontal wave speed and acceleration in the centre points of the underwater tower elements (see paragraph on hydrodynamic conversion). The wave generation is based on Airy s theory, in which the horizonal wave speed and acceleration on any underwater level are related to the elevation of the water surface via the wave length and the water depth. The wave length is assumed to be related to the wave 12 ECN-E

23 2 TURBINE SYSTEMS frequency via the dispersion relation, so that the wave speeds and accelerations only have a frequency-dependency on the surface elevation, which is purely deterministic. Time-domain realisations of the wave speeds and accelerations in the center points of the underwater tower elements are obtained by Fourier synthesis of the weighted issues of the water surface elevation spectrum. The weighting consists of frequency-dependent multiplication, which depends on the specific location of an element s center point. As mentioned before in the paragraph on hydrodynamic conversion, the weighting function that is used for the acceleration realisations is extended with the McCamy-Fuchs correction factor. This correction factor adapts the inertia coefficent in the hydrodynamic load model to the rate of diffraction [14]. Electric grid rigidity The frequency and voltage of the grid is assumed to be constant. Therfore dynamic behaviour (grid faults, grid variations, transients) are left out of consideration) Gravity The gravitation yields periodic gravity coordinates in the rotating coordinate systems of the rotor blades and the drive-train. These gravity coordinates have a sine/cosine-dependency on the rotor azimut angle. For the rotor blades, the rotor azimut is shifted over the azimuthal separation between the rotor blade axes and the azimut reference axis. ECN-E

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25 3 PERIPHERAL DEVICES In this chapter, simulation blocks for the peripheral devices are dealt with. These blocks realise control actions and power losses, governed from the control system and the turbine dynamic systems respectively. Their implementation heavily depends on the specific manufacturing concepts. Nevertheless, basic building blocks are supplied for pumps, motors and friction generating devices 3.1. The servo and power loss behaviour of the gearbox system (3.2), generator system (3.3), yaw system (3.4), pitch system (3.6), brake system (3.5) are implemented in the peripheral devices. 3.1 General systems Induction motor Many turbine systems contains a drive system to drive a pump, fan or to rotate a mechanical device. In most cases an industrial drive is an standard induction motor. The dynamics of this machine are quite fast in relation to the driven process, therefore a high order model is of less interest for real time process simulation. Additionally, the parameters for such a model are difficult to obtain. In this paragraph a direct grid connected induction motor model is derived. Because of it s dependency of terminal voltage and frequency, it is also possible to use it for frequency controlled motors (voltage and frequency control) or soft started motors. The model is derived from stationary equations and it s parameters are calculated by using nameplate and catalogue data only. The model takes two speed directions into account and also the specific high starting torque behaviour (due to it s rotor construction). Regeneration is left out of consideration. For actual values of the supply voltage U s and frequency ω s and shaft speed Ω, the model calculates the motor torque, T motor current, I s, the power losses Q s. The stationary torque-speed relationship of an induction machine is described by the formula of Kloss [19] T (U s, ω s ) = K t U s 2 s s ωs 2 k s 2 + s 2 (3.1) k where, K t is a motor torque constant, U s is the terminal voltage, ω s is the radian terminal frequency. The slip, s, is defined as the difference between the synchrouneous speed Ω syn and the shaft speed Ω relative to the synchrouneous speed. s = Ω syn Ω Ω syn (3.2) The synchroneous speed depends of the motor construction (e.g. for a 50Hz 2-pole machine 3000rpm, 4-pole machine 1500rpm) and at this speed the slip becomes zero. The slip at which maximum torque occurs, is named s k. From Kloss formula, K t and s k can be expressed relative to (known) rated values K t = 2 ω2 rat U 2 rat s k = s k,rat ωrat ω s k,rat = s rat µ (1 + µ 2 1) (3.3) in which µ is defined as the ratio between maximum torque and rated torque at rated motor supply. From the complex stator current [19] the stator current I s and magnetising current I m ECN-E

26 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems can be derived (length of complex vector): I s (U s, ω s ) = I s m σ 2 +σ 2 s 2 k s 2 +s 2 k (3.4) I m (U s, ω s ) = U s ω s L s In eq.(3.4), both the the stator induction, L s, and the scattering factor σ are unknown quantities. The scattering factor σ can be calculated from the working factor cosϕ s at rated conditions. The working factor cosϕ s can be derived from the complex stator vector by division of its real part and length: s rat s k,rat (1 σ) cosϕ s = (s 2 rat + s2 k,rat )(s2 rat + σ2 s 2 k,rat ) (3.5) From eq.(3.5), the scattering factor σ can then be isolated: σ = 1 s2 rat +s2 k,rat s 1 (cosϕ s,rat ) 2 cosϕ s,rat rat s k,rat (3.6) 1 s2 rat +s2 k,rat (cosϕ s 2 s,rat ) 2 rat The stator induction, L s then follows from eq.(3.4) at rated conditions: s 2 I m,rat = σ I s,rat rat +s 2 k,rat s 2 rat +σ2 s 2 k,rat (3.7) U L s = s,rat I m,rat σ ω s,rat Kloss formula neglects the rotor skin effect which is responsable for high starting torques and accompanying high starting currents. For these effects an additive correction between s=1 and s=s k has been calculated, that causes an exact fit for both torque and current at standstill. T x = K t K tx U s 2 (s s ωs 2 k ) αt I x = I m K ix (s s k ) α i (3.8) The parameters K tx, K ix and α t, α i are determined such that an exact fit is caused at standstill (s=1). The power losses of the motor (due to slip) can be significant. These losses can be calculated from the difference between extracted grid power and mechanical shaft power. Finally, the motor shaft speed is dynamically calculated from eq.(3.9): Ω = 1 (T J motor T load )dt (3.9) shaft The inertia J shaft is the total inertia of both the motor and load. The load torque T load is the total torque as caused by the load (e.g. pump) and the speed dependent viscous losses. The described motor model can also be used to compose a two-speed motor, using tow models and a switching device. Fig.(3.1) shows the stationary charactersitics of an induction motor: Centrifugal pump Turbines contains many pumps for system lubrication and cooling purposes. The stationary model of a centrifugal pump is based on the non lineair pressure-flow characteristic, and its 16 ECN-E

27 3 PERIPHERAL DEVICES Src:E:\vdhooft\Projects\PrSimDev\v1\ps\brkDrvCurve1.ps; :06:24 Motor torque (o=nom) Motor Current (o=nom) Torque [*Tnom] Motor Speed [rpm] Current[*Inom] Motor Speed [rpm] 2 Motor Loss (o=nom) 0.8 Motor CosPhi (o=nom) PowLoss [*Pnom] CosPhi [ ] Motor Speed [rpm] Motor Speed [rpm] Figure 3.1: Stationary characteristics of induction motor speed dependency. The required parameters can then determined easily from typical pump data. A parabolic relationship between pressure and flow is a conservative assumption for pressure [20]. Furthermore, the speed dependency has been incorporated such that the pressure changes in a square way with the speed and the flow in a proportional way. The pump pressure p pmp is then written as: p pmp (Ω pmp, φ pmp ) = K 1 φ 2 pmp + K 2 Ω 2 pmp Ω 2 pmp,rat (3.10) For rated conditions (Ω pmp =Ω pmp,rat ), the pump constants K 1 and K 2 can be calculated at two typical points of the parabolic characteristic: (p pmp,rat,φ rat pmp) and (0,φ max pmp). The latter implies a theoretical/virtual point at which maximum flow occurs without remaining pressure. K 1 = p pmp,rat φ 2 pmp,rat φ2 pmp,max K 2 = K 1 φ 2 pmp,max (3.11) At different pump speeds the working point of the pump depends of the system behaviour, in general the system has a square relationship between pressure and flow (hydraulic system resistance). The required electric power P pmp Ppmp rat = prat pmp φ rat pmp (3.12) η drv ECN-E

28 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems should be deliverd by the electric pump drive. Here the efficiency η drv includes the viscous friction of both the motor and pump. The equilibrium speed at which the pump generates rated pressure and flow is found on the non linear power curve of the induction motor. The implementation of the model is based on flow generation (output) dependent of the actual system pressure (input) and pump shaft speed (input) (fig.(3.2). Figure 3.2: Simulink implementation of pump model Fig.(3.3) shows a result of a pump characteristic driven by a two speed motor (gearbox oil pump). Src:E:\vdhooft\Projects\PrSimDev\v1\ps\GbxPmpCurve.ps; :06:37 40 Stat. Flow Pressure (r:system, b:pump)( :rat,.:min, :max) 6 Pump power, loss, total (.,, ) 35 Power [kw] Oil pressure [bar] Torque [Nm] Pump torque, loss, total (.,, ), Motor (Lo Hi), (r) Motor current (Lo Hi) Oil flow [lit/min] Current [A] Shaft Speed [rpm] Figure 3.3: Stationary characteristics of pump and two speed pump drive 18 ECN-E

29 3 PERIPHERAL DEVICES Coulomb friction Coulomb or dry friction is a discontinuous phenomenom around standstill [4]. The mechanical brake uses dry friction to keep the turbine rotor in standstill position and the yaw brakes use it to keep the nacelle in position. The value of Coulomb friction torque is always opposite to the driven torque. This constant (speed independent) friction is due to rough surfaces and depends on the direction of motion. At standstill there s also another related phenomenon: stick friction. Stick friction is extra friction which should be overcome first to start motion. It is caused by sticking due to mechanical pressure after long time of standstill. The model is described by transitions between three speed states: negative, zero and positive. When zero speed, the friction equals the driving torque and it becomes constant when the stick friction is exceeded. When the driving torque exceeds the stick friction level (StckFric), the friction immediately decreases to the coulomb friction level (ClmbFric). For numerical implementation reason (hysterisis) the speed has been quantisised before the state is determined for use in the state machine, fig.(3.4). Stick friction is defined as a fraction of Coulomb friction. Figure 3.4: Simulink implementation of Coulomb friction ECN-E

30 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems 3.2 Gearbox system (GBX) Description The primary function of a gearbox concerns mechanical conversion from low speed, high torque at the slow speed shaft towards low torque, high speed at the fast speed shaft. Good operation of the gearbox between its design limits, requires proper oil circulation to achieve cooling (or heating) and lubrication of the internal gears, shaft and bearings ( mesh ). Lubrication causes minimisation of gear wearing and reduction of friction losses, while the same oil flow transports the unavoidable losses towards the cooling system. The mechanical losses in a gearbox are classified in speed dependent losses and load dependent losses. Speed dependent losses are mainly caused by windage friction on the gears due to turbulent oil mist of gears and bearings ( windage losses ) and oil fling off due to rotating gears ( churning losses ). Load dependent losses are mainly caused by friction of the gear tooths ( sliding friction ) and entrainment and compression of the oil ( rolling friction ). The mechanical losses of the internal and external bearings of the gears are speed ( windage ) and load dependent ( sliding and rolling ) too. The efficiency of spur-gear-systems at partial and full load have been investigated experimentally by [6] and described in [5]. Fig.(3.5) shows a significant dependancy of power losses of speed and power. At low speed the sliding loss Figure 3.5: Spur gear system power losses breakdown for part and full operation [6] accounts for most of the system loss. At higher speeds, the sliding loss becomes less important, and the rolling loss and bearing losses become significant. Sliding loss is most important at high torque levels. Windage losses are less important, but at high speed they also contribute to the total loss. To achieve optimal lubrication and heat transfer of the mesh, cooled oil is splashed via an oil labyrint over the gears at a certain pressure. The hot oil is collected in the oil sump of the gearbox. The oil flow is realised by an (speed controlled) oil pump which sucks the hot oil and presses it through a heat exchanger, which extracts heat to a secundary (water) system. Besides the gearbox and the heat exchanger, oil filtering equipment and valves causes hydraulic 20 ECN-E

31 3 PERIPHERAL DEVICES resistances in the oil circulation system Definition The gearbox cooling and lubrication system is shown in fig.(3.6). Sec. circuit Prim. circuit Heat exchanger ThermoValve Gearbox Filter M Pump Sump Figure 3.6: Gearbox cooling and lubrication system The following behaviour, subsystems and components are included: gearbox losses (par ) speed and load dependent mesh power losses speed dependent front and rear bearing power losses gearbox torque loss pump drive (par ) two speed controlled induction motor mechanical rotation pump speed and motor current pump and drive power losses oil circulation (par ) oil pump oil flow circuit ECN-E

32 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems oil filter pollution thermostatic bypass valve oil pressure at gearbox oil inlet heat exchanger (par ) heat exchange between oil (gearbox) and water (generator) temperature of in-/outlet oil and water Obviously, the gearbox system interacts with other turbine systems fig.(3.7) Because the tem- Figure 3.7: Interactions of gearbox cooling and lubrication system perature of the cooling water, (θ wu, θ wy ), is heated up by the oil in the heat exchanger and the water flow, φ w, is caused by the water pump, the gearbox oil cooling system interacts with the generator water cooling system (see par. 3.3). The power disspation losses are dependent of the actual slow shaft torque, T gbx, and slow shaft speed, Ω gbx, as calculated in the structural dynamic system. The oil cooling capacity is related to the gearbox inlet oil temperature θ ou and therefore interacts with the gearbox thermic model (see par ). The turbine control system commands at least the pump speed of the two speed drive, while several detections and measurements will be used for monitoring and control, e.g. oil pressure, oil flow, pump speed, valve position and gearbox temperatures. Power (400V/50Hz) is supplied for the two-speed oil pump drive. A detailed Simulink implementation is shown in fig.(3.8) Gearbox losses The gearbox losses are acting as a heat power source for the gearbox thermic model. Because the dynamics of this model are rather slow, it is sufficient to compose a stationary loss model for the operating torque and speed range of the gearbox. The model has been derived from investigations as described in [6], it was concluded that the power loss percentage of a single spur-gear-system relative to the full load power loss, depends of torque and speed as illustrated in fig.(3.9) Although, a wind turbine gearbox consists of multiple stages which can be planetary, normalisation of the curves in fig.(3.9) will be a suitable base for modelling the gearbox losses. A 2D torque loss percentage table has been composed from fig.(3.9). The table elements are dependent of speed and torque relative to the gearbox design speed and torque, which are usually 22 ECN-E

33 3 PERIPHERAL DEVICES Figure 3.8: Simulink implementation of gearbox cooling and lubrication system overdimensioned in practice. For this reason the table is scaled such that the rated gearbox efficiency, matches at rated turbine speed and torque. Additionally, the rated gearbox torque loss is defined as: Tgbx,loss rat = Cgbx T Tgbx rat + Cgbx Ω Ω rat gbx (3.13) All speed and torque quantities are related to the rotor side of the gearbox (slow shaft equivalent). The overall speed and torque dependent gearbox losses then become: ( ) T gbx,loss (Ω gbx, T gbx ) = Fgbx loss (Ω rel gbx, Tgbx) rel Cgbx T Tgbx rat + Cgbx Ω Ω gbx In eq.(3.14), F loss gbx (Ωrel gbx, T rel gbx ) is the loss table, containing loss percentages, which are depen- and speed Ωrel gbx : dent of the rated relative values of the gearbox torque T rel gbx (3.14) T rel gbx = T gbx /T rat gbx (3.15) Ω rel gbx = Ω gbx /Ω rat gbx (3.16) ECN-E

34 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 3.9: Percentage of spur-gear-system power loss at par-load conditions relative to fullload power loss [Anderson,1980]. The total gearbox torque losses T gbx,loss consist of the mesh loss, the slow shaft bearing loss and the fast shaft bearing loss. The latter two were assumed to be speed dependent only. T gbx,loss (Ω gbx, T gbx ) = T mesh,loss (Ω gbx, T gbx ) + T ssb,loss (Ω gbx ) + T fsb,loss (Ω gbx ) T ssb,loss (Ω gbx ) = C Ω ssb Ω gbx (3.17) T fsb,loss (Ω gbx ) = C Ω fsb Ω gbx Gearbox power losses Q gbx,loss are easily determined by multiplying torque losses with the gearbox speed Ω gbx (slow shaft equivalent). The gearbox loss model requires four parameters. The loss table Fgbx loss has been read from fig.(3.9) and scaled for gearbox design speed and torque. The gearbox torque loss constant Cgbx T, should be estimated as a fraction relative to the rated gearbox torque T rat gbx. The gearbox torque loss constant Cgbx Ω can then be derived when eq.(3.13) is set equal to (1 ηrat gbx ) T gbs rat, where ηgbx rat is the gearbox efficiency at rated turbine operation. Cgbx Ω = T gbx rat ( ) Ω rat 1 ηgbx rat Cgbx T gbx (3.18) The bearing loss gains Cbrg,fss Ω, CΩ Trat brg,sss, should be estimated as fractions, f relative to the total gearbox losses at rated Tgbx,loss rat. gbx,loss brg,fss, f Trat gbx,loss brg,fss C Ω brg,fss = C Ω brg,sss = ( ( 1 η rat gbx 1 η rat gbx ) ) T rat gbx Ω rat gbx T rat gbx Ω rat gbx f Trat gbx,loss brg,fss (3.19) f Trat gbx,loss brg,sss 24 ECN-E

35 3 PERIPHERAL DEVICES Fig.(3.10) shows the design results for where the gearbox design speed and torque amounts 1.5 and 1.25 times the rated speed respectively. The gearbox efficiency was here set to 96%. It Src:E:\vdhooft\Projects\PrSimDev\v1\ps\GbxLossTq.ps; :29:19 Tq=c; <b r c m k> Tq=c; <b r c m k> Tq=c; <b r c> [ ]*T rat [ ]*T rat x [SSS,FSS,SSS+FSS] T loss,total [*Trat] T loss,bearing [*Trat] Tq=c; <b r c m k> [ ]*T rat Tq=c; <b r c> 6 x [SSS,FSS,SSS+FSS] 10 3 Efficiency [ P loss,total [*Prat] P loss,bearing [*Prat] Speed [*N rated ] Speed [*N rated ] Speed [*N rated ] Figure 3.10: Gearbox torque (two upper left)losses, power losses (two lower left) and efficiency (right), for torque levels [0.1,0.25,0.5,1.0,1.25] Tgbx rat and speed range [0..1.5] Ωrat gbx. The bearing losses are shown separately at the right side of the left plots shows that the total torque and power losses increase with torque and speed, or, the efficiency becomes worse when when operating at part load. At rated torque and speed, the efficiency equals the rated efficiency (here 96%). Fig.(3.11) shows the Simulink implementation of the gearbox loss model as previously described. The input signals are actual gearbox torque and speed as generated from the structural dynamic model. The output signals, gearbox torque and power losses, are sent to the structural dynamic model and the thermic model respectively Gearbox oil pump drive The gearbox oil pump is driven by a two speed induction motor. Dependent of the actual torque T sys (eq.(3.20)) the stationary motor model as described in par results in a pump speed. The electric drive is commanded by voltage supply and frequency and a high/low speed control command Gearbox oil circulation The gearbox oil circulation model is shown in fig.(3.13). It consists of an oil pump, the system load and a thermostatic valve. Fig.(3.12), shows the interaction between these components and the interface with other (gearbox) systems. The additional block oilheat represent the the calculation of actual heat power of the oil flow (see eq.(3.27) which is sent to gearbox heat ECN-E

36 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 3.11: Simulink model of gearbox power and torque losses exchanger (par ) and the thermic gearbox model. The gearbox oil sump temperature θ gbx,out = θ vlv,in and the heat exchanger outlet temperature θ hex,out. The pump drive speed n pmp is generated by the pump drive model. The required drive torque of the oil pump is calculated (eq.(3.20)) in trqcalc and used to load the pump drive (par ). T sys = p pmp φ pmp ω pmp (3.20) Oil pump Dependent of the actual pump pressure p pmp and shaft speed, n pmp, the pump generates an oil flow φ pmp through the system. The stationary pump model has been described in par System load A linear model of the system load is shown in fig.(3.6). The system working point (p rat at n rat pmp, is determined by the total hydraulic system resistance, R sys : pmp,φ rat pmp) R sys = R rat filt (1 + µ filt ) + R vlv + R hex δ vlv + R gbx (3.21) 26 ECN-E

37 3 PERIPHERAL DEVICES Figure 3.12: Simulink model of gearbox oil circulation system φ pmp φ sys R filt R vlv R hex φ c p filt p vlv p hex p pmp C sys p gbx R gbx n pmp Figure 3.13: Linear model of gearbox oil system load In eq.(3.21), the hydraulic filter resistance is dependent of the pollution degree, µ filt of the filter and the heat exchanger resistance depends on the position of the thermostatic bypass valve, δ vlv. Following the modelling approach of par and using a pressure capacitor C sys which represents the (first order) pressure dynamics of the system, the oil ciculation system can be decribed as: 1 ( ) p pmp = φ pmp φ sys φ 0 pmp dt p 0 pmp C sys p pmp = φ sys R sys (3.22) ( ( ( ))) p φ 0 pmp = max 0, φ rat sys 1 p0 The static system counter pressure (pre-pressure) has been incorporated as a non linear element φ 0 pmp, which causes full leakage at zero pressure and no leakage above p pmp,0. The value of the hydraulic system resistance, R sys is determined from a square relationship between the system pressure and the system flow at the rated operation point (p rat pmp,φ rat pmp). At rated it is assumed that the oil flow passes the heat exchanger and the filter is not polluted. R rat sys = p rat pmp/φ rat pmp (3.23) ECN-E

38 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Based on estimated fractions, f Rrat sys filt, f Rrat sys vlv, f Rrat sys hex, the different component resistances can be calculated: R filt = f Rrat sys filt Rsys rat R vlv = f Rrat sys vlv Rsys rat R hex = f Rrat sys hex Rrat sys (3.24) R gbx = Rsys rat Rfilt rat Rvlv rat Rhex rat The system capacity, C sys is rather small, because of the open system (atmospheric pressure in gearbox). A time constant, τ sys, is used to determine its value: C sys = τ sys /R rat sys (3.25) Thermostatic bypass valve Dependent of the oil inlet temperature, θ vlv,in, the thermostatic bypass valve directs the oil either via the heat exchanger (S vlv = open ) or to the gearbox (S vlv = closed ). A valve transition ( open / close ) will take some time, τ vlv and the resulting the oil temperature θ vlv,out will be a temporarily weighted average of the heat exchanger outlet temperature θ hex,out and θ vlv,in. Therefore, the valve is modelled as a dynamic non linear element, consisting of an output limited integrator (open/close), which is driven by the result of a temperature hysteresis, δ θvlv,in, to avoid shuttling. S vlv = 1 τ vlv δ θvlv,in = δ θvlv,in dt ] , if : θ vlv,in > θ sw vlv + θhys vlv 1, if : θ vlv,in < θ sw vlv θhys vlv θ vlv,out = S vlv θ hex,out + (1 S vlv ) θ vlv,in (3.26) The value of the valve position S vlv is limited between -1 ( closed ) and +1 ( open ) and the temperature switch level is defined by θvlv sw and the hysteresis width θhys vlv. The values of the time constant τ vlv, can be estimated Gearbox heat exchanger The gearbox oil (primary medium) exchanges heat with the generator cooling water (secundary medium), see fig.(3.14)). Using the simple modelling approach of par , this device comprises two energy storages (thermic capacitors, C 1,C 2 ), which exchange heat to each other, G 12,G 21 (heat transfers) and to their surroundings G 1s,G 2s (heat leakages). Both heat storages are charged or decharged by the heat power of their flow φ f : Q 1φ = ρ 1 c 1 φ 1 (θ 1y θ 1u ) Q 2φ = ρ 2 c 2 φ 2 (θ 2u θ 2y ) (3.27) In eq.(3.27), c and ρ are the specific heat and density of the medium, respectively. The different temperature signs at the primary and secundary side imply heat transfer from primary to secundary side, although this is arbitrary. The heat exchanger model then results in: θ 1y = 1 C 1 ( Q 1φ Q 1s Q 12 ) dt + θ 0 1y θ 2y = 1 ( ) Q C 12 Q 2s Q 2φ dt + θ2y 0 (3.28) 2 28 ECN-E

39 3 PERIPHERAL DEVICES φ 1 θ 1u φ 2 θ 2u G 12 C 2 θ s G 1s G 2s θ s C 1 G 21 θ 1y θ 2y Figure 3.14: Gearbox heat exchanger ) Q 1s = G 1s (θ 1 θ s ) Q 2s = G 2s (θ 2 θ s ) Q 12 = G 12 (θ 1 θ 2 (3.29) where, θ 1 and θ 2 are the average inlet/outlet temperatures of the primary and secundary medium, respectively. The values of the thermic conductivities are determined from the equilibrium state at rated conditions. This implies: Q rat 1φ Q rat 1s Q rat 12 = 0 Q rat 12 Q rat 2s Q rat 2φ = 0 (3.30) If the heat leakage to the surrounding at rated is assumed to be a fraction of the heat flow transfer: Q rat 1s = f Qrat 1φ Q rat 1s 2s = f Qrat 2φ 2s Q rat 1φ Q rat 2φ, (3.31) the secundary temperature difference between inlet and outlet can be expressed as a fraction of the primary temperature difference by using eq.(3.27) and eq.(3.30): ) ( ) c 1 ρ 1 φ rat θ2y rat θ2u rat 1 (1 f Qrat 1φ 1s ( ) = ) θ1u rat θ1y rat (3.32) If the rated temperatures,θ1u rat, θrat 1y, θrat 2u c 2 ρ 2 φ rat 2 (1 + f Qrat 2φ 2s and the rated flows,φrat 1, φrat 2 are known, the secundary outlet temperature θ2y rat can be calculated as a consequence. The heat conductivities G 1s,G 2s,G 12 can now be calculated by using using eq.(3.30) through eq.(3.31) G 12 = Q rat 1φ (1 f Qrat 1φ 1s θ1 rat θ2 rat G 1s = Qrat 1φ f Qrat 1φ 1s θ1 rat θs rat G 2s = Qrat 2φ f Qrat 2φ 2s θ2 rat θs rat ) = Q rat 2φ (1 + f Qrat 2φ 2s θ1 rat θ2 rat ) (3.33) (3.34) (3.35) ECN-E

40 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems The values of the thermic capacitors C 1 and C 2 can be calculated from their physical volumes V of both thermic storages : C 1 = c 1 ρ 1 V 1 C 2 = c 2 ρ 2 V 2 (3.36) Fig.(3.15) shows the Simulink implementation of the gearbox heat exchanger model. The input signals are the actual the oil (primary) and water (secundary) inlet temperatures, the surrounding temperature as generated by the thermic model, the oil flow heat power as determined in the oil circulation model and the water flow as generated by the generator system. The output signals, the actual outlet temperatures are sent to the oil and water circulation models, while heat leakage to the surrounding is sent to the thermic model. The primary and secundary side are similar and Figure 3.15: Simulink model of gearbox heat exchanger shown in fig.(3.16) and fig.(3.17), respectively Figure 3.16: Simulink model of primary (oil) side of the gearbox heat exchanger 30 ECN-E

41 3 PERIPHERAL DEVICES Figure 3.17: Simulink model of the secundary side of the gearbox heat exchanger ECN-E

42 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems 3.3 Generator system (GEN) Description The primary function of a generator concerns conversion of mechanical power to electrical power. Due to conversion losses, the rotor and stator of the generator will heat up. Therefore, a water cooling system is installed to extract and transport the dissipation losses from the generator to a radiator on the nacelle roof. The dissipation losses in the generator are classified in stator/rotor copper losses and stator/rotor iron losses, windage loss and bearing friction losses. For a well designed generator (no saturation), the iron losses are much smaller than the copper losses and therefore neglected. The windage and bearing losses are speed dependent and the dissipated copper losses in each phase are proportional with the square of the phase current. Besides self ventilation of the air gap, the stator phases of the generator are water cooled. The cooling water transfers the generator heat to an air cooled radiator on the roof of the nacelle, The radiator transfers the heat to the surrounding. An electric driven water pump supplies the cooled water to the gearbox heat exchanger before it enters the generator inlet (see par. 3.2) Definition The generator cooling system is shown in fig.(3.18). Generator stator Sec. circuit (oil) Prim. circuit (water) Prim. circuit (water) Sec. circuit (air) M Pump Pressure holder Figure 3.18: Generator cooling system The following behaviour, subsystems and components are included: generator losses (par ) speed dependent windage loss speed dependent front and rear bearing power losses 32 ECN-E

43 3 PERIPHERAL DEVICES generator stator copper losses generator rotorr copper losses pump drive (par ) constant speed controlled induction motor mechanical rotation pump speed and motor current pump and drive power losses water circulation (par ) water pump water flow circuit water pressure at pump outlet radiator (par ) heat exchange between water (generator) and surrouding temperature of radiator outlet water The oil/water heat exchanger is not part of the generator system but it was modelled in the gearbox system (par ) Obviously, the generator system interacts with other turbine systems fig.(3.19). Because the Figure 3.19: Interactions of generator cooling system inlet temperature of the generator cooling water, (θ wu, θ wy ), is heated up by the oil in the gearbox heat exchanger and the water flow φ w of the water pump, the generator water cooling system interacts with the gearbox oil cooling system (see par. 3.2). The power dissipation losses are dependent of the generator speed, Ω gen, (Structural dynamics system) and the stator currents (Electric conversion). The water cooling capacity of the radiator is related to the generator outlet water temperature θ wy and therefore interacts with the generator thermic model (par ). Several detections and measurements will be used for monitoring and control, e.g. water pressure, water flow, pump speed and water temperatures. Power (400V/50Hz is supplied for the water pump drive. A detailed Simulink implementation is shown in fig.(3.20) ECN-E

44 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 3.20: Simulink implementation of generator cooling system Generator losses The generator power losses are acting as a heat power source for the generator thermic model (par Because the dynamics of this model are rather slow, it is sufficient to compose a stationary loss model for the operating speed Ω gen and phase currents I s,ph, I r,ph of the generator. Q RCu = 3 Ir,ph 2 Rph r Q SCu = 3 Is,ph 2 Rph s Q BrgF = C BrgF Ω gen (3.37) Q BrgR = C BrgR Ω gen 34 ECN-E

45 3 PERIPHERAL DEVICES The windage loss do not contribute to heat dissipation but cause air flow. It has a centrifugal nature and is relevant for mechanical torque loss only. T gen,loss = T BrgF + T BrgR + T vent T vent = C vent Ω gen Ω gen Ω rat gen (3.38) The loss constants C BrgF, C BrgR and C vent are calculated from the rated generator efficiency ηgen rat and rated mechanical shaft power Q rat gen,mech. The loss fraction of the bearings related to the rated mechanical losses f Trat gen,loss BrgF, f Trat gen,loss BrgR and the electrical resistances Rr ph, Rs ph are assumed to be known. Q rat gen,loss = Q rat gen,mech (1 ηrat gen) Q rat gen,mloss = Q rat gen,loss Qrat RCu Qrat SCu TBrgF,loss rat = (Q rat gen,mloss /Ωrat gen) f Trat gen,loss BrgF TBrgR,loss rat = (Q rat gen,mloss /Ωrat gen) f Trat gen,loss BrgR T rat vent,loss = (Q rat gen,mloss /Ωrat gen) T rat BrgF,loss T rat BrgR,loss C BrgF = T rat BrgF,loss /Ωrat gen C BrgR = T rat BrgR,loss /Ωrat gen C vent = T rat vent,loss /Ωrat gen (3.39) Fig.(3.21) shows the Simulink implementation of the generator losses. The input signals are generator speed from the structural model and the generator phase currents form the electric conversion system Generator water pump drive The generator water pump is driven by an induction motor. Dependent of the actual torque T sys the stationary drive model as described in par results in a pump speed. The electric drive is commanded by voltage supply and frequency Generator water circulation The generator water circulation model is shown in fig.(3.18). It consists of an oil pump and the system load. Fig.(3.22), shows the interaction between these components and the interface with other (generator) systems. Water pump Dependent of the actual pump pressure p pmp and shaft speed, n pmp, the pump generates an water flow φ pmp through the system. The stationary pump model has been described in par System load A linear model of the system load is shown in fig.(3.23). ECN-E

46 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 3.21: Simulink implementation of generator power and torque losses The system working point (p rat resistance, R sys : pmp,φ rat pmp) at n rat pmp, is determined by the total hydraulic system R sys = R hex + R rad + R gen (3.40) which consists of three hydraulic filter resistances: the water side of the gearbox heat exchanger, the radiator and the generator cooling tubes. Following the modelling approach of par and using a pressure capacitor C sys which represents the (first order) pressure dynamics of the system, the water circulation system can be decribed as: 1 ( ) p pmp = C φ sys pmp φ sys φ 0 pmp dt p 0 pmp p pmp = φ sys R sys φ 0 pmp = max ( 0, φ rat sys ( ( ))) 1 p p0 (3.41) The static system counter pressure (pre-pressure) has been incorporated as a non linear element φ 0 pmp, which causes full leakage at zero pressure and no leakage above p pmp,0. The value of the hydraulic system resistance, R sys is determined from a square relationship 36 ECN-E

47 3 PERIPHERAL DEVICES Figure 3.22: Simulink model of generator water circulation system φ pmp φ sys R hex R rad φ c p hex p rad p pmp C sys p gen R gen n pmp Figure 3.23: Linear model of generator water system load between the system pressure and the system flow at the rated operation point (p rat pmp,φ rat pmp) R rat sys = p rat pmp/φ rat pmp (3.42) Based on estimated fractions, f Rrat sys hex, f Rrat sys rad, the different component resistances can be calculated: R hex = f Rrat sys hex Rrat sys R rad = f Rrat sys rad Rrat sys (3.43) R gen = Rsys rat Rhex rat Rrat rad The system capacity, C sys is rather high, because of the pressure accumulator. A time constant, τ sys, is used to determine its value: C sys = τ sys /R rat sys (3.44) Generator radiator The generator water (primary medium) exchanges heat with the air at the roof of the nacelle (secundary medium), see fig.(3.24)). Using the simple modelling approach of par , this ECN-E

48 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems φ 1 θ 1u G 1s K 1s V w θ s C 1 θ 1y Figure 3.24: Generator radiator device comprises only one energy storage (thermic capacitor, C 1 ), which exchange heat to the surrounding G 1s, K 1s V w. The thermic capacitor is charged or decharged by the heat power of water flow φ 1 : Q 1φ = ρ 1 c 1 φ 1 (θ 1y θ 1u ) (3.45) In eq.(3.45), c and ρ are the specific heat and density of the medium (water), respectively. The heat exchanger model then results in: θ 1y = 1 ( ) C Q 1 1φ Q 1s dt + θ1y 0 ) ) Q 1s = (G 1s + K 1s V w,roof (θ 1 θ s (3.46) where, θ 1 is the average inlet/outlet temperatures of the water. The values of the thermic conductivities are determined from the equilibrium state at rated condition: Q rat 1φ = Q rat 1s (3.47) If f Qrat 1φ 1s is defined as a assumed fraction due to heat radiation (independent of the wind speed) of the radiator and the rated temperatures θ1 rat, θrat s and flow φ rat 1 are also known, the thermic conductivities can be calculated: G 1s = f Q 1s rat 1φ Q rat 1φ θ1 rat θrat s K 1s = (1 f Q 1s rat 1φ θ1 rat θrat s ) Q rat 1φ V rat w,roof (3.48) The values of the thermic capacitor C 1 is calculated from its physical volume V of the thermic storage : C 1 = c 1 ρ 1 V 1 (3.49) (3.50) Fig.(3.25) shows the Simulink implementation of the generator radiator model. The input signals are the actual water (primary) and the surrounding temperature as generated by the thermic model and the water flow heat power as determined in the water circulation model. The water outlet temperature is sent to the water circulation model. 38 ECN-E

49 3 PERIPHERAL DEVICES Figure 3.25: Simulink model of generator radiator ECN-E

50 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems 3.4 Yaw system (YAW) Description The nacelle and rotor are aligned to the wind direction with the yaw sytem. Therefore, the nacelle is connected to the tower structure by a yaw bearing, which enables the possibility to rotate the nacelle with respect ot the fixed tower. An active but simple on/off controlled drive mechanism is commanded from the turbine control system. Often, multiple electric drives are applied (6-8), each mechanically connected via a yaw drive gearbox and a pinion gear to the gear ring. The gear ring is bolted to the tower wall, usually inside, and the driveframes are fixed connect to the nacelle frame. The turbine controller compares the deviation (yaw misalignment) between the wind direction, as measured on the top of the nacelle, and the fixed mounted zero degree wind turbine axis. Dependent of the yaw misalignment the yaw drives are commanded to turn the nacelle in clockwise and counter-clockwise direction towards its alligned position. Because of the heavy construction yawing is usually a quiet process (yaw speed) to avoid additional loading due to acceleration/decelleration and tilting. To keep the nacelle at its aligned position, a number of (hydraulic) yaw brakes clamp the nacelle to the brake disc, which is part of the tower top flange. The yaw brakes are also commanded by the turbine controller, during yawing they can be full or partly released to realise smooth motion (friction damping). To prevent for cable twist, the amount of cable turns due to yawing are counted. When the number of cable turns become are too many, the turbine control system will determine a good opportunity (windspeed, production state) and command the yaw system to untwist the power cable Definition The yaw system is shown in fig.(3.26). It consists of a oil supply unit to operate the yaw brake and multiple yaw drive units, only one is drawn. The hydraulic system is a pressure system, in case of no oil pressure, the yaw brake is released. The pressureholder or accumulator is a passive device for maintaining constant pressure at the system side. The following behaviour, subsystems and components are included: Yaw brake oil pressure (par ) pump drive * grid connected induction motor * mechanical rotation * shaft speed and motor current * pump and drive power losses oil circulation * oil pump * oil flow circuit * unidirectional valve * control valve * overpressure valve * oil pressure and flow at pump and brake cylinder Yaw nacelle (par ) 40 ECN-E

51 3 PERIPHERAL DEVICES Gear ring M Yaw Motor Yaw gearbox Tower wall Pinion gear pressure holder piston Brake disc Brake caliper Brake cylinder Unidirectional valve 1 yaw drive system M E-motor Pump Overpressure valve oil tank Control valve Figure 3.26: Yaw system yaw mechanics * multiple yaw brakes * brake friction torque and power * nonlinear coulomb and stick friction yaw drives * multiple grid connected induction motors * shaft speed and motor currents * motor yawing torque * viscous torque and power losses Yaw sensors (par ) The cable untwist subsystem is left out of consideration. Obviously, the yaw system interacts with other turbine systems fig.(3.27) The total yawing torque T yaw,ctrl to the structural dynamic system. is calculated from yaw load torque, T yaw,load, as calculated in the structural dynamic system and the nett driving yawing torque. The latter is generated by the yaw motors taking the Coulomb counter torque loss into account. This friction requires the actual angular yawing speed Ω yaw. Both the nacelle position (with respect to fixed reference) and the wind direction (wind vane) are measured tot determine the misalignment angle for control purposes (clockwise=positive direction). The yaw motors are on/off and CW/CCW commanded by supply voltage contactors. The turbine control system commands the yaw motor contactors, oil pump running and yaw brake activation, while several detections and measurements will be used for monitoring and control of the oil pump (e.g. oil system pressure, oil flow, pump speed, brake valve position.) and yaw motors (e.g. motor speed, motor current, yaw rate, yaw angle). A more detailed Simulink implementation is shown in fig.(3.28) ECN-E

52 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 3.27: Interactions of yaw system Figure 3.28: Simulink implementation of yaw system Yaw brake oil pressure The yaw brake oil pressure is generated by an oil circulation system which is driven by an oil pump system. Yaw oil pump drive The pump of the yaw brake system is driven by a on/off controlled grid connected induction motor. Dependent of the actual system torque, T sys, the stationary drive model as described in par results in a pump speed. The electric drive is commanded by constant voltage and frequency supply. The on/off switch command depends on the actual system pressure p sys level 42 ECN-E

53 3 PERIPHERAL DEVICES with repect ot an upper and lower level limit. Yaw brake oil circulation The yaw brake oil circulation model consists if two subsystems: the oil pump and the system load. In fig.(3.29), the mutual interactions and the interface to the other (brake) systems are shown. Figure 3.29: Simulink implementation of yaw brake oil circulation system The pump drive speed n pmp is generated by the pump drive model (par ), while the required oil pump load torque is calculated in eq.(3.51) from actual pump pressure, p pmp, flow, φ pmp and angular speed, ω pmp. T sys = p pmp φ pmp ω pmp (3.51) The system pressure p sys is responsible for the state of the mechanical yaw brake. Above a lower limit (opening the control valve) the brake piston will start pushing the linings against the yaw brake ring. Below this limit a spring will push the linings from the ring. Oil pump Dependent of the actual pump pressure p pmp and shaft speed, n pmp, the pump generates an oil flow φ pmp through the system. The stationary pump model has been described in par System load Based on the modelling approach from par , the system load model is shown in fig.(3.30). The model contains two pressure storages, pump pressure p pmp and system pressure p sys, which are separated by an unidirectional valve. The pressure of both storages decharge slowly due to leakages. The maximum pressure at the pump side is guaranteed via a overpressure valve that opens when an upper level is exceeded and closes below a lower level. The pressure at the system side can be discharged quickly via a control valve. The unidirectional valve requires a minimum pressure to open and causes resistance when charging the system side. The valves are discontinuous elements which have two states (open/close) and flow constraints (min/max). The ECN-E

54 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems φ pmp φ uv R uv φ sv φ pc φ pl p uv φ sc φ sl p φ po p pmp p sys n pmp C p R pl R po C s R sl R sv Figure 3.30: Linear model of yaw brake oil system load load system is described in eq.(3.52) - eq.(3.55) and shown in fig.(3.30), where the conductivity G is equal to the reciproce of the resistance R. p pmp = 1 C φpc dt + p 0 p pmp φ pc = φ pmp φ pl φ po φ uv φ pl = G pl p pmp φ po = min(φ max po, δpo G po p pmp ) (3.52) The overpressure valve in eq.(3.52) is modelled as a pump pressure dependent switch, δ po, with hysteresis to avoid shuttling and a maximum flow φ max po : δ po = { +1, if : ppmp > p sw pmp + p hys pmp 0, if : p pmp < p sw pmp p hys pmp (3.53) The unidirectional valve is modelled as a pressure drop dependent switch, δ uv, which opens gradually, G uv when the pressure drop exceeds a minimum level (p uv =p pmp -p psys ) only: p uv = p pmp p sys φ uv = min(φ max uv, max(0, δ uv G uv (p uv p 0 uv)) (3.54) p sys = 1 C φsc dt + p 0 s sys φ sc = φ uv φ sl φ sv φ sl = G sl p sys φ sv = min(φ max sv, δ sv G sv ) (3.55) The control valve in eq.(3.55) is modelled as a switch, using an external open/close control command δ sc, and a maximum flow φ max sc : Brake model parameter values The values of the system conductivities and capacities are determined on their rated values, p rat pmp,φ rat pmp, p rat uv, oil leakage flows (often specified in component data sheets), φ rat pl, φrat sl, φ rat po, and estimation of time constants, τ p, τ s. Because of the accumulator and oil volume, the time constant at the system side τ s will be much larger (over 10 times) than the time constant τ p φ rat sc 44 ECN-E

55 3 PERIPHERAL DEVICES Figure 3.31: Simulink implementation of yaw brake oil pump load system at the pump side. G pl = φ rat pl /prat pmp G sl = φ rat sl /(p rat pmp p rat uv ) G uv = (φ rat pmp φ rat pl )/prat uv ) G po = φ rat po /p rat pmp G sc = φ rat sc /(p rat pmp p rat uv ) C p = τ p G pl (3.56) C s = τ s G sl At the stationary equilibrium point the pump covers the system losses. Usually the pump will be switched off by the control system (and switch on again, to charge the system as soon as the pressure comes below the lower limit). The stationary equilibrium point (p eq pmp, φ eq pmp) of the pump and system load can be derived from their stationary transfer functions (see also eq.(3.57)): p eql pmp = R p(1+g uv R s ) 1+G uv (R p +R s ) φeql pmp = Ka eql φ eql pmp (3.57) p eql pmp = K eql 1 (φeql pmp) 2 K eql 3 (neql pmp) 2 Substitution of both equations results in the pump and system equilibrium point: ( ) φ eql 1 pmp = 2 K K 1 a Ka 2 4 K 1 K 3 (n eql pmp) Yaw drive system p eql pmp = K a φ eql pmp p eql sys = G uv R s 1+G uv R s p eql pmp p 0 uv (3.58) The yaw drive system consists of three identical mechanically coupled induction motor drives and three mechanical brakes (fig.(3.32)). ECN-E

56 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 3.32: Simulink implementation of yaw nacelle drive system Yaw nacelle drives The total yaw drive torque Tyaw,drv hss and power losses Q yaw,loss are equal to the sum of the individual torque and power losses of each yaw motor. The motor model is bi-directional and has been described in par before. The motors are supplied via a two contactors: CW and CCW. Both directions exclude each other and are commanded from the turbine control system. The netto torque as delivered to the mechanical yaw ring is determined from the the drive torque of the yaw drives Tyaw,drv hss, the counteracting and speed dependent viscous loss T hss loss,viscous, the yaw inertia torque Tyaw,inertia o f the yaw motors and the yaw moment via the structural dynamic systems. The inertia of each yaw motor is square transfered from the high speed shaft to the slow speed yaw ring. The inertia of the nacelle is already incorporated in the structural model. T hss yaw,drv = T hss yaw,drv1 + T hss yaw,drv2 + T hss yaw,drv3 T hss loss,viscous = C Ω yaw Ω hss yaw T yaw,inertia = Jyaw,drv1 hss Ω hss yaw +...Jyaw,drv2 hss Ω hss yaw +...Jyaw,drv3 hss Ω hss yaw (3.59) Q yaw,loss = Q yaw,loss1 + Q yaw,loss2 + Q yaw,loss3 Mechanical yaw brakes The mechanical yaw brake model consists if two subsystems: generation of braking torque, and the non linear friction: stiction at standstill and speed independent coulomb friction. In fig.(3.33), the mutual interactions and the interface to the other (brake) systems are shown. Hydraulic system pressure, p sys (par ), will cause a constant braking torque, T brk, at the 46 ECN-E

57 3 PERIPHERAL DEVICES Figure 3.33: Simulink implementation of the yaw braking torque system braking disc which is opposite in direction to the high speed shaft torque. Braking torque The braking torque, T brk, is generated by a friction force, F fric, at a yaw ring radius R ring. The friction force is a consequence of the normal force F n and the friction coefficient f fric of the brake linings on the yaw ring. The normal force F normal is caused by a force due to the piston with surface A piston in the hydraulic cylinder with pressure p sys and a counterforce from a soft passive spring F spring. F normal = A piston p sys F spring F fric = f fric F normal (3.60) T brk = F fric R ring The values of the brake spring and piston surface are derived from the brake design torque T dsgn brk, the hydraulic pressure at which braking starts pbrkon sys, the friction factor f fric and the disc radius R ring. F dsgn spring = T dsgn brk /(f fric R disc ) (3.61) A piston = F dsgn spring /pbrkon sys Coulomb friction The braking force acts as a constant force F fric due to Coulomb friction. At standstill there s not only Coulomb friction, but the stick friction as well. The non linear friction models has been described in par Yaw sensors To commmand the yaw system, the nacelle misalignment should be measured. The misalignment angle is defined as the difference between the wind direction angle and the nacelle reference position. A positive direction implies clockwise when looking toward the front of the rotor. The measured wind direction is low pass filtered. ECN-E

58 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems 3.5 Brake system (BRK) Description Since almost all moderne turbines have (three)independent active pitch systems, the mechanical brake is mostly used as a parking brake. Therefore this component is no longer part of the primary safety system and it s size and thermic dimensions are drasticcally reduced. The mechanical brake consists of a single brake disc which is mounted on the rotating high speed shaft of the gearbox. The brake calliper with brake pads (linings) are mounted on teh gearbox housing. The fail safe brake system is hydraulic actuated. The constant system pressure is maintained by an accumulator and realised via a electric driven oil pump. The high oil pressure in hydraulic cylinder generates a counterforce greater than the springforce to keep the braking linings from the disc. When the control valve is activated (for a parking brake around standstill, after aerodynamic braking) the pressure fades away, the stiff spring releases and the linings of the calliper pushes with constant force against the outer ring of the disc. The friction torque decellerates the shaft speed, and dissipates heat power during braking (see par ) Definition The brake system is shown in fig.(3.34). The pressureholder or accumulator is a passive device for maintaining constant pressure at the system side. The following behaviour, subsystems and piston Brake cylinder Brake disc pressure holder Unidirectional valve Brake caliper M E-motor Pump Overpressure valve Control valve oil tank Figure 3.34: Brake system components are included: pump drive (par ) grid connected induction motor mechanical rotation 48 ECN-E

59 3 PERIPHERAL DEVICES shaft speed and motor current pump and drive power losses oil circulation (par ) oil pump oil flow circuit unidirectional valve control valve overpressure valve oil pressure and flow at pump and brake cylinder mechanical brake (par ) brake friction torque and power nonlinear coulomb and stick friction Obviously, the brake system interacts with other turbine systems fig.(3.35) The coulomb and Figure 3.35: Interactions of brake system viscous friction losses during braking T brk, are dependent of the actual slow shaft torque, T shf, and slow shaft speed, Ω shf, as calculated in the structural dynamic system. The turbine control system commands on/off brake activation and pump running, while several detections and measurements will be used for monitoring and control, e.g. oil system pressure, oil flow, pump speed, brake lining temperatures. Power (400V/50Hz) is supplied for the oil pump drive. A more detailed Simulink implementation is shown in fig.(3.36) Brake oil pump drive The oil pump of the brake system is driven by a on/off controlled grid connected induction motor. Dependent of the actual system torque, T sys, the stationary drive model as described in par results in a pump speed. The electric drive is commanded by constant voltage and frequency supply. The on/off switch command depends on the actual system pressure p sys level with repect ot an upper and lower level limit. ECN-E

60 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 3.36: Simulink implementation of brake system Brake oil circulation The brake oil circulation model consists of two subsystems: the oil pump and the system load. In fig.(3.37), the mutual interactions and the interface to the other (brake) systems are shown. The pump drive speed n pmp is generated by the pump drive model (par ), while the Figure 3.37: Simulink implementation of brake oil circulation system 50 ECN-E

61 3 PERIPHERAL DEVICES required oil pump load torque is calculated in eq.(3.62) from actual pump pressure, p pmp, flow, φ pmp and angular speed, ω pmp. T sys = p pmp φ pmp ω pmp (3.62) The system pressure p sys is responsible for the state of the mechanical brake. Below a lower limit (opening the control valve) the spring of the brake will start pushing the linings against the disc. Oil pump Dependent of the actual pump pressure p pmp and shaft speed, n pmp, the pump generates an oil flow φ pmp through the system. The stationary pump model has been described in par System load Based on the modelling approach from par , the system load model is shown in fig.(3.38). The model contains two pressure storages, pump pressure p pmp and system pressure p sys, which φ pmp φ uv R uv φ sv φ pc φ pl p uv φ sc φ sl p φ po p pmp p sys n pmp C p R pl R po C s R sl R sv Figure 3.38: Linear model of brake oil system load are separated by an unidirectional valve. The pressure of both storages decharge slowly due to leakages. The maximum pressure at the pump side is guaranteed via a overpressure valve that opens when an upper level is exceeded and closes below a lower level. The pressure at the system side can be discharged quickly via a control valve. The unidirectional valve requires a minimum pressure to open and causes resistance when charging the system side. The valves are discontinuous elements which have two states (open/close) and flow constraints (min/max). The load system is described in eq.(3.63) - eq.(3.66) and shown in fig.(3.38), where the conductivity G is equal to the reciproce of the resistance R. p pmp = 1 C φpc dt + p 0 p pmp φ pc = φ pmp φ pl φ po φ uv φ pl = G pl p pmp φ po = min(φ max po, δpo G po p pmp ) (3.63) The overpressure valve in eq.(3.63) is modelled as a pump pressure dependent switch, δ po, with hysteresis to avoid shuttling and a maximum flow φ max po : δ po = { +1, if : ppmp > p sw pmp + p hys pmp 0, if : p pmp < p sw pmp p hys pmp (3.64) ECN-E

62 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems The unidirectional valve is modelled as a pressure drop dependent switch, δ uv, which opens gradually, G uv when the pressure drop exceeds a minimum level (p uv =p pmp -p psys ) only: p uv = p pmp p sys φ uv = min(φ max uv, max(0, δ uv G uv (p uv p 0 uv)) (3.65) p sys = 1 C φsc dt + p 0 s sys φ sc = φ uv φ sl φ sv φ sl = G sl p sys (3.66) φ sv = min(φ max sv, δ sv G sv The control valve in eq.(3.66) is modelled as a switch, using an external open/close control command δ sc, and a maximum flow φ max sc : Figure 3.39: Simulink implementation of brake oil pump load system Brake model parameter values The values of the system conductivities and capacities are determined on their rated values, p rat pmp,φ rat pmp, p rat uv, oil leakage flows (often specified in component data sheets), φ rat pl, φrat sl, φ rat po, and estimation of time constants, τ p, τ s. Because of the accumulator and oil volume, the time constant at the system side τ s will be much larger (over 10 times) than the time constant τ p at the pump side. φ rat sc G pl = φ rat pl /prat pmp G sl = φ rat sl /(p rat pmp p rat uv ) G uv = (φ rat pmp φ rat pl )/prat uv G po = φ rat po /p rat pmp G sc = φ rat sc /(p rat pmp p rat uv ) C p = τ p G pl C s = τ s G sl (3.67) 52 ECN-E

63 3 PERIPHERAL DEVICES At the stationary equilibrium point the pump covers the system losses. Usually the pump will be switched off by the control system (and switch on again, to charge the system as soon as the pressure comes below the lower limit). The stationary equilibrium point (p eq pmp, φ eq pmp) of the pump and system load can be derived from their stationary transfer functions (see also eq.(3.68)): p eql pmp = R p (1+G uv R s ) 1+G uv (R p +R s ) φeql pmp = Ka eql φ eql pmp (3.68) p eql pmp = K eql 1 (φeql pmp) 2 + K eql 3 (neql pmp) 2 Substitution of both equations results in the pump and system equilibrium point: ( ) φ eql 1 pmp = 2 K K 1 a Ka 2 4 K 1 K 3 (n eql pmp) 2 p eql pmp = K a φ eql pmp p eql sys = G uv R s 1+G uv R s p eql pmp p 0 uv (3.69) Mechanical brake The mechanical brake model consists if two subsystems: generation of braking torque, and the non linear friction: stiction at standstill and speed independent coulomb friction. In fig.(3.40), the mutual interactions and the interface to the other (brake) systems are shown. Low hydraulic Figure 3.40: Simulink implementation of braking torque system system pressure, p sys (par ), will cause a constant braking torque, T brk, at the braking disc which is opposite in direction to the high speed shaft torque. This friction torque converts kinetic energy to heat power (see par ) Braking torque The braking torque, T brk, is generated by a friction force, F fric, at a disc radius R disc. The friction force is a consequence of the normal force F n and the friction coefficient f fric of the brake linings on the disc. Because of the fail safe design, the normal force F normal is caused ECN-E

64 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems by a stiff passive spring F spring and a counter force due to the piston with surface A piston in the hydraulic cylinder with pressure p sys. F normal = F spring A piston p sys F fric = f fric F normal (3.70) T brk = F fric R disc The values of the brake spring and piston surface are derived from the brake design torque T dsgn brk, the hydraulic pressure at which braking starts pbrkon sys, the friction factor f fric and the disc radius R disc. F dsgn spring = T dsgn brk /(f fric R disc ) (3.71) A piston = F dsgn spring /pbrkon sys Coulomb friction Because of the constant spring force, the braking force acts as a constant force F fric due to Coulomb friction. At standstill there s not only Coulomb friction, but the stick friction as well. Dependent of the actual speed the braking torque generates a heat power Q brk = T brk Ω brk (see par ) The non linear friction models has been described in par ECN-E

65 3 PERIPHERAL DEVICES 3.6 Pitch system (PIT) Electric blade pitching Electric actuators for pitch control use to be driven by a DC-motor or a vector controlled synchronous machine with permanent magnet rotor (AC-drive). In the context of control, the field controlled AC-drive can be assumed to act as a DC-motor. In order to verify pitch control algorithms, this section focusses on the actuation dynamics of this system [3]. The next paragraphs provide the working principles of this class of electric pitch speed servo actuators (par ) and a feedback scheme for the effective servo behaviour (par ) Working principles Fig.(3.41) shows the configuration of a pitch speed servo system with AC-drive. The effect of blade torsion has not been included in this figure. The feedback block for speed servo control the effective AC-motor current in an inner loop and the rotor speed in an outer loop. The current control loop of the AC-motor compensates for changes in the electro motive force (EMF) caused by rotor speed variation. The resulting motor torque T m accelerates the pitch-wise blade rotation via a gearbox with transmission ratio i. The load inertia J p as felt by the motor amounts to J m + J pb /ι 2, with J m the moment inertia of the rotor of the AC-motor and J pb the pitch-wise inertia of the rotor blade. The friction torque T fric caused by the pitch bearings counteracts this acceleration. The modelling of the transient dynamics in the electric pitch servo behaviour is based on [21]. Figure 3.41: Pitch speed servo system with vector controlled AC-motor An important issue in pitch actuation is the Coulomb friction in the pitch bearing, which results from the bending moment M bend on the pitch bearing. In a wide range of applications the Coulomb friction moment amounts to 1% to 2% of the bending moment. To assume proportionality satisfies when considering the impact of bearing friction on the overall actuation. Since a gearbox with transmission ratio i is included between the drive and the pitch axis, the friction torque T fric on the drive amounts to T fric = 1 i ( ) M bend (3.72) ECN-E

66 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems The friction torque always counteracts the motion. At constant blade bending moment, it will change of sign when the pitching direction reverses. This implicates that the motor torque has to bridge over the Coulomb friction value twice in order to reverse the pitching direction. Thus, a delay is introduced in the pitch speed servo that depend on the rate of change of the motor torque and the level of Coulomb friction. The motor torque results from the feedback control loops and the motor behaviour. Fig.(3.42) shows a prevailing feedback scheme for a speed controlled servomotor. Figure 3.42: Feedback scheme for electric pitch speed servo system The speed and DC-current feedback loops have both proportional-integral (PI) functionality and are implemented in a micro controller. The latter introduces very small data handling delays (τ n v, τ I v). The P- and I-gains in the speed and current loops are K n p, K n p and K I i, KI i. The DC-current loop emits a voltage value U r m to the motor, which causes a change in the motor current I m via first order dynamics characterised by gain 1/R s and time constant τ s, being the ratio between inductance L s and resistance R s. Altogether, the current feedback loop is such fast that it can be assumed to behave ideally within the speed feedback loop (I m I r m). The variations in the EMF-voltage (E m ) by the rotor speed (ω) are relatively slow and are compensated for via the DC-current loop Effective pitch speed servo behaviour The scheme in fig.(3.42) can be used for simulation of the electric pitch servo behaviour. However in the design of the pitch control loops, an as simple as possible linear formulation is required that comprises the friction behaviour. Since the fast inner loop makes I m to be approximately equal to Im, r the motor torque T m can be assumed to depend directly upon the PI-action on the motor speed error ω r ω via the machine constant K t, which maps the current to the torque. This is depicted in the feedback scheme for the effective pitch speed behaviour. Fig.(3.43) shows a block diagram that can be used for modelling the servo behaviour in the design of the pitch control loops. The influence of the Coulomb friction at reverse of the pitching direction is modelled by a delay of τv Clmb seconds in this scheme. An example for the pitch speed servo behaviour without reversing the pitching direction is given in fig.(3.44). This concerns an AC-drive for a rotor blade with length of ca. 30m. The 56 ECN-E

67 3 PERIPHERAL DEVICES Figure 3.43: Feedback scheme for effective electric pitch speed servo behaviour design data for PI-feedback, current to torque conversion and rotor inertia have been used in the determination of the amplitude and phase characteristics for the transfer function from the pitch speed setpoint to the pitch speed without reverse of the pitching direction (solid lines). Figure 3.44: Amplitude and phase characteristics pitch speed servo behaviour without reverse of pitching direction The non-solid lines represent approximations by respectively a second order system with very small delay (dash) and by just a pure delay (dash-dot). The second order system has cut-off frequency ω pt0 equal to 300rad/s, an exponential damping rate β pt of 0.5 and delay τpt of s. The effective pure delay τ pteff equals 0.005ms. Note that the amplitude of the pure delay remains 1 for all frequencies (dash-dot line in amplitude characteristics). If the pitching direction reverses, then the response of the pitching speed includes the bridging over by the motor torque of twice the Coulomb friction torque T fric. In the model for the effective pitch servo behaviour, this has been modelled via the extra delay τv Clmb. A maximum value for this delay can be derived if the maximum friction moment is known and if a minimum pitch speed setpoint value θ set min is assumed. The integral gain Kn i, together with the current to torque conversion factor K t then effectuates the following rate of change T m in the motor ECN-E

68 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems torque: The required time τ Clmb v T m = K t K n i i for bridging 2T fric is then estimated by: τ Clmb v = K t K n i 2T fric i θ set min (3.73) θ set min (3.74) The replacing models listed below can then be used for including the pitch servo behaviour in control design. These are described by the governing equations: 1 ω 2 pt 0 d2 dt 2 ( θ(t)) + 2β pt ω θ(t) + θ(t) = θ set (t τ pt τv Clmb ), (3.75) pt0 and θ(t) = θ set (t τ pteff τ Clmb v ), (3.76) A realistic minimum pitching speed of e.g o /s typically yields values for the additional delay τv Clmb significantly smaller then 0.1s. If much lower pitching speed values would be desired, then it is recommended to provide a much higher speed setpoint value during the bridging-overinterval of twice the Coulomb friction. This compensation should be implemented with care in order to avoid over-compensation. This would yield transients in the pitching speed from the temporarily high value to the desired value during a very short period after the reverse of the pitching direction. 58 ECN-E

69 4 TURBINE DYNAMIC SYSTEMS In this chapter simulation blocks for the turbine dynamic systems are dealt with. These blocks realise the principal conversion flow in a wind turbine system as well as the thermic behaviour. The involved models are highly generic. The aero- and hydrodynamic conversion behaviour, the structural dynamic behaviour of the rotor, main shaft, gearbox, generator, support structure, the thermic and the electric conversion behaviour are implemented in the turbine dynamic systems. 4.1 Structural dynamic system (SDS) The functionality of the Structural Dynamic System (SDS) consists of: Real-time simulation of the structural dynamic behaviour of the rotor, main shaft, gearbox, generator, and support structure via a state space model that is driven by demodulated input signals after they have been reduced with their mean values. Providing measurement and process signals from the state space model that are modulated if they pertain to a rotating structure and augmentated with their mean values to - Peripheral Devices - Aerodynamic Conversion System (ACS); - Hydrodynamic Conversion System (HCS); - Wind Generation (WIN) Making available parameters and initial conditions that result from equilibrium assessment for itself and for - Aerodynamic Conversion System (ACS); - Hydrodynamic Conversion System (HCS); - Wind Generation (WIN); - Wave generation (WAV). Figures 4.1 and 4.2 show the signals that drive the SDS and that are provided by the SDS. The latter enable feedback control, simulation of peripheral devices and aero- and hydroelastic interation. These figures also show that the core of the SDS is established by three blocks, which are simulated as one integrated state space model. This state space model results from the application of TURBU Offshore in the so called structural mode [22]. The blocks represent structural models for the so called components, viz. rotor: composed of blade blocks identified by sysd, syse, sysf drive train: block identified by sysr support structure: block identified by syss ECN-E

70 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.1: Wind, wave, control and loss signals to Structural Dynamic System from neighbouring simulation blocks Since these blocks provide linearised simulation of structural dynamic behaviour, their in- and output signals are freed from the mean values. This is visualised in the figures by subtraction and addition operations with overlined symbols like tgnr_1 and Ω or where the symbol µ, which stands for mean value, is added to signal descriptions. 60 ECN-E

71 4 TURBINE DYNAMIC SYSTEMS Figure 4.2: Measurement signals from Structural Dynamic System to neighbouring simulation blocks For so called small signal analysis, that is to say the examination of the dynamic deformation at relatively small variations of input (and output) signals around their mean values, this linear approach satisfies. However, at process simulation a number of conditions involves large signal variations, which in general requires a non-linear approach. Although the majority of the ECN-E

72 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems non-linear behaviour lies in the aero- and hydrodynamic conversion, which is essentially nonlinearly implemented in the process simulator, two non-linear phenomena are relevant in view of the structural dynamic behaviour: blade stiffening by centrifual forces that depend on the rotor speed; coupling of in- and out-of-plane blade vibration modes by (twist-wise) blade setting that depend on the pitch angle. This blade stiffening slightly alters the vibration frequencies of the blades while the blade mode coupling affects the exchange of (strong) out-of-plane aerodynamic damping forces with (weak) in-plane ones. In order to account for these non-linearities, different sets of linear simulation blocks are included in the process simulator. Two of those sets are active, viz. those that surround the actual working conditions. The linear structural models from TURBU work with time-invariant coefficients in so called fixed-frame coordinates. The transformation from rotating to stand still frames allows for very fast simulation but requires input demodulation and output remodulation while application is restricted to wind turbines with three blades and more. Features like prebend, shear offset, and average deformation are catered for. Figure 4.3 shows in the right-hand lower corner, on the basis of the fore-aft tower deformation, that the mechanical modelling is based on a co-rotational formulation for beam elements. Each element has a local Figure 4.3: Assemblage of substructures from elements coordinate system. The orientation of this coordinate system results from all (bottom-up) foregoing angular displacements and the local additive angular displacements. The latter occur in the entry point of the concerning element. These entry points also may accommodate degrees 62 ECN-E

73 4 TURBINE DYNAMIC SYSTEMS of freedom (DOFs) for linear displacements. The span from the entry point after the DOFs to the exit point of an element is a rigid bar (undeformable). The bending and torsion behaviour of the tower and rotor blades is modelled via these rigid elements with DOFs in the entry points for springs and dampers. Componenent models are created for the rotor blades, the drive-train, and the support structure. Next to DOFs for bending and torsion, the rotor blades allow DOFs for flap- and leadwise hinges, and full span blade pitching. Also additional support structure DOFs apply; for roll, tilt and yaw of the nacelle, and the corresponding translations, as well as for full flexibility of the foundation. DOFs in the drive-train allow for torsion and bending of the rotor shaft constrained by a main bearing, for rotor speed variation, and for co-axial gearbox house rotation. In the following three subsections is described how the linear structural models of the components are obtained and implemented. In the last section is described how the mean values are determined and how the linear models, together with these mean values, are used for process simulation Rotor The rotor includes three blades and the hub. The model of the hub is added to the drive-train model because of its axi-symmetry. Each rotor blade is modelled via an assemblage of two submodels. Consider the model for the rotor blade D as depicted in figure 4.4. Figure 4.4: Blade model as assemblage of submodels The two submodels, typed as D f and D p, model the structural behaviour of the blade flange and the blade profile & structure. The (structural) blade model consists of N elements: one element models the flange behaviour via submodel D f ; the remaining N 1 elements model the deformation behaviour of the rotor blade via submodel D p (typical N = 14). All N blade elements are loaded by concentrated gravity forces and aerodynamic forces and torques. The structural dynamic behaviour of the rotor blade is modelled under the assumption of massless elastic beam elements and concentrated point masses to which moments of inertia are attributed. In figure 4.5 five of such beam elements of length S are distinguished. Consider the bending deformation at the end of a beam element from concentrated force and torque loads f and t along unit vectors e D 0 z and e D 0 x in that location for invariant stiffness ECN-E

74 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.5: Structural blade model EI over an element (beam bending equivalence). Exactly the same deformation is obtained with a concentrated angular and linear spring in the mid-span location with stiffness EI/S and 12EI/S 2 respectively. In our model, the massless elastic beams are replaced by these springs and rigid bars. The effective bending stiffness EI is based on the maintenance of similar rotation per moment characteristics (the bending moment is assumed almost constant along the beam element). It then holds EI = S / ( s e s s 0 EI(s) ds ) (4.1) Two orthogonal bending spring pairs are located in the mid-span location of each of the five beam elements in order to model blade bending in two directions. Also, springs are added for the torsional deformation. The incremental rotations and translations in the angular and linear springs are the degrees of freedom (DOFs) in the structural model. So far, the Holzer-Myklestad method as described by Bielawa in [23] is followed, except for a slight reallocation of mass and the added torsion springs. Figure 4.5 shows that the rotor blade model consists of the 5 elements D 2... D 6 that set up the blade profile & structure D p and an additional element D 1 that is used for for the flange D f. The flange element D 1 transduces the rigid body motion from the end of the blade root to the entry point of the first blade profile & 64 ECN-E

75 4 TURBINE DYNAMIC SYSTEMS structure element D 2. The flange element D 1 allows for motion relative to the blade root, via pitching and flap and leadwise hinges. For the sake of equally structured equations of motions for all elements, the blade mass beteen 0 and 1 2 S of the blade span S b is allocated to D 1. The inertia properties of elements S 2... S 5 and S 6 are derived from integrals over span intervals { 1 2 S, 3 2 S}... { 7 2 S, 9 2 S} and { 9 2 S, S b}. Note that the mid-span locations of the model elements D 2... D 5 correspond with the end points of the first four beam elements according to Holzer & Myklestad. In order to maintain as much as possible the principle of beam bending equivalence, the spanwise distributed aerodynamic loads are mapped to concentrated loads in: mid-span locations of elements D 2... D 5 ; end point of element D 6 ; entry point of element D 1. The mapping procedure is based on equal deformation in the end points of massless elastic beam elements, just as mentioned above for the dimensioning of the springs. Before the model equations can be derived, the structural model is cast into a co-rotational formulation in such a way that the angular bending springs are located on the elastic axis. An exception concerns element D 1 and those that are preceeded by elements for which explictely out of plane and in-plane configuration angles are defined (coning angle(s), δ 3 -angle(s)). In D 1 the location of angular springs coincides with the pitch axis while in the latter elements the location of these springs follows from the configuration angles. This allows for coning, flap and leadwise skewness, prebend and the inclusion of the averaged deformed state in the equations of motion. Figure 4.6 shows in the upper part the configuration of model element D 3 via the spanwise elastic (y-)axis and the neutral bending axes for edge and flatwise bending (x- and z-axis). The edge- and flatwise orientation are obtained by nose-up rotation over the structural pitch angle relative to the in-plane and out of plane orientation (lead- and flapwise). The local coordinate system e D 3 1,2,3 coincides with the elastic x-, y- and z-axis of D 3 and is used for the formulation of the equations of motion in the entry point D3. The lower part of the figure shows (i) the shear centre in the cross section with the entry point D3, (ii) the mid-span cross section with the aerodynamic conversion point Da 3 for the concentrated aerodynamic force and torque loads and (iii) the cross section in which the centre of gravitiy D3 is located. The chord line is nose-up positive rotated over the aerodynamic pitch angle relative to the in-plane orientation. The effect on deformation through the shear axis offset from the elastic axis is modelled as follows: the linear bending springs and the torsion spring are moved to the shear centre over place vector r ES while the angular bending springs remain located in the elastic centre. Elements D 2... D 6 are all treated in the same way. The equations of motion for the blade elements are derived by Newton s law related to the rate of change in the linear impulse and in the angular impulse relative to the entry point of each element. For element D 3 these (vector) equations look like: D 3 h- D 3 = RS t D 3 + D 4t D 3 + D 3 r D 3 D 4f D 3 + a t Da 3 + D 3 r Da 3 a f Da 3 + D 3 r D 3 g f D 3 (4.2) p D 3 = RS f D 3 + D 4f D 3 + a f Da 3 + g f D 3 The responsive loads RS t D 3 and RS f D 3 are visco-elastic loads; in the blade flange, the torque component along the y-axis results from the pitch actuator. ECN-E

76 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.6: Configuration and center points of 3 rd blade model element What is referred to as the rate of change of angular impulse actually is the biased angular impulse (h- is used instead of h), which excludes the term that is set up by the linear acceleration of the reference point D 3. The loads and kinematic variables (accelerations, velocities and positions) are expressed in coordinates along the local coordinate systems of the elements. These coordinates depend upon the angular linear degrees of freedom (φ D p and ρ D p) and input variables to the concerning subcomponent (v D p). For the profile and structure subcomponent D p, the latter are the kinematic variables from the flange subcomponent D f and the distributed aerodynamic force and torque loads from the Aerodynamic Conversion System. The coordinate expressions for the loads and kinematic variables in the equations of motion are split up in mean values and variations. The mean-value-expressions are used for the determination of the average deformation while the variation-value-expressions are used for the derivation of the linear models for the (small signal) structural dynamic behaviour. The visco-elastic bending and torsion torques in D3 are simply proportional to the angular DOFs. Since all DOFs are modelled in the entry point of the blade elements, the translation of the shear centre relative to the foregoing element depends on the linear as well as on angular bending DOFs. A heuristic approximation tells that for the total relative translation δ ˆρ D 3 holds: δ ˆρ D3 x ˆρ D3 y ˆρ D3 z = δ ρ D3 x ρ D3 y ρ D3 z + δ φ D3 x φ D3 y φ D3 z D 3 r Ds 3 (4.3) 66 ECN-E

77 4 TURBINE DYNAMIC SYSTEMS The elastic linear bending loads are obtained from a slightly modified expression for δ ˆρ D 3 in order to take into account the average rotation increments. The heuristic expressions for the elastic responsive loads in D3 are: s D 3 r x 0 0 ˆφ D3 x RS t D 3 = 0 s D 3 r y 0 ˆφ D3 y RS f D 3 = s D s D 3 r z t x s D 3 t y s D 3 t z ˆφ D3 z ˆρ D3 x ˆρ D3 y ˆρ D3 z + D 3 r Ds 3 RS f D 3 The viscous reactions are obtained by replacing φ x by φ x and s rx by d rx etc. When the equations of motions for all elements in D p are expressed in DOFs and input variables, obtained from the vector-coordinate expressions along the local coordinate systems, they are linked to each other. This yields the matrix/vector differential equation for D p as a massdamper-spring formulaton: [ ] [ ] [ ] ϕ D p ϕ D p ϕ D p M D p + D D p + S D p = G D p v D p (4.5) ϱ D p ϱ D p The model equations for the output variables of D p are obtained by the linear expansion of the output vector y D p in the DOFs and their first and second derivatives and the submodel-input variables. After elimination of the dependency on the second derivatives of the DOFS via the state equations, the following expression for y D p is obtained: [ ] [ ] ϕ D p ϕ D p y D p = L D p + H D p + K D p v D p (4.6) ϱ D p The output vector y D p provides elements normal and leadwise speeds and their setting angles to the Aerodynamic Conversion System and it provides force and torque loads in the exit point of the blade flange. The same model derivation procedure applies for the flange D f, although only one element applies, viz D 1. This yeilds the following mass-damper-spring formulation: [ ] [ ] [ ] ϕ D f ϕ D f ϕ D f M D f + D D f + S D f = G v D f ϱ D f ϱ D f ϱ D f ϱ D p ϱ D f ϱ D p ϱ D f (4.4) [ ] [ ] rcl (4.7) ϕ D f ϕ D f y D f = L D f + H D f + K D f v D f First order state space representations of these subcomponent models are obtained by introduction of position- and speed-associated state subvectors in accordance with: [ ] [ ] z D p ϕ D p p = ; z D p ϕ D p s = z D f p = ϱ D p [ ϕ D f ϱ D f ] ; z D f s = ϱ D p [ ϕ D f ϱ D f ] (4.8) Now compose the state vectors of the pursued first order state space representations by stacking the position and speed subvectors: [ ] [ ] D z D p z p D s z f = ; z z D D s f = (4.9) p s z D f s ECN-E

78 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems The following state space models can then be derived ((D q = D p, D q ): ż D q = y D q = 0 I ( A D q p C D q p A D q s C D q s z D q + ( 0 B D q ) z D q + K D q v D q ) v D q (4.10) with: ( A D q 1 p = M q) D S D q ( A D q 1 s = M q) D D D q C D q p = H D q C D q s = L D q (4.11) The subcomponent models can be connected to each other. This makes the interaction variables between the flange and the profile & structure subcomponent to vanish and yields the pursued first order state space representation of the structural blade model: ż D = A D z D + B D v D y D = C D z D + K D v D (4.12) Drive Train The drive-train is modelled via submodels R f and R r for the generator rotor and the rotor shaft & hub. Figure 4.7 shows the interactions of the drive-train model R with the support structure, rotor blades and the environment. Figure 4.7: Drive-train as assemblage of submodels 68 ECN-E

79 4 TURBINE DYNAMIC SYSTEMS The rotor shaft & hub R r is loaded by a concentrated gravity force and by the force and torques from the rotor blades in the rotor centre R c r. The rotor centre is the location on the shaft axis with the shortest distance to the centre of the pitch bearing. The mass and inertia moments of the rotating part of the gearbox are added to the masses of the shaft & hub. The transmission ratio i gb of the gearbox governs that the fraction 1/i gb of the outgoing co-axial torque is fed into the generator rotor R f. The remaining outgoing loads are transmitted to the gearbox house. All load exchange for R r is assumed to occur in the rotor centre, which allowed under the assumption of rigid nacelle behaviour. An extern (co-axial) generator torque is emitted to the generator rotor R f. This torque acts as a responsive torque that facilitates variable speed operation; it acts in opposite direction upon the support structure S. Since the load exchange for R f only concerns a torque, it may be located anywhere under the assumption of rigid nacelle behaviour; we chose the rotor centre for this (fast shaft) torque loading. The mass and non-coaxial inertia moments of R f are added to those of the nacelle S n. The desired first order state space representations for submodels R r and R f are obtained from the mass-spring-damper formulations in the same way as described for the blade flange and profile & structure. When the two submodels of the drive-train R are connected to each other via the interaction variables according to figure 4.7, the desired first order state space representation for R is obtained: d dt(z R ) = A R z R + B R v R y R = C R z R + K R v R Modelling details for the shaft and for the generator rotor are listed in the two following paragraphs. Rotor shaft & hub submodel The rotor shaft & hub is considered to behave deformable as concerns the rotor shaft and undeformable as concerns the hub. Figure 4.8 shows the physical layout and the vector variables in the model element layout of R r. Axi-symmetry is assumed for the shaft. The elasticity of the shaft is modelled via up to six DOFs with visco-elastic responsive loads in the shaft s entry point Rr. We chose to let coincide this entry point with the exit point Rr in the rotor centre Rr. c Together with the inclusion of the main bearing in the bending behaviour, the chosen location of the entry and exit point necessates to include cross couplings between the linear displacement and the angular visco-elastic reaction in a bending direction and vice versa. The mass, inertia moments and location of the centre of gravity and the bending, torsion and axial stiffness follow from (weigthed) integration of the shaft cross section data over the shaft-axis. The coordinate system independent vector expressions for the equations of motion of the rotor shaft & hub element R r are the following: R r h- R r = RS t R r + D 1 t D 0 + E 1 t E 0 + F 1 t F 0 + Rc r r R r g f R r p Rr = RS f D 3 + D 1 f D 0 + E 1 f E 0 + F 1 f F 0 + g f R r (4.13) The mass-spring-damper formulation in the DOFs of the shaft and input variables is obtained by considering the vectorcoordinates along the coordinate system of the shaft. Input variables come from the generator rotor (rotor speed variation), rotor blades (torque and force loads) and support structure (modulated kinematic variables), as well as external periodic gravitation inputs in the rotating frame of reference. ECN-E

80 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.8: Equation of motion governing vector variables for rotor shaft & hub submodel The expressions for the elastic responsive loads in Rr are (read s R r tr x for s trx and φ R r x for φ x etc.): ( ) strx 0 0 φ x RS t R r = 0 s try 0 φ y ρ x 0 0 s ttyz ρ y 0 0 s trz φ 0 s ttzy 0 z ρ z (4.14) RS f R r = s ft x 0 0 ρ x 0 s fty 0 ρ y φ x 0 0 s fryz φ y 0 0 s ftz ρ 0 s z frzy 0 φ z The bending stiffness values (y and z orientation) are determined under the following constraints, visualised in figure 4.8 (see [22], Chapter 3): reponsive force & torques in rotor centre; zero linear displacement in main bearing centre; zero angular and linear displacement at slow gearbox exit. 70 ECN-E

81 4 TURBINE DYNAMIC SYSTEMS Generator shaft submodel The generator rotor is considered to behave undeformable while a co-axial visco-elastic or controllable angular DOF is allowed. Only the co-axial angular behaviour is taken into account in the submodel. The equation of motion is based on the real fast shaft angular speed. The co-axial rotational speed of the generator rotor is the sum of the corresponding rotational speed of the nacelle, the variation ϕ R f x of the rotor itself (the DOF), and the average rotational speed transformed via the gearbox, that is to say i gb Ω. The generator speed variation is transmitted to the slow shaft of the gearbox, that is to say δω = ϕ R f x /i gb Support structure The support structure is modelled via four submodels. The two submodels S f and S t establish the structural and hydrodynamic behaviour of the tower just as the flange and blade & profile submodels D f and D p cater for the structural and aerodynamic behaviour of the rotor blade D. The ond-element submodel S n is used for including the nacelle. Finally, the gearbox house is a non-rotating subcomponent and is added as the fourth submodel to the support structure. Since it physically belongs to the drive-train it is typed R h. The (structural) support structure model consists of M elements (gearbox house not included): the first element models the foundation behaviour via submodel S f, the elements 2 up to M 1 model the deformation behaviour of the tower via submodel S t, and the last element models the nacelle behaviour via submodel S n (typical M = 15). All M blade elements are loaded by concentrated gravity forces. For an offshore turbine, the underwater tower elements and the foundation element are also loaded by concentrated hydrodynamic forces and torques. The gearbox house R h just feeds through all loads from the drive-train except the co-axial torque. An extern co-axial support torque is emitted to the gearbox house R h, which facilitates gearbox house rotation; this torque acts as a responsive torque and acts in opposite direction upon the nacelle. The mass and inertia moments of R h are added to those of the nacelle S n. An extern yawing torque is emitted to the nacelle S n. This responsive torque facilitates dynamic yawing; it acts in opposite direction upon the top of the tower S t. Forces and torques in three directions can be added as independent loading on the gearbox house, located in the rotor centre. These are non-responsive loads; they are to be considered as loads such as caused by the wind or gravitation. TURBU provides values for these loads that just compensate for the average gravitation loads in the standstill frame, average loads that result from the periodic gravity loading of the blades in the rotating frame. Forces and torques in three directions can also be added as independent loading on the tower, located in the centre of the tower top (= yaw bearing centre). These non-responsive loads can be used for simulation of damper devices in the tower top. It must be remarked that all reaction loads on these devices as well as the inertia effects perpendicular to their effecutation orientation are not included in the structural behaviour of the support structure. The desired first order state space representations are obtained from the mass-spring-damper formulations in the same way as described for the blade flange and profile & structure. When the four submodels of the support structure S are connected to each other via the interaction variables according to figure 4.9, the desired first order state space representation for S is obtained: d dt(z S ) = A S z S + B S v S y S = C S z S + K S v S ECN-E

82 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.9: Support structure as assemblage of submodels Modelling details for the gearbox house, nacelle and tower and foundation are listed in the three following paragraphs. Gearbox house submodel R h The gearbox house is considered to behave undeformable while a co-axial visco-elastic or controllable angular DOF is allowed. Only the co-axial angular behaviour is taken into account in the submodel. The full relative rotation φ R h x applies in the equation of motion for R h whereas the rotational speed of the slow gearbox shaft is augmented with the fraction ((i gb 1)/i gb ) of ϕ R h Nacelle submodel S n The nacelle is considered to behave undeformable while visco-elastic DOFs in all six directions are allowed in the connection point with the tower top (=yaw bearing centre). The yaw DOF can also be specified controllable. In that case the external yaw torque replaces the visco-elastic responsive behaviour in the yaw DOF. Except the co-axial moment of inertia, all inertia properties of the the gearbox house R h and generator rotor R f are added to those of the nacelle. For the generator rotor this also involves gyrocopic effects because of the average rotation that amounts to i gb Ω. Foundation and tower submodels S f, S t x. 72 ECN-E

83 4 TURBINE DYNAMIC SYSTEMS The submodels for the foundation and the tower are established from equations of motion that are obtained in the same way as those for the structural submodels X f and X p of the rotor blades. The model equations are basicly the same, but are less complicated because we assume that in the tower axis: the shear and elastic axis coincide; the concentrated hydrodynamic loads affect. In addition there is no average motion of the support structure, so the equations of motion for an element S k do not include terms derived from ω I ω because they are of second order. The masses of the underwater elements may be augmented with the enclosed water mass Integrated structural dynamic model In this subsection it is described in the following following paragraphs how the mean values are determined (equilibrium assessment); an integrated structural model is set up from the linear component models; a set of such integral structural models is used for simulation in a wide range of operating conditions. Equilibrium assessment The driving variables for the equilibrium conditions are the mean values of the wind speed, the rotational speed, yaw misalignment angle and the pitch angles; the latter are assumed to be equal for the three rotor blades. The variables that define the equilibrium conditions are equilibrium driving variables; mean induction speeds in the rotor annuli; mean values of angular and linear DOFs Actually, the mean yaw angle misalignment and pitch angles define the mean values of the yaworiented DOF in the nacelle and the pitch-oriented DOF in the flange of the blades. All mean loads on the wind turbine and mean displacements are derived from these equilibrium defining variables. In TURBU an iteration procedure has been implemented for the determination of the equilibrium. Each iteration consists of four steps. One step pertains to the aerodynamic equilibrium, the remaining three steps to the structural equilibrium of the rotor blades, drive-train and support structure. The structural equations of equilibrium are derived from the general equations of motion like the ones listed in expression 4.2 for the rotor blades. The aerodynamic equations of equilibrium correspond to those adopted by Lindenburg and Schepers in [24]. Herein is assumed that only the lift forces contribute to the setting of the aerodynamic equilibrium. The equation for the axial and tangential induction speed U i and V i then looks like (number of blades B = 3): 1 ρc 2 L(φ a )cs U l cos β Un 2 + Ul 2 = 2π B ρr rf p2u i U tr (4.15) 1 ρc 2 L(φ a )cs U n Un 2 + Ul 2 = 2π B ρr rf p2v i U tr The correction factor F p as adopted by Prandtl accounts for both tip and root effects. The expressions for U tr and F p are generally known and can a.o. be found in [25], [24] and [22]. ECN-E

84 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems The transportation speed U tr is the vector sum of the axial, lateral and vertical wind speed in the rotor annulus; the axial speed is diminished with the axial induction. In dynamic conditions the average over the wind speeds on the three blade elements that rotate in the annulus are considered for the determination of U tr. The iteration procedure is preceeded by setting the mean values of the angular DOFs of the blades equal to the configuration values, derived from the cone and pitch angle and skewness and prebend specifications. The mean values of the angular DOFs of the drive-train and support structure, as well as the mean values of the linear DOFs of all substructures are set equal 0. The mean deformation values of all DOFs are initially set to 0. The sum of the configuration and the mean deformation values of the DOFs set up the average shape of the components. The following four calculation steps are performed sequentially in each iteration: solution of the BEM-equations that yield the axial and tangential induction speed values in the rotor annuli for the rotor layout that belongs to the lately determined average shape of the rotor, drive-train and support structure; solution of the steady-state rotor blade impulse equations that yield the mean deformation values of the angular and linear DOFs for the lately determined induction state of the rotor annuli and average shape of drive-train and support structure (note: non-zero average accelerations in the rotating-frame); solution of the steady-state drive-train impulse equations that yield the mean deformation values of the co-axial DOFs for the rotor loading that belongs to the lately determined induction state and average shape of rotor and support structure solution of the steady-state support structure impulse equations that yield the mean deformation values of the angular and linear DOFs for the loading that belongs to the lately determined induction state and average shape of the rotor and drive-train All solution procedures except that for the the drive-train consist of iteration procedures by theirselves. The solution procedure for the BEM-equations may also be directed by the user to calculate the required pitch angle for achieving nominal power. The equilibrium driving parameter then is the nominal power instead of the pitch angle. In that case the BEM solution procedure consists of nested iterations in which the outer loop is governed by pitch angle search. Because of axi-symmetry the mean values of the bending DOFs of the drive-train remain 0. All equilibrium values of loads, deformations, induction speeds and structural speeds are stored as fields in Matlab structure variable pequilib. Linking component models and handling in fixed-frame coordinates The integrated structural dynamic model is available as a set of state-space model matrices that is obtained by linking together the component models for the structural behaviour of the rotor bladesd, E and F, the drive-train R and the support structure S. This matrix set is extended with name lists, mean-value lists and demodulation/modulation matrices for the input and output signals. All those matrices and lists are stored as fields in the Matlab strcuture variable syst. Figure 4.10 shows the matrix equations for the state space representations of the component models and the interaction signals as well as the driving signals from the aero- and hydrodynamic and electric conversion system and the peripheral devices. The interface between the drive-train and support structure involves the rotor azimut angle Ψ, which amounts to the time-integral of the slow shaft speed Ω; the interfaces between the rotor blades and the drive-train involve the azimut seperation between the rotor shaft and the blades, which amount to 0, 120 o and 240 o respectively. 74 ECN-E

85 4 TURBINE DYNAMIC SYSTEMS Figure 4.10: Interdependency of state space models for the substructures of the wind turbine The equations of motion for the integrated model have coefficients that are periodic in azimut angle Ψ: ż = A(Ψ) z + B(Ψ) v y = C(Ψ) z + K(Ψ) v (4.16) Since we assume wind turbines with 3 blades, polar symmetry exists in the rotor layout. The socalled multi-blade transformation, see Coleman & Feingold [12], is applied for the elimination of the periodic coefficients in the full system equations. The only price to be payed for it consists in modulation of the wind speed variations before they enter the transformed system equations via input vector ɛ and in modulation of the transformed system output variables in η in order to retransform them to along the original coordinate systems. Of course, these transformation only pertain to variables in rotating frames of reference (rotor, drive-trian). The following linear time invariant model formulation with modulated input preprocessing and output postprocessing then ECN-E

86 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems applies: ɛ = Tv 1 cm (Ψ) v q = A cm q + B cm ɛ η = C cm q + K cm ɛ y = T ycm (Ψ) η (4.17) The invariant input, state transition, output and feedthrough matrix in this model is obtained from the corresponding periodic matrices in the earlier mentioned model via the Coleman transformation matrices T zcm (Ψ), T vcm (Ψ) and T ycm (Ψ) on the state, input and output vector respectively, by performing the transformation for an arbitrary fixed value ψ of the rotor azimut angle. The fixed-frame system matrices are obtained as follows from the (partially) rotating system matrices: A cm = Tz 1 cm ( ψ) ( A( ψ) T zcm ( ψ) T zcm ( ψ) ) B cm = Tz 1 cm ( ψ) B( ψ) T vcm ( ψ) C cm = Ty 1 cm ( ψ) C( ψ) T zcm ( ψ) K cm = Ty 1 cm ( ψ) K( ψ) T vcm ( ψ) (4.18) The Coleman transformation involves the mapping of any corresponding quantities on the rotor blades D, E and F to multi-blade coordinates via the 3 3 matrix kernel T kerx : T kerx (Ψ) = 1 sin(ψ) cos(ψ) 1 sin(ψ π) cos(ψ π) 1 sin(ψ π) cos(ψ π). (4.19) It also involves the mapping of x, y and z-coordinates of any vector on the rotor shaft to multiblade coordinates via the 3 3 matrix kernel T kerr : T kerr (Ψ) = cos(ψ) sin(ψ) 0 sin(ψ) cos(ψ). (4.20) It should be noted that periodic coefficients due to external forces (gravity and uniform wind loading) are not eliminated by the Coleman transformation. The parametric excitations terms due to gravity and wind loads can generally be ignored for all but extremely flexible blades. Reduction of the model order The submodels of the blade profile & structure D p and of the tower S t are reduced in order before they are connect with the other submodels of the blades and support structure respectively. For this we apply the method proposed by Hurty [11], [26], which has been made more easily applicable by Craig & Bampton [10]. This method allows for reduction of the overall model order from typcially 600 to 100 without any loss of accuracy in frequencies up to 5 Hz or even higher. This enormously accelerates the matrix/vector processing as stated in Eq This improvement in computational efficiency is required for real-time process simulation. Scheme for non-linear simulation of structural dynamics As mentioned in the introductory part of this section, both the rotor speed and the pitch angle cause observable non-linear behaviour when large signal variations are considered. In general, these two determining variables would require a 2-dimensional grid of linear models in order to choose the most appropriate linear model, dependent on the actual rotor speed and pitch angle. However, the operation of a variable speed pitch to vane wind turbine is characterised in partial load by a varying rotor speed and nearly constant pitch angle whereas in full load the rotor 76 ECN-E

87 4 TURBINE DYNAMIC SYSTEMS speed hardly varies and the pitch angle is adapted over several tens of degrees in order to limit the captured aerodynamic power. Further, the pitch angle will be changed from feathering to working position during start-up while the rotor speed will still be very low. These operational characteristics allows for basing the choice of the most appropriate linear model in respectively start-up, partial and full load on only the pitch angle, only the rotor speed and again only the pitch angle. Thus, a 1-dimensional grid of linear models satisfies. Initially the next provisions are to be made: Derive equilibrium conditions and linear structural models, in a start-up and a partial and a full load working point range from driving conditions established by triples of wind speed, rotor speed and pitch angle. Note that the pitch angle values in the start-up working point range monotonously decrease while they monotonously increase in thefull load range; the rotor speed will monotonously increase in the partial load range. This yields the three sets of structure variable pairs with linear models and equilibria: {(syst s1, pequilib sl )... (syst sl, pequilib sl )} {(syst p1, pequilib p1 )... (syst pm,, pequilib p m )} {(syst f1, pequilib f1 )... (syst fn,, pequilib f n )} for working points that belong to three sets of equilibrium driving variables: {(Ūs 1, Ω s1, Θ s1 )... (Ūs l,, Ω sl, Θ sl )} {(Ūp 1, Ω p1, Θ p1 )... (Ūp m,, Ω pm, Θ pm )} {(Ūf 1, Ω f1, Θ f1 )... (Ūf n,, Ω fn, Θ fn )} Alltogether, l + m + n linear structurul models syst W i and equilibra pequilib W i are created for working points that are set up by driving-variable triples: {(Ū W 1, Ω W 1, Θ W 1 )... (Ū W l+m+n, Ω W l+m+n, Θ W l+m+n )} Map the time-continuous state space representations from TURBU to time-discrete representations for use in the process simulator and rewrite the input demodulation matrix and output modulation matrix as a 3-term series. In addition, create a second set of output and feedthrough matrices that is limited to the to be controlled output signals in vector y c, being the slow shaft speed and the pitch angles and yaw angle and dito speeds. This yields the matrices for state vector update and output signal computation of the following scheme (integer time index k stands for k t with the discretisation time): ɛ k = (T 1 v cm,d + cos Ψ k T 1 v cm,c + sin Ψ k T 1 v cm,s ) v k q k+1 = F cm q k + Γ cm ɛ k η k = C cm q k + K cm ɛ k η c k = C c cm q k + K c cm ɛ k (4.21) y k = (T ycm,d + cos Ψ k T ycm,c + sin Ψ k T ycm,s ) η k y c k = (T c y cm,d + cos Ψ k T c y cm,c + sin Ψ k T c y cm,s ) η c k with F cm = e Acm T Γ cm = t 0 e Acm ( T τ) B cm dτ (4.22) ECN-E

88 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Derive the upward and downward deviation of the state vector of each linear structural model when the conditions for the surrounding working points apply (for working points W 1 and W l+m+n this only concerns the upward and downward deviation respectively; A cm is used as short form for A W i cm, etc): get q W i 1,i such that: 0 = A cm q W i 1,i + B cm T 1 v cm,d ( v W i 1 v W i) T 1 y cm,d (ȳ W i 1,c ȳ W i,c ) = C c cm q W i 1,i + K c cm T vcm,d ( v W i 1 v W i) get q W i+1,i such that: 0 = A cm q W i+1,i + B cm T 1 v cm,d ( v W i+1 v W i) T 1 y cm,d (ȳ W i+1,c ȳ W i,c ) = C c cm q W i+1,i + K c cm T vcm,d ( v W i+1 v W i) (4.23) The last step in the initialisation procedure concerns the simultaneous solution of the steady state formulation of the state transition vector equation ( q = 0 0 = A q + B v) and of the output vector equation for the to be controlled signals in y c. This is required because the state transition matrix A is rank deficient for the states associated with the outputs in y c. The output vector equation provides the missing number of equalities in order to be able to compute the state deviations q W i 1,i and q W i+1,i to the surrounding working points. The non-linear simulation scheme that has to be carried out each discretisation (sample) time t involves the following steps: Select the working points W idn and W iup for the linear models that surround the actual working conditions: if Ω < Ω p1 and Θ > Θ sl then find i dn, i up such that Θ W i dn Θ Θ W iup elseif Ω p1 Ω < Ω f1 and Θ < Θ f1 then find i dn, i up such that Ω W i dn Ω Ω W iup else find i dn, i up such that Θ W i dn Θ Θ W iup end Initialise state vectors of linear models relative to surrounding working points in case of change of working of conditions: if i dn = i pr dn 1, and so i up = i pr dn, then q W i dn k W i pr dn = qk q W i pr dn 1,ipr dn ; q W i up k W i pr dn = qk elseif i dn = i pr dn + 1, and so i dn = i pr up, then 78 ECN-E

89 4 TURBINE DYNAMIC SYSTEMS q W i up k = q W i pr up k q W i pr up+1,i pr up ; q W i dn k = q W i pr up k Update the state vector and compute the output vector of the actually surrounding models: ɛ W i dn k = (T 1 v cm,d + cos Ψ k T 1 v cm,c + sin Ψ k T 1 v cm,s ) (v k v W i dn ) ɛ W i up k = (T 1 v cm,d + cos Ψ k T 1 v cm,c + sin Ψ k T 1 v cm,s ) (v k v W i up ) q W i dn k+1 = F W i dn cm q W i dn k + Γ W i dn cm ɛ W i dn k q W i up k+1 = F W i up cm q W i up k + Γ W i up cm ɛ W i up k η W i dn k = C W i dn cm q W i dn k + K W i dn cm ɛ W i dn k (4.24) η W i up k = C W i up cm q W i up k + K W i up cm ɛ W i up k y W i dn k = ȳ W i dn + (T ycm,d + cos Ψ k T ycm,c + sin Ψ k T ycm,s ) η W i dn k y W i up k = ȳ W i up + (T ycm,d + cos Ψ k T ycm,c + sin Ψ k T ycm,s ) η W i up k Derive the output vector y k of the structural dynamic system as a weighted average of the output vectors of the linear models in the surrounding working points of the actual working conditions: if Ω < Ω p1 and Θ > Θ sl then y k = Θ Θ W i up Θ W i dn Θ W i up elseif Ω p1 Ω < Ω f1 and Θ < Θ f1 then y k = Ω Ω W i dn Ω W i up Ω W i dn y W i dn k + Θ W i dn Θ Θ W i dn Θ W i up y W i up k + Ω W i up Ω Ω W i up Ω W i dn y W i up k (4.25) y W i dn k (4.26) else y k = Θ Θ W i dn Θ W i up Θ W i dn y W i up k + Θ W i up Θ Θ W i up Θ W i dn y W i dn k (4.27) end When the real-time simulation is started, the index values i pr dn and ipr up for the working conditions of the previous time step do not yet exist. In the first simulation time-step the state vectors q W i dn k and q W i up k are initialised as 0. ECN-E

90 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems 4.2 Aerodynamic conversion (ACS) The Aerodynamic Conversion System (ACS) provides the distributed aerodynamic normal and lead forces and aerodynamic pitching torques to the three rotor blades of the Structural Dynamic Systems (SDS). The governing equations are based on Blade Element Momentum (BEM) theory extended with a dynamic term for the evoluation of the wake. This approach has been adopted by Snel and Schepers and elaborated in the so called ECN s Differential Equation model [13]. The BEM-equations for the equilibrium conditions have already been mentioned in the subsection on the integrated structural model. The axial impulse equation for each rotor annulus is extended with a dynamic term for the axial induction speed U i. A rotor annulus is fit on each of N triples of corresponding elements on the rotor blades. Within an annulus no spatial variation of U i is modelled. This is a small simplification of the implementation that has been carried through by Lindenburg in PHATAS [24]. In PHATAS, instantaneous axial induction variations per rotor blade can occur in addition to annulus-uniform first order dynamic behaviour. These instantaneous induction variations arise from oblique inflow conditions and local blade deformations. The latter are neglected in the process simulator while the former are included in the blade effective wind speeds as pure azimuth dependent additions in accordance with the equation scheme 5.76 in the paragraph on periodic axial induction variation in subsection The simplification reduces a coupled set of algebraic and differential eqations for the axial induction per annulus to just one differential equation per rotor annulus. The tangential induction speed V i is assumed to react instantaneously on changing conditions. This makes the Aerodynamic Conversion System to consist of N differential and N algebraic equations. Each equation pair for an annulus then models the annulus-uniform induction behaviour; the annulus-average axial and tangential induction speed variations are thus internal variables in the Aerodynamic Conversion System. The model equations for an annulus W k are the following: ρ 4πr W k r W krf W k a ρ 4πρr W k r W k F W k p d dt(u W k (X=D,E,F ) i ) = ρ 4πr W k r W k F W k p 1 2 ρcx k L (φx k a ) cs X k U X k l cos β X k V W k i U W k tr = (X=D,E,F ) 1 2 ρcx k L U W k i U W k tr + (U X k n (φx k a ) cs X k U X k n ) 2 + (U X k l ) 2 (U X k n ) 2 + (U X k l ) 2 (4.28) The angle of attacks φ X k a for all blade elements, corrected for blade cone and flap, are output signals of the Structural Dynamic System. The annulus-average transportation speed U W k tr depends on the axial, tilt and yaw components of the (blade effective) wind speed and the annulus-uniform axial induction speed by: tr = U D k tr + U E k tr B U W k with U X k tr = (U X k + U F k tr ax F X k p (4.29) U X k i ) 2 + (Ū X k yaw) 2 + (Ū X k tilt )2 The influence of axial induction variations on the transportation speed as caused by oblique inflow is catered for via the (undisturbed) axial wind speed component, just a the rotationally sampled turbulence by the rotor blades and the effect of tower shear and shadow: U X k ax (t, Ψ) = Ūax + ũ X (t) + tow u X ax(ψ) + shr u X ax(ψ) + obl u X i ax (Ψ) (4.30) The azimuth dependent influences will cancel in averaging over the rotor blades in the expression for the annulus transport speed U W k tr 80 ECN-E

91 4 TURBINE DYNAMIC SYSTEMS The following expressions hold for the annulus-average F W k p to Prandtl (accounts for both tip and root effects): of the correction factor according F W k p = F D k p + F E k p B with + F F k p F W k p = 2 π arccos(e (R rw m)π/d W k ) 2 π arccos(e (rw m R root)π/d W k ) d W k = 2πrW m B U W k tr ( Ω r W m ) 2 + (U W k tr ) 2 (4.31) The intermediate transportation speed U W k tr, for unknown Prandtl number, is determined as: tr = U D k tr + U E k tr + U F k tr U W k with U X k tr = with U X k ax given by Eq B (U X k ax U X k i (4.32) ) 2 + (Ū X k yaw) 2 + (Ū X k tilt )2 The following expression holds for the radius dependent weighting function F W k a : F W k a = 2π 0 2π { 1 (r W k /R) cos ψ [1 + (r W k /R) 2 2(r W k /R) cos ψ] 1.5 } dψ (4.33) The normal and leadwise wind speed U X k n and U X k l relative to blade element X k follow from the axial wind speed U X k ax (t) by Eq. 4.30, the periodic tangential wind speed obl u X tg by Eq and the normal and leadwise speeds of X k itself, which are output signals of the Structural Dynamic System: U X k n (t) = U X k ax (t, Ψ) v X k n (t) U X k l (t) = obl u X tg (Ψ) + vx k tg (t) (4.34) Note the sign convention in the periodic tangential variation and blade element speeds; the orientations comply with the axis-directions of the local coordinates system on the blade element. Further, the tangential element speed v X k tg includes the rotation driven blade speed. The leadwise and normal forces per unit span on each blade element, as well as the aerodynamic pitch torque, are output signals to the Structural Dynamic System. For element X k these are determined as: q X k f n = 1 2 ρcx k L (φx k a ) c U X k l (U X k n ) 2 + (U X k l ) 2 q X k ρcx k D (φx k a ) c U X k n f l = 1 2 ρcx k L (φx k a ) c U X k n (U X k n 1 2 ρcx k D (φx k a ) c U X k l q X k t t = 1 2 ρcx k M (φx k a ) c 2 ( (U X k n (U X k n ) 2 + (U X k (U X k n ) 2 + (U X k l ) 2 l ) 2 ) 2 + (U X k l ) 2) ) 2 + (U X k l ) 2 (4.35) These loads are in accordance with local coordinate system on the blade element, which implies that the force loads are downwind and leadwise positive while the aerodyamic pitching torque is nose-down positive. q X k t t ECN-E

92 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems The expressions for the axial wind speed variations and the mean values in axial, sideways and vertical directions are given in the section on wind speed generation in the chapter on external influences. All involved mean values are obtained from the equilibrium specifying structure variables pequilib W i as provided by TURBU in the derivation of the linear structural models for the simulation of the Structural Dynamic System (SDS). The mean values that are used in the expressions above are obtained by interpolation between mean values that pertain to the surrounding working points of the actual working conditions. The interpolation is the same as the one applies in the determination of the output vector y of the SDS, as listed in the scheme for non-linear simulation in subsection ECN-E

93 4 TURBINE DYNAMIC SYSTEMS 4.3 Hydrodynamic conversion (HCS) The Hydrodynamic Conversion System provides the fore-aft and sideways water forces per unit span to the Structural Dynamic System. The governing equations are based on Morison s hydrodynamic load model, which includes both drag and mass forces [?]. The hydrodynamic drag loads depend on the water current, horizontal wave speeds and tower element speeds. The mass loads depend on horizontal wave accelerations and tower element accelerations. The drag loads are proportional on the square of the vector sum of the water current, wave and tower speed via the drag-coefficient The mass loads are proportional to the vector sum of the wave and tower accelerations via the inertia coefficient. Assume that the wave and current direction are clockwise rotated over respectively γ w and γ c rad relative to the fore-aft axis of the tower base (view from helicopter). The adopted expressions for the distributed force loading q S k w on the center point of underwater element S k in the direction of the waves and for the loading q S k w perpendicular to the waves are: q S k w = 1 2 ρ w π 4 (DS k) 2 (C M ẇ S k a S k fa cos(γ w) a S k sd sin(γ w))+ C V 1 2 ρ w D S k w S k cos(γ w γ c ) + w S k v S k fa cos(γ w) v S k sd sin(γ w) ( w S k cos(γ w γ c ) + w S k v S k fa cos(γ w) v S k sd sin(γ w)) q S k w = 1 2 ρ w π 4 (DS k) 2 (+a S k fa sin(γ w) a S k sd cos(γ w))+ C V 1 2 ρ w D S k w S k sin(γ w γ c ) + v S k fa sin(γ w) v S k sd cos(γ w) (4.36) ( w S k sin(γ w γ c ) + v S k fa sin(γ w) v S k sd cos(γ w)) The wave speed and acceleration values are provided by the wave generation system (WAV) in the next chapter. The fore-aft and sidways tower speed and accelerations (v S k fa, vs k sd, as k fa, as k sd ) are provided by the structural dynamic system (SDS), as described in the first section of this chapter. The fore-aft and sideways forces per unit span, which are are inpuut to the SDS, are obtained from the above listed expression by: q S k fa = q S k w cos(γ w ) q S k w sin(γ w) (4.37) q S k sd = q S k w sin(γ w ) + q S k w cos(γ w) The inertia coefficient C M in the mass-term of Morison s equation deviates from its basic value (2) when diffraction of waves occurs. The diffraction level depends on the ratio between tower diameter and wave length (D/L). The McCamy-Fuchs correction is adopted for the dependency of the inertia coefficient on the ratio D/L. Since the wave length unambiguously depends on the wave frequency via the dispersion relation, the inertia coefficient can be expressed as function of the wave frequency ω for a certain tower diameter and water depth. The inertia coefficient is not taken into account as to the mass loads from the tower acceleration. So, a frequency dependent scaling of only the wave acceleration satifies to take into account the effect of diffraction. This scaling factor can be simply included as a multiplier in the realisation scheme of the horizontal wave acceleration, which we did. Thus, the variation of the inertia coefficient is completely removed from Morison s equations and is yet accounted for by a modified realisation scheme for the wave acceleration. All involved mean values and orientation angles are obtained from the equilibrium specifying structure variables pequilib W i as provided by TURBU in the derivation of the linear structural models for the simulationo fhe Structural Dynamic System (SDS). The mean values that are used in the expressions above are obtained by interpolation between mean values that pertain to the surrounding working points of the actual working conditions. The interpolation is the same as the one applies in the determination of the output vector y of the SDS, as listed in the scheme for non-linear simulation in subsection ECN-E

94 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems 4.4 Thermic conversion (TCS) Gearbox system Description For control and monitoring purposes the temperatures at different gearbox locations are usually measured. These temperatures are a result of mechanical gearbox losses (par ). Usually the gearbox consists of multiple stages. The required gear ratio is usually achieved by three separate stages, each with ratios of 1:3.. 1:5. The high torque stages are often planetary (epicyclic), while the high speed stages are mostly parallel shaft. The gears can be spur (teeth parallel to the gear axis) or helical (teeth skewed to the gear axis). The gearbox is situated in the nacelle and conditioned by the cooling and lubrication system (par. 3.2). The dissipated heat in the gear mesh is transferred to the oil (splash) and via the oil mist (air/oil mixture) to the housing. The oil flow transfers the absorbed heat tot the heat exchanger (par and the housing transfers heat to the nacelle surrounding. The heat dissipation of the bearings is transferred partly to the oil, the housing and the surrounding. Definition The thermic behaviour of the gearbox is shown in fig.(4.11). There s no distiction made between Figure 4.11: Thermic gearbox system the three gear stages, so the gearbox model is approached as one equivalent thermic system.. The dynamics are determined by five different thermic capacities (mesh, oil, housing and two bearings) and the heat conductivities (mutual and ambient). The conductivities and capacities are related to the physical dimensions and material properties of the gearbox, therefore, the dynamics are rather slow (large time constants). The main heat source is the dissipated heat power in the mesh, which flows away both through the cooling oil flow and the oil mist. The oil (mist) exhanges heat with the surrouding via the gearbox housing. The front and rear bearings dissipate heat due to viscous friction (speed dependent) and exchange heat with the gearbox and surrounding. The gearbox model should comprise the following features: power dissipation and thermodynamic behaviour of the mesh heat extraction and thermodynamic behaviour of oil flow 84 ECN-E

95 4 TURBINE DYNAMIC SYSTEMS heat transfer and thermodynamic behaviour of the housing to surrounding power dissipation, heat transfer and thermodynamic behaviour of bearings temperatures of mesh, oil, housing, bearings Obviously, the gearbox system interacts with other turbine systems (fig.(4.12)). The gearbox Figure 4.12: Simulink model of interactions of thermic gearbox system power losses Q gbx,loss and the extracted heat by the oil flow Q φoil are calculated at the gearbox system (par. 3.2). The oil outlet temperature is returned to the gearbox cooling system (par. 3.2). Other temperatures of the thermic gearbox system are available for monitoring and control. The heat power of the housing to the nacelle interior is used to calculate the nacelle interior temperature, which is an input for the thermic gearbox system. Thermic gearbox model The subsystems of the thermic gearbox model are shown in fig.(4.14) The thermic gearbox model is described below. For convenience, subscript notations are used in the model equations: m (mesh), o (oil mixture), h (housing), s (surrrounding), ssb (slow shaft bearing), fsb (fast shaft bearing). θ m = 1 ( Q C m,loss Q mo) dt + θm 0 m θ o = 1 ( ) Q C mo Q oh Q φo dt + θo 0 o θ h = 1 (Q C oh Q hs ) dt + θh 0 (4.38) h θ ssb = θ fsb = 1 ( Q ssb,loss Q ssb,o Q ssb,h Q ssb,s) dt + θssb 0 C ssb 1 C fsb ( Q fsb,loss Q fsb,o Q fsb,h Q fsb,s) dt + θfsb 0 The nett heat power exchange to the thermic capacitors of the mesh, C m, the oil/oilmist C o, the housing C h and the bearings C ssb,c fsb, will cause accompanying temperatures θ m, θ o, θ h and θ ssb,θ fsb. The heat power exchanges between the capacitors are driven by the actual temperature differences and determined by the conductivities in between. Q mo = G mo (θ m θ o ) ECN-E

96 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.13: Simulink model of thermic gearbox sub-systems Q oh = G oh (θ o θ h ) (4.39) Q hs = G hs (θ h θ s ) Q ssb,x = G ssb,x (θ ssb θ x ) with : x {o, h, s} (4.40) Q fsb,x = G fsb,x (θ fsb θ x ) with : x {o, h, s} (4.41) The values of the thermic conductivities are determined from the equilibrium state at rated conditions: Q rat m,loss Q rat mo = 0 Q rat mo Q rat oh Q rat φo = 0 Q oh Q hs = 0 (4.42) Q rat ssb,loss Q rat ssb,o Q rat ssb,h Q rat ssb,s = 0 Q rat fsb,loss Q rat fsb,o Q rat fsb,h Q rat fsb,s = 0 If the rated heat power loss of the mesh,q rat m,loss, and rated heat power of the oil flow, Qrat φo, and 86 ECN-E

97 4 TURBINE DYNAMIC SYSTEMS the rated temperatures of the mesh φ rat m, the oil φ rat o and the housing φ rat h known, the conductivities G mo, G oh, G hs can be calculated: are assumed to be G mo = Q rat m,loss θ rat m θ rat o G oh = Qrat mo Q rat φ o G hs = θo rat θ rat h θ rat h Q rat oh θrat s (4.43) To determine the conductivities of the bearings, arbitrary fractions relative to the rated bearing heat power loss Q rat ssb,loss and Qrat fsb,loss are used. For both the slow shaft bearing and fast shaft bearing this results in: G ssb,x = f Qrat ssb,loss ssb,x Q rat θssb rat θrat x ssb,loss G fsb,x = f Qrat ssb,loss ssb,x Q rat θssb rat θrat x ssb,loss, with : x {o, h, s}, with : x {o, h, s} (4.44) The values of the thermic capacitors are calculated from their physical volumes, V, specific heat c and density ρ. These volumes are derived from the gearbox radii and length of a cylinder model and the total gearbox mass (fig.(4.16)). The bearing volumes can be determined from their physical diameters and thickness. C x = c x ρ x V x with : x {o, h, s} C y = c y ρ y V y with : y {ssb, fsb} (4.45) Thermic implementation model Fig.(4.14) shows the Simulink implementations of the sub-systems of the thermic gearbox model: mesh, oil-mixture, housing. The implementation of the gearbox bearings are shown in fig.(4.15). ECN-E

98 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.14: Simulink model of thermic gearbox sub-systems: mesh (upper), oil (middle) and housing (lower). 88 ECN-E

99 4 TURBINE DYNAMIC SYSTEMS Figure 4.15: Simulink model of thermic gearbox sub-systems: slow shaft bearing (upper) and fast shaft bearing (lower). ECN-E

100 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Generator system Description The generator consists of a stator (fixed to the turbine frame) and a rotor (rotating) and in between an air gap. Both the stator and the rotor have three windings (120 degrees spatially shifted), which consists of copper windings surrounded by iron. The (cast iron) generator housing encapsules the outer part of the stator iron. Each winding causes heat dissipation ( copper loss ) which is caused by the winding resistance and the actual phase current. For control and monitoring purposes the temperatures in each winding of the stator and rotor are usually met. The generator is situated in the nacelle and conditioned by the water cooling circuit (par. 3.3) and air ventilation. The dissipated heat in the rotor windings is transferred to the ventilated air-gap and to the stator, while the dissipated heat in the stator windings is transferred also to the water cooling circuit. The water flow transfers the absorbed heat tot the heat exchanger (par ) and the housing heat to the nacelle surrounding. The heat dissipation of the generator bearings is transferred partly to the housing and partly to the surrounding. Definition The thermic behaviour of the generator is shown in fig.(4.16). The dynamics are determined Figure 4.16: Thermic generator system by thermic capacities (stator, rotor airgap, water, housing and two bearings) and the heat conductivities (mutual and ambient). The conductivities and capacities are related to the physical dimensions and material properties of the generator, therefore, the dynamics are rather slow (high time constants). The main heat source is the dissipated heat power in the stator and rotor 90 ECN-E

101 4 TURBINE DYNAMIC SYSTEMS windings. Iron losses in stator and rotor are taken for granted. Both the copper stator and rotor windings are encapsuled in iron and thus indirectly cooled. The stator winding is cooled via the housing, air gap (self) ventilation and water cooling. The rotor only transfers heat via the air gap (to the stator and ventilation). The front and rear bearings dissipate heat due to viscous friction (speed dependent) and exchange heat with the generator housing and generator surrounding. The generator model should comprise the following features: power dissipation and thermodynamic behaviour (copper/iron) of each rotor winding section power dissipation and thermodynamic behaviour (copper/iron) of each stator winding section heat extraction due to water cooling of stator iron thermodynamic behaviour of air-gap and air ventilation heat transfer and thermodynamic behaviour of the generator housing to surrounding power dissipation, heat transfer and thermodynamic behaviour of bearings temperatures rotor windings, stator windings, bearings, housing, cooling water (inlet/outlet) Obviously, the generator system interacts with other turbine systems (fig.(4.17)). Figure 4.17: Simulink model of interactions of thermic generator system The generator power losses Q gen,loss and the extracted heat by the water flow Q φwtr are calculated at the generator system and the water outlet temperature is returned to the generator cooling system (par. 3.3). Other temperatures of the thermic generator system are available for monitoring and control. The heat power from the housing to the nacelle interior is used to calculate the nacelle interior temperature, which is an input for the thermic generator system. Thermic generator model The subsystems (stator, rotor, airgap, water, housing) of the thermic generator model are shown in fig.(4.19) The thermic generator model is derived below. For each specific stator and rotor (iron) section, there s heat exchange to its neighbour section, if, a temperature difference between the sections exists. However, the stationary temperature between these sections will be homegeneous θ FeS, θ FeR. For convenience, subscript notations are used in the model equations: R (rotor), S (stator),sur (surrrouding) Cu (copper winding), Fe (iron mass), A (air-gap), H (housing), Bfs ECN-E

102 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.18: Simulink model of thermic generator sub-systems (front-side bearing), Brs (rear-side bearing). The thermic model of the stator and rotor sections i, are described in eq.(4.47) and eq.(4.48). θ RCu,i = θ RFe,i = 1 ( Q RCu,i,loss Q RCuFe,i) dt + θrcu,i 0 1 Q RCuFe,i Q RFeA,i Q RFe,ij dt + θrfe,i 0 C RCu,i C RFe,i ) Q RCuFe,i = G RCuFe,i (θ RCu,i θ RFe,i ) Q RFe,ij = G RFe,ij (θ RFe,i θ RFe,j with: i {1, 2, 3} and j i i,j (4.46) 92 ECN-E

103 4 TURBINE DYNAMIC SYSTEMS θ SCu,i = θ SFe,i = 1 C SCu,i 1 C SFe,i ) Q SCuFe,i = G SCuFe,i (θ SCu,i θ SFe,i ) Q SFe,ij = G SFe,ij (θ SFe,i θ SFe,j ( Q SCu,i,loss Q SCuFe,i) dt + θscu,i 0 Q SCuFe,i Q SFeA,i Q SFe,ij Q SFeHx Q SFeWx dt + θsfe,i 0 with: i {1, 2, 3} and j i C Bfs i,j (4.47) The model equations of the air-gap, housing, bearings and water cooling are described in eq.(4.49) and eq.(4.50) (y=outlet, u=inlet): θ Ay = 1 (Q C SA + Q RA Q ASur ) dt + θay 0 A θ H = 1 (Q C SFeH + Q BfsH + Q BrsH ) dt + θh 0 H 1 ( θ Bfs = Q Bfs,loss Q BfsH Q BfsSur) dt + θbfs 0 (4.48) θ Brs = θ W = 1 C Brs 1 C W ( Q Brs,loss Q BrsH Q BrsSur) dt + θbrs 0 ( Q SFeW Q Wφ) dt + θw 0 ) Q SA = G SA (θ SFe θ A ) Q RA = G RA (θ RFe θ A ) Q ASur = G ASur (θ Ay θ Au ) Q SFeH = G SFeH (θ SFe θ H Q HSur = G HSur (θ H θ Sur ) Q BfsH = G BfsH (θ Bfs θ H ) (4.49) Q BrsH = G BrsH (θ Brs θ H ) Q BfsSur = G BfsSur (θ Bfs θ Sur ) Q BrsSur = G BrsSur (θ Brs θ Sur ) Q SFeW = 3 ) G SFeW,i (θ SFe,i θ Wy i=1 The heat transfer of the air-gap ventilation to the surrounding, is assumed to be proportional to the generator speed G ASur = K ASur Ω g. The heat transfer between stator and rotor via the air-gap uses the mean air-gap temperature θ A mean stator iron temperature θ SFe and mean rotor iron temperature θ RFe : θ A = 1 ) (θ 2 Ay + θ Au θ SFe = 1 3 θ RFe = θ SFe,i i=1 3 θ RFe,i i=1 (4.50) ECN-E

104 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems The total heat power transfer of the generator to the nacelle surrounding, Q Sur, is equal to the sum of Q BfsSur, Q BfsSur and Q HSur. The values of the thermic conductivities are determined from the equilibrium state at rated conditions. The stationary temperatures of all thermic storages are assumed to be known at rated. For the rotor (eq.(4.52)) and stator (eq.(4.54)) sub-systems this implies: Q rat RCu,i,loss Q rat RCuFe,i = 0 Q rat RCuFe,i Q rat RFeA,i i,j with: i {1, 2, 3} and j i Q rat RFe,ij = 0 (4.51) Q SCuFe,i Q SFeA,i i,j Q SCu,i,loss Q SCuFe,i = 0 (4.52) Q SFe,ij Q SFeH,i Q SFeW,i = 0 (4.53) with: i {1, 2, 3} and j i The rated dissipated power of each rotor winding Q rat RCu,i,loss is assumed to be known. Furthermore, at rated condition, there s no heat exchange between to rotor iron neighbour sections: Q RFe,ij =0, Q SFe,ij =0. The heat conductions G RFe,ij are assumed to be equal between all sections and are determined as a fraction f Qrat RFe,ij RFe,ij of Q rat RCu,i,lossper Kelvin temperature difference. The rotor conductivities can then be determined as given in eq.(4.55) G RCuFe,i = Q rat RCu,i,loss θ rat RCu,i θrat RFe,i G RFe,ij = 1 2 f Qrat RFe,ij RFe,ij Q rat RCu,i,loss (4.54) with: i {1, 2, 3} and j i At rated, the total conductivity of the rotor to the airgap can be determined: Q rat RFeA = G RFeA = 3 i=1 Q rat RCu,i,loss Q rat RFeA θrfe rat θrat A The stator conductivities can be calculated similar to the rotor: (4.55) G SCuFe,i = Q rat SCu,i,loss θ rat SCu,i θrat SFe,i G SFe,ij = 1 2 f Qrat SFe,ij SFe,ij Q rat SCu,i,loss with: i {1, 2, 3} and j i The power transfer of the stator to the air-gap is a little more complicated, because there s heat transfer to the housing and cooling water too (see last equation in eq.(4.54)). The heat transfer of each stator section to the housing Q SFeH,i is assumed to be equal, even as the heat transfer of each section to the cooling water Q SFeW,i. To determine the stator conductivities, the heat transfer of the stator to the air-gap is assumed to be a equal for each stator section and a fraction f Qrat SCu,i,loss SFeA,i of the dissipated stator winding power 94 ECN-E

105 4 TURBINE DYNAMIC SYSTEMS Q rat SCu,i,loss. Using the heat power of the cooling water, Qrat SFeW,i it is then possible to determine the heat conductivity from the stator to the housing, G SH, to the cooling water G SFeW, and to the airgap G SFeA as shown below: Q rat SFeA,i = f Qrat SCu,i,loss SFeA,i Q rat SCu,i,loss SFeW,i = ρ W c W φ rat W (θwy rat θwu) rat (1 f Qrat SCu,i,loss SFeA,i Q rat Q rat SFeH,i = Q rat SCu,i,loss ) Q rat SFeW,i Q rat SFeH = G SFeH = Q rat SFeW = G SFeW = Q rat SFeA = G SFeA = 3 i=1 Q rat SFeH,i Q rat SFeH θsfe rat θrat H 3 i=1 Q rat SFeW,i Q rat SFeW θsfe rat θrat Wy 3 i=1 Q rat SFeA,i Q rat SFeA θsfe rat θrat A = 3 G SFeH,i = 3 G SFeW,i = 3 G SFeA,i (4.56) The determination of the resting thermic conductivities which are related to the surrounding and the bearings are quite straighforward. Q rat SFeA + Q rat RFeA Q rat ASur = 0 Q rat SFeH + Q rat BfsH + Q rat BrsH = 0 Q rat Bfs,loss Q rat BfsH Q rat BfsSur = 0 (4.57) Q rat Brs,loss Q rat BrsH Q rat BrsSur = 0 Q rat SFeW Q rat W = 0 The rated bearing power losses Q rat Bfs,loss,Qrat Brs,loss are assumed to be known and a fraction f Qrat Bfs,loss, f Qrat Brs,loss BrsH is transferred to the generator house. BfsH G ASur = K ASur Ω rat g G BfsH = f Qrat Bfs,loss BfsH Q rat Bfs,loss θbfs rat θrat H G BrsH = f Qrat Brs,loss BrsH Q rat Brs,loss G BfsSur = G BrsSur = θbrs rat θrat H (1 f Qrat Bfs,loss BfsH = Qrat SFeA + Qrat RFeA θbfs rat θrat Sur (1 f Qrat Brs,loss BrsH θ rat Brs θrat Sur θay rat θrat Au ) Q rat Bfs,loss ) Q rat Brs,loss (4.58) ECN-E

106 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems The values of the thermic capacitors are calculated from their physical volumes, V, specific heat c and density ρ. These volumes are derived from the generator radii and length of a concentric cylinder model as shown in fig.(4.16). Due to the symmetry the phase sections of the stator or rotor have equal volumes (one third of total). The bearing volumes can be determined from their physical diameters and thickness. C x = c x ρ x V x with : x {H, A, W, SF e, RF e, SCu, RCu, Brs, Bfs} (4.59) Thermic generator implementation Fig.(4.19) through fig.(4.21)shows the Simulink implementations of the sub-systems of the thermic generator model: stator. The implementation of the gearbox bearings are shown in fig.(4.20). 96 ECN-E

107 4 TURBINE DYNAMIC SYSTEMS Figure 4.19: (lower). Simulink model of thermic generator sub-systems: stator (upper) and rotor ECN-E

108 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.20: Simulink model of thermic generator sub-systems: airgap (upper) and water cooling (lower). 98 ECN-E

109 4 TURBINE DYNAMIC SYSTEMS Figure 4.21: Simulink model of thermic generator sub-systems: generator house including bearings (upper) and detailed level of stator phase section 1 (similar for other stator/rotor sections)(lower). ECN-E

110 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Brake system Description For thermic modelling of the mechanical brake, only the thermic behaviour of the disc and linings are of relevance. After activation of the brake, stiff spring pushes the linings of the calliper with a constant force against the outer ring of the disc (see also section par. 3.5). The friction torque decellerates the shaft speed and power dissipation will heat up the disc (and linings) during braking. The friction heat is transferred by conduction from the outer ring to the inner ring and the whole disc is cooled due to radiation and convection. For monitoring purposes the temperature of the linings are measured [27] [25]. When a parking brake is (accidentlly) applied during full operation it will result in significant heat production and certainly cause great damage to the brake. Definition The thermic behaviour of the mechanical brake is shown in fig.(4.22). The dynamics are deter- Figure 4.22: Thermic generator system mined by the thermic capacities of the braking track (outer ring) and the inner ring of the disc) and the heat conductivities (mutual and ambient). The heat source is the dissipated braking power during braking, which is developed at the braking area. Because of the disc rotation the temperature of the braking lining and the outer ring is assumed to be the same. The temperature increase of the inner ring will follow the outer ring rapidly via heat conduction, however, a fraction of the braking power has already transferred to the surrounding. The brake model should comprise the following features: power dissipation and thermodynamic behaviour of the braking track thermodynamic behaviour inner ring of the disc heat transfer between the disc rings and to the surrounding temperatures of the inner ring and brake lining (=outer ring) 100 ECN-E

111 4 TURBINE DYNAMIC SYSTEMS Figure 4.23: Simulink model of interactions of thermic brake system Obviously, the brake system interacts with other turbine systems (fig.(4.23)). The brake power losses Q brk,loss are calculated at the brake system (par. 3.5). The disc temperatures are available for monitoring and control. The heat power from the brake disc to the nacelle interior is used to calculate the nacelle interior temperature, which is an input for the thermic brake system. Thermic brake model The thermic model is based on two thermic storages (inner disc, outer disc) which exchange heat power to each other and to the surrounding. The heat power source (brake loss) is activated when braking starts (see par. 3.5). The subscript notations in the model equations are : o (outer ring of the disc), i (inner ring of the disc), s (surrouding). The temperatures of the disc θ o, θ i are assumed to be homogeneous. The thermic model of the brake disc is described in eq.(4.61) θ o = 1 ( ) Q C brk,loss Q os Q oi dt + θo 0 o θ i = 1 (Q C oi Q is ) dt + θi 0 i Q oi = G oi (θ o θ i ) (4.60) Q os = G os (θ o θ s ) Q is = G is (θ i θ s ) Thermic brake model parameter values The parameters for the thermic model are derived from the maximum temperature of the outer disc track, θo max, when braking under design conditions. These design conditions are appointted as the design brake torque, speed and temperature: T dsgn brk,ωdsgn brk θdsgn brk. Although, there s no difference between a safety brake model and a parking brake model, the latter requires much lower design values. The time domain model from eq.(4.61) is transformed to Laplace domain and described in eq.(4.62) by using the Laplace operator s : θ o (s) = H θo Q brk (s) Q brk (s) + H θo θ i (s) θ i (s) (4.61) θ i (s) = H θi θ o (s) θ o (s) If we define for the outer ring G o =G os +G oi, τ o =C o /G o and for the inner ring G i =G is +G oi, τ i =C i /G i, the transfer functions can be written as: H θo Q brk (s) = 1 G o τ o s ECN-E

112 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems H θi θ o (s) = G oi G i τ i s H θo θ (s) = G oi 1 i G o 1 + τ o s (4.62) The transfer function H θo Q (s) implies the direct first order temperature increase due to brake brk loss power. Because the inner disc temperature increase is a first order response (H θi θ o ) on θ o, backward response from the θ i to θ i is initially of less importance. An expression for the maximum temperature increase and the time to achieve this value, can be derived from a inverse talud response on H θo Q (s). An inverse talud response is used to approach the power decreases during braking (constant braking torque, linear speed decrease) and it characterised by brk the initial braking power and the braking time to standstill: ( ) s α Q brk (s) = Q 0 brk s 2 where : α = 1/t brk (4.63) Partial fraction expansion of H θo Q (s) Q brk brk (s), inverse Laplace transformation to the time domain and equating the time derivative of the result to zero, results in the maximum temperature increase time t max θ and value θo max : t max θ = τ o ln ( α τo 1 + α τ o θo max = Q0 brk (1 α t max G o = Q0 brk τ o C o ) θ ) (4.64) ( ( )) α τo 1 + α τ o ln 1 + α τ o (4.65) If the braking time tot standstill t brk is determined from the total drive train inertia, initial speed and braking torque, the time constant τ o, can be determined implicitely from eq.(4.65). The values of the thermic capacitors C o and C i or derived from the material density ρ and heat coefficient c and their pertaining volumes V ( radii, disc thickness): C x = c x ρ x V x with : x {o, i} (4.66) If we assume that a fraction f Q o os of the outer disc ring power and a fraction f Q i is ring power is transferred to the surrouding by radiation and convection: of the inner disc G os = f Q o os G os G is = f Q (4.67) o is G is all parameters can be determined using the previous results τ o, C o and C i : G o = C o τ o G i = f Q o os ) G o G oi = (1 f Q o os ) G o (4.68) G is = f Q i is G 1 f Q i oi is Although, these results are quite accurate, there s a deviation of the maximum temperature in relation to the full model, which is caused by neglecting the inner disc indirect effects. To achieve an accurate solution, an iterative loop can be programmed. Starting with the previous result of G o this parameter can be updated such that the error between the desired θo max and the actual value is minimised. 102 ECN-E

113 4 TURBINE DYNAMIC SYSTEMS Thermic brake implementation Fig.(4.24) shows the Simulink implementations of the brake model Fig.(4.25), shows a thermal Figure 4.24: Simulink model of thermic brake system. design result of a mechanical brake when brake at design conditions (left hand side) and at half braking power (right hand side). ECN-E

114 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Src:E:\vdhooft\Projects\PrSimDev\v1\ps\BrkDsgn.ps; :06:18 Brake temperatures; <r b b > = [outer, inner, discavg] Brake temperatures; <r b b > = [outer, inner, discavg] at tbrk=21.3s, Pbrk=1626.1kW at tbrk=10.6s, Pbrk=813.0kW Tmp [dgc] time [s] Tmp [dgc] time [s] Brake power to surrounding; <r b b > = [OS, IS, OIS] at tbrk=21.3s, Pbrk=1626.1kW 400 Brake power to surrounding; <r b b > = [OS, IS, OIS] at tbrk=10.6s, Pbrk=813.0kW Power [kw] 200 Power [kw] time [s] time [s] Figure 4.25: Stationary thermic characteristics of mechanical brake 104 ECN-E

115 4 TURBINE DYNAMIC SYSTEMS 4.5 Electric conversion (ECS) Modelling in a dq0-reference frame Introduction The models of all electrical components are derived in the dq0-reference system. To obtain these models from the standard abc-models, the Park Transformation is used. This transformation is well-known from its use in dynamic models of electrical machines. The electrical signals, i.e. voltages, currents and magnetic fluxes, are transformed to a stationary rotating reference frame. As this stationary frame is chosen to rotate with the grid frequency, all voltages, currents and fluxes in the dq0-reference frame are constant in steady state situations. Therefore, modelling in the dq0-reference frame is expected to increase the simulation speed significantly, as a variable step-size simulation program can apply a large time step during quasi steady-state phenomena. Since balanced conditions are assumed (equal phase conditions), the 0-component can be disregarded. For a description of the way in which models of different electrical components in the dq0- reference frame are obtained is referred to [15]. In this chapter, the models of the doubly fed induction generator (DFIG) will be described, together with the power electronic converter model. The control of the generator will also be described. This is followed by a description of the models for other electrical components in the wind turbine: the cable and three way transformer. This chapter concludes with the implementation of the models in Simulink and a simulation example. Park Transformation In the study of power systems, mathematical transformations are often used to decouple variables, to facilitate the solution of difficult equations with time-varying coefficients, or to refer all variables to a common reference frame [28]. Probably the most well-known, is the method of symmetrical components, developed by Fortescue. This transformation is mostly used in its time-independent form and applied to phasors, when it is used in electrical power system studies [29]. Another commonly-used transformation is the Park transformation, which is well-known from the modelling of electrical machines. The Park transformation is instantaneous and can be applied to arbitrary three-phase time-dependent signals. The electrical signals are transformed to a stationary rotating reference frame. As this stationary frame is chosen to rotate with the grid frequency, all voltages and currents in the dq0-reference frame are constant in steady state situations. For θ d = ω d t + ϕ, with ω d angular velocity, t the time and ϕ initial angle, the Park transformation is given by: [x dq0 ] = [T dq0 (θ d )] [x abc ] (4.69) with: and [x dq0 ] = [x abc ] = x d x q x 0 x a x b x c (4.70) (4.71) ECN-E

116 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems and with the dq0-transformation matrix T dq0 defined as: ( ) cos θ d cos θ d 2π 2 ( 3 ) [T dq0 (θ d )] = 3 sin θ d sin θ d 2π 3 and its inverse given by: [T dq0 (θ d )] 1 = ( ) cos θ d + 2π ( 3 ) sin θ d + 2π cos θ d sin θ ( ) ( d ) cos θ d 2π 3 sin θ d 2π ( ) ( 3 ) cos θ d + 2π 3 sin θ d + 2π (4.72) (4.73) The positive q-axis is defined as leading the positive d-axis by π/2, as can be seen from figure Some additional properties of the Park transformation can be derived. As the transformation is orthogonal: [T dq0 (θ d )] [T dq0 (θ d )] 1 = [T dq0 (θ d )] [T dq0 (θ d )] T = [I] (4.74) Figure 4.26: Relationship between abc and dq With equation 4.74 it can be shown that the Park transformation conserves power and therefore is a valid transformation. The power conservation principle can then be shown as follows: P (t) = [v abc ] T [i abc ] [ = [ T dq0 (θ d )] 1 T [v dq0 ]] [ Tdq0 (θ d )] 1 [i dq0 ] = [v dq0 ] T [ [ T dq0 (θ d )] 1] T [ Tdq0 (θ d )] 1 [i dq0 ] = [v dq0 ] T [ T dq0 (θ d )] [ T dq0 (θ d )] 1 [i dq0 ] = [v dq0 ] T [i dq0 ] (4.75) Note that by replacing the factor 2/3 by a factor 2/3 in (4.72) and (4.73) the transformation will be amplitude-invariant, implying that the length of the current and voltage vectors in both abc- and dq0-reference frame are the same, however in that case the conservation of power is lost. This amplitude-invariant transformation is mostly used in modelling of electrical machines [29]. 106 ECN-E

117 4 TURBINE DYNAMIC SYSTEMS The voltages and currents in the dq0-reference frame are constant in steady-state situations. Note that non-fundamental harmonics are correctly transformed as x a, x b and x c are time signals, including all harmonics. In steady state a non-fundamental frequency component with frequency ω h will appear as a sinusoidal signal with frequency (ω h -ω d ) in the dq0-domain. The highest frequency that can be represented accurately in the dq0-frame depends on the time step that is used. With electric machines the d-axis is mostly chosen along the stator flux, which implies that i q corresponds to real power and i d to reactive power (see (4.76)), since ideally v d = 0. In general the voltages will be phase shifted with respect to the d-axis which means that active and reactive power cannot be related directly to the d and the q axis component (v q 0). The instantaneous active and reactive power can be obtained directly from the voltages and currents in the dq0-reference system [Aka 84]: P = v d i d + v q i q Q = v q i d v d i q (4.76) Doubly-Fed Induction Machine model In this section a description will be given of the Doubly-Fed Induction Generator model. The Doubly-Fed Induction Generator has a converter connected to the rotor instead of to the stator windings. The advantage is that variable speed operation of the turbine is possible with a much smaller and therefore also much cheaper converter. The power rating of the converter is often about 1/3 of the generator rating. A schematic drawing of a wind turbine with Doubly-Fed Induction Generator is shown in figure First a description of the generator model will be given. Section describes the controller model. Figure 4.27: Wind turbine with Doubly-Fed Induction Generator Generator model A dq reference frame is chosen to model the doubly-fed induction generator (see section 4.5.1). The model that is used is well known [30], [28]. The generator convention will be used, which means that the currents are outputs instead of inputs and real power and reactive power have a positive sign when they are fed into the grid. Using the generator convention, the following set of equations results: ECN-E

118 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems with v ds = R s i ds ω s ψ qs + dψ ds dt v qs = R s i qs + ω s ψ ds + dψqs dt v dr = R r i dr ω r ψ qr + dψ dr dt v qr = R r i qr + ω r ψ dr + dψqr dt (4.77) ψ ds = (L s + L m ) i ds L m i dr ψ qs = (L s + L m ) i qs L m i qr ψ dr = (L r + L m ) i dr L m i ds ψ qr = (L r + L m ) i qr L m i qs (4.78) with v the voltage, R the resistance, i the current, ω s and ω r the stator and rotor electrical angular velocity respectively, L m the mutual inductance, L s and L r the stator and rotor leakage inductance respectively and ψ the flux linkage. The indices d and q indicate the direct and quadrature axis components of the reference frame and s and r indicate stator and rotor quantities respectively. All voltages, currents and fluxes are functions of time. The electrical angular velocity of the rotor, ω r, equals: ω r = ω s pω m (4.79) with p the number of pole pairs and ω m the mechanical angular velocity. The electromagnetic torque of the generator is given by: T e = p (ψ dr i qs ψ qr i ds ) (4.80) The synchronously rotating dq reference frame is oriented with the direct d-axis along the stator flux vector position: so ψ qs = 0. This implies that in steady state v ds 0. In this way a decoupled control between the electrical torque and the rotor excitation current is obtained. In steady state the vectors of all stator and rotor voltage, current and flux, as seen by an observer fixed to the stator, rotate with the same speed. During transients, the angles between the vectors change and the speeds differ accordingly. The angle of the stator flux vector can be calculated as: θ s = ω s dt (4.81) The reference frame as seen by an observer on the rotor is rotating with the electrical frequency of the rotor ω r. The angle of the rotor can be obtained as: θ r = ω r dt = (ω s pω m )dt (4.82) With the dq0-transformation used in (4.72) the active power delivered by the stator is given by: and the reactive power by: P s = v ds i ds + v qs i qs (4.83) Q s = v qs i ds v ds i qs (4.84) Due to the chosen reference frame, ψ qs and v ds are zero. Therefore the reactive power and the active power delivered by the stator can be written as: P s = v qs i qs = v qs ( Lm L r + L m ) i qr (4.85) and: Q s = v qs i ds = ω s ( (L s + L m ) i ds L m i dr ) i ds (4.86) 108 ECN-E

119 4 TURBINE DYNAMIC SYSTEMS As the stator current is equal to the supply current, it can be assumed that it is constant. If the frequency is also constant, the reactive power is proportional to the direct component of the rotor current i dr : Q s = K 1 + K 2 i dr (4.87) With constants K 1 and K 2 : and K 1 = ω s (L s + L m ) i 2 ds (4.88) K 2 = ω s L m i ds (4.89) Converter model Introduction As can be seen from figure 4.27, the Doubly-Fed Induction Generator, has a converter connected to its rotor windings. The generator side converter is used to control the rotor currents of the machine. With this rotor currents, the active power (or indirectly the rotational speed) and reactive power of the machine can be controlled according to (4.83)-(4.84). The grid-side converter draws or supplies power to the rotor-side converter and is operated to keep the DC-link voltage constant. For the doubly-fed induction generator it should be possible to transport power in both directions, and therefore a back-to-back converter consisting of two Voltage Source Converters (VSCs) and a DC link is used. The converter is shown in figure The DC link separates the two Voltage Source Converters, which can be controlled independent of each other. Therefore, only one converter has to be considered. To obtain sinusoidal line currents, a filter can be placed between the converter and the grid. The phase voltages are referred to the node n. The value of the arbitrary voltage reference node n depends on the circuit configuration. It should not be confused with the neutral. The line voltages can be derived from the phase voltages. For example the voltage v ab1 is: v ab1 = v an1 v bn1 (4.90) Figure 4.28: Back-to-back converter In the following sections the grid side converter (inverter) is described. The machine side converter (rectifier) is operated in a slightly different fashion. The controller of the grid side converter will be based on a dq0-reference frame, implementing the vector control method. All signals will be constant in steady-state and therefore PI controllers can be used to realise the reference values without steady-state errors. A triangular carrier based Pulse Width Modulation scheme is used to control the switches of the converter. The controller is based on two control loops. The inner loop is a current controller, which get its reference from the outer loop controller, which can be for example a reactive power or torque controller. A block diagram of a PWM converter with a vector controller is shown in figure ECN-E

120 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.29: Scheme of PWM Voltage Source Converter with controller Switching Function Concept The switching function concept has been used to model the converter [31]. Using this concept, the power conversion circuits are modeled according to their functions, rather than to their circuit topologies. The switching function concept will be described shortly with reference to the circuit configuration of a VSC as shown in figure 4.30 and the type of voltages that are generated at the AC side. It is well known that with voltage source converters pulsating voltages are generated at the AC side. In figure 4.33 an example of the voltage V an is shown. This voltage is obtained by alternatively switching the upper and the lower switch in phase a. The ON/OFF control signals for the switches are generated by Pulse Width Modulator (PWM). An example of the principle of such a modulator is shown in figure 4.33, where the desired output voltage V ref is compared with a triangular carrier. Whenever V ref >V tri the upper switch is closed, and when V ref >V tri, the lower switch is closed. In this way the output waveform has the same shape as the output signal of the comparator. For each phase leg a separate modulator is used, where the reference voltages are displaced over 120 or 240 degrees respectively. The output voltage of a phase can mathematically be described as the product of the logical output signal of the comparator, also called switching function SF a of phase a, and the DC link voltage [31]: V an = SF a V d (4.91) The voltage V an can also be obtained with the circuit from figure 4.31, where controllable voltage sources are applied instead of switches in phase legs. The controllable voltage sources are controlled by the same signals as with the phase leg after multiplication by V d. The mode of figure 4.31 is obtained then. The switching functions can be expressed as Fourier series. SF = A n sin (nωt) (4.92) n=1 It can be shown that in the lower frequency range the frequency components of SF V d and V an are equal if the frequency of the carrier is sufficiently large [32]. When the complete PWM-operation, or even the switching functions, of the VSC has to be taken into account, the model of the converter becomes complicated, and simulation will become very slow. If the filter is designed well, the higher harmonics that are generated by the switching process will be attenuated. It can be shown that, with a well-designed filter, in the lower frequency range the frequency components of the reference voltage and the practical obtained voltage are equal 110 ECN-E

121 4 TURBINE DYNAMIC SYSTEMS Figure 4.30: Voltage source converter Figure 4.31: Voltage source converter, switching function equivalent if the switching frequency is sufficiently large [32]. A further assumption is that the dc-link voltage V d is constant. In reality this isn t true, causing some higher frequency terms in the output signals. The resulting model is then shown in figure The whole system can then be replaced by a system, creating sinusoidal waveforms, exactly equal to the reference waveforms. One should be aware that this is only valid for frequencies far below the resonance frequency of the filter. In case of a grid-connected converter, with a grid-frequency of 50Hz, this requirement will be met and the model can be used for applications like voltage regulation, as long as normal grid operation is assumed. Figure 4.32: Voltage source converter, sinusoidal voltages equivalent ECN-E

122 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.33: Pulse Width Modulation (l) and switching function (r) for one phase of the voltage source converter Converter model behaviour during voltage disturbances It has been explained in the previous section that if the filter is designed well, the higher harmonics that are generated by the switching process will be attenuated. With a well-designed filter, in the lower frequency range the frequency components of the reference voltage and the practical obtained voltage are equal if the switching frequency is sufficiently large, i.e. f s >> f 0 (4.93) with f s the switching frequency and f 0 the fundamental harmonic of the voltage. During fast phenomena the voltage will also have higher harmonic terms and the condition (4.93) will no longer be valid. When there is no current control or voltage control applied, the harmonics will not be present in the reference voltages and the voltage that is made by the converter is still a good representation of the reference voltage. When control is applied, the reference voltage will also have the higher harmonics in most cases, and the representation between the voltage that is made and the reference voltage isn t correct any longer. To investigate whether the models based on the switching function concept can be used during disturbances the reduced model has been compared to a reference model. The SimPowerSystems Blockset [33] of Matlab has been used to obtain this reference model of the converter. The universal bridge model with IGBTs has been used. This block of the SimPower Systems Blockset implements a 3-phase bridge converter with 6 IGBT switches with antiparallel diodes. RC-snubber circuits are included in the IGBT-models. Typical parameters such as rise and fall times and voltage drop can be defined in the model. The voltages and currents are measured and transformed to the dq0-reference system. Sample-and-hold circuits are implemented in the measurement loops. The measured voltages and currents are filtered with low-pass filters with a cut-off frequency of 200 Hz. Ordinary PI controllers are used to obtain the desired currents. The reduced model described in the previous section, has been compared to the full model. A three-phase balanced voltage dip of 70% (the RMS value of the grid voltage is reduced to 30% of its pre-fault voltage) has been simulated. The grid voltage is shown in figure The converter current supplied to the grid is shown in figure The behaviour of the reduced model of the converter has been compared to the behaviour of the full converter. In order to make comparison easier, the currents have been compared in the dq0-reference frame. These current are constant in steady-state situations, which makes it easier to compare. The d-axis current for the reduced and the full model are shown in figure 4.36: on average the currents of the two models are the same and the initial peaks at the moment of the step in voltage are also equal. The difference is due to the switching of the converter in the full model. When the switching frequency is sufficiently high, these high-frequency terms will be attenuated by a filter. More information on the comparison can be found in [34]. 112 ECN-E

123 4 TURBINE DYNAMIC SYSTEMS Figure 4.34: Ideal voltage dip Figure 4.35: Converter current Figure 4.36: Converter current in d-axis for complete model (solid line) and reduced model (dashed line) ECN-E

124 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.37: Modes of operation of the DFIG system Control of the DFIG converters The Doubly Fed Induction Generator can be operated in four modes: motor and generator operation at subsynchronous as well as supersynchronous speed (figure 4.37). In a wind turbine only generator operation is intended but this still requires bi-directional operation of the rotor converter. In this section the control of the converter is described. In the DFIG system four control parameters are available: the d- and q-current in the rotor and the d- and q-current from the converter to the grid. The d- and q-current in the rotor are controlled by the machine side converter (the rectifier, although not always operating as a rectifier). The d- and q-current from the converter to the grid are controlled by the grid side converter (inverter). The current components are controlled to realise four system control objectives: the electromagnetic torque and the reactive power of the stator of the induction machine are controlled by master controllers which operate on the rectifier current control loops; the DC link voltage and the reactive power of the inverter are controlled by master controllers which operate on the inverter current control loops. Control of the DFIG rectifier The DFIG rectifier control then consists of two decoupled master-slave loops, see figure The inner loops determine the voltage applied to the rotor. The performance of these loops are improved by cross-voltage compensators, which cancel the effect of the stationairy flux contributions in the voltage equations. Although the induction machine is a multi-input-multioutput system with strong coupling, the use of the Park transformation and the cross-voltage compensation reduces the system to a decoupled two-input-two-output system. This makes it relatively easy the determine suitable control settings. The current controller proportional and integral action will be determined by the Internal Model Control (IMC) method [35, 36], which will be described first. The generic layout of a single input-single output (SISO) controller is 114 ECN-E

125 4 TURBINE DYNAMIC SYSTEMS Figure 4.38: Control of the rectifier of the DFIG system Figure 4.39: Internal model control (IMC) method, reduced given in figure The objective is to keep the difference between the controlled signal C and the reference signal R as small as possible. The signal L is a disturbance operating on the controlled signal C. The IMC method assumes the availability of an approximate process model G p. The process controller then consists of a controller G c in series with the process and an approximate process model parallel to the process, see figure The total controller G c, see figure 4.39 then equals: G c G c = 1 G G (4.94) c p If a desired system response is defined based on the transfer function of the process G p, G c contains the reciprocal of G p and the poles of G c equal the zeros of G p (pole-zero cancellation). If G p has positive zeros, this will lead to a G c with positive poles, which is undesirable (open loop unstability). Therefore G p is split in a part with positive G p+ and negative zeros G p and only the part with negative zeros is included in G c. The IMC controller is chosen as follows: G c = 1 G p τ c s (4.95) ECN-E

126 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.40: Internal model control (IMC) method If G p = G p and L = 0 the closed loop transfer function is: C R = G p G c 1 + G c(g p G p ) = G pg c = τ c s (4.96) So, this choise results in a first order response with unity gain. The only parameter in the controller design is time constant of the response τ c. DFIG model to be controlled The IMC method was used to design the current controllers of the DFIG. The current controllers are the inner loops of the master-slave control loops on the electromagnetic torque and the reactive power of the stator. The current loops set the rotor voltages generated by the rotor side converter. So the process input signals are the two rotor voltages v dr and v qr. The relation between rotor voltages and currents i dr and i qr (process inputs and outputs, respectively) are: v dr = d R r i dr L r dt i dr ω r ψ qr (4.97) v qr = d R r i qr L r dt i qr + ω r ψ dr (4.98) (4.99) The transfer functions between the voltages and the rotor currents become relatively simple if the cross-coupling terms are subtracted before the currents are controlled: v dr = d R r i dr L r dt i dr (4.100) v qr = d R r i qr L r dt i qr (4.101) (4.102) The controller sets the v dr and v qr and the cross-coupling terms are then added to the output of the controller to get the correct process input signals again. The process transfer function for both current controller is: G p = G p = i 1 v = (4.103) R r + L r s This transfer function does not include positive zeros, so it can be used directly to generate the total controller G c. 116 ECN-E

127 4 TURBINE DYNAMIC SYSTEMS G c = 1 1 G p 1 + τ c s = R r + L r s 1 + τ c s (4.104) G c = G c 1 G c G p = R r τ c s L r τ c (4.105) This is equal to the result in Erao-2, exept for the minus signs. The reason for this difference is a different sine convention in the flux equations, also leading to the opposite sign of the induction matrix of the induction machine. Control parameter values Equation specifies the control parameters according to the IMC method. The R r and L r values are given, the τ c can be chosen. Since we have experimented with the individual settings of the proportional and integral terms in the controllers, I will first go back to this IMC result and test various τ c settings. Using a time constant τ c = 0.01 s, the controller settings are: The controller settings are: for both d and q current directions. L r = (4.106) R r = (4.107) K p = L r τ c = 1.85 (4.108) K i = R r τ c = 0.52 (4.109) k p = 0.2 (4.110) k i = 3.2 (4.111) Control of the DFIG inverter The grid side converter (inverter) is controlled in a similar way as the DFIG machine side converter (rectifier), viz. by adjusting the amplitude and the phase of the output AC voltage, through the d- and q-components of the voltage, the d- and q-components of the current to the grid are controlled. The setpoints of the current components are determined by setpoint for the power and the reactive power. The setpoint for power to the grid is determined by a master control loop on the DC link voltage. The reactive power level can be chosen freely, the actual reactive power is limited by the maximum current of the inverter. Figure 4.41 illustrates the control of the grid converter. The main components are two PID-controllers and two crossvoltage compensators. The cross-voltage compensators result from the fact that the electrical components are described in dq-coordinates instead of the abc values of the individual phases [15] and compensate for the cross coupling of the dq voltage equations. The d- and q-currents currents are determined by the voltage difference over a small inductance at the grid side of the converter. The setpoints for the currents are determined from the power and reactive power setpoints. The converter control generates converter output voltages. The controller design will be explained with reference to the converter diagram in figure A vector-control approach is used for the supply side converter, with a reference frame oriented along the grid voltage vector (dq voltages and currents). The reference frame enables independent control of the active and reactive power flowing between the converter and the grid. ECN-E

128 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.41: Control of the grid side converter Figure 4.42: Three-phase full-bridge Voltage Source Converter The voltage balance across the inductors and resistors in figure 4.42 is: v a = v an v agn = L f dia dt + R f i a v b = v bn v bgn = L f di b dt + R f i b (4.112) v c = v cn v cgn = L f dic dt + R f i c With the Park transformation this equation can be transformed to the dq reference frame: v d = R f i d + L f di d dt + ω el f i q v q = R f i q + L f diq dt ω el f i d (4.113) The last term in both equations causes a coupling of the two equations, which makes it difficult to control both currents independently. This was also observed in the control of the rectifier and the same solution can be applied here. The last terms can be considered as a disturbance on the controller. Reference voltages to obtain the desired currents can be written as: with: v d = v d + ω el f i q v q = v q ω e L f i d (4.114) v d = R f i d + L f di d dt v q = R f i q + L f diq dt (4.115) Treating the cross-related terms as a disturbance, the transfer function from voltage to current of (4.115) can be found as (for both the d- and the q-component): G p (s) = 1 L f s + R f (4.116) 118 ECN-E

129 4 TURBINE DYNAMIC SYSTEMS Using the Internal Model Control Method (see section 4.5.4) to design the current controllers yields: k p = L f τ c ; k i = R f τ c (4.117) The active and reactive power delivered by the converter are given by: P = v dg i d + v qg i q Q = v qg i d v dg i q (4.118) with the d-axis of the reference frame along the stator-voltage position, v q is zero and as long as the supply voltage is constant, v d is constant. The active and reactive power are proportional to i d and i q then. DC-link controller The capacitor in the dc-link is an energy storage device. Neglecting losses, the time derivative of the stored energy must equal the sum of the instantaneous rotor power P r and grid power P g : de c dt = 1 2 C d ( vdc 2 ) = P r P g (4.119) dt This equation is nonlinear with respect to v dc. To overcome this problem a new state-variable is introduced: Substituting this in (4.119) gives: W = v 2 dc (4.120) 1 2 C dw dt = P s P g (4.121) which is linear with respect to W. The physical interpretation of this state-variable substitution is that the energy is chosen to represent the dc-link characteristics [37]. With the dq-reference frame of the current controller along the d-axis, (4.121) can be written as: 1 2 C dw dt = P s v d i d (4.122) and the transfer function from i d to W is then found to be: G (s) = 2v d sc (4.123) As this transfer function has a pole in the origin control will be difficult. An inner feedback loop for active damping will be introduced [37]: i d = i d + G a W (4.124) With G a the active conductance, performing the active damping, and i d the reference current provided by the outer control loop, see figure Substituting (4.124) into (4.122) gives: 1 2 C dw dt = P s v d i d v d G a W (4.125) ECN-E

130 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Which is shown in figure The transfer function from i q to W becomes [37]: G 2v d (s) = (4.126) sc + 2v d G a Using Internal Model Control and since (4.126) is a first-order system, the following controller is proposed: F (s) = α d 1 s G (s) = α dc α dg a (4.127) 2v d s This is a PI-controller. A suitable choice is to make the inner loop as fast as the closed-loop system [37]. When the pole of G (s) is placed at -α d the following active conductance is obtained: This gives the following PI-controller parameters [37]: G a = α dc 2v d (4.128) k p = α dc 2v d, k i = α2 d C 2v d (4.129) The controller is completed by a feedforward term from P s to i q. Figure 4.43: DC-link controller structure Other electrical component models The models of the electrical components in a wind turbine are completed by a cable and a transformer model. Cable model The general equations relating voltage and current in a cable recognize the fact that all impedances of a cable are uniformly distributed along the cable. The single-phase equivalent circuit of the lumped cable model is shown in figure Figure 4.44: Single-phase equivalent circuit of a transmission line 120 ECN-E

131 4 TURBINE DYNAMIC SYSTEMS In the model used in the turbine, the two shunt capacitances are taken together on one side of the cable segment. The cable model then consists of a three-phase shunt capacitor and a three-phase series resistance and inductance. The dq0-models of a three-phase shunt capacitor and a three-phase series resistance and inductance are described in [15]. The voltage across the resistance and inductance is given in by: v d = v d2 v d1 = R a i dl + L a di dl dt ωl ai ql v q = v q2 v q1 = R a i ql + L a di ql dt + ωl ai dl (4.130) v 0 = v 02 v 01 = (R a + 3R g ) i 0 + (L a + 3L g ) di 0 dt The current through the shunt capacitances is given in by: i dc1 = C dv d1 dt i qc1 = C dv q1 dt i 0c1 = C dv 01 dt ωcv q1 + ωcv d1 (4.131) The resulting cable model equivalent circuits for the d-, q-, and 0-axis are shown in figure Figure 4.45: Cable model equivalent circuits in dq0-coordinates Note that it is assumed that the shield of the cables is grounded, which is true in most cases. The shunt capacitor C in the cable model thus represent the capacitance between cable and shield. The 0-model for the resistance and inductance assumes that a ground return exists. As the cable shield is grounded the ground return exists and the 0-model of figure 4.45 can be used. The 0-model is only relevant for unbalanced conditions however. ECN-E

132 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Transformer model Wind turbines are connected to the grid by a three-phase transformer. The transformer model to be used will be described in this section. A single-phase equivalent circuit of a three-phase two-winding transformer is shown in figure Normally the magnetising current i m is small and can be neglected. The model of figure 4.46 can then be reduced to the model shown in figure 4.47, with R=R 1 +a 2 R 2 and L=L 1 +a 2 L 2, with a=n 1 /N 2. Figure 4.46: Single-phase equivalent circuit of a transformer Figure 4.47: Transformer equivalent circuit with magnetising current neglected The voltage-current relationship of the single-phase equivalent transformer of figure 4.47 can be written as: v 1 = Ri 1 + L di 1 dt + av 2 (4.132) with R = R 1 + a 2 R 2 and L = L 1 + a 2 L 2. A capacitor will be added to the primary side of the transformer. The capacitance can be seen as representing the winding capacitance of the transformer and is primarily needed for numerical reasons. The effect on the results is small. The dq0-model of the transformer is shown in figure The zero-sequence part of the model depends on the type of transformer (star-start, star-delta, etc.), in this case it represents a stardelta transformer with a grounded star at the primary side [38]. The voltage equations for the dq0-model are: i dc1 = C dv d1 dt v d1 av d2 = Ri dr1 ωli qr1 + L di dr1 dt i qc1 = C dv q1 dt v q1 av q2 = Ri qr1 + ωli dr1 + L di qr1 dt i 0c1 = C dv 01 dt v 01 = Ri 0r1 + L di 0r1 dt ωcv q1 (4.133) (4.134) + ωcv d1 (4.135) (4.136) (4.137) (4.138) 122 ECN-E

133 4 TURBINE DYNAMIC SYSTEMS i 0c2 = 0 (4.139) (4.140) Figure 4.48: Equivalent transformer model for d, q and zero sequence componenents Implementation of the DFIG model in Simulink Figure 4.49 gives an overview of the DFIG model in Simulink. A three-winding transformer model connects the invertermodel and the stator of the induction machine model to the grid. The torque and reactive power controllers generate the d- and q-voltage of the induction machine rotor. Figure 4.50 shows the induction machine model. The main blocks evaluate the voltage equations, the electomagnetic torque, the slip and the mechanical speed. A signal bus is used to have easu access to all relevant signals. Figures 4.51 and 4.52 show the master-slave control loops for torque and reactive power in the stator. The DC link including its control is shown in figure Figure 4.54 shows the model of the inverter, consisting of the current controller and a grid impedance. The transformer and cable model in figure 4.55 and 4.56 are similar, due to the same basic components: resistance, inductance and capacitance Response of the DFIG system to a setpoint change and a disturbance Figure 4.57 demonstrates the DFIG model for a step in the torque setpoint from 0 to 1 MNm (low speed shaft side). The figure shows the electromagnetic torque T e (high speed shaft side), the speed of the generator wm (which is almost constant since the mechanical torque is increased to the same level as the torque setpoint), the stator power P s, the stator voltage vs, the stator ECN-E

134 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.49: Simulink implementation of the DFIG Figure 4.50: Simulink implementation of the induction generator current is and the stator reactive power Qs. The new torque value is reached in two seconds without overshoot. The reactive power oscillates and returns to zero in two seconds. Figure 4.58 shows the behaviour of the converter and DC link. The figure shows the DC voltage Udc, the DC current at the inverter side Idc, the difference between the inverter and the rectifier 124 ECN-E

135 4 TURBINE DYNAMIC SYSTEMS Figure 4.51: Simulink implementation of the electromagnetic torque control Figure 4.52: Simulink implementation of the stator reactive power control DC current, the rotor voltage vr, the rotor current ir and the power transferred to the grid by the converter P rconv. The DC voltage remains practically constant while the DC current decreases to about 270 A. The rotor voltage shows a peak of about 70 V and the rotor increases from about Figure 4.53: Simulink implementation of the DC link ECN-E

136 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.54: Simulink implementation of the inverter Figure 4.55: Simulink implementation of the transformer 330 A to 850 A. The initial rotor current is not zero although the torque and the DC current are. The current corresponds to the reactive current in the rotor which enables zero stator reactive power. The rotor power decreases to about -0.3 MW. The speed is subsynchronous and the DFIG is in generator operation, so the rotor converter draws power from the grid. 126 ECN-E

137 4 TURBINE DYNAMIC SYSTEMS Figure 4.56: Simulink implementation of the cable Figure 4.57: DFIG model response to a step in the electromagnetic torque setpoint Figure 4.59 shows the response of the DFIG model to a 33% dip of 0.2 s. in the grid voltage. The setpoint of the electromagnetic torque and the mechanical torque are constant. The electromagnetic torque and the stator power oscillate violently. Similar oscillations occur in the stator current and the stator reactive power. After the voltage has recovered, the oscillation quickly damp out. In figure 4.60 the reaction of the converter system to the voltage dip is shown. The DC link voltage and current are not as severly affected by the dip as the stator current. The rotor voltage however shows peaks which exceed the capability of the converter and demonstrate the need for ECN-E

138 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 4.58: DFIG model response to a step in the electromagnetic torque setpoint a voltage ride through controller. Figure 4.59: DFIG model response to a grid voltage dip 128 ECN-E

139 4 TURBINE DYNAMIC SYSTEMS Figure 4.60: DFIG model response to a grid voltage dip ECN-E

140 130 ECN-E

141 5 EXTERNAL INFLUENCES In this chapter simulation blocks for the external influences are dealt with. These blocks realise the generation of excitation signals from the wind, waves, grid and gravitation. The involved models are highly generic. The external influences are the wind, the waves and current, and the behaviour of the grid and gravitation. Wind and waves are treated in the next two sections, grid and gravity are briefly discusses. 5.1 Wind Generation (WIN) A compact real-time simulation setting is obtained by the use of so called blade effective wind speed signals instead of a complete three-dimensional wind field. The three wind speed signals are designed such that they cause realistic blade root loads; the loads are similar to those that arise when a rotating rotor blade samples a homogeneous turbulent wind field for longitudinal turbulence only, affected by wind shear and tower shadow. Also oblique inflow is accounted for, viz. (i) in the way the longitudinal turbulence is sampled and (ii) by the appearance of periodic tangential wind velocity components through the average sideway and vertical wind speed in the rotor plane; in-plane turbulence is not taken into account. In the following three subsections it is described how 3D-turbulence is handled in respect of blade effectiveness, how tower shadow and wind shear are accounted for, and how yaw misalignment and inclination in the wind field are carried through in the blade effective turbulence signals and periodic signal components. In the fourth subsection the overall realisation scheme for the blade effective turbulence signals in case of variable rotor speed and oblique inflow is presented Blade effective turbulence Fig.(5.1) visualizes the rotational sampling of the wind field by the tips of the three rotor blades D, E and F. A point on a rotating blade experiences the wind velocity along a spiral-shaped trajectory (helix) in the wind cylinder. As a matter of fact, the longitudinal wind velocity component continuously drives disks in the wind cyclinder down the x-axis. Thus, a rotating point experiences the intersection of a helix with the rotor plane; in the figure, the rotating blades move forward along the x-axis at increasing rotor azimut angle Ψ. Since the three blades have an azimut separation of 120 o ( 2 π), each blade tip samples along a 3 specific helix. The derived blade effective turbulence signal effectively represents a weighted average of helix realisations over radial coordinate r for a blade. Actually, the three blade effective turbulence signals are obtained as a realisation-set for a 3 3 power spectrum matrix. This spectrum matrix is set up by the auto and cross power spectra of the r-dependent helices on and between the blades. At a constant rotational speed, the time separation between these helices is equal to the revolution time ( 2π Ω ) divided by the number of blades (3). Consequently, the same non-moving location in the wind cylinder is sampled every 2π seconds. The latter is of importance when 3 Ω oblique inflow is considered in the context of a limited number of additional helix realisations. The following three paragraphs deal with (a) rotational wind sampling by a point on a rotor blade, (b) the mapping of spectral properties of the wind speed in rotating points to those that are effective for three rotating blades, and (c) the algorithm for obtaining a time-domain realisation of three blade turbulence signals from the spectra. Rotational wind sampling by a point on a rotor blade For a disk A in a wind cylinder with homogenous turbulence, a time-dependent Fourier expan- ECN-E

142 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 5.1: The sampling of helices in the wind field by three rotor blade tips. sion holds for the wind speed u on radius r in azimut angle ψ [39]: u A (t, ψ, r) = p= e jpψ û A p (t, r), û A p (t, r) = 1 2π 2π 0 e jpφ u A (t, r, φ) dφ (5.1) The Fourier coefficients û A p (t, r), the rotational modes, are time-dependent. A rotational mode is a harmonic basis function over the circle on radius r of the wind field. The time-dependency represents the evolution of the amplitude and phase of such a rotational mode, which mainly occurs in frequencies below 0.1Hz 2. Fig.(5.2) shows that blade D samples the turbulence on time t a = 0 in disk A a of the wind cyclinder for 0 azimut. On time t b = 2 3 π Ω this occurs for azimut 2π in disk A 3 b. Note that disk A b has moved downward the x-axis to the former position of disk A a, viz. the rotor plane! For the two time instants t a and t b the following azimutal expansions apply for the experienced modes 2 In a linearised approach, tower shadow and wind shear can be considered as the 0 Hz contents of the rotational 132 ECN-E

143 5 EXTERNAL INFLUENCES Figure 5.2: The sampling of the radius-r helix by rotor blade D wind speed on radius r in the rotor plane on respective azimut position ψ a and ψ b : u Aa (t a, ψ a, r) = u A b(t b, ψ b, r) = e jpψa û Aa p (t a, r), û Aa p (t a, r) = 1 p= e jqψ bû A b q (t b, r), û A b q (t b, r) = 1 q= 2π 2π 2π 0 2π 0 e jpφ u Aa (t a, r, φ) dφ e jqα u A b (t b, r, φ) dα (5.2) Since the wind disk A a coincides with the rotor plane on time t a and A b does on t b, we can omit the wind disk superscript under the notion that every variable is manifest in the rotor plane. Because of the rotional speed Ω, the time and azimutal position are related. When we assume constant speed Ω, the wind speed u Aa (t a, ψ a, r) becomes the rotationally sampled wind speed ũ D (t a, r) for blade D for ψ a = Ω t a. We then can rewrite the expressions above as: ũ D (t a, r) = ũ D (t b, r) = p= q= e jp Ωtaû p (t a, r), û p (t a, r) = 1 2π e jq Ωt b û q (t b, r), û q (t b, r) = 1 2π 2π 0 2π 0 e jpφ u(t a, r, φ) dφ e jqα u(t b, r, α) dα (5.3) At the end of this paragraph it will be proved that, in case of homogeneous turbulence, constant speed and a non-oblique oriented rotor plane, the rotationally sampled wind speed ũ D (t a, r) for blade D on radius r is a purely stationary process in linearised sense (effect of blade motion ECN-E

144 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems variations neglected). The general expressions for the auto covariance function C and power spectrum S of the stationary process ũ D r, the short form of ũ D (t a, r), are: CũD r ũ D r (τ) = E [ũ D r (t + τ) ũ D r (t) ] only depends on time-shift τ (5.4) SũD r ũ D(ω) = e jωτ CũD r r ũ D(τ)dτ no time-dependency r Under the conditions mentioned above, the cross power spectrum between u 1 and u 2 in locations (ψ 1, r) and (ψ 2, r) of the rotor plane only depends on the (radiated) frequency ω and the distance d between the locations [39]: S u2 u 1 (ω) = γ(d(ψ 2, ψ 1, r), ω) S uu (ω) (5.5) The auto power spectrum S uu (ω) is equal in each location and the coherence function γ(d, ω) is real-valued. The distance d only depends on the radius r and the azimut difference ψ 2 ψ 1 : It will be derived below that the power spectrum of ũ D r spectra of successive rotational modes: d(ψ 2 ψ 1, r) = 2 r sin(ψ 2 ψ 1 ) (5.6) SũD r ũ D r (ω) = p= can be expressed as a series of shifted Sûpr û pr (ω p Ω) (5.7) The main point of the derivation procedure is the use of the azimut expansions 5.3 for t a = t and t b = t + τ in the expression for the covariance function of ũ D r. Further, it will be pointed out that the rotional mode spectra can be computed in a straightforward way from the cross power spectrum S u2 u 1 (ω, d) as expressed in Eq. 5.5: 2π Sûpr û pr (ω) = 1 2π S uu(ω) e j pφ γ(d(φ, r), ω)dφ (5.8) 0 Proof of pure stationarity and derivation of spectral properties for ũ D r First, it will be proved that the rotational modes are orthogonal and purely stationary for all combinations of radii r 2 and r 1. This implies that the covariance function only depends on time-shift τ when it is non-zero: E [û qr2 (t + τ) û pr1 (t) Cûpr2 û pr1 (τ) if p = q ] (5.9) 0 if p q The derivations stated below yield: Cûpr2 û pr1 (τ) = e j ωτ 1 2π S uu(ω) 2π 0 e j pφ γ(d(φ, r 2, r 1 ), ω)dφ dω (5.10) The Fourier transform of this covariance function for r 2 = r 1 = r yields expression 5.8 for the rotational mode auto spectra on radius r. Then, the covariance function between two wind speeds ũ b 2 r2 and ũ b 1 r1 sampled by blades b 2 and b 1 is derived from the rotational mode expansions and appears to be purely stationary: [ E ũ b 2 r2 (t + τ) ũ b 1 r1 (t) ] Cũb2 r2 ũ b 1 (τ) (5.11) r1 The derivations stated below yield: Cũb2 r2 ũ b 1 (τ) = r1 e j ωτ p= e j p (ψ b 2 ψ b1 ) Sûpr2 û pr1 (ω p Ω)dω (5.12) 134 ECN-E

145 5 EXTERNAL INFLUENCES The Fourier transform of this covariance function for b 2 = b 1 and r 2 = r 1 yields expression 5.7 for the auto spectrum of ũ D r. Now the stationarity proof will be stated as well as the derivation of the required spectral properties. The covariance function of the rotational mode pairs {û qr2, û pr1 } is derived from the covariance function C u2 u 1 between u 1 and u 2 in locations (ψ 1, r 1 ) and (ψ 2, r 2 ) of the rotor plane for varying azimut angles ψ 2 and ψ 1. For the assumed non-oblique rotor plane and homogeneous stationary turbulence it holds for C u2 u 1 : C u2 u 1 (t, τ, ψ 2, ψ 1 ) = E [u 2 (t + τ, ψ 2 ) u 1 (t, ψ 1 ) ] homogoneous stationarity C u2 u 1 (τ, d(ψ 2 ψ 1, r 2, r 1 )) (5.13) So the covariance function only depends on time shift τ and distance d between u 1 and u 2. In the case for two points on the same radius, the distance d only depends on the azimut difference ψ 2 ψ 1. However, now the cosine-rule applies for expressing the distance in the two different radii r 1 and r 2 and the azimut difference: d(ψ 2 ψ 1, r 2, r 1 ) = r2 2 + r2 1 2 r 2 r 1 cos(ψ 2 ψ 1 )) (5.14) By definition it holds for the covariance function for {û qr2, û pr1 }: Cûqr2,û pr1 (t, τ) = E [û qr 2 (t + τ) û pr 1 (t) ] (5.15) We now carry through the time-dependent Fourier coefficient expressions for the rotational modes by Eq. 5.3 for t a = t, r = r 1 and t b = t + τ, r = r 2 in the definition above. The Fourier coefficient expressions for the rotational modes then become: û qr2 (t + τ) = 1 2π 2π 0 e jqψ2 u 2 (t + τ, ψ 2 ) dψ 2 ; û pr1 (t) = 1 2π So that the definition for the covariance function can be rewritten as 2π 2π 0 e jpψ1 u 1 (t, ψ 1 ) dψ 1 (5.16) 2π Cûqr2,û pr1 (t, τ) = 1 4π e j (q p)ψ2 e j p(ψ2 ψ1) E [u 2 2 (t + τ, ψ 2 ) u 1 (t, ψ 1 ) ] dψ 1 dψ (5.17) Because of the assumed homogeneous stationarity, from scheme 5.13 it follows that the inner covariance function E[.] in Eq can be replaced by C u2 u 1 (τ, d(ψ 2 ψ 1, r 2, r 1 )); this removes the dependency on t and implies pure stationarity. As concerns the azimut-dependency, this replacement will make the integrands only depend on ψ 2 and ψ 2 ψ 1. The inner integral over ψ 1 can be replaced by an integral over ψ 2 ψ 1 by changing the integration bounds: 2π 2π ψ2 (.)dψ 1 = (.)d(ψ 1 ψ 2 ) = (.)d(ψ 2 ψ 1 ) (5.18) 0 ψ 2 ψ 2 2π We now carry through these replacements in the expression for the covariance function of the rotational modes and set (ψ 2 ψ 1 ) equal to φ. This yields: Cûqr2,û pr1 (τ) = 1 4π 2 2π 0 ψ2 ψ 2 2π ψ2 e j (q p)ψ2 e j p φ C u2 u 1 (τ, d(φ, r 2, r 1 ))dφdψ 2 (5.19) Since the two integrands that depend on φ are both periodic in 2π, the integral over φ yields an equal result for any ψ 2 in the integration bounds (see Equation 5.14 for distance d(φ, r 2, r 1 )). Thus, it is allowed to substiture ψ 2 by 2π in the integration bounds of the inner integral and so to seperate integration over ψ 2 from integration over φ. This yields Cûqr2,û pr1 (τ) = 1 4π 2 2π 0 e j (q p)ψ2 dψ 2 2π 0 e j p φ C u2 u 1 (τ, d(φ, r 2, r 1 ))dφ (5.20) Note that the integral over ψ 2 yields 2π if p = q and 0 otherwise. This proves the orthogonality of the modes. Finally, the covariance function can be written as: Cûqr2,û pr1 (τ) = δ pq 1 2π 2π 0 e j p φ C u2 u 1 (τ, d(φ, r 2, r 1 ))dφ (5.21) ECN-E

146 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems with Kronecker delta δ pq, which equals 1 if p = q and 0 otherwise. Since the covariance function of u 2 and u 1 is the inverse Fourier transform of the cross power spectrum S u2 u 1 (ω), which can be expressed as follows for homogeneous stationary turbulence in a non-oblique rotor plane (see Eq. 5.5, 5.6): S u2 u 1 (ω) = γ(d(ψ 2 ψ 1, r 2, r 1 ), ω) S uu (ω), (5.22) the covariance function of rotational mode pair {û pr2, û pr1 } can be expressed as: Cûpr2,û pr1 (τ) = e j ω τ 1 2π S uu(ω) 2π 0 e j p φ γ(d(φ, r 2, r 1 ), ω)dφdω (5.23) Note that the integrand for ω, with e j ω τ excluded, represents the cross power spectrum of the rotational mode pair {û pr2, û pr1 }: Sûpr 2 û pr 1 (ω) = e j ω τ Cûpr 2 û pr 1 (τ)dτ = 1 2π S uu(ω) 2π 0 e j pφ γ(d(φ, r 2, r 1 ), ω)dφ (5.24) The covariance function of the wind speed pair {ũ b2 r 2, ũ b1 r 1 }, sampled by blades b 2 and b 1 on radii r 2 and r 1, is finally expressed as a series of inverse Fourier transforms of the cross power spectra of rotational mode pairs {û pr2, û pr1 }. Because of the orthogonality of the (purely stationary) rotational modes, the rotationally sampled wind speeds are also purely stationary. The covariance function is defined by: Cũb2 r2 ũ b 1 (t, τ) = E [ ũ b2 r1 r 2 (t + τ) ũ b1 r 1 (t) ] (5.25) The azimut expansions for the wind speeds sampled by blades b 2 and b 1 on radii r 2 and r 1 on time t + τ and t follow from Eq. 5.3 by setting t a = t, r = r 1 and t b = t + τ, r = r 2. Further, the azimut angles Ωt a and Ωt b are replaced by Ψ b1 + Ωt and Ψ b2 + Ω(t + τ), with Ψ b1 and Ψ b2 the azimutal offsets of blades b 1 and b 2 to the reference azimut angle Ψ. This yields: ũ b2 r 2 (t + τ) = q= e j q (Ψ b 2 + Ω(t+τ)) û qr2 (t + τ) ; ũ b1 r 1 (t) = p= e j p (Ψ b 1 + Ωt) û pr1 (t) (5.26) The expansions for the rotionally wind speeds are carried through in the definition for the covariance function: Cũb 2 r2 ũ b 1 r1 (t, τ) = q= p= e j (q p) (Ψ b 1 + Ωt) e j q (Ψ b 2 Ψ b1 ) e j q Ωτ E [ û qr2 (t + τ) û pr1 (t) ] (5.27) Since the rotational modes are orthogonal, the dependency on t via harmonic function e j (q p) (Ψ b 1 + Ωt) vanishes because for q = p it is equal to 1 for all t. Thus the covariance function only depends on time shift τ, which implies that the rotationally sampled wind speeds on the rotor blades are purely stationary. So elimination of all equal-0 terms yields: Cũb 2 r2 ũ b 1 r1 (t, τ) = Cũb 2 r2 ũ b 1 r1 (τ) = p= e j p (Ψ b 2 Ψ b1 ) e j p Ωτ Cûpr2 û pr1 (τ) (5.28) The Fourier transform of the modulated covariance function by e j p Ωτ yields the cross power spectrum of the mode-pair, shift in frequency p Ω: It then holds that: and Sũb 2 r2 ũ b 1 r1 (ω) = Cũb 2 r2 ũ b 1 r1 (τ) = e j ωτ e j p Ωτ Cûpr2 û pr1 (τ)dτ = Sûpr2 û pr1 (ω p Ω) (5.29) e j ωτ Cũb2 r2 ũ b 1 e j ωτ p= r1 (τ)dτ = e j p (Ψ b 2 Ψ b1 ) Sûpr2 û pr1 (ω p Ω)dω (5.30) p= e j p (Ψ b 2 Ψ b1 ) Sûpr2 û pr1 (ω p Ω)dω (5.31) 136 ECN-E

147 5 EXTERNAL INFLUENCES Spectral properties of blade effective wind speed signals In this paragraph, the 3 3 power spectrum matrix is derived for the so called blade effective wind speed signals in a non-oblique oriented rotor plane for homogeneous turbulence and constant rotational speed. This spectrum matrix is derived from the cross spectra between wind speeds on the blades on a finite number of radial locations between the root and the tip (blade elements). Under the stated conditions, the cross spectrum between the wind speeds on blade elements {b 2, r 2 } and {b 1, r 1 } is expressed as follows in the rotational modes for radii r 2 and r 1 (see prevoius paragraph): Sũb 2 r2 ũ b 1 r1 (ω) = p= The cross power spectra of the rotational modes are then given by: e j p (Ψ b 2 Ψ b1 ) Sûpr2 û pr1 (ω p Ω)dω (5.32) 2π Sûpr2 û pr1 (ω) = 1 2π S uu(ω) e j pφ γ(d(φ, r 2, r 1 ), ω)dφ (5.33) 0 with S uu (ω) the auto spectrum of the wind speed in any fixed location of the rotor plane and γ(d, ω) the coherence function. The distance d in γ only depends on the azimut difference φ between two fixed locations, characterized by radii r 2 and r 1 and azimut angles ψ 2 and ψ 1 : d(φ, r 2, r 1 ) = r2 2 + r2 1 2 r 2 r 1 cos φ, with φ = ψ 2 ψ 1. (5.34) The applied formulations for the (two-sided) auto power spectrum of the wind speed in a fixed location and for the coherence function were those proposed by the Engineering Science Data Unit (ESDU) [40] and by Kaimal e.a. [?]: σ S uu (ω) = w 2 L 2 1/ V w (1+6 L 1 ω/(2π V (σ w)) 5/3 w = I 15 (15+a V w) (a+1) ) 8.8 d (ω/(2π (5.35) Γ(ω, d) = e V w)) 2 +(0.12/3.5 Λ) 2 It will be proved below that the elements Sũbn ũ bl (ω) of the power spectrum matrix for the blade effective wind speeds depend as follows on the blade element related cross spectra (assume N equal-span elements from root to tip with centre radii r 1... r N ; ũ bn r j and ũ b l r i ) are the rotationally sampled wind speeds on radius r j and r i of blade b n and b l respectively): Sũbn ũ bl (ω) = N N r p j rp i S ũ bn r j=1 i=1 j ũ b l (ω) r i N N r p j r p i j=1 i=1 (b n, b l = 1, 2, 3) (5.36) where the value of exponent p determines which type of blade root loading from the turbulent wind field is approximated best: p = 0: optimal for blade root leadwise force; p = 1: optimal for wind field blade root axial force and leadwise moment; p = 2: optimal for blade root flapwise moment. Proof for optimal composition of cross spectra for blade effective wind speeds The derivation of the cross power spectra between the blade effective wind speeds is based on mappings from wind speed to load variations. Two approximations will be made in the windto-load mappings in order to be able to derive the pursued spectra. These approximations are ECN-E

148 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems similar in equilibrium- and dynamic-wake conditions. We choose to base the wind-to-load mappings on equilibrium-wake conditions. This allows for the use of power and thrust coefficient data, which yields very compact expressions. Assume blade effective wind speed variation ũ bn on blade b n, n = 1... B, which is of course invariant over the blade radius. For the belonging leadwise moment Tl b n in the rotor centre then holds (number of blades B = 3; mean wind speed Ū)): T lbn = 1 2 ρ π R2 /B C pn (Ū + ũ b n ) 3 Ω The span-average power coefficient C pn is assumed for blade b n. (5.37) Now consider the N element specific wind speed variations u bn,r j on blade b n. This yields the following expression for the belonging leadwise moment T N l n : T N l bn = N j=1 1 ρ 2π r 2 j r/b C pn,j (Ū + ũ b n,r j ) 3 The element specific power coefficients {C pn,j, j = 1... N} apply on blade b n. Ω (5.38) We now try to find a realisation algorithm for 3 time series {Tl N n (t), b n = 1, 2, 3}. This algorithm should provide statistical equivalency between the leadwise moment variations for the 3 time series and those caused by the 3N time series {{ũ bn,rj, j = 1... N}, b n = 1, 2, 3}. We choose to derive the realisations {ũ bn (t)} from Fourier synthesis on the spectral matrix Sũb ũ b (see next paragraph): Sũb1 ũ b1 (ω) Sũb1 ũ b2 (ω) Sũb1 ũ b3 (ω) Sũb ũ b (ω) = Sũb2 ũ b1 (ω) Sũb2 ũ b2 (ω) Sũb2 ũ b3 (ω) (5.39) Sũb3 ũ b1 (ω) Sũb3 ũ b2 (ω) Sũb3 ũ b3 (ω) In this case the statistical equivalency is achieved when the following equality constraint on the linearised lead moment dependency on wind speed variations is met: S Tlbn ũ bn ũ bn T lbl (ω) = S ũ Nj=1 Tl N ũ bl bn T N bl ũ bn rj ũ Ni=1 lbl bn r j ũ bl r i ũ bl r i (ω) (5.40) The first approximation we make is that the power coefficint dependency on λ and θ is invariant over the blade radius and does not depend on the blade. This implies: Ω R C pn,j C p (λ n,i, θ n ) with λ m,i = (Ū+ũ bn,r j ) (5.41) C pn C p (λ n, θ n ) with λ n = Ω R (Ū+ũ bn ) The second approximation is that we assume equal aerodynamic efficiency over the blade radius while the blade effective wind speed sizes the power coefficient. Thus, the expression for the leadwise torque in element specific wind speeds can be approximated as follows: T N l bn N j=1 1 ρ 2π r 2 j r/b C p (λ n,i, θ n ) [Ū + ũ b n,r j ] 3 N j=1 Ω 1 ρ 2π r 2 j r/b C p (λ n, θ n ) [Ū + ũ b n,r j ] 3 Ω (5.42) For the variation δtl N bn caused by only wind speed varations then holds in linearised sense: 1 N ρ 2πr δtl N 2 j r/b [C p ( λ n, θ n ) 3Ū 2 + Cp λ n λn, θ λn bn = n U Ū 3 ] Ω, Ū ũ bn,r Ω j (5.43) j=1 138 ECN-E

149 5 EXTERNAL INFLUENCES Since N π R 2 = 2πr j r (5.44) j=1 a similar expression can be derived for the variation δt lbn in the leadwise torque caused by the blade effective wind speed variation: Nj=1 1 ρ 2πr 2 j r/b [C p ( λ n, θ n ) 3Ū 2 + Cp λ n λn, θ λn δt lbn = n U Ū 3 ] Ω, Ū ũ bn (5.45) Ω We now require that δt lbn and δt N l bn are statistically equivalent, which implies: S δtlbn δt lbn (ω) = S δt N lbn δt N l bn (ω) (5.46) When the above mentioned expressions for δt lbn and δt N l bn are carried through in this equality requirement, Equation 5.36 is obtained for the cross power spectrum between the effective wind speeds on blade b n and b l for p = 1. Since the axial forces on the blade elements are proportional to the blade radius at equal aerodynamic efficiency, just as the leadwise torque in the rotor centre, the same expression for the cross power spectrum applies for the axial force in the rotor centre. The leadwise force is constant over the blade radius at equal aerodynamic efficienty, which implies that the proportionality of the variation expressions in r j vanishes. Thus p = 0 applies for the optimal leadwise force approximation. The flapwise moment in the rotor centre cause by an axial force on a blade element is proportional to the squared radius at equal aerodynamic efficienty. This imples that the variation expressions become quadratic to r j so that p = 2 applies for the opitmal flapwise moment approximation. We use p = 1 in the generation of blade effective wind speeds. This means the optimal approximation for the thrust and driving torque. However, the turbulence in the outer part of the rotor is weighted somewhat too light as concerns the blade root flap moment and in the inner part somewhat too heavy. Since the effect of rotational sampling grows at increasing radius, because the coherence becomes weaker, the flapwise moment will be slightly underestimated; in contrary, the leadwise force will be slightly overestimated. Realisation algorithm for blade-effective homogeneous turbulence signals A realisation algorithm as proposed by Shinozuka [41] is applied. Such a realisation algorithm is also included in the ECN 3D-wind field simulation program SWIFT [16]. Here, it is applied on the 3 3 spectral matrix Sũ b ũ b with elements obtained from Equation So, it will yield time series {ũ bn (t p ), b n = 1, 2, 3}, with t p = 0, t,..., (2 m 1) t for non-oblique, constant speed operation in homogeneous turbulence. The algoririthm comprises the next steps (operator stands for complex conjugate transpose): 1 Get the Cholesky decomposition S ũ b ũ b of the two-sided cross power spectrum Sũb ũ b in M f 1 equidistant frequency points (M f = 2 m 1 ): Sũb ũ b (ω k ) = S ũ b ũ b (ω k ) S ũ b ũ b (ω k ) with ω k = k 2π 2M f t, k = 1... M f 1 (5.47) 2 Get M f 1 column vectors e jφ k from random phase angles φ bn,k between 0 and 2π : [ ] e j φ 1,1... e j φ 1,M f 1 e jφ 1... e jφ M f 1 = e j φ 2,1... e j φ 2,M f 1 (5.48) e j φ 3,1... e j φ 3,M f 1 ECN-E

150 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems 3 Establish 3 randomly phased re-arranged two-sided harmonic row vectors {s ũ bn (ω k )} of length 2M f with ω k = k 2π 2M f t (b n = 1, 2, 3): {s ũ 1 (ω k ), k=0, 1,..., M f 1, M f, (M f 1),..., 1} {s ũ 2 (ω k ), k=0, 1,..., M f 1, M f, (M f 1),..., 1} = {s ũ 3 (ω k ), k=0, 1,..., M f 1, M f, (M f 1),..., 1} (5.49) [ 0 {S ũ b ũ (ω b k ) e jφ k, k=1... M f 1 } 0 {S ] ũ b ũ (ω b k ) e jφ k, k=m f } 4 Obtain the blade effective wind speed realisations {ũ bn (t)} by inverse Fourier transformation for b n = 1, 2, 3 of the harmonic row vectors {s ũ bn (ω k )}: ũ bn (p t) = 2M f 1 1 2M f t e j ω k p t s ũ bn (ω k ) k =0 for p = M f 1 (5.50) Since with {ω k, k =0...2M f 1} = { k 2π 2M f t, k=0, 1,..., M f 1, M f, (M f 1),..., 1} e j k 2π 2M f t p t = e j (2M f +k) 2π p t 2M f t the expression for the Fourier synthesis can be rewritten as: ũ bn (p t) = 2M f 1 1 2M f t k =0 e j k 2π 2M f t p t s ũ bn (ω k ) (5.51) In the expressions above, the rotational speed is assumed constant, say Ω. The belonging rotor azimut Ψ then simply evolves in time p t as: Ψ(p t) = Ω p t (5.52) It the rotational speed varies with δω, the rotor azimut angle Ψ(t) obeys: Ψ(p t) = Ω p t p t + δω(τ)dτ (5.53) 0 The blade effective turbulent wind speed realisations for varying rotor speed Ω + δω are also obtained from the realisation algorithm mentioned above. However, the time-index p t in the right hand side of Equation 5.51 is replaced by p t in such a way that: Ω p t = Ω p t p t + δω(τ)dτ (5.54) 0 So, the values for {ũ bn, b n = 1, 2, 3} that are used in the time point p t are set equal to the (in advance obtained) realisation values in that time point which is shifted 1 Ω p t 0 δω(τ)dτ s ahead of p t: ũ bn (p t) = 2M f 1 1 2M f t j ω e k (p t+ 1 Ω k =0 p t 0 δω(τ)dτ) s ũ bn (ω k ) (5.55) It will be proved below that this approach to rotational sampling with variable speed gives a good approximation in frequencies just above 0p and around 1p, 2p, 3p,... (p is rotational frequency). 140 ECN-E

151 5 EXTERNAL INFLUENCES A more accurate approximation is obtained by interpolation between two subsequent helices. Assume azimut deviation δψ relative to Ψ in time point p t. On the B existing helices for constant rotor speed Ω, the deviating rotor azimut Ψ = Ψ + δψ corresponds with the time points p t + 1 Ω (δψ + n 2π B ) for n =..., 2, 1, 0, 1, 2,.... So wind speed values in azimut Ψ are available in those helix time points. The wind speed in the target time point p t is then obtained from interpolation between wind speed values in the nearest surrounding helix time points, that is to say the time points p t + 1 Ω (δψ + n 2π B ), with n = n lo, n lo +1, such that: p t + 1 Ω (δψ + n lo 2π B ) p t p t + 1 Ω (δψ + (n lo + 1) 2π B ) This approximation is further carried through in the turbulence generation scheme for oblique inflow as described in the paragraph on modified rotationally sampled turbulence of subsection Proof for accurate wind sampling at varying rotor speed near qp-frequencies Consider the rotationally sampled wind speed ũ D r (t) on radius r of blade D in the rotor plane. Its value in time t, when azimut angle Ψ by Equation 5.53 for varying rotor speed Ω + δω applies, can always be expressed in the rotational modes from Eq. 5.3 in a straightforward way (see [42]): ũ D r (t) = e jq( Ω p t+δψ)û p t qr (t) with δψ = δω(τ)dτ (5.56) q= 0 An equally structured realisation algrotithm as for the blade effective wind speeds can be used for the rotational modes û qr (t). Since the rotational modes are orthogonal the Cholesky decomposition yields a diagonal matrix, with elements { Sûqr ûqr (ω k), q=..., 2,1,0,1,2,...} on the diagonal. When we carry through the realisation algorithm by Equations 5.47 up to 5.51 in the Fourier series for ũ D r (t) the following expression is obtained: ũ D r (p t) = q= e j qδψ 2M f 1 1 2M f t k =0 e j (q Ω p t+ω k p t) Sûqr û qr (ω k ) e j φ q,k (5.57) As a matter of fact, it is allowed to consider the inner summation term over k as a spectral representation. In a spectral representation, a modulation factor like e j q Ω p t can be replaced by a frequency shift over q Ω rad/s in the spectral basis function Sûqr ûqr (ω k ) e j φ q,k. The rotationally sampled wind speed can then be expressed as (interchanging the summation sequence is always permitted): ũ D r (p t) = 2M f 1 1 2M f t k =0 q= e j qδψ e j ω k p t Sûqr û qr (ω k q Ω) e j φ q,k (5.58) Now, we will tailor Equation 5.55 to an expression for the approximated sampled wind speed ũ D r (p t) at varying rotational speed. Its power spectrum for constant rotational speed is a sum of shifted spectra of the rotational modes: SũD r ũ D r (ω) = q= Sûqr û qr (ω q Ω) (5.59) The approximation ũ D r (p t) is then expressed as follows (note that only one set of random phases {φ k } applies because of the spectral relationship between the modes and sampled wind ECN-E

152 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems speed, and not the infinite number of sets over q {{φ q,k }}): ũ D r (p t) = which can be rewritten as: ũ D r (p t) = 2M f 1 1 2M f t e j ω k (p t+ 1 Ω δψ) k =0 2M f 1 1 2M f t k =0 q= q= Sûqr û qr (ω k q Ω) e j φ k (5.60) e j ω k Ω δψ e j ω k p t Sûqr û qr (ω k q Ω) e j φ k (5.61) The expressions 5.58 and 5.61 are statistically equivalent if ω k = q Ω. This implies that the approximation 5.55 for the blade effective wind speed realisations at varying rotor speed is accurate near frequency 0 and integer multiples qp of the average rotational frequency. Since the power spectra of the rotational modes decay fast at increasing frequency, the modal expansion of the sampled wind speed proves that most spectral energy is near 0 and around these qpfrequencies. Therefor, we conclude that the adapted approximation 5.55 satisfies in case of process simulation and controller evaluation. The blade effective turbulent wind speed realisations {ũ bn (p t), b n = 1, 2, 3} are determined as the inverse Fourier transforms of the randomly phased two-sided harmonic row vetors {s ũ bn (ω k )} (see Equations 5.47 up to 5.54). ũ bn (p t) = 2M f 1 1 2M f t j ω e k (p t+ 1 Ω k =0 p t 0 δω(τ)dτ) s ũ bn (ω k ) (5.62) The time-shift p t 1 Ω 0 δω(τ)dτ in the inverse Fourier transformation accounts for the rotor speed variation δω in the sampling of the turbulence Periodic wind speed variations by tower shadow and wind shear Wind shear and tower shadow are accounted for via azimuth-dependent axial wind speed variations shr u X r a and tow u X r a on 2/3 blade radius (r) that result from application of IEC-accepted method of exponential wind shear modelling; 3D potential stream theory for a semi-infinite dipole by [24] (pg 67). The 0 th -order approximations x X r, ỹ X r and z X r of the place vector coordinates from the tower top centre to the rotating 2 3 R point X r along the coordinate system of the wind plane amount to: x X r ỹ X r z X r ( cos (Ψ) = φyw sin φ yw 0 sin φ yw cos φ yw ( cos φtl 0 sin φ tl sin φ tl 0 cos φ tl ) ) [ cos φtl 0 sin φ tl ( cos Ψ X sin Ψ X 0 sin Ψ X cos Ψ X ) ] ( d n )+ [ sin φcn cos φ cn 0 ] 2 3 R (5.63) Herein, d n represents the distance from the rotor center to the centre of the tower top. The y-coordinate points in lateral direction while the z-coordinate points downward. Note that a negative x-coordinate value means that the blade point is situated upwind relative to the tower centre. 142 ECN-E

153 5 EXTERNAL INFLUENCES The misalignment angle φ yw is positive at clockwise rotation at view from top to bottom; the effective tilt angle φ tl, caused by tilting and slope, is positive when the hub is lifted up; the cone angle φ cn is positive at downwind blade coning. We first give the expression for the periodic wind speed variation by wind shear since it is used in the expression for the tower influence: ( ) shr u X r a (Ψ) = cos φ yw cos φ tl ( ζ Xr Z hub ) α 1 Ūu. (5.64) The exponent α caters for the terrain roughness. For the azimut dependent height ζ Xr of the 2 3 radius point holds: ζ Xr (Ψ) = Z hub z X r(ψ) + sin φ tl d n (5.65) The expression for the axial wind speed variation by tower shadow then becomes: tow u X r a (Ψ) = (cos φ tl cos φ yw Ūu + shr u Xr a (Ψ)) Dtow 2 8( x 2 k + ỹ2 k ) ( ( x2 k ỹ2 k x 2 k + ỹ2 k ) (1 + z k ) d + x2 k z k d 3 ) (5.66) Herein, D tow represents the average tower diameter between the tower top and 2 3 R b below the top. The azimut dependent variation obl u X r by the yaw and tilt component of the mean wind velocity follows straightforward from the transformation matrices from the wind plane to the rotor blade elements Wind speed affection by oblique inflow Oblique inflow occurs at yaw misalignment or tilting of the rotor shaft relative to the longitudinal wind speed direction. In the process simulator, oblique inflow causes purely or nearly periodic wind speed variations on the rotor blades because of: azimut-dependent delayed turbulence; yaw and tilt components of the mean wind velocity; modelled azimut-dependent axial induction variation. In the adopted linear approach to turbulence modelling, the average wind speed Ū transports the turbulence along the longitudinal axis in accordance with Taylor s frozen wave hypothesis [43]. In oblique inflow conditions, the distance between a point on a rotating blade and its projection in a plane perpendicular to Ū (longitudinal separation) depends periodically on the azimut angle. Thus, a blade is alternately located relatively downwind and upwind. This implies that, in case of sudden variations in the rotor-uniform wind speed, a rotor blade feels the old value in the more downwind locations and the new value in the more upwind locations. So, oblique inflow causes the turbulence to be azimut-dependent delayed relative to the rotating blades. This causes nearly periodic wind speed variations on the blades. In the first paragraph of this subsection is described how this modified rotationally sampled turbulence is catered for in the blade effective wind speed signals. Modified rotationally sampled turbulence The realisation algorithm presented in subsection yields three series of longitudinal wind turbulence values. The realisations can be considered as weighted averages of the wind speeds ECN-E

154 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems on the three sets of helices that correspond with the trajectories of element centres of the blades in the wind cylinder, the element centre helices. The note of blade helix is introduced as being a kind of average of the element centre helices for a blade. The blade effective rotationally sampled wind speed values lie on the blade helices. Element centre and blade helices apply at constant rotational speed Ω, non-oblique inflow conditions and homogeneous turbulence for average longitudinal wind speed Ū. The blade motion variations cause second order effects and are not taken into account in the treatment of the turbulence. At oblique inflow a blade is alternately located downwind and upwind relative to its blade helix. In Figure 5.3 it is visualised that the downwind location of the blade relative to its helix is maximal if the azimut angle coincides with the orientation angle ψ obl of the oblique inflow. Figure 5.3: Azimut dependent longitdunial separation by oblique inflow For the thus defined oblique inflow conditions, a point on 2R of blade X is shifted over 3 δxx r obl meters ahead of the helix for the blade element on 2 R, so upwind, given by: 3 δx X r obl = 2 3 R tan ξ 2 3 R cos(ψx Ψ obl ) (5.67) 144 ECN-E

155 5 EXTERNAL INFLUENCES The average obqliue orientation angle Ψ obl is the four-quadrant arctangent of the tilt and yaw component of the average wind velocity: arctan( Ūtilt/Ūyaw ) if Ūtilt > 0 Ūyaw > 0 arctan( Ūtilt/Ūyaw ) + Ψ 1 obl = π if 2 Ūtilt > 0 Ūyaw < 0 (5.68) arctan( Ūtilt/Ūyaw ) + π if Ūtilt < 0 Ūyaw < 0 arctan( Ūtilt/Ūyaw ) + 3π if 2 Ūtilt > 0 Ūyaw > 0 Let the oblique wind velocity components Ūtilt and Ūyaw correspond with yaw angle φ yw and tilt angle φ tl and let Ū iax, 2 3 R be the average axial induction speed on 2 R. Now, define the 3 average transporation speed Ūturb of the turbulence relative to the rotor plane as: Ū turb = cos φ tl cos φ yw Ū Ūi ax, 2 3 R (5.69) For the oblique inflow angle ξ 2 3 R on radius 2 R then holds: 3 Ū 2 ξ 2 3 R = arctan tilt + Ū yaw 2 cos φ tl cos φ yw Ū (5.70) Ūi ax, 2 3 R Assume constant rotor speed with evolving azimut angle Ψ. Due to oblique inflow and Taylor s frozen wave hypothesis, a blade X experiences on 2R for azimut angle Ψ the wind speed that 3 occurs δt X r obl = δxx r obl /Ūturb seconds ahead in time, relative to the time point that belongs to Ψ on the helix of blade X. By use of the above mentioned expressions for δx X r obl, ξ 2 R and Ūturb, 3 the forward time shift δt X r obl can be expressed by: δt X r obl = Ū 2R 2 3 tilt + Ū yaw 2 (cos φ tl cos φ yw Ū Ūi cos(ψx Ψ obl ) (5.71) ax, 2 3 R)2 On the B blade helices the azimut angle Ψ corresponds with the time points p t + n 2π B Ω for n =..., 2, 1, 0, 1, 2,.... So wind speed values in azimut Ψ are available in those helix time points. The wind speed in the target time point p t+δt X r obl is then obtained from interpolation between wind speed values in the nearest surrounding helix time points, that is to say the time points p t + 1 Ω (n 2π B ), with n = n lo, n lo +1, such that p t + n lo 2π B Ω p t + δtx r obl p t + (n lo+1) 2π B Ω It is clear that the wind speed for the same azimut angle, so in a fixed location, is seen every 2π seconds at B blade helices. At mb blade helices (m integer) this sample frequency for B Ω the wind speed in a fixed location amounts to 2π seconds. Since at oblique inflow interpolation applies between wind speed values on two sample time points, the sample time must be mb Ω small enough for successful reonstruction of the maximum relevant wind speed frequency. This maximum frequency pertains to the power spectrum S uu (f) of the wind speed in fixed locations in the wind speed cylinder. The frequency reconstruction is based on interpolation between two samples only. In that case, it is required to oversample the maximum relevant frequency approximately by a factor 6. Thus, the highest frequency that is accurately catered for amounts to mb Ω 6 2π Hz. So, for a maximum frequency of f max Hz, the number of blade helices is constrained to: ( 12π mb Ω f max 360 ) N[rpm] f max (5.72) ECN-E

156 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems This yields mb 6 for a maximum frequency of 0.2 Hz at a rotor speed of 12 rpm. Note that for mb = 6 at 12 rpm the distortion in the frequency reconstruction between 0.2 Hz and 0.6 Hz (half the sample frequency) will increase, while above 0.6 Hz aliasing will happen. The latter implies that frequencies of 0.7 and 0.8 Hz will be reconstructed as frequencies of 0.5 and 0.4 Hz in the interpolated wind speed signal, etc. However, the spectrum for the wind speed in a fixed locations has very low energy density in frequencies above 0.2 Hz. Periodic tangential wind speed variations Assume clockwise-positive yaw angle φ yaw at top-down view and hub-liftup-positive tilt angle φ tilt. The undisturbed wind velocity components with normal ( ax ), lateral ( yaw ) and vertical ( tilt ) orientation relative to the rotor plane are expressed in the average longitudinal wind speed Ū u as follows: Ū ax Ū yaw Ū tilt = ( cos φtl 0 sin φ tl sin φ tl 0 cos φ tl ) ( cos φyw sin φ yw 0 sin φ yw cos φ yw ) Ū u 0 0 (5.73) The already used average axial component and the yaw and tilt component then become: Ū ax = cos φ tilt cos φ yaw Ūu Ū yaw = sin φ yaw Ūu (5.74) Ū tilt = sin φ tilt cos φ yaw Ūu The tangential periodic windspeed component obl u X t depend on the radius. It is expressed as follows: by oblique inflow for blade X does not obl u X t = sin Ψ X Ūyaw + cos Ψ X Ūtilt ) = (sin Ψ X sin φ yaw cos Ψ X sin φ tilt cos φ yaw Ūu (5.75) When positive, obl u X t reduces the relative leadwise wind speed on the rotor blade. Periodic axial induction variation The periodic axial induction variation obl u X r i ax by oblique inflow follows from the linearised parametrisation by Glauert or by Schepers and Vermeer ([44]; emperical 2-harmonics engineering model ). The value on 2 R is considered to be typical, and carried through as the axial induction 3 variation for blade X. In linearised sense, the periodic axial induction speed obl u X r i ax by oblique inflow is a purely periodic external input. This is the case because we neglect the influence of the variation in the oblique-inflow angle ξ 2 R on the periodically varying axial induction, which is very small. For 3 the azimut dependent obl u X r i ax then holds: 15π 64 2 tan( ξ 2 3 R/2) cos(ψx Ψ 3 obl ) (a) obl u X r i ax (Ψ) = Ūi ax, 2 3 R a 1 cos(ψ X Ψ obl 1 Ψ)+ (5.76) a 2 cos(2(ψ X Ψ obl ) 2 Ψ) (b) The oblique inflow angle ξ 2 3 R and azimut offset angle Ψ obl for the oblique orientation are determined by Eq and The parameters in the right hand factor in the expression for obl u X r i ax are the same as applied in PHATAS in function f skew in [24]. Option (a) implies the parametrisation by Glauert while (b) implies a 2-harmonics engineering model based on measured oblique inflow effects by Schepers and Vermeer [44]. 146 ECN-E

157 5 EXTERNAL INFLUENCES Blade effective turbulence realisations for variable speed and oblique inflow The influence of both the variable rotor speed and the oblique inflowing turbulence is carried through in the realisation scheme via interpolation between two subsequent blade helices. It was argued in the subsection on oblique inflow that the extension of the number of blade helices with an integer multiple, so from B to mb, is required for achieving sufficient accuracy at interpolation. In that case helices exist for the real blades as well as for (m 1)B intermediate virtual blades. Let the whole set of helices be denoted by {H n, for n = 1... mb}. Each helix has its own azimut angle Ψ Hn (t), which jumps with 2π rad from helix n to n + 1. Identify the helices mb with respect to a certain blade by H(ν) X, which implies for ν = 0 the true-blade helix of X and for ν 0 a neighbouring helix, whether virtual or real. It is clear that the first and second true-blade helices H(0) D and HE (0) are identifical to H 1 and H m+1 etc. A neighbouring helix H(ν) X lies upwind relative to the true-blade helix if ν > 0 and downwind if ν < 0. It is clear that the first blade helix H(0) D has no downwind neighbouring helices. This lack is catered for by considering helix H mb for reduced azimut angle Ψ HmB 2π rad as the closest downwind neighbour H( 1) D, etc. First, a tutorial realisation scheme is given for the blade effective turbulence on blade E for B = 3 and mb = 6 at variable speed and oblique inflow under certain assumptions. Afterwards the realisation scheme is generalised so that it pertains to any blade X without constraints. Blade effective turbulence on blade E The realisation algorithm of subsection provides blade effective turbulence values for three rotor blades without oblique inflow and at constant rotor speed. Without increase of complexity, this algorithm can be extended for the generation of six blade effective turbulence time series. Expression 5.51 then yields time series {ũ bn (p t)} for b n = , which, for B = 3 are the wind speeds on three pairs of a true-blade and virtual-blade helix. For convencience, assume ũ bn available for any time instant t instead of at discrete time samples p t. Because of the constant rotor speed Ω, the time series {ũ bn } can be expressed as a function of the azimut angles {Ψ Hn } of the helices, which grow with a constant Ω rad/s (b n, n = ): ũ bn (t) = ũ bn (Ψ Hn ) with Ψ Hn = Ωt + (n 1) 2π mb (5.77) We now want to use the provided wind speed series ũ bn for the derivation of wind speed values for the second blade E at varying rotor speed and oblique inflow. For reason of clarity, the blade-associated notation ũ bn of the available series is replaced by the helix-associated notation u H E (ν) for blade E with n = ν+3. Let ũ E be the effective wind speed experienced by blade E. Assume that on time-instant t q the rotor azimut angle lies δψ rad ahead of Ωt q and that due to an upwind blade position by tilt and yaw a wind speed value δt seconds ahead applies via Taylor s hypothesis. This means that ũ E will be equal to the wind speed value that is constructed in azimut angle Ψ E = Ψ H3 +δψ for the time instant t q δt. The azimut angle Ψ E on helix H(0) E corresponds with the non-actual time instant t H 3 for which the wind speed value u H E (Ψ E ) exists: (0) Ψ E = Ψ H3 (t H3 ) (Eq. 5.77) = ΩtH π mb The azimut angle Ψ E is related to the actual time instant t q by: (5.78) Ψ E = Ψ H3 (t q ) + δψ (Eq. 5.77) = Ωtq + 2 2π mb + δψ (5.79) ECN-E

158 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems The time instant t H3 for azimut angle Ψ E on helix H(0) E then obeys: Ωt H π mb = Ωt q + 2 2π mb +δψ t H 3 = t q + δψ Ω (5.80) The azimut angle Ψ E on more downwind located helices H( 1) E and HE ( 2) corresponds with the non-actual time instants t H2 and t H1 for which the wind speed values u H E (Ψ E ) and ( 1) u H E (Ψ E ) exist: ( 2) Ψ E (Eq. 5.77) = Ψ H2 (t H2 ) = ΩtH π mb Ψ E = Ψ H1 (t H1 ) (Eq. 5.77) = ΩtH π mb (5.81) It still holds for the actual time instant t q that Ψ E = Ωt q + 2 2π mb + δψ. The later time instants t H2 and t H1 that belong to azimut angle Ψ E on helices H( 1) E and HE ( 2) then obey: Ωt H π mb = Ωt q + 2 2π mb + δψ t H 2 = t q + δψ Ω + 1 2π mb Ωt H π mb = Ωt q + 2 2π mb + δψ t H 1 = t q + δψ Ω + 2 2π mb (5.82) The azimut angle Ψ E on more upwind located helices H(+1) E and HE (+2) the non-actual time instants t H4 and t H5 for which the wind speed values u H E (+1) u H E (Ψ E ) exist: (+2) Ψ E = Ψ H4 (t H4 ) Ψ E = Ψ H5 (t H5 ) (Eq. 5.77) = ΩtH4 1 2π mb (Eq. 5.77) = ΩtH5 2 2π mb corresponds with (Ψ E ) and (5.83) It still holds for the actual time instant t q that Ψ E = Ωt q +2 2π mb +δψ. The earlier time instants t H4 and t H5 that belong to azimut angle Ψ E on helices H(+1) E and HE (+2) then obey: Ωt H π mb = Ωt q + 2 2π mb + δψ t H 4 = t q + δψ Ω 1 2π mb Ωt H π mb = Ωt q + 2 2π mb + δψ t H 5 = t q + δψ Ω 2 2π mb (5.84) Thus, wind speed values for the azimut angle Ψ E are available for time-instants t H1... t H5, specified by equations 5.80, 5.82 and We now assume that the applying time-instant t q δt is in the interval between t H5 and t H1 ; note that t Hn 1 > t Hn. Say that t q δt is enclosed by t H3 and t H2, then the wind speed value ũ E for blade E on time t q is obtained by interpolation between wind speed values on the helices H E ( 1) and HE (0) for azimut angle ΨE : ũ E (t q ) = t H 2 (t q δt) t H2 t H3 u H E (Ψ E ) + (t q δt) t H3 (0) t H2 t H3 u H E (Ψ E ) (5.85) ( 1) A realisation scheme for the wind speed on the second blade E for mb = 6 and B = 3 could look like: Determine the time-shift δt caused by oblique-inflow as (ahead of actual time t q ; see also Eq for Ψ obl, etc.): δt = 2 3 R Ū 2 tilt + Ū 2 yaw (cos φ tl cos φ yw Ū Ūi ax, 2 3 R)2 cos(ψe Ψ obl ) Determine the azimut variation δψ ahead of the azimut angle for constant rotor speed as: δψ = Ψ E ( Ωt q π) 148 ECN-E

159 5 EXTERNAL INFLUENCES Find helix-offset values ν 0 and ν 1 relative to the 3rd helix that belongs to blade E such that (ν 1 = ν 0 + 1): t q + δψ Ω ν 0 2π mb t q δt and t q + δψ Ω ν 1 2π mb t q δt Reconstruct wind speed on blade E for time-instant t q by interpolation between wind speed values on helices H 3+ν0 and H 3+ν1 with ũ E (t q ) = t H 3+ν0 (t q δt) t H3+ν0 t H3+ν1 u H3+ν1 (Ψ E ) + (t q δt) t H3+ν 1 t H3+ν0 t H3+ν1 u H3+ν0 (Ψ E ) t H3+ν0 = t q + δψ Ω ν 0 2π mb and t H3+ν1 = t q + δψ Ω ν 1 2π mb Blade effective turbulence on any blade X Now consider an arbitrary blade X for a B-bladed rotor and an arbitrary multiple m of B helices. Let blade counter i x be such that X = D, E,... for i x = 1, 2,.... The true-blade helix for X is then identified by H (ix 1)m+1 and its neighbouring helices by H (ix 1)m+1+ν for ν = ±1, ±2,..., which can be either virtual or true. From the previous paragraph in this subsection it is clear that the wind speed values on neighbouring helices are used for construction of the wind speed value on a blade for a certain timeinstant. In the case that X = D and the downwind helix to D is to be used for this purpose, the wind speed on helix H 0 for the actual azimut angle Ψ D would apply. However this helix does not exist, but it would contain exact the same wind speed value as H mb does contain for the azimut value that is shifted over 2π rad back from Ψ D. So it holds: u H0 (Ψ D ) = u HmB (Ψ D 2π) which can be generalised to: u Hn (Ψ X ) = u Hmodulo(n,[1,mB]) (Ψ X +floor( n 1 mb ) 2π) with modulo(n,[1,mb]), floor( n 1 mb ) =. n+2mb, 2 if 2mB 1 n mb n+mb, 1 if mb 1 n 0 n, 0 if 1 n mb n mb, +1 if mb+1 n 2mB n 2mB, +2 if 2mB+1 n 3mB. (5.86) The on-line realisation scheme for the wind speed u X on blade X for time-instant t q is as follows: Determine the time-shift δt caused by oblique-inflow as (ahead of actual time t q ; see also Eq for Ψ obl, etc.): δt = 2 3 R Ū 2 tilt + Ū 2 yaw (cos φ tl cos φ yw Ū Ūi ax, 2 3 R)2 cos(ψx Ψ obl ) (5.87) ECN-E

160 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Determine the azimut variation δψ ahead of the azimut angle for constant rotor speed as: δψ = Ψ X ( Ωt q + (i x 1) 2π B ) (5.88) Find helix-offset values ν 0 and ν 1 relative to the helix no. (i x 1)m+1 for blade X such that (ν 1 = ν 0 + 1): t q + δψ Ω ν 0 2π mb t q δt and t q + δψ Ω ν 1 2π mb t q δt (5.89) Map desired no. s (i x 1)m ν 0 and (i x 1)m ν 1 of enclosing helices to available helix no. s n 0 and n 1 and adapt the azimut angles to be used in the applying helices H n0 and H n1 accordingly: n 0 = modulo((i x 1)m ν 0, [1, mb]) n 1 = modulo((i x 1)m ν 1, [1, mb]) Ψ X H n0 = Ψ X + floor( (ix 1)m+1+ν 0 1 mb ) 2π Ψ X H n1 = Ψ X + floor( (ix 1)m+1+ν 1 1 mb ) 2π (5.90) Find time-index value pairs {p (0) 0, p (1) 0 } and {p (0) 1, p (1) 1 } such that the azimut angles Ψ X H n0 and Ψ X H n1 are enclosed by azimut values pairs for which wind speed values have been generated via the realisation algorithm in subsection 5.1.1: Ωp (0) 0 t Ψ X H n0 Ωp (0) 1 t Ψ X H n1 Ωp (1) 0 t Ωp (1) 1 t (5.91) Estimate wind speed value on helix H n0 and H n1 from the available wind speed values in azimut angles { Ωp t}: û Hn0 (Ψ X Hn 0 ) = û Hn1 (Ψ X Hn 1 ) = Ωp (1) 0 t ΨX Hn 0 Ω t Ωp (1) 1 t ΨX Hn 1 Ω t u Hn0 ( Ωp (0) u Hn1 ( Ωp (0) t) + ΨX Ωp (0) Hn 0 0 t 0 Ω t t) + ΨX Ωp (0) Hn 1 1 t 1 Ω t u Hn0 ( Ωp (1) 0 t) u Hn1 ( Ωp (1) 1 t) (5.92) Reconstruct wind speed on blade X for time-instant t q by interpolation between the estimated wind speed values on helices H n0 and H n1 for the appropriate azimut angles: with ũ E (t q ) = t H n0 (t q δt) t Hn0 t Hn1 û Hn1 (Ψ X Hn 1 ) + (t q δt) t Hn1 t Hn0 t Hn1 û Hn0 (Ψ X Hn 0 ) (5.93) t Hn0 = t q + δψ Ω ν 0 2π mb and t Hn1 = t q + δψ Ω ν 1 2π mb (5.94) 150 ECN-E

161 5 EXTERNAL INFLUENCES The blade effective wind speed signals have been designed in the conditions as listed in Table 6.1. These conditions also did apply in the time-domain simulations of the next section. parameter description value dimension V w mean wind speed 16 [m/s] f 0 mean rotational frequency 0.29 [Hz] α shear coefficient 0.12 [-] d n distance between hub and tower top centre 5.92 [m] d tow average tower diameter between top and 2R 3 b 1.71 [m] α n tilt angle of nacelle 5 [dg] α c blade cone angle 0 [dg] I 15 turbulence intensity at 15 m/s 0.16 [-] a adapts turbulence to mean wind speed by IEC [-] Λ turbulence length scale parameter 21 [m] L 1 integral scale parameter longitudinal wind speed 170 [ m] Table 5.1: Conditions for blade effective wind speed variations The plots in figures 5.4 and 5.5 show auto power spectra and a cross power spectrum of the contribution by turbulence in the blade effective wind speeds. Both plots contain the design spectra (solid lines) and the measured spectra (squares and circles); the latter were derived from the realisation via FFT and Daniel Windowing, i.e. averaging a fixed number of rough spectrum points (periodogram ordinates); see e.g. [45]. Figure 5.4: Auto power spectra of blade and rotor effective wind speed ECN-E

162 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems As expected, the auto power spectra of the blade effective wind speed signals shows peaks around np-frequencies, whereas the average of the three blade signals only shows a peak around the first mbp-frequency (higher frequencies were not included in the plot). The dashed line in the plotboxes represents the power spectrum of the wind speed in a fixed location. Figure 5.5: Cross power spectrum of blade effective wind speeds The cross power spectrum shows a phase-shift that grows to multiples of 120 o at multiples of the rotational frequency; the steps of 360 o result from the modulo 2π behaviour of the applied phase retrieving function. The measured spectra match well to the design spectra, which proves the validity of the realisations. Measured cross spectra converge much slower and thus require longer realisations for validity checks, especially the phase values in high frequencies (see lower box in Figure 5.5. The design spectra were obtained from only four elements per blade while the number of rotational modes taken into account (limits for q in Eq.?? amounted to ±9). The sensitivity of the design spectra to enhancing the number of elements of rotational modes appeared very weak. The realisations involved samples at 50 Hz sample frequency. The first drive-train frequency of 1.85 Hz (ca 6.5p) was thus sufficiently well excited; this is the highest eigenfrequency in the adopted setting for scoping the aeroelastic control concepts. 5.2 Wave Generation (WAV) The external drivers for the hydrodynamic loads are the waves. In accordance with the hydrodynamic load model of Morison, the hydrodynamic loads are derived from the relative horizontal 152 ECN-E

163 5 EXTERNAL INFLUENCES speed and acceleration of the water that hits the tower. This approach has been adopted in the Process Simulator. However, concentrated loads in the centre points of the underwater tower elements apply instead of distributed loads-per-unit-span. The latter complies with Morison s load model. In this section a realisation algorithm is presented. This provides wave speed and acceleration realisations in the center points of the underwater tower elements. The wave generation is based on Airy s theory, in which the horizonal wave speed and acceleration on different underwater levels have a frequency dependent relation with the elevation of the water surface. This relation is purely deterministic. Thus, it satisfies to obtain a realisation for the water surface elevation only and to derive from it the needed wave speed and acceleration realisations. These topics are dealt with in the following two subsections Realisation algorithm for water surface elevation The algorithm by Shinozuka [41] that is used for obtaining wind speed realisations is also used for elevation realisations of the water surface. Now, it is applied on the scalar auto power density spectrum S ηη (ω) of the water surface elevation. It will yield an expression for the time series η(t p ) with t p = 0, t,..., (2 m 1) t. The algoririthm comprises the next steps : 1 Get the square root S ηη of the two-sided cross power spectrum S ηη in M f 1 equidistant frequency points (M f = 2 m 1 ): S ηη(ω k ) S ηη(ω k ) = S ηη (ω k ) with ω k = k 2π 2M f t, k = 1... M f 1 (5.95) 2 Get a row vector from M f 1 random phase angles φ k between 0 and 2π : [ e jφ 1 ]... e jφ M f 1 (5.96) 3 Establish a randomly phased re-arranged two-sided harmonic row vector {s η(ω k )} of length 2M f with ω k = [ k 2π 2M f t : {s η(ω k ), k=0, 1,..., M f 1, M f, (M f 1),..., 1} [ 0 {Sηη(ω k ) e jφ k, k=1... M f 1 } 0 {Sηη(ω ] (5.97) k ) e jφ k, k=m f } ] = 4 Obtain the water surface realisation η(t) by inverse Fourier transformation of the harmonic row vector {s η(ω k )}: η(p t) = 2M f 1 1 2M f t e j ω k p t s η(ω k ) k =0 for p = M f 1 (5.98) with {ω k, k =0...2M f 1} = { k 2π 2M f t, k = k=0, 1,..., M f 1, M f, (M f 1),..., 1} Since e j k 2π 2M f t p t = e j (2M f +k) 2π p t 2M f t the expression for the Fourier synthesis can be rewritten as: (5.99) η(p t) = 2M f 1 1 2M f t k =0 e j k 2π 2M f t p t s η(ω k ) (5.100) ECN-E

164 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems The applied formulation for the two-sided auto power spectrum S ηη (ω) of the water surface elevation is the one adopted by Pierson & Moskowitz (see a.o. [?], pg 166): S ηη (ω) = g 2 e 0.74 ( g U ω )4 ω 5 (5.101) The formulation is in the radiated frequency ω (ω 0) while g represents the gravitation and U the wind speed. The significant wave height H sig is related to the average wind speed U by: 3.11 H sig = 0.74 ( g ; (5.102) U ) Wave speed and acceleration from elevation realisation In accordance with Airy s theory the following expressions for the horizonal wave speed w and acceleration ẇ in the water surface elevation η in a wave frequency ω l hold (see [?], pg 84): w ωl (t, z) = ω l cosh(2π/λ l (d z)) sinh(2π/λ l d) ẇ ωl (t, z) = ω 2 l cosh(2π/λ l (d z)) sinh(2π/λ l d) η ωl (t) η ωl (t + π 2ω l ) (5.103) while the the wave length λ l depends on the (radiated) wave frequency ω l, the gravitation g and waterdepth d in accordance with: ω l 2 = 2πg ( ) 2 πd tanh. (5.104) λ l λ l These relationships can also be considered in the frequency domain. The wave frequency ω l then becomes the independent frequency variable ω. The elevation and wave motions in underwater centre points {Sk e } are then related via transfer functions, which represent amplitude ratio s and phase shifts in any frequency ω. w Se k(ω) = ω ẇ Se k(ω) = jω ω 2π cosh( λ( ω ) (d k)) zse sinh( 2π λ( ω ) d) η(ω) 2π cosh( λ( ω ) (d k)) zse sinh( 2π λ( ω ) d) η(ω) (5.105) The distance z Se k between the water surface and an underwater center point is obtained from the sum of the water depth d and the (negatively valued) mean coordinate along the z-axis in the hydrodynamic coordinate system e H (.) of the place vector S 1 r Se k from the sea bottom to the point Sk e: ν 1 z Se k = d + ( S r k) Se H 3 = d + υ=1 G ΦS υ (3,:) S υ r S υ + G ΦS ν (3,:) S ν r Se k (5.106) The transfer functions from the elevation to the wave speed and acceleration can theoretically be transformed into linear filter expressions in the time domain, genericly denoted by convolution integrals: w(t) = ẇ(t) = g (t τ) η(τ) dτ g (t τ) η(τ) dτ (5.107) 154 ECN-E

165 5 EXTERNAL INFLUENCES The impulse-reponse-functions g (t) and g (t) in the filter expressions for the wave speed and acceleration w(t) and ẇ(t) are equal to the inverse Fourier transforms of the transfer functions: g (t) = g (t) = e jωt ω e jωt jω ω 2π cosh( λ( ω ) (d z)) sinh( 2π λ( ω ) d) dω 2π cosh( λ( ω ) (d z)) sinh( 2π λ( ω ) d) dω (5.108) It would be very cumbersome to obtain realisations w(t) and ẇ(t) in this way, although in a certain sense they are derived from the elevation η(t) in a straightforward way. Instead of filtering η(t), the wave motion realisations w(t) and ẇ(t) are directly derived from the random phase angles {φ k }. This is accomplished by carrying through the transfer functions in steps 3 and 4 of the realisation algorihtm for η(t) in the previous paragraph. Reviewed steps 3 and 4 in realisation algorihtm of η(t) for horizontal wave speed and acceleration w Se k(t) and ẇ Se k(t) 3 w, ẇ Establish randomly phased re-arranged two-sided harmonic row vectors {s w Se k (ω k )} of length 2M f for underwater centre points {Sk e} while ω k = [ {s (ω k ), k=0, 1,..., M f 1, M f, (M f 1),..., 1} ] = w Se k [ 0 { ω k cosh( 2π λ( ω k ) (d zse k )) 2π sinh( 0 { ω k cosh( 2π k 2π 2M f t : λ( ω k ) d) S ηη(ω k ) e jφ k, k=1... M f 1 }... λ( ω k ) (d zse k )) 2π sinh( λ( ω k ) d) S ηη(ω k ) e jφ k, k=m f } ] (5.109) Establish randomly phased re-arranged two-sided harmonic row vectors {s ẇ Se k (ω k )} of length 2M f for underwater centre points {Sk e} while ω k = [ {s (ω k ), k=0, 1,..., M f 1, M f, (M f 1),..., 1} ] = ẇ Se k [ 0 {jω k ω k cosh( 2π λ( ω k ) (d zse k )) 2π sinh( 0 { jω k ω k cosh( 2π k 2π 2M f t : λ( ω k ) d) S ηη(ω k ) e jφ k, k=1... M f 1 }... λ( ω k ) (d zse k )) 2π sinh( λ( ω k ) d) S ηη(ω k ) e jφ k, k=m f } ] (5.110) 4 w, ẇ Obtain wave speed and acceleration realisations {w Se k(t)} and {ẇ Se k(t)} for underwater centre points {Sk e } by inverse Fourier transformation of the harmonic row vectors {s (ω w Se k )} and {s (ω k ẇ Se k )}: k w Se k(t) = ẇ Se k(t) = 2M f 1 1 2M f t k =0 2M f 1 1 2M f t k =0 e j e j k 2π 2M f t p t s (ω w Se k ) for p = M f 1 k k 2π 2M f t p t s (ω ẇ Se k ) for p = M f 1 k with {ω k, k =0...2M f 1} = { k 2π 2M f t, k = k=0, 1,..., M f 1, M f, (M f 1),..., 1} (5.111) ECN-E

166 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems 5.3 Grid (GRD) The turbine is connected to the electric grid. Although, the frequency and voltage of the grid vary in practice (grid control, incidents, switching transients), the grid is assumed to be rigid. This implies that the voltage and frequency are assumed to be constant. 5.4 Gravity (GRV) The gravity causes a periodic effect on rotating structures. Dependent of the actual azimut angle the periodic acceleration effects tot the drive train and rotor blades are calculated. 156 ECN-E

167 6 WIND TURBINE PROCESS SIMULATION 6.1 Wind turbine simulation model In accordance with the process simulator lay-out as shown previously in fig.(2.1), all the simulation blocks as developed in chap. 3, chap. 4 and chap. 5 are programmed in Simulink blocks to make time domain simulation possible. The Simulink implementation has been done modular and uses elementary blocks, this makes automated compilation to real time code with Real Time Workshop (par. 6.4) quite easy. The full Simulink scheme is shown in fig.(6.1) The structural dynamic system (par. 4.1) is the core of this model. It interacts to other turbine systems by external input forces and torques and output positions, speeds and accellerations. Internally it comprises all structural dynamic interaction between the strucural systems: rotor, drive train, tower and foundation. The periodic influences on the drive train and rotor blades are azimuth dependent generator by the gravitation system. (par. 5.4) The aerodynamic conversion block (par. 4.2) is driven by the generated blade effective wind speed (par. 5.1). Because oblique inflow is included it is possible to simulate misalignment situations of the rotor. Based on the normal/leadwise blade speeds and the setting angle of the blade elements, the normal/leadwise aerodynamic forces and aerodynamic pitch torques are calculated and send to the structural dynamic system. The hydrodynamic conversion block (par. 4.3) is driven by wave speed and acceleration (par. 5.2). Based on tower speeds and acceleration, the foreaft and sideway hydrodynmaic forces for each under water tower element are calculated. The peripheral devices (par. 3), gearbox lubrication and cooling, generator cooling etc.), affect the turbine behaviour by realisation of demanded setpoints and commands from the turbine control system. The signal measurements and detections (acceleration, speed, position, angle, temperature pressure etc.) are also incorporated. Therefore many kinematic quantities from the structural dynamic system and temperatures from the thermic conversion system (par. 4.4) are sent to the peripheral devices. The actuators are controlled (e.g. pitch control) and commanded (e.g. pumps and contactors on/off) by the turbine control system. The output of the peripheral devices are additional torques (e.g. pitch torque, yaw torque) and loss torques. These torques are taking into account in the structural dynamic system. The branching (contactors, fuses) of the main supply (par. 5.3) in order to supply the different devices is also incorporated. For process simulation the turbine control system is usually (hardwired) connected to the real time computer, but for testing purposes it is incorporated in the Simulink scheme. The measurement and detection signals are used in the turbine control system to determine control signals and commands. The electric and mechnical power losses are supplied to the thermic conversion system (par. 4.4). In this system the power loss causes temperature changes of different components. To take windage and self ventilation into account the speed is also an input. Because of the presence of heat exchangers (generator water, gearbox oil, radiator), there s also interaction with the peripheral devices of heat power and temperatures. The electric conversion system (par. 4.5) realises the electric torque demand of the turbine control system. The currents and voltages are sent to the peripheral devices (electric loss, measurements). The system (transformer) is connected to the rigid assumed electric grid (par. 5.3). 6.2 Data input and processing The value of the model parameters (fig.(6.1)) are obtained from different sources. Both the linear state space model and the pertaining parameters of the structural dynamic system, are ECN-E

168 Real-time Process Simulator for Evaluation of Wind Turbine Control Systems Figure 6.1: Simulink implementation of process simulator. generated by a specific application of TURBU [8]. The data required to perform this is turbine and blade design data. The model order can be reduced and the input- output signals can be configured in advance. The processed input data of TURBU is also used to parameterise the non linear aerodynamic conversion system (blade profile coefficients and rotor data) and the non linear hydrodynamic conversion system (tower element geometry). Wind, wave, grid and gravity data is obtained from site properties. 158 ECN-E