MODELING AND SIMULATION OF A HYBRID WIND-DIESEL MICROGRID
VINCENT FRIEDEL
Master of Science Thesis in Electric Power Systems at the School of Electrical Engineering
Royal Institute of Technology Stockholm, Sweden, June 2009
—
In collaboration with the Georgia Institute of Technology
XR-EE-ES 2009:007
Abstract
Some communities in remote locations with high wind velocities and an unreliable utility supply, will typically install small diesel powered generators and wind generators to form a microgrid. Over the past few years, microgrid projects have been developed in many parts of the world, and commercial solutions have started to appear. Such systems face specific design issues, especially when the wind penetration is high enough to affect the operation of the diesel plant.
The dynamic behavior of a medium penetration hybrid microgrid is investigated. It consists of a diesel generator set, a wind-generator and several loads. The diesel engine drives a 62.5 kVA synchronous generator with excitation control. The fixed-speed wind turbine drives a 60 kW cage rotor induction generator. The microgrid can be connected to the utility grid but can also run as an isolated system. The total load of the microgrid is about 100 kVA which varies during the day, and consists of static and dynamic loads, including an induction motor.
The excitation controller and speed controller for the diesel’s synchronous generator are designed, as well as the power control of the wind turbine, and the controller for capacitor banks and dump load. The system is modeled and simulated using PSCAD.
The study evaluates how the power generation is shared between the diesel generator set and the wind generator, the voltage regulation dur- ing load connections, and discusses the need of battery energy storage, the system ride- through-fault capability and frequency control, partic- ularly at times when the utility is disconnected and the microgrid is run as an independent isolated power system. The results of several case studies are presented.
Keywords: Power Systems, Hybrid microgrid, Wind-Diesel system, PSCAD modeling
Acknowledgment
I wish to thank Prof. R. G. Harley for hosting my Master’s Thesis work in his lab, for his guidance and availability at every steps of this project, and for reviewing and commenting on my final report.
I am also thankful to all the members of the Power Electronics Lab at Georgia Tech for welcoming and helping me, especially Yi Du who assisted my first steps with PSCAD.
I would like to thank Katherine Elkington from the Royal Institute of Technol- ogy for her comments and for reviewing my report.
I am grateful for the opportunity I have been given to do this Master’s Project in two countries, between Sweden and the United States, and I wish to thank Mehrdad Ghandhari, my examiner, as well as the Royal Institute of Technology in Stockholm, Sweden, and the Georgia Institute of Technology in Atlanta, USA.
List of Symbols
Diesel generator set
Vb Rated RMS Line-to-Neutral Voltage V
Ib Rated RMS Line Current A
ωb Base Electrical Frequency rad/s
H Inertia s
Ta Armature Time constant s
Xd D-axis synchronous reactance p.u.
Xd′ D-axis transient reactance p.u.
Tdo′ D-axis transient open-circuit time constant s
Xd′′ D-axis subtransient reactance p.u.
Tdo′′ D-axis subtransient short-circuit time constant s
Xq Q-axis synchronous reactance p.u.
Xq′′ Q-axis subtransient reactance p.u.
Tqo′′ Q-axis subtransient open-circuit time constant s
TE Exciter time constant s
KE Exciter constant related to self-excited field -
SE Exciter Saturation Function -
KA Voltage Regulator gain -
TA Voltage Regulator amplifier time constant s
VRM IN Voltage Regulator limiter p.u.
VRM AX Voltage Regulator limiter p.u.
KF Voltage Regulator stabilizing circuit gain - TF i Voltage Regulator stabilizing circuit time constant s τ1 Dead Time of the diesel’s speed governor s K Actuator Gain of the diesel’s speed governor p.u.
τ2 Actuator time constant diesel’s speed governor s Kd Droop gain diesel’s speed governor p.u.
r Line resistance p.u.
g Line shunt conductance p.u.
l Line inductance p.u.
c Line shunt capacitance p.u.
R Cable resistance Ω/km X Cable inductance Ω/km
Loads parameters
PL Power load p.u.
QL Reactive power load p.u.
UL Voltage magnitude p.u.
TW aterP ump Water pump torque p.u.
ωim Water pump speed p.u.
sr Induction machine rated slip p.u.
ηr Induction machine efficiency at rated load p.u.
Wind turbine parameters
JRotor Rotor moment of inertia kg·m2
JBlades Blade moment of inertia kg·m2
JGenerator Generator moment of inertia kg·m2
n Gearbox ratio -
Pw Wind Turbine output power p.u.
ρ Air density kg/m3
U Wind speed m/s
A Swept area m2
Cp Power coefficient p.u.
λ Tip-speed ratio -
β Blade pitch angle. deg.
θp Motor angular position deg.
J Inertia of the blade and the motor s
B coefficient of viscous friction of the pitch mechanism -
K spring constant of the pitch mechanism -
k slope of the torque/voltage curve of the pitching system - v(t) voltage applied to the pitching motor terminals p.u.
m slope of the torque/speed curve of the pitching motor -
Qp pitching moment on the pitching system s
Dump loads parameters
PDump Active dump load p.u.
PW ind Active wind power p.u.
PDieselM IN Minimum diesel load p.u.
PLoad Active load demand p.u.
Utility grid connection parameters
VHV Voltage on the high side of the transformer p.u.
SSC Short-circuit power of the utility p.u.
ZHV Equivalent impedance on the high voltage side p.u.
VLV Voltage on the low side of the transformer p.u.
ZLV Equivalent impedance on the low voltage side p.u.
RT R Resistance on the low voltage side of the transformer p.u.
XT R Inductance on the low voltage side of the transformer p.u.
Contents
1 Introduction 1
1.1 Background . . . 1
1.2 Purpose of the project . . . 1
1.3 Literature review . . . 2
1.4 Introduction to PSCAD . . . 3
2 Modeling of the system 5 2.1 Introduction . . . 5
2.1.1 Description of the Microgrid . . . 5
2.1.2 Methodology . . . 5
2.2 Diesel Generator Set . . . 6
2.2.1 Synchronous generator . . . 6
2.2.2 Excitation system . . . 8
2.2.3 Diesel Engine . . . 11
2.2.4 Voltage and Frequency droop control . . . 13
2.3 Lines . . . 15
2.4 Loads . . . 17
2.4.1 Static Loads . . . 17
2.4.2 Dynamic Load . . . 18
2.5 Soft-Starter . . . 20
2.5.1 Building of the soft-starter . . . 20
2.5.2 Operating modes . . . 21
2.6 Wind Turbine . . . 23
2.6.1 Design choices . . . 23
2.6.2 Components of the wind turbine model . . . 24
2.6.3 Estimation of the wind turbine inertia . . . 25
2.6.4 Aerodynamics of the wind turbine . . . 28
2.6.5 Power control . . . 32
2.6.6 Wind model, gusts and turbulences . . . 34
2.7 Dump Load . . . 36
2.7.1 Need for a dump load . . . 36
2.7.2 Dump load control . . . 36
2.8 Capacitor banks . . . 37
2.9 Grid Integration . . . 38
3 Simulations and Results 41 3.1 Case study . . . 41
3.2 Diesel Generator Set Alone . . . 43
3.2.1 Step connection of the static loads . . . 43
3.2.2 Starting of the induction motor . . . 44
3.3 Wind turbine and Utility . . . 48
3.3.1 Wind Turbine Connection . . . 48
3.3.2 Step connections of loads . . . 48
3.3.3 Wind Speed steps . . . 50
3.3.4 High wind with stall or pitch controlled turbines . . . 50
3.3.5 Utility disconnection . . . 52
3.4 Wind Turbine and Diesel Generator Set . . . 54
3.4.1 Wind turbine connection . . . 54
3.4.2 Step connection of loads . . . 56
3.4.3 Wind speed steps . . . 59
3.5 Wind turbine, Diesel Generator Set and Utility . . . 61
3.5.1 Utility connection and disconnection . . . 61
3.5.2 Step connection of loads . . . 62
3.6 Three-phase faults . . . 65
3.6.1 At the diesel generator set terminal . . . 65
3.6.2 At the wind turbine terminal . . . 67
3.6.3 At the loads . . . 68
3.6.4 Conclusion . . . 70
4 Conclusion and Future Developments 71 4.1 Conclusion . . . 71
4.2 Future Work . . . 72
A PSCAD Models 73 B Particle Swarm Optimization 77 C Wind Turbine Aerodynamical Model 83 C.1 WTPerf input file example . . . . 83
C.2 Polynomial regression . . . 85
C.3 Fortran Model . . . 87
D Capacitor Bank controller 91
Bibliography 95
Chapter 1
Introduction
1.1 Background
In some remote locations, or in locations of a weak utility grid, distributed gen- eration offers a higher reliability, by providing on-site generation from many small energy sources. Typical distributed generation (DG) sources range from a few kilo- watts to a few megawatts. The disadvantage of DG sources is usually their higher costs.
Typical DG sources include hydro power, combustion engines, small wind turbines and photovoltaic systems.
Many hybrid wind-diesel systems are in operation around the world [1], [2]. These systems offer different penetration levels, with a large choice of technical solutions.
This study models a medium penetration hybrid microgrid which includes renew- able penetration of about 50 % of the load. The wind power allows a reduction of the diesel generator rating. Systems both with and without battery energy storage are commercially available. Different options are studied to ensure that the power quality requirements are matched, including design options in the wind turbine power controller, the installation of capacitor banks to correct the power factor, or dump load to ensure the power balance in the system.
1.2 Purpose of the project
This project has been carried out under the supervision of Prof. R. G. Harley, in the School of Electrical and Computer Engineering at the Georgia Institute of Tech- nology, in Atlanta, USA.
The dynamic behavior of the medium penetration hybrid system in Figure 1.1 is studied. The microgrid consists of a diesel generator set, a wind generator and sev- eral loads. The excitation controller for the synchronous generator of the diesel, as
Figure 1.1. Hybrid wind-diesel Microgrid
well as the power control of the wind turbine, are designed. The system is modeled and simulated using PSCAD.
The study investigates how the power generation is shared between the diesel gen- erator set and the wind generator, the voltage regulation during load connections, and discusses the need of battery energy storage, the system ride- through-fault ca- pability and frequency control, particularly at times when the utility is disconnected and the microgrid is run as an independent isolated power system.
1.3 Literature review
This microgrid project deals with many aspects of power systems, including electric machines, excitation systems, diesel engine, wind turbines, soft-starters and grid integration, and more particularly the specific aspects of isolated hybrid micro sys- tems. Manufacturers’ data often omit some parameters, which are then estimated based on typical values. The following books and papers have served as reference during this project.
• Microgrids : Much work on isolated systems has already been reported, but it is often case oriented, and it is difficult to import methods from one project to another. However, a review of studies related to isolated systems with Wind Power appears in Isolated Systems with Wind Power from the Risø National Laboratory [2]. The most reliable systems are simple concepts, though there is currently a tendency to include energy storage and power electronics. More and more systems take advantage of the surplus energy using some sort of storage such as water pumping, heating and cooling. The EU project “Micro- grids” [17] proposes a benchmark low-voltage network, and discusses control and safety methods in these systems.
1.4. INTRODUCTION TO PSCAD
• Power systems : Anderson’s [3] and Kundur’s [4] books have been used as reference books in the modeling of the synchronous generator, its exciter and speed governor. Both provided typical parameters values and design rules.
Furthermore, KTH compendiums [5], [15] and [16] gave the basis for electrical machines and power systems analysis.
• Wind power : Wind-diesel systems [26] and Wind Energy Explained [19]
present wind turbine design and a review of hybrid systems design issues.
Some of the main issues are the dump load for the surplus energy in the sys- tem (because of a required minimum load of the diesel or a surplus of wind energy), the need of energy storage, the possibility to start and stop the diesel generator (continuous operation implies simplicity and reliability while inter- mittent operation enables fuel savings) and the need to supply reactive power when the diesel engine is stopped, since wind turbines frequently use induction generators.
1.4 Introduction to PSCAD
PSCAD/EMTDC is an industry standard software for studying the transient be- havior of electrical networks. EMTDC performs the electromagnetic transients cal- culations while PSCAD provides the graphical interface.
PSCAD/EMTDC offers a large database of built-in components, and allows the user to define his own models, either using other built-in components or coding a component in Fortran.
Both methods are used during this study. Some components, like the electrical generators, are taken from the software library, while others, like the governors or the wind turbine aerodynamical model, are user-defined.
Chapter 2
Modeling of the system
2.1 Introduction
2.1.1 Description of the Microgrid
This project investigates the dynamic behavior of a medium penetration hybrid system or microgrid consisting of a diesel generator set, a wind generator and several loads as shown in Figure 1.1. The diesel engine drives a 62.5 kVA synchronous generator with excitation control. The wind turbine drives a 60 kW cage rotor induction generator. The microgrid can be connected to the utility grid but can also run as an isolated system. The total load of the microgrid is about 100 kVA which varies during the day, and consists of different kinds of loads, including an induction motor.
2.1.2 Methodology
Each component of the microgrid is modeled and tested independently in PSCAD.
The first component to be modeled is the diesel generator set. This first modeling, and the first simulations with the diesel generator set connected to an infinite bus are used as an introduction to PSCAD. Then, the loads and the lines are modeled by themselves while first connected to the infinite bus, and then to the diesel gen- erator set in order to check if the voltage requirements are fulfilled. A wind turbine aerodynamical model is then designed based on the blade performance in order to get closer to real small turbine performances.
Finally, the whole microgrid is assembled. A series of case studies are carried out, including the different operation modes of the system, i.e. diesel generator set only, wind turbine and diesel generator set, wind turbine and utility grid, and finally wind turbine, diesel generator set and utility grid. The results of these simulations are presented in Chapter 3.
2.2 Diesel Generator Set
The modeling of the diesel generator set is the first step of the microgrid modeling.
The purpose of this section is to introduce the diesel generator set and describe the modeling of each of its components.
The diesel generator set has to be controlled to maintain the frequency and voltage of the system while the microgrid is running in islanded mode. In this mode, it is also the only reactive power supplying component, as the wind turbine, modeled with a squirrel-cage induction generator, always consumes reactive power.
Figure 2.1. Principal components and controls of a diesel generator set.
A diesel generator set comprises a diesel combustion engine driving a synchronous electrical generator and are often used when a general power grid is not available, as a primary or auxiliary power supply [6, 19]. The model of the diesel generator set in PSCAD thus comprises a 62.5 kVA synchronous generator, an excitation system, and a diesel engine plus governor. The diesel engine and the synchronous generator rotate at the same mechanical speed, i.e. 1800 rpm, and no gearbox is used.
2.2.1 Synchronous generator PSCAD Model
Synchronous machines are common in Power Systems. They are used to convert the mechanical energy supplied by the prime mover into electrical power. Their electrical frequency is proportional to the mechanical speed and is obtained by:
ωe= p
2ωm (2.1)
where p is the number of poles, ωe and ωm respectively the generator’s electrical and mechanical speeds.
The dynamic model of the synchronous generator is designed using the built-in synchronous generator in PSCAD. As usual, the mathematical description of the
2.2. DIESEL GENERATOR SET
Table 2.1. Dynamic parameters for the Marathon make synchronous generator.
Parameters Value Unit Description
Vb 277.1 V Rated RMS Line-to-Neutral Voltage Ib 75.2 A Rated RMS Line Current
ωb 377 rad/s Base Electrical Frequency
H 0.09 s Inertia
Ta 0.01 s Armature Time constant Xd 1.875 p.u. d-axis synchronous reactance Xd′ 0.132 p.u. d-axis transient reactance
Tdo′ 0.68 s d-axis transient open-circuit time constant Xd′′ 0.11 p.u. d-axis subtransient reactance
Tdo′′ 0.0096 s d-axis subtransient short-circuit time constant Xq 1.875 p.u. q-axis synchronous reactance
Xq′′ 0.228 p.u. q-axis subtransient reactance
Tqo′′ 0.0658 s q-axis subtransient open-circuit time constant
generator is based on Park’s transformation [3], thus projecting the variables on the direct axis of the field winding, on the quadrature axis and the neutral axis. Among other dynamic calculations, the model determines the torque and uses the dynamic equation:
∆ ˙ω = Tm−Te−D∆ω
J (2.2)
where Tm is the mechanical torque, Te is the electrical torque, D is the damping of the generator, ω is the shaft speed and J the inertia.
The modeling is based on a Marathon 62.5 kVA, 4 pole machine. Its mechani- cal speed is 1800 rpm at 60 Hz and its parameters are shown in Table 2.1. It is more convenient to normalize all the parameters, and that is done as described in [3], and explained below. Note that the rated power and voltage of the synchronous generator are chosen as the base values for the whole microgrid.
Sb1Φ= Generator rated per-phase VA = 20.8 kVA (2.3) UbLN = Generator rated line-to neutral RMS terminal voltage = 277.1 V (2.4)
Ib = Sb1Φ
UbLN = 75.2 A (2.5)
Zb= UbLN
Ib = 3.69 Ω (2.6)
Some of the generator’s parameters have not been provided by the manufacturer:
Xq, Tdo′′, Xq′′ and Tqo′′. These parameters are therefore estimated according to [7]:
Xq≈Xd (2.7) as the generator is modeled as a round rotor generator,
Tdo′′ = Xd·Td′ ·Td′′
Xd′′·Tdo′ (2.8)
Xq′′= 2X2−Xd′′ (2.9)
where X2 is the negative sequence reactance, and X2 = 0.169 p.u., and
Tq′′≈Td′′ (2.10)
Tqo′′ = Xq
Xq′′ ·Tq′′ (2.11)
The inertia in seconds H is derived from the given inertia J in kg·m2 according to Equation 2.12
H =
1 2Jω2
RVA (2.12)
where ω is the shaft speed in rad/s and RVA the synchronous generator rated rating in VA.
2.2.2 Excitation system
The generator excitation system consists of an exciter and a voltage regulator, shown in Figure 2.2. A comprehensive description of such systems is provided in [3] and [4].
The purpose of this part of the modeling is to simulate a typical manufacturer’s exciter. Though simple excitations systems may be implemented in the PSCAD system, the IEEE provides standardized mathematical models which are designed to represent specific commercial systems. These systems, intended for use in com- puter simulations, are described by Kundur in [4] and have been updated by the IEEE in [8].
The exciter used with the chosen Marathon generator is a rotating and brushless system. It is modeled as the AC5A transfer function presented in Figure 2.2. The transfer function can be divided in two parts: the exciter itself and the voltage regulator. The automatic voltage regulator (AVR) contains a stabilizing feedback loop. The description of each symbol and typical values given by the IEEE are shown in Table 2.2.
As the manufacturer’s parameters are not available, most of these typical values from Table 2.2 are used. However the stabilizing feedback loop is optimized, as described by Kundur [4]. Note that according to [9], a critical selection of these
2.2. DIESEL GENERATOR SET
Figure 2.2. Transfer function of the AC5A excitation system model [8].
Table 2.2. Excitation system model parameters.
Symbol Typical Value Description
TE 0.8 Exciter time constant
KE 1.0 Exciter constant related to self-excited field
SE - Exciter Saturation Function
SE(EFD1) 0.86 -
EFD1 5.6 -
SE(EFD2) 0.5 -
EFD2 4.4 -
KA 400 Regulator gain
TA 0.2 Regulator amplifier time constant
VRMIN -7.3 -
VRMAX 7.3 -
KF 0.03 Regulator stabilizing circuit gain
TF1, TF2, TF3 1.0, 0, 0 Regulator stabilizing circuit time constant
parameters is not necessary, and that stability can be maintained over a significant range even when using the typical parameter values. However, the exciter has to be optimized to provide an acceptable steady-state error.
The optimized system, consisting of the synchronous generator set and its exci- tation system, is tested by applying a step increase in the voltage reference VREF. The stabilizing components KF, TF1, TF2 and TF3 are optimized using the Particle Swarm Optimization algorithm, which is described in Appendix B. The simulation is computed by PSCAD, and the PSO algorithm is implemented in MATLAB. The cost function CF of this PSO is a linear function of the overshoot (OS), the settling time (ST) at a 2% band and the integral of the difference between the reference and terminal voltages (Ar), as defined in Equation 2.13 and Figure 2.3.
Figure 2.3. Overshoot, settling time and area of voltage response.
CF = a · OS + b · ST + C · Ar (2.13)
0 1 2 3 4 5
0 0.2 0.4 0.6 0.8 1 1.2
Time (s)
Terminal Voltage (pu)
Figure 2.4. Response of the 10 particles at the sixth iteration of the PSO algorithm.
Each particle has four elements: KF, TF1, TF2, TF3. The initial values of the elements and thus the positions of these particles in the solution space have a strong influence on the rapidity of convergence and on the ability to find the optimal solution. Nevertheless, these positions are chosen randomly within given limits.
Table 2.3 shows the initial range and final optimal value of each element when all the particles have converged to the same solution. The disturbance is a step change from 0.9 to 1.0 p.u. in the reference voltage. Figure 2.4 shows the response to the reference voltage Vref step change at the sixth iteration of the algorithm, when the response is assumed satisfactory. The final parameters in Table 2.3 are selected.
2.2. DIESEL GENERATOR SET
Table 2.3. Initial range and final position of the particles.
Symbol Initial Range Final Value
KF [0;0.1] 0.0683
TF1 [0.5;1.5] 1.1413 TF2 [0;1.0] 0.4645 TF3 [0;0.5] 0.0037
2.2.3 Diesel Engine
Internal Combustion Engine Model
The diesel engine is used to provide power to the generator and control the speed of its shaft by means of a governor. An Internal Combustion Engine is available in PSCAD Library. It takes the shaft speed and the fuel intake as inputs, and supplies a mechanical torque Tm as output. This built-in model is customizable, and the parameters of Table 2.4 are entered. The diesel engine rating is sized about 25 percent larger than the electrical generator in order to support overload. It rotates at the same speed as the synchronous generator, and thus no gearbox is needed.
The PSCAD model also allows to study the effect of misfired cylinders, but this option is not used in this study.
Table 2.4. Input parameters of the Internal Combustion Engine.
Parameter Value Unit
Engine rating 80 MW
Machine rating 62.5 MVA
Engine speed rating 1800 rpm
Number of cylinders 6 -
Number of engine cycles Four stroke -
Misfired cylinders No -
PSCAD generates the output mechanical torque from an input cylinder torque/angle curve. Many diesel engines operate on a four-stroke cycle, and the typical torque- angle characteristic curve is presented in a shape as seen in Figure 2.5, [10]. This curve data is entered into the PSCAD model. The four events during this cycle are the intake, compression, power and exhaust strokes. During intake and exhaust, the torque production is negligible compared to the compression and power strokes.
During compression, the torque is negative to increase the gas pressure, and the explosion occurs during the power stroke. It corresponds to the highest peak in Figure 2.5.
Each cylinder offers the same torque-angle characteristic, with an angle difference between cylinders depending on the number of cylinders. Finally, the total diesel
−400 −200 0 200 400
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Angle (deg.)
Torque (p.u.)
Figure 2.5. Produced torque of one cylinder in a cycle.
engine torque is equal to the sum of these individual torques. Because of the sharp- ness of the power stroke, the resultant torque of a four-cylinder engine contains significant ripples. These ripples are somewhat smoothed by the engine inertia, but are still large enough to cause vibrations. A flywheel may be added in order to in- crease the smoothing, or an engine with more cylinders offering a smoother torque may be used. In this study, a six-cylinder engine is chosen.
The torsional effect on the shaft is neglected, as the system is small and mechanically stiff. A rigid body diesel generator set is therefore assumed and the inertia of the engine is added to the synchronous generator’s inertia. A typical inertia of a 75 kW diesel engine is about 1 kg·m2, which gives H ≈ 0.24 s. Adding a flywheel, a typical inertia for the whole diesel engine and generator set is about 0.5 s [11].
Speed Controller
The diesel engine speed is controlled by its fuel intake which in turn is regulated by a governor as shown in Figure 2.6. In this microgrid, the speed controller of the diesel generator set is responsible for the system frequency whenever the microgrid is run isolated from the main grid.
Typical diesel governors are described in [11] and [12]. The dead time τ represents the time required for each cylinder to receive the fuel, since not all the cylinders are in that position at the same time. The actuator, which produces the fuel flow, is also represented by a time constant τ2. A limiter is added, as the fuel intake can not be negative, and also has a maximum value. The mechanical torque Tm is transmitted to the synchronous generator. The typical values used in the modeling
2.2. DIESEL GENERATOR SET
Figure 2.6. Governor model in PSCAD.
are shown in Table 2.5.
Table 2.5. Typical governor data [11], and Proportional controller gain.
Parameter Value Unit Description
τ 0.02 s Dead Time
K 1.0 p.u. Actuator Gain
τ2 0.05 s Actuator time constant Kd 50.0 p.u. Droop gain
2.2.4 Voltage and Frequency droop control
The diesel engine is responsible for the system’s frequency and voltage control. The main task of these controllers is to take care of the active and reactive power sharing between the sources, using only the local information (voltage and frequency).
This is done using an active power versus frequency droop, and a reactive power versus voltage droop [13, 14], as illustrated in Figure 2.7. Using this solution, the micro-source only uses the local information to adjust its power production.
This facilitates the expansion of the microgrid, since each micro-source controls its production based on the local voltage and frequency. The power setpoint of each individual source can be set independently, and each micro-source has the ability to autonomously adjust its output following a disturbance.
In Figure 2.7, the frequency versus active power and voltage versus reactive power droops are illustrated. The power setpoints P0 and Q0 determine the power gener- ation at nominal frequency f0 and voltage V0, when the microgrid is connected to the utility grid.
Figure 2.7. Frequency vs active power and voltage vs reactive power droops.
Proportional-integral (PI) or proportional-integral-derivative (PID) controllers are typically used in governors of diesel generator sets. However, these controllers do not allow a speed droop, and a proportional (P) controller is used. The frequency is allowed to change as a function of the active power demand. When the active power demand increases, the diesel generator set speed decreases slightly and reaches a new steady-state. When the active power demand decreases, less kinetic energy is extracted from the generator, which accelerates.
In this microgrid, the gain of the governor is designed to allow a 4 % frequency droop, as shown in Figure 2.7. This droop is responsible for the active power sharing between the active power sources, which are the wind turbine, the diesel generator set and the utility grid.
If the voltage regulator of the diesel generator set does not allow a voltage droop, then there might be reactive power circulating between the utility grid and the diesel generator set at times when both are connected to the microgrid. This re- active power flow would be caused by the difference between the reference voltage of the synchronous generator’s exciter and the actual terminal voltage set by the utility grid.
A voltage vs reactive power droop is implemented as follows [13]: the voltage refer- ence is set to be a function of the reactive power output from the diesel generator, within an acceptable percentage (in this study ± 2 %) around 1.0 p.u.
2.3. LINES
2.3 Lines
In a power system, electric energy is transmitted from power plants to consumers via lines, cables and transformers [16]. A classical single-phase model of a symmet- rical three-phase line is shown in Figure 2.8.
Figure 2.8. Model of a line with distributed quantities.
This power line model has a resistance r and an inductance l, corresponding respec- tively to the resistivity of the conductor and the magnetic flux surrounding the line.
The shunt parameters (shunt conductance g and shunt capacitance c) represent the leakage currents in the insulation and the electric field between the lines. These quantities are distributed along the line.
Figure 2.9. Lumped parameters of a medium line model.
For short and medium length lines, this distribution along the line can however be neglected, and it is possible to calculate the total resistance and inductance of the line as lumped parameters. The π-equivalent model of a medium line with lumped parameters is shown in Figure 2.9.
Figure 2.10. Lumped parameters of a short line model.
In a microgrid, the lines are usually tens to hundreds of meters, and thus the short line model shown in Figure 2.10 is used, neglecting the shunt capacitance, as shown in [18].
In this study, the wind turbine is assumed to be 500 m far from the substation, the diesel generator set connected through a 20 m cable, and the loads are 250 m from the substation. The voltage level throughout the microgrid is 480 V.
Typical impedance values of overhead, twisted, Al. cables are shown in Table 2.6.
Table 2.6. Impedance data for low-voltage cables [18]
Cable type R (Ω/km) X (Ω/km)
Twisted cable, 4×50 mm2, Al 0.642 0.100 Twisted cable, 4×120 mm2, Al 0.255 0.096 Twisted cable, 4×150 mm2, Al 0.208 0.096
Note that cables offering lower resistance may be more expensive, but they may limit losses and voltage drop. Here, 4×120 mm2 cables are chosen, and the cable parameters are:
Table 2.7. Impedance of the system’s cables.
Cable Length (m) R (p.u.) X (p.u.)
Wind Turbine to Substation 500 0.0346 0.013 Diesel Generator Set to Substation 20 0.0014 0.00052
Substation to loads 250 0.0173 0.0065
2.4. LOADS
2.4 Loads
The microgrid in Figure 1.1 includes three different kinds of loads: lighting or heating, computers and a water pump. These loads are represented respectively by a constant impedance characteristic, a constant power characteristic, and an induction motor. Each type of load amounts to about 33 kVA, and may vary during the day. The total maximum load is thus 100 kVA. The load modeling is further detailed in this section and is divided between static (lighting/heating, computers) and dynamic loads (induction machine).
2.4.1 Static Loads
A description of static load models is given in Chapter 3 of [15]. A general repre- sentation of static loads, taking the frequency dependency into account is given.
PL= PZIP+ κPL0 2
X
k=1
kpk
UL
UL0
mpk
(1 + Dpk∆f)
!
(2.14)
QL = QZIP+ κQL0 2
X
k=1
kqk
UL
UL0
mqk
(1 + Dqk∆f)
!
(2.15) Simpler models are also widely used, as the exponential and the so-called ZIP mod- els, composed of constant impedance (Z), constant current (I) and constant power (P). The exponential models for active and reactive loads are expressed as:
PL= PEXP = PL0
UL
UL0
mp
and QL= QEXP= QL0
UL
UL0
mq
(2.16) The exponents mp and mq are the parameters of this model, and can be set as follows:
• mp = mq = 0 : constant power characteristic
• mp = mq = 1 : constant current characteristic
• mp = mq = 2 : constant impedance characteristic
The static loads of the microgrid are represented using the exponential model of Equations 2.16. Both the constant power and the constant impedance are repre- sented with a 0.9 power factor. Note that other loads such as a dump load may be added, when the wind power exceeds the load demand.
Among the loads in the system, the computers are likely to have the tightest volt- age variation tolerance. The Information Technology Industry Council (ITIC) gives voltage sag tolerance of typical computers in the ITIC Curve shown in Figure 2.11.
The voltage of the microgrid will thus have to obey these clearing times, which are summarized in Table 2.8, and this table will be taken as reference especially when the microgrid is islanded.
Figure 2.11. ITIC curve.
Table 2.8. Voltage tolerance for computers.
Voltage Sag Duration (s) V < 0.7 0.02 0.7 < V < 0.8 0.5 0.8 < V < 0.95 10
1.1 < V < 1.2 0.5 1.2 < V < 1.4 0.03
1.4 < V 0.01
2.4.2 Dynamic Load
The dynamic load is a 34 kW induction motor, which drives a water pump. A squirrel-cage rotor model is used, and its parameters are entered in PSCAD using the so-called EMTP Type 40 format, in which the parameters are entered based on the steady state torque-slip curve, as shown in Table 2.9. The water pump impeller is a square-law device and its torque TWaterpump is proportional to the square of its speed ωim, such that
TWaterpump = C · ω2im (2.17)
where the parameter C is tuned so that the motor reaches full torque at rated slip.
The motor rated slip is sr = 0.233 p.u., and its efficiency at full load is ηr = 0.92.
2.4. LOADS
Thus, C can be calculated as follows:
C = 1
(1 − sr)2 · 1 ηr
= 1
(1 − 0.0233)2 · 1
0.92 = 1.14 (2.18)
Table 2.9. Induction machine parameters.
Parameter Unit Description
Rated Power kW 34
Rated RMS Voltage V 277
Rated speed rpm 1758
Efficiency % 92
Power Factor - 0.82
Locked Rotor current p.u. 6.887 Starting Torque p.u. 2.5043 Break down Torque p.u. 2.9217
2.5 Soft-Starter
2.5.1 Building of the soft-starter
Although the use of power electronics is preferably avoided because of their cost or harmonics they introduce, soft-starters may be used to reduce the voltage drop along the supply feeder during induction machines’ starting. In the present system, soft-starters may be required to start the water pump, or during the connection of the wind turbine [28]. These applications are discussed later in this report.
A soft-starter is implemented by 6 thyristors installed in an anti-parallel config- uration, as shown in Figure 2.12. Its control circuit is shown in Figure 2.13. A snubber RC-circuit limits the rate of change of the voltage across each thyristor.
Figure 2.12. Model of a soft-starter in PSCAD.
Here a simple current limiting soft-starter is modeled. The firing angles are con- trolled in order to limit the voltage drop. In Figure 2.12, Ea2 refers to the voltage of phase a and its zero-crossings are taken as reference to calculate the angle at which the forward thyristor in phase a is triggered. The reverse and forward thyristors triggerings have a phase difference of 180 degrees, and forward thyristor triggerings are separated by 120 degrees.
Figure 2.14 shows the control signals determining the firing angle for Ia, governing the forward thyristor on phase a. Different control methods may be implemented to limit the starting current. Here the firing angle is simply decreased linearly with
2.5. SOFT-STARTER
Figure 2.13. Control circuit of the soft-starter.
time, and the slope of this decrease is tuned to minimize the voltage drop at the motor terminals.
Figure 2.14. Control signal of the soft-starter.
2.5.2 Operating modes
Depending on the firing angle, four operating modes may take place [28]:
• Open circuit. When the firing angle is greater than 150 degrees, the thyristors will not conduct when they are triggered, and thus the soft-starter appears as an open-circuit.
• If the firing angle is decreased, the soft-starter enters a mode where either two or none of the thyristors are conducting.
• If the firing angle is further decreased, it increases the conduction interval, and either 3 or 2 thyristors are conducting.
• If the conduction interval is extended further (i.e. the firing angle is de- creased), the soft-starter will not change the voltage and will transmit all the energy.
These operating modes are illustrated in Figure 2.15, showing the current Ia in the forward thyristor in phase a.
16.075 16.08 16.085 16.09 16.095
−1.5
−1
−0.5 0 0.5 1 1.5
Time (s)
Current (pu)
(a) Firing angle: 150 degrees
18.485 18.49 18.495 18.5 18.505 18.51
−1
−0.5 0 0.5 1
Time (s)
Current (pu)
(b) Firing angle: 135 degrees
23.94 23.95 23.96 23.97 23.98
−3
−2
−1 0 1 2 3
Time (s)
Current (pu)
(c) Firing angle: 90 degrees
31.45 31.46 31.47 31.48
−1.5
−1
−0.5 0 0.5 1 1.5
Time (s)
Current (pu)
(d) Firing angle: 0 degree Figure 2.15. Current waveform in phase a for different firing angles.
2.6. WIND TURBINE
2.6 Wind Turbine
2.6.1 Design choices
The wind turbine chosen in the microgrid of Figure 1.1 is a 60 kW fixed-speed turbine driving a cage-rotor induction generator. In this section, different aspects of wind turbines are discussed.
Induction generator
The electrical generator converts mechanical power into electrical power. Different kinds of generators may be used for wind turbines. Some small turbines use DC generators, but the most common types are synchronous and induction generators.
Induction generators have usually a simple construction, and are relatively cheaper.
They also simplify the connection and disconnection from the grid. Different types of induction generators are used, such as a cage rotor type, a wound rotor with variable rotor resistance type, or a doubly fed slip ring type.
However, when induction generators are used in small or isolated electrical networks, special measures must be taken in order to supply reactive power or maintain the voltage stability.
Figure 2.16. EW50 50 kW wind turbine [23]
Operating scheme
Wind turbines with induction generators may operate at a nearly constant rotor speed or at variable speed. In both cases, below rated wind speed, the goal is to maximize the energy production, while power has to be limited above rated speed.
However, variable speed turbines require the use of power electronics, which in- creases the cost, and may introduce harmonics in the system. In this study, a
constant speed turbine is chosen. Both a stall-regulated and a pitch-controlled tur- bines are modeled.
The blades of a stall-regulated wind turbine are designed to intrinsically regulate the wind power production. They are optimized so that their efficiency drops at higher wind speeds.
In the case of a pitch-controlled turbine, the blade angle can be controlled. Be- low rated wind speed, the blade angle is kept constant and aims to maximize the power production. Once rated wind speed is reached, this blade angle is increased in order to limit the aerodynamical torque.
Both a stall-regulated and a pitch-controlled model are simulated in PSCAD, and these are discussed later in the report.
Starting of wind turbines
There are mainly two methods to start a fixed-speed wind turbine. The first one is to run its generator as a motor until rated speed is reached, and then it switches to generator mode. The second one is to release the brakes and let the aerodynamical force accelerate the rotor, until its speed approaches the rated speed, and then connect the generator to the grid. Connection methods have been described in [27].
2.6.2 Components of the wind turbine model
In PSCAD, the wind turbine model consists of an electrical machine, a wind source and a turbine aerodynamical model. In this study, the electrical generator is a cage rotor induction machine. The aerodynamical model takes the blades’ performance into account, and if pitch-control is used, another block is used to model the blade pitch control. The gearbox is not represented in PSCAD, and all the parameters have either to be transferred to the high-speed side or to be calculated in the per- unit system.
The induction generator’s modeling is based on manufacturer’s data, like the water pump induction machine described in Section 2.4.2. Its parameters are detailed in Table 2.10, and its torque speed curve is shown in Figure 2.17.
The wind turbine used in the microgrid is a 60 kW turbine. Since the goal here is to build a realistic model, the goal is to approach the data given by manufacturers.
Two turbines giving a good amount of information have been found. The first one is a WES 80 kW turbine [22]. It is a 2-blade, variable-speed turbine with a rotor diameter of 18m, and its output power is regulated both by a power electronic con- verter and pitch angle. The second one is a EW 50 kW [23], 3 blade, stall-regulated turbine with a rotor diameter of 15 m. Both offer similar per-unit power curves,
2.6. WIND TURBINE
0 600 1200 1800 2400 3000 3600
−4
−3
−2
−1 0 1 2 3 4
Speed (rpm)
Torque (pu)
Figure 2.17. Torque-Speed Curve of the induction machine.
with a cut-in speed (at which the wind turbine starts to generate power) of about 4 m/s, rated wind speed (when it reaches its rated power) of 13 m/s and cut-out speed (at which it is disconnected from the system) of 25 m/s. Both turbines use a gearbox, offering ratios from 20 to 28. The development of the wind turbine aero- dynamical model is detailed in Section 2.6.4.
The pitch controller which regulates the wind turbine’s power and the wind source are explained in Section 2.6.5 and Section 2.6.6.
Table 2.10. Wind turbine induction generator parameters.
Parameter Unit Description
Rated Power kW 60
Rated RMS Voltage V 277
Rated speed rpm 1758
Power Factor - 0.88
Locked Rotor current p.u. 6.6783
Starting Torque p.u. 2.4
Break down Torque p.u. 2.6
Inertia kg·m2 0.52
2.6.3 Estimation of the wind turbine inertia
The inertia time constant has a strong impact on the transients of wind turbines.
However, this parameter is typically not reported by manufacturers, and an estima- tion is necessary. Examples of inertia time constants are [29, 30] 3.5 s for a 150 kW turbine, 4.64 s and 5.19 s respectively for a GE 1.5 MW and a GE 3.6 MW turbine.
In this study the induction generator inertia is known, and a simplified model of the rotor is analyzed in order to estimate the mass distribution along the blades and calculate the inertia. Note that the gearbox does not appear in PSCAD, and all values have either to be transferred to the high-speed side or to be directly cal- culated in the per-unit system.
The WES80 inertia will serve as model, since the manufacturer provides a fair amount of informations. This turbine has 2 blades. The gear ratio is 20, the gener- ator rotates at 1800 rpm, and the blades rotate at 90 rpm. The total rotor diameter is 18 m, and each blade is 7.8 m long. The rotor without the blades has a diameter of 2.4 m. Each blade weighs 86 kg, and the total rotor weighs is about 900 kg.
This rotor is illustrated in Figure 2.18.
Figure 2.18. Rotor model
The generator’s inertia is known, equal to 0.52 kg·m2 and its rotor weighs about 100 kg [24]. The generator rotor is considered as the only part of this system ro- tating at 1800 rpm. So the group consisting of the blades and the rotor hub weighs 900 − 100 = 800 kg and is rotating at 90 rpm.
The rotor is approximated by a homogeneous cylinder with a 2.4 m diameter. Ac- cording to the previous estimations, it weighs 800 − (2 · 86) = 628 kg and its inertia is calculated as
Jcyl = 1
2 ·M · R2 = 1
2·628 · 1.22= 452 kg · m2 (2.19)
2.6. WIND TURBINE
Mass distribution and inertia of the blades
The blade chord varies from 500 mm at the tip to 625 mm near the center of inertia. A much simplified model of the blade is used, considering that the chord varies linearly (Figure 2.19) and the weight is uniform along the blades. In that case, the blade density is calculated as
ρSurface = 86 · 2
7.8 · (0.625 + 0.5) = 19.6 kg/m2 (2.20)
Figure 2.19. Blade model.
Then, the weight w of a section at a distance r from the bottom of the blade and of length δr is estimated as
w(r, δr) = ρSurface·δr · Chord(r) = 19.6 · δr ·
0.625 − r 7.80.125
(2.21) Finally, the inertia of each blade is expressed as
J1Blade=Z 7.8
0
ω(r, δr) · (r + 1.2)2 (2.22)
J1Blade= 19.6
−0.016
4 r4+ 0.5865
3 r3+ 1.4779
2 r2+ 0.9r
7.8 0
(2.23) J1Blade= 2547 kg · m2 J2Blades= 2 · J1Blade= 5094 kg · m2 (2.24) Calculation of the total wind turbine inertia
To summarize, there are three components in this simplified system. Two of them are rotating at 90 rpm and their inertia is JRotor = 452 kg·m2 and JBlades = 5094 kg·m2 respectively. The induction generator is rotating at 1800 rpm and its inertia is JGenerator= 0.52 kg·m2. Seen from the high speed side, the equivalent moment of inertia of this system is
Jtotal= JRotor+ JBlades
n2 + JGenerator (2.25)
where n is the gearbox ratio.
Jtotal= 452 + 5094
202 + 0.52 = 14.38 kg · m2 (2.26) Finally, the inertia time constant of the whole system is
H =
1
2Jtotalω2
RVA =
1
214.381800·2π60 2
80000 = 3.19 s (2.27)
where RVA is the rated power of the turbine, and ω is the shaft speed.
The value 3.2 s will be used in the study.
2.6.4 Aerodynamics of the wind turbine PSCAD built-in model
Figure 2.20. Power Curves: PSCAD model (squares) and actual turbine
The prebuilt wind governor in PSCAD offers two models, one suitable for horizontal- axis turbines with two blades (called MOD 5), and the other one for horizontal-axis turbines with three blades (MOD 2), based on two papers in IEEE Transactions on Power Apparatus and Systems. Some parameters can be set: the generator rated power, the machine angular speed, rotor radius, air density, gear box efficiency and the gear ratio.
The first test is to plot the power curve (Figure 2.20) of the PSCAD model in order to validate the aerodynamic model, before studying the dynamics of the system and optimizing the pitch-controller.
User-defined aerodynamical model
Since the power curve of the PSCAD model does not match the selected model, a second aerodynamical model is developed. This work is based on Chapter 3 of [19].
The output power from a wind turbine is given by Equation 2.28.
2.6. WIND TURBINE
Pw = 1
2ρU3ACp(λ, β) (2.28)
where Pw is the output power in kW, ρ is the air density in kg/m3, U the wind speed in m/s, A the swept area in m2, Cp the power coefficient in p.u., β the blade pitch angle in degrees, and λ the tip-speed ratio defined as
λ = ω · R
U (2.29)
where ω is the rotor rotational speed, U is the wind speed and R is the rotor radius.
The National Renewable Energy Laboratory website [25] provides different codes helping to predict rotor performances and the Cp values are found from WTPerf.
This software simulates the blades’ performance based on many input parameters like the number of blades, rotor radius, hub height, blade chord and twist distribu- tions. An example of a WTPerf input file is shown in Annex C.1.
The goal is to approach an actual turbine’s performance. This is done by “tuning”
the blade chord and twist and testing the blade’s performance thanks to WTPerf.
The optimal chord distribution is given by the equation [20]
C(r) = 1 n· 8
9 · CL
2πR 1
λopt
qλ2opt(Rr)2+49 (2.30) where CL is the lift coefficient in p.u., R the rotor radius in m, λopt the design tip-speed ratio and n the number of blades.
For small tip-speed ratios, a large number of blades is usually recommended (8 to 24 blades for a tip-speed ratio equal to 1). For ratios larger than 4, 1 to 3 blades are recommended.
The optimal twist angle distribution can also be determined as (see [19])
φ = arctan 2 3λRr
!
(2.31) For example, the parameters in Table 2.11 are obtained for a 3-blade turbine, with a 7.5 m rotor radius, and a tip-speed ratio equal to 6.5.
The optimal distributions given by Equation 2.30 and Equation 2.31 are approx- imated by linear segments to be entered as input in WTPerf. The software then calculates the power coefficients for different tip-speed ratios. Figure 2.21 shows Cp(λ, β) curves for different blade pitch angles. Note that the highest efficiencies are obtained with the smallest angles in Figure 2.21.
Table 2.11. Optimal chord and twist angles.
r/R Twist Angle (deg.) Chord (m)
0.1 45.73 3.23
0.2 27.15 2.06
0.3 18.87 1.46
0.4 14.38 1.12
0.5 11.59 0.91
0.6 9.70 0.76
0.7 8.34 0.65
0.8 7.31 0.57
0.9 6.50 0.51
1.0 5.86 0.46
Figure 2.21. Curves of power coefficient Cp(λ, β) versus tip-speed ratio, computed with WTPerf
WTPerf provides a table of discrete values of Cp for different values of the wind speed and the blade angle. However, an approximating function is required to build the model in PSCAD. A two-dimensional polynomial regression (approximation) is computed (see the MATLAB code in Appendix C.2) as seen in Figure 2.22 and finally the power coefficient as a function of the tip-speed ratio λ and the pitch angle β is obtained and used to build the aerodynamical model in PSCAD.
Final model
This aerodynamic model of the wind turbine is based on the same principles as the prebuilt model in PSCAD. It calculates the output torque and power transmitted to the electrical generator based on the wind speed Uwind, the shaft speed ω and the blade angle β.
2.6. WIND TURBINE
Figure 2.22. Polynomial regression of the power coefficient function, computed with MATLAB
First, the tip-speed ratio λ is calculated based on the rotor radius R, the shaft angular velocity ω and the wind speed Uwind:
λ = ωR Uwind
(2.32) Then the Power coefficient is estimated based on the polynomial approximation P (λ, β) previously explained:
Cp = P (λ, β) (2.33)
The wind power in the swept area is
PW ind = 1
2ρAU3 (2.34)
The wind mechanical power generated by the wind turbine is calculated as
Pm = Cp·PW ind (2.35)
Finally, the mechanical torque transmitted to the induction generator is Tm= Pm
ω (2.36)
Two models are built: a stall-regulated turbine, taking the Power curve of EW50 as reference (see Figure 2.23), and a pitch controlled turbine, taking the WES80 power curve as reference (see Figure 2.24).
Figure 2.23. Power curve of the stall regulated turbine, model (squares) and refer- ence
Figure 2.24. Power curve of the pitch controlled turbine, model (squares) and reference
2.6.5 Power control Stall control
In the case of a stall regulated turbine, the blades are designed to reduce their own efficiency in high winds. The blades have a fixed pitch, and no power controller is required.
Pitch control
The pitch controller is of paramount importance in a pitch-controlled constant-speed wind turbine. The design of the pitch actuator is based on the proposed system in [19], which gives the transfer function of a simple pitch mechanism driven by an AC motor. It includes moments from spring and viscous frictions. The differential equation for this system is
2.6. WIND TURBINE
J ¨θp+ B ˙θp+ Kθp= kv(t) + m ˙θp+ Qp (2.37) where θp is the angular position of the motor in degrees, J is the inertia of the blade and the motor in seconds, B is the coefficient of viscous friction, K is the spring constant, k is the slope of the torque/voltage curve for the motor mechanism system, v(t) is the voltage applied to the motor terminals in p.u., m is the slope of the torque/speed curve and Qp is a pitching moment due to aerodynamic forces that disturb the system.
The differential equation leads to the following transfer function:
Θps = 1
Js2+ (B − m)s + K(KΘp,ref(s) + Qp(s)) (2.38) Reference [19] gives typical values for the parameters. J = 161 s, B − m = 14 and K = 1.
One of the main limitations of such pitch controlled turbines is the fact that the pitch mechanism is very slow. A rate limiter has thus been added, limiting the blade angle rate change to ± 5 deg/s as in [31]. Finally, the blade angle itself is limited between 1 and 25 degrees to replicate the model’s performance.
Figure 2.25. Pitch mechanism and pitch controller in PSCAD.
Typical pitch controllers are of the PI or PID type. Both are evaluated and op- timized using Particle Swarm Optimization, and a PID is finally selected as it increases the system’s responsiveness, and its stability, especially in higher winds.
Using Particle Swarm Optimization, the power output response to a wind speed step is optimized. The optimized parameters are the PID parameters: the propor- tional gain Kp, the integral time constant Ti and the derivative time constant Td. Starting ranges and final values are shown in Table 2.12. Figure 2.26 shows the response to a wind speed step after the tenth iteration of the algorithm. In that case, the first overshoot remains high in any case, as a result of the slow response of the pitch mechanism.
