OPTIMISATION CONTROL STRATEGY TO
IMPROVE TOTAL FARM POWER
PRODUCTION
Dissertation in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE WITH A MAJOR IN ENERGY TECHNOLOGY WITH FOCUS ON WIND POWER
Uppsala University
Department of Earth Sciences, Campus Gotland
Jun Liang, Tan
DEVELOPMENT OF A PITCH BASED WAKE
OPTIMISATION CONTROL STRATEGY TO
IMPROVE TOTAL FARM POWER
PRODUCTION
Dissertation in partial fulfilment of the requirements for the degree of
MASTER OF SCIENCE WITH A MAJOR IN ENERGY TECHNOLOGY WITH FOCUS ON WIND POWER
Uppsala University
Department of Earth Sciences, Campus Gotland
Approved by:
Supervisor, Stefan Ivanell
Examiner, Simon-Philippe Breton
Abstract
Department of Earth Science
Masters In Wind Power Project Management
Development of a pitch based wake optimisation control strategy to improve total farm power production
by Jun Liang Tan
In this thesis, the effect of pitch based optimisation was explored for a 80 turbine wind farm. Using a modified Jensen wake model and the Particle Swarm Optimisation (PSO) model, a pitch optimisation strategy was created for the dominant turbulence and atmospheric condition for the wind farm.
Acknowledgements
The author would like to thank Dr. Lei, Liu for his invaluable help and support when the author was working under his guidance in the Goldwind Science and Technology.
The author also like to thank Maria Klemm and Liselotte Alden from Uppsala University for their advice during the thesis period and for the academic support that they have rendered. Without the guidance and support from the above-mentioned professors, this thesis would not have became a reality.
Besides the academic and technical support provided by Goldwind Science and Technology, the author will like to thank Xu Len, Pan for the logistical support during the internship. It would have been impossible to have such an enriching or educational internship otherwise. Also, the various interns at Goldwind Science and Technology have made the internship period an enjoyable experience. The author will like to especially thank Nicholas Reckham for intro-ducing him to outdoor climbing and for the party celebrating the birth of Tiramisu. Tunak Tunak Tun.
Lastly, a big thank you to Katarzyna Tempes and Henry L.E Kot for their moral support during the long dark nights leading to the completion of the thesis report. You guys are awesome.Kocham go, ale kocham cie bardziej.
Abstract ii
Acknowledgements iii
Contents iv
List of Figures vii
List of Tables xii
Nomenclature xv
1 Introduction 1
1.1 Introduction . . . 1
1.2 Hypothesis . . . 3
2 Literature Review 4 2.1 Yaw based wake model . . . 4
2.2 FLOw Redirection and Induction in Steady state (FLORIS) model . . . 6
2.3 Determination of inflow direction and the turbine coordinates . . . 7
2.4 The modified Jensen wake model . . . 8
2.4.1 Wake deflection by yaw . . . 9
2.4.2 Wake expansion with consideration to the different wake zones . . . 10
2.4.3 Single wake using the modified Jensen model . . . 11
2.4.4 Multiple wakes using the modified Jensen model . . . 12
2.5 Power calculation model . . . 14
2.6 Pitch based optimisation model . . . 15
2.7 Conversion of CT into axial induction factor . . . 16
2.8 Particle Swarm Optimisation . . . 17
2.9 CFD simulations using OpenFOAM . . . 20
3 Methodology 22 3.1 Outline of pitch-based optimisation model . . . 22
3.2 Verification of parametric values . . . 25
3.3 Wake deflection in pitch-based optimisation model . . . 26
3.4 Determination of wake area coverage ratios and wake decay constants . . . 26
3.5 Verification of wake parametric values using CFD . . . 28
Contents v
3.6 Processing of wind farm production data . . . 29
3.7 Power optimisation using PSO . . . 31
3.8 Convergence and time study . . . 34
4 Application of methodology and results 36 4.1 Verification of parametric values . . . 36
4.1.1 Anomalous results due to CFD model deficiencies . . . 38
4.1.2 Wake expansion and velocity deficit parametric values . . . 40
4.1.3 Wake deflection parametric values . . . 43
4.2 Model verification and data collection process . . . 46
4.2.1 Model verification using the FLORIS model . . . 46
4.2.1.1 Result - Model verification using the FLORIS model . . . 48
4.2.2 Data collection and processing . . . 50
4.2.2.1 Error quantification based on the circular bootstrap method . 52 4.3 Verification of dynamic predictive capability with production data . . . 57
4.3.1 Results from the verification . . . 57
4.4 Optimisation of an 80 turbine wind farm . . . 62
4.4.1 Results obtained from the unoptimised wind farm . . . 63
4.4.2 Results obtained from the optimised wind farm . . . 68
4.4.3 Convergence and time study . . . 74
4.4.4 Effects of using a smoothed pitch response curve. . . 81
4.4.5 Observations from the optimisation of the 80 turbine wind farm . . . . 85
5 Discussion 87 5.1 Model’s dynamic predictive capabilities . . . 87
5.1.1 Uncertainty of measured dynamic response . . . 88
5.1.2 Effects of turbulence intensity and atmospheric stability on modelled results . . . 88
5.1.3 Effects of directional sectors on power prediction . . . 90
5.1.4 Effects of wake travel time on power prediction . . . 91
5.2 Deficiency of the Particle Swarm Optimisation method . . . 93
5.3 Implementation of the pitch based optimisation strategy . . . 96
5.4 Coupling of pitch and yaw based optimisation model . . . 98
6 Conclusion 102 6.1 Future work . . . 102
6.1.1 Cost saving and increase in power production from the reduction of fatigue loads . . . 102
6.1.2 Improvements to the dynamic predictive capabilities of the pitch opti-misation model . . . 104
6.1.3 Pre-generating the optimised control settings . . . 106
6.1.4 Convergence study using another optimisation algorithm . . . 108
6.2 Conclusion . . . 108
A Tables showing the ratio of wake area coverage for both the FLORIS and
B Optimised control strategy for a 19 wind turbine array 113
B.1 Generation of the CP and CT functions . . . 113
B.2 Results - Implementation of the optimisation algorithm . . . 120
B.3 Convergence study for the PSO algorithm . . . 122
B.4 Time study for the PSO algorithm . . . 127
B.5 Discussion on the optimisation process for the 19 turbine array . . . 129
C Curve fit information for turbine type 2, 3 and 4 131 C.1 Curve fit information for turbine type 2 . . . 131
C.2 Curve fit information for turbine type 3 . . . 132
C.3 Curve fit information for turbine type 4 . . . 133
List of Figures
1.1 Graphs showing the global cumulative annual installed wind capacity and the forecast for the growth of the global wind energy market for 2016 - 2020. . . 1 2.1 Yaw based optimisation strategy proposed by Gebraad P.M.O, et. al. Source:
Gebraad et al., 2014. . . 5 2.2 Schematic showing the FLORIS model and the steps used for the optimisation
of the farm using game theory optimisation. Source: Gebraad et al., 2014. . . . 6 2.3 The diagram on the left shows the farm layout based on the conventional
Carte-sian coordinate system, while the diagram on the right changes the farm’s coor-dinate systems with respect to the incoming wind direction. The latter method is more suitable as it allows for further calculations without additional trigono-metric conversion. . . 7 2.4 Diagram showing the wake propagation in a wind farm and the wake dependency
of the individual wind turbines. In this particular scenario, the wake calculations will only be done for two downstream turbines. . . 7 2.5 Diagram showing how yawing an upwind turbine can result in a flow deflection
in the opposite direction to the yaw rotation. Source: Gebraad et al., 2014. . . 9 2.6 Diagram showing an example of a turbine rotor under the effects of both partial
and multiple wake shadowing. Source: Lebioda et al., 2005. . . 12 2.7 Graph showing the relationship of both CT and CP with the pitch angle and
TSR for a reference offshore wind turbine provided by NREL. Source: Jonkman et al., 2009. . . 15 2.8 Graph showing the relationship between CT and a using both the momemtum
theory and the Glauert empirical relationsip. Source: Manwell et al., 2009. . . 17 2.9 Graph showing the thrust coefficient as predicted by the BEM method . . . 19 2.10 Diagram showing the overview of OpenFOAM programming structure. Source:
OpenFOAM, 2016. . . 20 2.11 Some of the commonly used elements for the mesh structure. Source:
Open-FOAM, 2016. . . 21 3.1 Flowchart showing the major steps in the pitch-based power optimisation model.
The subroutines used in step 4 and 5 are outlined separately for clarity. . . 23 3.2 Flowchart showing how the proposed pitch-based power optimisation model will
integrate with a yaw-based optimisation model. The process steps highlighted in yellow are based on yaw optimisation and are not included in the pitch-based optimisation model. . . 24 3.3 Plot showing the intersection area between one turbine rotor and five upwind
wakes. The intersection area is indicated on the plot. . . 28
3.4 Plot showing the torque experienced by the generator during various wind speeds. Kopt is selected such that the maximum Cp value is attained for each
wind speeds. Source: L. Johnson, 2006 . . . 33 3.5 Graph showing an example of Gbest values predicted by the PSO algorithm
using as the number of iterations increases. . . 34 4.1 Diagram illustrating the difference between UM ean and Uy for the 6 m/s
simu-lation. . . 37 4.2 Simulation results showing the flow fields for both the 6 and 8 m/s input wind
velocity. . . 37 4.3 Velocity profiles with a separation distance of 1-3D. The anomalous behaviour
can be observed in these velocity profiles. . . 38 4.4 Velocity profiles with a separation distance of 4-10D. These velocity profiles
shows the expected behaviour that is in line with the modified Jensen wake model. . . 39 4.5 Diagram illustrating the acceleration effect that the gap has on the wind velocity
immediately behind the turbine rotor. The wind velocity that flows through the gap (illustrated by the red arrows) are accelerated through the convergence nozzle due to the laws of mass conservation. . . 39 4.6 Comparison of the wind velocity profiles predicted by the modified Jensen wake
model and the CFD simulations for an input wind velocity of 6 m/s and the parametric values shown in table 3.1. . . 40 4.7 Comparison of the wind velocity profiles predicted by the modified Jensen wake
model and the CFD simulations for an input wind velocity of 8 m/s and the parametric values shown in table 3.1. . . 41 4.8 Comparison of the wind velocity profiles using the revised parametric wake
expansion and velocity deficit values for an input wind velocity of 6 m/s. . . 42 4.9 Comparison of the wind velocity profiles using the revised parametric wake
expansion and velocity deficit values for an input wind velocity of 8 m/s. . . 42 4.10 Comparison of the wind velocity profiles using the revised parametric wake
expansion and velocity deficit values for an input wind velocity of 8 m/s. . . 43 4.11 Comparison of the wind velocity profiles predicted by the modified Jensen wake
model and the CFD simulations for an input wind velocity of 6 m/s and the revised parametric values. . . 45 4.12 Comparison of the wind velocity profiles predicted by the modified Jensen wake
model and the CFD simulations for an input wind velocity of 8 m/s and the revised parametric values. . . 45 4.13 Figures showing both the unrotated and the rotated configuration of the wind
turbines. . . 47 4.14 Plot showing the different wake zones as predicted by the proposed wake model.
The blue lines represents the boundary of the mixing wake zone, while the red and green lines represent the far and near wake zones respectively. . . 48 4.15 Schematic showing some of the systems that are responsible for the data
collec-tion process. . . 50 4.16 Flowchart illustrating the entire data collection and processing process. The
List of Figures ix
4.17 Diagram illustrating the circular bootstrap process. The circles on the left represents the time seried measured wind and power production values. The size of the circle represents the total energy production within 10 minutes, while the direction and length of the arrow represents the wind direction and the magnitude of the wind velocity respectively. The block length is set to 30 minutes, or 3 sets of time seried data, and the bootstrap sample length is set to 1 hour, or 2 block length. Source: Nygaard, 2015. . . 55 4.18 An example of the histogram produced using the bootstrap method. The dash
line at the 16th and the 84th empirical percentile is chosen to represent 1 stan-dard deviation, which is the industry stanstan-dard for the confidence interval of a wake model. Source: Nygaard, 2015. . . 55 4.19 Wind farm array that is used for the verification of the wake and power model.
The numbering of the wind turbine is intentionally set from 48 - 68, as the array is a part of a bigger wind farm. . . 58 4.20 Plot comparing the values of the incoming wind speed, for each of the 19
tur-bines, obtained by the model and the actual farm data. . . 58 4.21 Plot comparing the values of the power production, for each of the 19 turbines,
obtained by the model and the actual farm data. . . 60 4.22 Plot showing the turbines’ layout within the wind farm. Refer to the legend to
distinguish between the different turbine types. . . 63 4.23 Graph showing the total farm power production for all wind directions at a
wind velocity of 7 m/s. . . 64 4.24 Graph showing the total farm power production for all wind directions at a
wind velocity of 11 m/s. . . 64 4.25 Graph comparing the individual turbine power production at a wind speed of
7 m/s and a wind direction of 23o and 113o. . . 65 4.26 Graphs showing the power production of each individual wind turbine at the
three rated wind speed and with a wind direction of 0o. The solid black line is drawn for y = 2.5M W , which is the rated power for the 80 turbines examined. 67 4.27 Graph comparing the total unoptimised and optimised farm power production
at a wind velocity of 7 m/s. . . 69 4.28 Graph comparing the total unoptimised and optimised farm power production
at a wind velocity of 11 m/s. . . 69 4.29 Graph showing the relative increase in total farm production across the full
spectrum of wind direction. The two dotted lines shows the position where wind direction = 23o and 203o. . . 70 4.30 Graphs showing the pitch response of turbine 1, 63, 55 and 56 (type 1, 2, 3,
4 respectively) at a wind velocity of 7 m/s across the full spectrum of wind direction. . . 71 4.31 Graphs showing the TSR of turbine 1, 63, 55 and 56 (type 1, 2, 3 and 4
respec-tively) at a wind velocity of 7 m/s across the full spectrum of wind direction. . 72 4.32 Graph showing the Gaussian curve fit of the pitch response for a wind direction
of 5o to 50o and a wind speed of 3 - 11 m/s for wind turbine 1. . . 73 4.33 Polar graph showing the normalised turbine power production distributed
ac-cording to the wind directions. The polar graph was obtained by normalising the total power output with the total number of turbines. . . 74 4.34 Graph comparing the total farm power output for both optimisation setting at
4.35 Graph comparing the total farm power output for both optimisation setting at
a wind speed of 11 m/s across the full spectrum of wind direction. . . 77
4.36 Graph comparing the relative power gain for both optimisation setting at a wind speed of 7m/s across the full spectrum of wind direction. . . 77
4.37 Graphs comparing the pitch angle of turbine 1, 63, 55 and 56 at a wind velocity of 7 m/s across the full spectrum of wind direction for both optimisation setting. 79 4.38 Graphs comparing the TSR of turbine 1, 63, 55 and 56 at a wind velocity of 7 m/s across the full spectrum of wind direction for both optimisation setting. . . 80
4.39 Graph showing the curve fitted pitch response of turbine 1 using the Gaus-sian function. The multi colour lines represents the optimised pitch response obtained at the 9 wind velocities, while the black line represents the fitted Gaussian function. . . 82
4.40 Graph showing a graphical plot of a 4th order Legendre polynomial function. . 84
4.41 Graph showing an example of how a piecewise continuous function can be used to curve fit a complex graph. Source:Bianchi et al., 2011 . . . 84
5.1 Plot showing the power production of a downstream turbine and a correspond-ing upstream turbine with the travellcorrespond-ing time taken into account. Source: Choi and Shan, 2013 . . . 92
5.2 Flowchart showing the inputs required for the pitch based optimisation model and the resulting outputs for the unified control algorithm. . . 100
6.1 Graphs showing how the potential solution would work. The turbine with the red cross represents the malfunctioning turbine and the turbines enclosed in the dashed box represents the turbines affects by the malfunctioning turbine. . . . 107
B.1 Curve fits for the CP values using the LOWESS, polynomial and linear inter-polant methods. . . 117
B.2 Curve fits for the CT values using the LOWESS, polynomial and linear inter-polant methods. . . 119
B.3 Plot showing the incoming wind velocity at each of the turbine within the array for both the optimised and unoptimised scenario. . . 120
B.4 Plot showing the power production at each of the turbine within the array for both the optimised and unoptimised scenario. . . 121
B.5 Plot showing the total farm power production predicted by the wake and power model for each optimisation iteration at a wind speed of 7 m/s and a wind direction of 13o. . . 123
B.6 Plot showing the schematic of a downwind turbine which is under the effect of a single upwind wake. In the two scenarios, the optimal operational parameters will not differ drastically, even though the area of wake coverage has increased. 124 B.7 Graphs showing the global best pitch angle setting for each turbine at every iteration step for a wind speed and direction of 7 m/s and 13o. . . 125
B.8 Graphs showing the global best TSR setting for each turbine at every iteration step for a wind speed and direction of 7 m/s and 13o. . . 126
C.1 Fitted CP curve for turbine type 2. . . 131
C.2 Fitted CT curve for turbine type 2. . . 132
C.3 Fitted CP curve for turbine type 3. . . 132
List of Figures xi
C.5 Fitted CP curve for turbine type 4. . . 133
2.1 Advantages and disadvantages of using the PSO algorithm. Source: AlRashidi and El-Hawary, 2009 . . . 18 2.2 Values of the constants and inertial factors used in equation 2.27 - 2.29. . . 19 3.1 Table showing the parametric values derived by Gebraad P.M.O, et. al. Source:
Gebraad et al., 2014. . . 25 3.2 Table showing the values of some of the variables describing the CFD simulation
model. These values were provided by Goldwind. . . 29 3.3 Table showing the values of some of the variables describing the flow conditions
and the turbine model using in the CFD simulation. . . 29 3.4 Example of a yearly wind velocity and direction probability table. The values
shown in the table is the probability values weighted for the entire range of possible wind velocities and directions. . . 30 3.5 Table showing the weighted average of the wind probability values across a wind
velocity range of 6 - 8 m/s and a wind direction range of 15o - 18o. Note that all the values add up to 100% as the range of probability values were weighted by the summation across the range of interest. . . 31 4.1 Table showing the revised parametric values describing the wake deficit and
expansion behaviour. . . 41 4.2 Table showing the revised parametric values obtained after the CFD verification. 44 4.3 Input parameters using the the model verification test case. Source: Gebraad
et al., 2014. . . 47 4.4 Coordinate of the six reference turbines that will be analysed in the test case.
Source: Gebraad et al., 2014. . . 47 4.5 Table showing the predicted incoming wind velocity at each turbine by both
the FLORIS and the proposed model. . . 48 4.6 Table showing the predicted power production at each turbine and the total
power production by both the FLORIS and the proposed model. . . 49 4.7 Table showing the velocity values for both the model’s prediction and the
mea-sured production data. . . 58 4.8 Table showing the power production values for both the model’s prediction and
the farm’s operational data. The wake and power model underestimated the total power production of the turbine array by 17.55%. . . 60 4.9 Table showing the geometrical specifications for the four different turbines found
in the wind farm. . . 62 4.10 Table showing some of the input atmospheric conditions used for the simulations. 63 4.11 Table comparing the results obtained from the power and wake models and the
results obtained from the FarmFlow CFD simulations. . . 68
List of Tables xiii
4.12 Table showing the maximum and minimum change in pitch angle and TSR between each degree change of wind direction. . . 71 4.13 Table showing the difference in annual and life time power generation between
the optimised and unoptimised wind farm. . . 73 4.14 Table showing the results for the annual power generation and the
computa-tional time for the optimisation at 150 iterations and particles (150 I and P) and 250 iterations and particles (250 I and P). . . 75 4.15 Table showing the maximum and minimum change in pitch angle and TSR
between each 1o change in wind direction for turbine 1, 63, 55 and 56. . . 78 4.16 Graph comparing the AEP results for the unoptimised, optimised (150 iterations
and particles) and the Gaussian curve fit scenario. . . 83 5.1 Table showing the percentage error between the predicted and the actual power
generation with respect to the width of the directional sector used for the Jensen wake model. The effects of turbulence intensity and atmospheric stability has been accounted for in the simulation. Source: Gaumond et al., 2012. . . 91 5.2 Table showing the estimated travelling time of a wake between two turbines
with a separation distance of 812 m. The travelling time is estimated using three different methods covered in the paper by J. Choi and M. Shan. Source: Choi and Shan, 2013 . . . 92 A.1 Table showing the ratio of wake area coverage for the near wake region as
predicted by the proposed model. . . 111 A.2 Table showing the ratio of wake area coverage for the near wake region as
predicted by the FLORIS model. . . 111 A.3 Table showing the ratio of wake area coverage for the far wake region as
pre-dicted by the proposed model. . . 112 A.4 Table showing the ratio of wake area coverage for the far wake region as
pre-dicted by the FLORIS model. . . 112 A.5 Table showing the ratio of wake area coverage for the mixing wake region as
predicted by the proposed model. . . 112 A.6 Table showing the ratio of wake area coverage for the mixing wake region as
predicted by the FLORIS model. . . 112 B.1 Table showing the goodness-of-fit statistics for the fitted CP curve. . . 116
B.2 Table showing the percentage error of the CP results from the fitted curve and
the actual CP,opt at βopt and ϕopt. . . 116
B.3 Table showing the goodness-of-fit statistics for the fitted CT curve. . . 118
B.4 Table showing the percentage error of the CT results from the fitted curve and
the actual CT ,opt at βopt and ϕopt. . . 118
B.5 Table showing the difference between the wind velocities at each wind turbine for the optimised and unoptimised scenario. . . 120 B.6 Table showing the difference between the power production at each wind turbine
for the optimised and unoptimised scenario. . . 121 B.7 Table showing the percentage difference in total farm power production before
B.8 Table showing the optimised power generation and incoming wind velocity at each turbine using two different configurations for the optimisation algorithm. I refers to the number of iterations and P refers to the number of particles used for the PSO algorithm. . . 127 B.9 Table showing the time taken for the two optimisation scenarios and the
per-centage difference in the total power production prediction by the two scenarios. 128 C.1 Table showing the goodness-fit-statistics for the fitted CP curve for turbine type
2. . . 131 C.2 Table showing the goodness-fit-statistics for the fitted CT curve for turbine type
2. . . 132 C.3 Table showing the goodness-fit-statistics for the fitted CP curve for turbine type
3. . . 132 C.4 Table showing the goodness-fit-statistics for the fitted CT curve for turbine type
3. . . 133 C.5 Table showing the goodness-fit-statistics for the fitted CP curve for turbine type
4. . . 133 C.6 Table showing the goodness-fit-statistics for the fitted CT curve for turbine type
Nomenclature
AEP Annual Energy Production GWh
CF D Computational Fluid Dynamics
CP Coefficient of power Non-dimensional
CT Coefficient of thrust Non-dimensional
F LORIS FLOw Redirection and Induction in Steady-state Gbest Global best solution (Particle Swarm Optimisation)
Kopt Optimal gain factor for generation Non-dimensional
M CP Measure, Correlate and Predict
Pbest Local best solution (Particle Swarm Optimisation)
P SO Particle Swarm Optimisation
RM SE Root mean squared error Non-dimensional
SSE Sum of squares due to error Non-dimensional
SSR Sum of squares of regression Non-dimensional
SST Total sum of squares Non-dimensional
a Axial induction factor Non-dimensional
ad,bd Wake deflection parametric coefficients Non-dimensional
au,bu Wake velocity deficit parametric coefficients Non-dimensional
ci Wake decay coefficient Non-dimensional
kd Coefficient describing the recovery of wake flow direction Non-dimensional
ke Wake expansion parametric coefficient Non-dimensional
me Wake expansion scaling coefficient Non-dimensional
MU Wake velocity deficit scaling coefficient Non-dimensional
s Particle position (Particle Swarm Optimisation) v Particle velocity (Particle Swarm Optimisation) w Inertia coefficient (Particle Swarm Optimisation)
β Rotor pitch angle o
ξ Angle of wake centreline o
ρ Air density kg/m3
Φ Mean wind direction o
ϕ Tip speed ratio (TSR) Non-dimensional
Introduction
1.1
Introduction
Wind energy has grown to become one of the major forms of renewable energy worldwide and has contributed significantly to the renewable energy production in many countries. The growth of wind power globally has been exponential, with significant growth observed in countries such as China, Germany and the UK. Unlike other forms of renewable energy such as geothermal and solar energy, wind energy can potentially be harnessed in most countries. Due to these reasons, wind energy is predicted to be one of the major sources of renewable energy in the world for the next decade[International Renewable Energy Agency, 2012].
(a) Graph showing the global cummulative annual installed wind capacity from 2000 - 2015. Source:
Global Wind Energy Council, 2016 (b) Graph showing the market forecast of the windenergy market from 2016 - 2020. Source: Global Wind Energy Council, 2016
Figure 1.1: Graphs showing the global cumulative annual installed wind capacity and the forecast for the growth of the global wind energy market for 2016 - 2020.
Due to the prevalence of wind energy, it is vital to examine ways to improve the power generation of existing or future wind farms. Studies have shown that the energy loss due to wake effects can account for up to 10% of the total power generated for onshore wind turbines and up to 15% for offshore wind turbines [Barthelmie et al., 2006] [Awerbuch and Morthorst,
Chapter 1. Introduction and hypothesis 2
2009]. Considering the amount of energy that wind power will contribute globally, these losses will significantly affect the total available power that can be harnessed by wind farms. In a study done by Larsen et. al., these losses can be reduced by operating upwind turbines at suboptimal operating conditions [Larsen et al., 2011]. This will result in an increase in the aerodynamic power that is available to downwind turbines. Although the power generated at the upwind turbines will be decreased, the total power generation by the wind farm will be increased. The extent of power improvement will very much depend on the existing wake losses and the geometrical layout of the wind farm. Moreover, this method will also decrease the turbulence intensity within the wake [Serrano Gonz´alez et al., 2015]. Consequently, the mechanical loads that are exerted on the wind turbines will be reduced, leading to an increase in the availability of the wind farm due to reduced fatigue loads. This will indirectly increases the annual power generation of the wind farm.
To achieve this, a few turbine control variables can be changed. Gonzalez et. al. explored the possibility of improving the total farm efficiency by controlling the individual pitch angles and tip speed ratios of each wind turbines in order to increase total farm production [Serrano Gonz´alez et al., 2015]. This pitch-based control strategy differs from the traditional “greedy” control strategy, where each wind turbine is operated at its design optimal pitch angle and tip speed ratio when wake effects are ignored. Through this pitch-based control strategy, the annual energy production (AEP) of the farm was shown to improve by 1.56% [Serrano Gonz´alez et al., 2015].
In the past few years, various studies have been done to explore the possibility of increasing farm power production by controlling the turbine yaw angle. Gebraad P.M.O et. al. conducted multiple studies to examine the effects of varying the turbines’ yaw angles [Fleming et al., 2014] [Gebraad and van Wingerden, 2014] [Gebraad et al., 2014]. Through these studies, it has been verified that the turbine yaw angles have a significant impact on a farm’s power production and fatigue loads.
1.2
Hypothesis
While both pitch and yaw based optimisation have be widely studied, there is currently no wake parametric model that combines both optimisation strategies. This will involve incorporating both optimisation strategies into a single unified optimisation model. In order to achieve this, both the yaw and pitch based optimisation models will have to rely on similar input conditions and assumptions.
The main focus of the thesis is the development of a pitch-based power optimisation model that can be used in tandem with the yaw based optimisation strategy. While the thesis will focus on pitch-based wake optimisation, the yaw based optimisation method will be discussed in chapter 2 and 3, as both optimisation method can be used concurrently to produce a better optimised result. However, due to time constraint, the yaw based optimisation method will not be applied in the control strategy examined in the thesis. The wake model used will be based on the modified Jensen wake model used in the FLORIS model and will involve three distinct wake regions that will be calibrated by a CFD study. Using the velocity generated from the wake model, the farm power production will be determined using a power model that will incorporate the aerodynamic properties of the wind turbines. Lastly, a particle swarm optimisation (PSO) algorithm will be used to obtain the optimal pitch and TSR settings for maximum farm power production.
Chapter 2
Literature Review
2.1
Yaw based wake model
The effects of wake has a detrimental impact on the fatigue loads and the power production of wind turbines, with wake losses accounting for up to 10% and 15% of the power generation of onshore and offshore wind farms respectively [Barthelmie et al., 2006]. Despite wake losses, most commercial wind farms follows a control strategy where individual wind turbines are operating at their design optimal control settings [Kulunk, 2002]. With this control strategy, the effects of wake are not considered, as the design optimal control settings assume no wake effects. While this is a simple control strategy, it does not account for the resultant wake effects during operation. Considering that the size and scale of modern wind farms are increasing, this control strategy is unattractive due to the greater prevalence of wake losses.
Currently, there are many models which aim to reduce the effects of wake losses by controlling the pitch and TSR of wind turbines. These models are often based on simplified parametric wake models such as the Jensen and Frandsen wake models [Bitar and Seiler, 2013] [Gonz´alez et al., 2013]. By using these parametric wake models, it is possible to predict the power production of each turbine for a given wind speed and direction. This allows for an optimi-sation strategy that priorities the total farm power generation over individual turbine power generation.
A yaw based control scheme has the advantage of influencing both the axial induction factor and the wake direction. This has the benefit of redirecting upwind wakes away from down-stream wind turbines. While this control scheme has been shown to work well in CFD models and in scaled wind turbine tests in wind tunnels, there is not sufficient data for a quantitative analysis [Knud A. Kragh, 2013].
In a study conducted by Gonz´alez-Longatt F, et. al., the responses of a wind farm are slow due to the time lag associated with the changes in wake parameters following a change in yaw direction [Gonz´alez-Longatt et al., 2012]. This makes a model-free global optimisation impractical for control implementation. Due to this lag time, it is difficult for the control algorithm to be fast enough to iteratively determine the optimal control setting on a real wind farm, as the controller will require time varying inputs such as the wind direction and velocity. Due to the deficiencies of a model-free global optimisation approach, a model approach with an optimisation algorithm will be more suitable, as a large amount of control settings can be determined in the model before the settings are applied onto the real world wind turbines [Gebraad et al., 2014].
In a study conducted by Gebraad P.M.O, et. al., this model approach was examined and an internal wind farm model is created after a CFD calibration study[Gebraad et al., 2014]. The internal wind farm model has the advantage of being able to predict the wake parameters after yaw deflection and will allow for optimisation. This approach is not dependent on farm measurements that will be affected by time lag. While computation was quicker, the model does not account for the realistic changes that occur in a wind farm. To account for this, the internal farm model is linked to actual measurements taken from the real world wind farm in a feedback loop. This will allow for a quicker iteration for the optimised turbine control settings. A schematic diagram of this control scheme is shown below.
Figure 2.1: Yaw based optimisation strategy proposed by Gebraad P.M.O, et. al. Source: Gebraad et al., 2014.
Chapter 2. Literature review 6
Jensen wake model used in the FLOw Redirection and Induction in Steady state (FLORIS) model [ ´Angel et al., 2010].
2.2
FLOw Redirection and Induction in Steady state (FLORIS)
model
The FLORIS model simulates the steady state behaviour of the wakes after yaw adjustments in a wind farm. Using the steady state effects of yaw misalignment, the energy production of the wind turbines can then be determined using the inflow conditions and the model parameters. The FLORIS model is created by combining the Jensen wake model and a model for wake deflection from yaw. A schematic diagram of the FLORIS model is shown below.
Figure 2.2: Schematic showing the FLORIS model and the steps used for the optimisation of the farm using game theory optimisation. Source: Gebraad et al., 2014.
The FLORIS model consists of four major steps, namely:
• Determination of turbine coordinates with respect to wind directions.
• Wake model to determine the flow conditions at each wind turbine with yaw corrections. • Turbine power model to determine total farm power production.
• Optimisation algorithm for the optimisation of the wind farm model.
2.3
Determination of inflow direction and the turbine
coordi-nates
As the wind direction will be changing constantly, the wakes created by the wind turbines will be changing in real time. Therefore, the model must be able to account for the turbines’ coordinates with respect to the wind direction. This is illustrated by the two diagrams below.
Figure 2.3: The diagram on the left shows the farm layout based on the conventional Cartesian coordinate system, while the diagram on the right changes the farm’s coordinate systems with respect to the incoming wind direction. The latter method is more suitable as it allows for further calculations without additional trigonometric conversion.
This step is important as it allows the optimisation model to determine the wake dependency (for instance, turbine 8 is affected by the wakes from turbine 1 in figure 2.4) in the wind farm. Using this information, the model will then calculate the inflow velocity for downwind turbines based on the upwind wake effects.
Figure 2.4: Diagram showing the wake propagation in a wind farm and the wake dependency of the individual wind turbines. In this particular scenario, the wake calculations will only be done for two downstream turbines.
Chapter 2. Literature review 8
is rotated by a direction that depends on the wind direction measured at the most upwind turbine.
Although this step is a simple trigonometric problem, the actual implementation is more complicated. As wind direction measurements are usually taken at the hub height of individual turbines, it is necessary to obtain a mean wind direction, Φ, for the wind farm in order to implement this process.
For the purpose of the FLORIS mode, the following steps are proposed by Gebraad P.M.O, et. al. [Gebraad et al., 2014]:
1. The first estimate for the inflow direction can be obtained by averaging the measured flow direction from each individual wind turbines, Φi. This wind direction can be calculated
using the horizontal and transverse velocity components, ( ¯ui) and ( ¯vi), measured at
hub height. Using this measure flow direction, the estimated wind direction can be determined using the equation below:
φ = 1 N N X i=1 Φi (2.1)
2. Using the estimated wind direction, the turbine positions in downwind/ crosswind coor-dinates can be calculated using a transformation matrix shown below.
" Xi Yi # = " cos(−Φ) − sin(−Φ) sin(−Φ) cos(−Φ) # " ¯Xi ¯ Yi # (2.2)
3. With the calculated coordinates, it is possible to determine the first upwind turbine by comparing the Yi values, assuming that the calculated mean wind direction is similar to
the measured wind direction at the first upwind turbine. This assumption is valid as long as the free stream wind direction is uniform across the wind farm.
f = arg minXiΦ = Φmeasuredf (2.3)
4. Step 2 and 3 are iterated until the value of Φ converges.
2.4
The modified Jensen wake model
For the purpose of clarity, the discussion on the wake model will be broken down into four different sections.
2.4.1 Wake deflection by yaw
Wake deflection is achieved by yawing an upwind turbine, which will induce a crosswind component that results in the deflection of the wake in an opposite direction to the yaw rotation.
Figure 2.5: Diagram showing how yawing an upwind turbine can result in a flow deflection in the opposite direction to the yaw rotation. Source: Gebraad et al., 2014.
As this wake deflection is caused by the thrust force component experienced by the wind turbine, the amount of deflection is a function of the thrust coefficient (CT) of the yawed
wind turbine. Assuming that the yaw is not misaligned with respect to the wind direction (i.e: γi = 0), the thrust coefficient can be related to the axial induction factor (ai) using the
following equation [Bianchi et al., 2011]:
CT(ai) = 4ai(1 − ai) (2.4)
Using this un-yawed CT value, the angle of the wake deflection can be determined from the
initial values of the wind turbine. This involves a heuristic relationship that relates the yaw angle of turbine i, γi, and the angle of the wake centreline, ξi, at a downstream location
(x > Xi) with the CT values defined above [ ´Angel et al., 2010].
ξi(x) ≈ ξinit(ai, γi)2 1 + 2kdx−XDii , where ξinit(ai, γi) = 1 2cos 2(γ i) sin(γi)CT(ai) (2.5)
Where Di is the rotor diameter of the turbine i, kd is the a model parameter that defines
the sensitivity of the wake deflection to yaw and ξinit is the initial wake angle at the rotor.
Assuming that the initial conditions are known, the only variable in the equation is kd. This
Chapter 2. Literature review 10
direction, Φ [Gebraad et al., 2014]. This accounts for the curved flow direction shown in figure 2.5.
To find the yaw induced lateral offset of the wake center with respect to the hub of turbine i, the tangent of the wake centreline angle can be integrated across x:
δyw,yaw,i(x) =
Z x−Xi
0
tan(ξi(x))dx (2.6)
This integral can be approximated by integrating the second-order Taylor series approximation, giving the following equation:
δyw,yaw,i(x) ≈ ξinit(ai, γi)[15[2kd[x−XDi i]+ 1]4+ ξinit(ai, γi)2] 30kd Di [ 2kd[x−Xi] Di + 1] 5 − ξinit(ai, γi)Di[15 + ξinit(ai, γi)2] 30kd (2.7) Apart from the wake deflection due to yaw, a small lateral wake deflection also occurs when the turbine is not yawed (γi = 0) [Fleming et al., 2014]. This deflection occurs due to the
vertical shear within the boundary layer and wake rotation. This rotation will occur in the opposite direction to the rotation of the wind turbine (ie: a clockwise turbine rotation will result in an anti-clockwise wake rotation). Due to this phenomenon, the low speed flow in the lower part of the boundary layer will be rotated up and to the right and the high speed flow in the upper part of the boundary layer will be rotated down and to the left. This will result in an increase in the velocity deficit at the right part of the wake (looking downstream), resulting in the wake deflecting to the right.
This rotation induced wake lateral offset can be parameterised by a simple linear function incorporating the downstream distance from the rotor [Gebraad et al., 2014].
δyw,rotation,i(x) = ad+ bd(x − Xi) (2.8)
Combining the rotation induced wake lateral offset and the yaw induced wake deflection, the following equation describing the position of the wake center of a turbine i at a downstream location, x > Xi, can be obtained:
yw,i(x) = Yi+ δyw,rotation,i(x) + δyw,yaw,i(x) (2.9)
2.4.2 Wake expansion with consideration to the different wake zones
realistic model of wake expansion will account for the faster velocity recovery at the edges of the wake [Sanderse, 2009].
In a method proposed by Gebraad P.M.O et. al., the Jensen wake model can be expanded to include this uneven velocity recovery by dividing the wake into three distinct wake zones. These wake zones will expand proportionally with the distance from the rotor, but will possess different expansion factors (see figure 2.5) [Gebraad et al., 2014]. The diameters of the wake zones behind turbine i can be determined by:
Dw,i,q(x) = max(Di+ keme,q[x − Xi], 0) (2.10)
⇒ Dw,i,q(x) = x1, x1> 0 0, Otherwise (2.11)
Index q = 1, 2, 3 denotes the different wake zones, Di being the rotor diameter of turbine i and
parameters me,q and ke are coefficients that defines the expansion of the different zones. The
wake zones are defined as “near wake” (q = 1), “far wake” (q = 2) and “mixing zone” (q = 3) respectively. The scaling parameter for the expansion of the near wake, me,1, is typically a
negative value, which forces the cross-sectional area of the near wake zone to taper to zero as the wake expands from the rotor (see figure 2.5). The values of me,q and ke are found by
comparing high fidelity CFD results with the modified Jensen wake model. This process step is shown in figure 2.2.
2.4.3 Single wake using the modified Jensen model
For the purpose of calculating the downwind velocities, a modified version of the Jensen wake model will be used. The Jensen model assumes that the time-averaged velocity in the far wake decays in a quadratic manner with wake expansion. This wake will eventually recover to the free stream velocity due to the effects of turbulence induced mixing past a distance from the rotor [Bianchi et al., 2011].
However, to better model the effects of wake decay, the wake is divided into three areas, as mentioned in the section above. This is done to decouple the effects of wake expansion and wake decay [Annoni et al., 2014]. Using this new method, the velocity deficit is now related to the distance from the rotor and the velocity profile behind turbine i can be expressed as:
Uw,i(x, y) = Ui[1 − 2aici(x, y)] (2.12)
Where Ui is the free stream velocity in front of the rotor. The wake decay coefficient, ci(x, y),
Chapter 2. Literature review 12
of the turbine. This wake decay coefficient is defined as [Gebraad et al., 2014]:
ci(x, y) = ci,1, if |r| ≤ Dw,i,12 (x) ci,2, if Dw,i,1(x) 2 ≤ |r| Dw,i,2(x) 2
ci,3, if Dw,i,22 (x) ≤ |r|Dw,i,32 (x)
0, if |r| > Dw,i,3(x)
2
(2.13)
Where r = y–yw,i(x).
The local wake decay coefficient for each zone can then be determined by:
ci,q(x) =
Di
Di+ 2kemU,q(γi)[x − Xi]
2
The coefficient mU,q is a parameter that determines how quickly the different wake zones will
decay. The wake decay rate can then be modified empirically to account for the yaw angle, γi, using the following relationship [Gebraad et al., 2014]:
mU,q(γi) =
MU,q
cos(aU + bUγi)
(2.14)
For q = 1, 2, 3 with model parameter MU,q, aU and bU.
2.4.4 Multiple wakes using the modified Jensen model
For the model to accurately model a realistic wind farm, the modified Jensen model has to account for both partial and multiple wake coverage. This is not a new concept and can be achieved by combining the wake effects by weighting their contributions based on the overlapping area between the wakes and the rotor [Lebioda et al., 2005] [M´echali et al., 2006].
Mathematically, this partial wake effect can be expressed as: ui = uo 1 − v u u t X k βi,k 1 −u2,k uo 2
Where ui is the downwind velocity after turbine i, uo is the freestream velocity, βi,k is the
ratio of overlapping area between the wake k and turbine i and u2,k is the velocity of wake k.
This equation accounts for both partial (ie: βi,k < 1). and full wake coverage (ie: βi,k = 1),
depending on the value of βi,k.
However, this conventional method is not suitable for a yaw-based control system as it does not account for the effects of yaw on the wake strength. Moreover, this method does not account for the different wake zones. In a study conducted by Gebraad P.M.O et. al [Gebraad et al., 2014], the effects of multiple wake can be accounted for by first considering estimating the effective wind speed by inverting the power equation and solving for uiusing the measured
turbine power, Pi, and yaw angle, γi. Next, the combined wake effects are added by weighting
the wake zones by the overlapping area ratio.
Therefore, the wake model will require a function that allows for the calculation for the over-lapping area. This overover-lapping area between a wake zone q of turbine i and a downstream rotor j is denoted as Aol,i,j,q. Next, the following sets will be used to determine the wake
dependency of the turbines. The following terms are used: U is the set of upstream wind turbines that are not influenced by wakes created by other wind turbines, F is the set of containing all the wind turbines considered by the model and D is the set of wind turbines that influenced by other turbines.
U = {j ∈ F | Aol,i,j,q = 0∀i ∈ F, q ∈ {1, 2, 3}} (2.15)
D = {j ∈ F | j 3 U } (2.16)
Assuming that the set U is tabulated in a matrix, u(j) is defined as the index of the turbine that has the largest overlap area with a turbine that is affected by wake j when compared with other turbines in set U . This can be expressed mathematically as:
u(j) = arg maxi∈U 3 X q=1 Aol,i,j,q ∀j ∈ D
Chapter 2. Literature review 14
Uj denoting the effecting wind speeds at turbine j.
Uj = f1(j)∀j ∈ U f2(j)∀j ∈ D (2.17)
Where the functions f1(j) and f2(j) are defined as:
f1(j) = 2Pj ρAjCP(aj, γj) 13 f2(j) = UU (j) 1 − 2 X i∈F :Xi<Xj ai 3 X q=1 ci,q(Xj)min Aol,i,j,q Aj , 1 2 0.5
2.5
Power calculation model
The equation describing the power generation of a single wind turbine is as shown below:
Pi=
1
2ρACP,iU
3
i (2.18)
From the power equation of a single wind turbine, the total power generated by a wind farm can be determined trivially by summing the total power contribution of all of the wind turbines.
PT otal =
X
Pi (2.19)
This equation is valid for the steady state operation of an idealised wind turbines. For most wind turbines, the value of CP will depend on the axial induction ratio, a. However, for a yaw
based control system, the yaw misalignment angle, γ, will also affect the CP value according
to wind tunnel experiments [Medici, 2005].
Moreover, as the equation does not account for losses, a loss factor, η, is introduced to account for turbine losses. This loss factor will depend on the turbine model used in the wind farm. Lastly, a non-dimensional empirical factor, Pp, will be used to in the CP calculation. This
factor is found to be 2 in wind tunnel experiment conducted by Medici D [Medici, 2005]. Using the factors listed above, the equation for CP can be expressed as:
CP(ai, γi) = 4ai[1 − ai]2η cos(γi)Pp (2.20)
However, Jonkman J, et. al. noted that the Pp factor is turbine dependent and should be
2.6
Pitch based optimisation model
Apart from yaw optimisation, it is also possible to optimise a wind farm based on the turbine pitch angle, β, and the tip speed ratio (TSR), ϕ. These two parameters are related to both the CT and the CP values of a wind turbine. By controlling the pitch angle response of each
individual turbine, it is possible to reduce the wake strength in a wind farm. This form of pitch control has shown to increase the AEP of wind farms by 0.4 - 4.2% [Barth et al., 2007].
(a) CP against TSR and blade pitch angle (b) CT against TSR and blade pitch angle
Figure 2.7: Graph showing the relationship of both CT and CP with the pitch angle and
TSR for a reference offshore wind turbine provided by NREL. Source: Jonkman et al., 2009.
While the modified Jensen wake model used in the FLORIS model is based on the axial induction factor (see equation 2.12), it is also possible to express the Jensen model based on CT. The equation below is the mathematical expression of the unmodified Jensen wake model,
where Uw refers to the downwind wake velocity, ro refers to the turbine rotor diameter and xo
refers to the separation distance between turbines [Jensen, 1983].
Uw = Ui[1 − (
ro
ro+ axo
)2(1 −p1 − CT)] (2.21)
From equation 2.21, it can be observed that the value of CT is directly related to the strength
of the wake produced by the turbine. This relationship is an important consideration when it comes to the optimisation of wind farms, as an increase in CT will lead to lower downwind
wind velocities.
Besides affecting the wake strength, modifying the TSR and blade pitch angle will also affect the power generating capability of the wind turbines. This can be seen in equation 2.18, where the power generation of a turbine is linked to the value of CP. Therefore, there is an optimal
Chapter 2. Literature review 16
In order to ensure that the pitch-based optimisation model can be integrate with the yaw-based optimisation model, the wake model should use the three wake system outlined in in Section 2.4.1. Also, instead of expressing the Jensen wake model in CT (see equation 2.21), the
axial induction ratio should be used instead (see equation 2.12). By ensuring that the wake equations are dependent on the same input variables, the pitch based optimisation model can be easily integrated into the FLORIS model. Using the conditions stated above, the modified Jensen model and the power calculation model, outlined in section 2.4 and 2.5 respectively, can be rewritten as:
Uw,i(x, y) = Ui[1 − 2ai(β, ϕ)ci(x, y)] (2.22)
Pi =
1
2ρACp,i(β, ϕ)U
3
i (2.23)
2.7
Conversion of C
Tinto axial induction factor
As mentioned in the preceding section, the modified Jensen wake model should utilise the axial induction factor instead of the CT values. However, the value of CT depends on the operational
parameters, β and ϕ, and will change during optimisation. Therefore, it is necessary to convert the CT values to the corresponding axial induction factors. Using the momentum theory, the
following equation can be obtained [Manwell et al., 2009]:
CT = 4a(1 − a) (2.24)
However, Equation 2.24 only works when the axial induction factor is equal or lower than 0.4, which corresponds to CT < 0.96. This is because the wind velocity in the far wake
will be negative at that point and the flow pattern through the wind turbine transits to turbulent, therefore violating the laminar flow assumption imposed by the momentum theory [Ackermann, 2005]. In order to account for the scenarios where CT > 0.96, the Glauert
empirical relation should be used to modify the equation [Manwell et al., 2009]. This empirical relation was obtained experimentally by Lock et. al. and is a result of a curve-fitted function from experimental data [Lock, 1926].
CT = 0.889 −
0.0203 − (a − 0.143)2
0.6427 (2.25)
CT = 0.96. To address this discontinuity problem, Buhl, Jr. proposed a method that
mathe-matically expressed the entire range of CT values with a tip loss factor, F [Buhl, 2005]:
CT = 4aF (1 − a), if CT ≤ 0.4 8 9 + (4F − 40 9)a + ( 50 9 − 4F )a 2, if C T > 0.4 (2.26)
Figure 2.8: Graph showing the relationship between CT and a using both the momemtum
theory and the Glauert empirical relationsip. Source: Manwell et al., 2009.
It is important to note that the empirical relation in equation 2.26 will likely lead to some inaccuracy, as the Glauert empirical relation is obtained from a best fit curve of scattered experimental results [Buhl, 2005]. However, it is not expected to have a significant impact on the overall power estimation, as the usual operating regime for most wind turbine is when CT < 0.96. This state is known as the windmill state and will accounts for most of the
operational regime of a wind turbine. For most wind turbines, the CT values will only exceed
0.96 when the wind speed is around the cut in velocity.
2.8
Particle Swarm Optimisation
The modified Jensen wake model allows for the calculation of the incoming wind velocities at downwind turbines using the initial conditions measured at the upwind turbines. Using these values, the total power generation of the wind farm can be obtained. However, this model has to be paired with a suitable optimisation algorithm that allows the model to determine the best operating parameters of each turbine for maximum power generation.
Chapter 2. Literature review 18
Table 2.1: Advantages and disadvantages of using the PSO algorithm. Source: AlRashidi and El-Hawary, 2009
Advantages Disadvantages
Requires little computational power Requires good initial solution for convergence Ideal for non-linear problems Does not guarantee convergence (meta-heuristic) Can be implemented in Matlab Convergence might occur at local maxima
The PSO algorithm was chosen because it can be easily implemented in Matlab and, as it is insensitive to objective functions, is suitable for the non-linear problem that is being examined in this thesis. However, it is important to understand the limitations of the algorithm. As it is meta-heuristic, it does not guarantee global optimal values and convergence is highly dependent on the choice of empirical constants [AlRashidi and El-Hawary, 2009].
The PSO algorithm is iterative and the optimal solution is found by comparing two variables: GBest, the unique global best set of variables, and PBest, the local best solution that is unique
to the particle. The algorithm will implement N number of particles, which represents N number of possible solutions per iteration. The first set of N solutions is generated using a predefined set of initial conditions and a random number generator. The algorithm output, which for the thesis is defined as the total power output by the wind farm, will be sent to a comparative function that will determine the set of control settings that will give the maximum power production. The single best solution for all the iterations will be stored as GBest, while
the local N best solutions will be stored as PBest. Both PBestand GBestcan potentially change
i times, where i is the total number of iterations. The figure below shows the schematic flow chart of the PSO algorithm process.
The equations of the PSO method is as follows and a table containing the constants used for the equation is shown in the table 2.2:
sk+1i = ski + vik+1, where (2.27)
vk+1i = wkvik+ c1,P SOr1× (Pbest,i− ski) + c2,P SOr2× (Gbest− ski), where (2.28)
wk = wmax−
wmax− wmin
kmax
× ki (2.29)
The values listed in table 2.2 changes depending on the problem that the PSO is examining and are empirical [Engelbrecht, 2005]. These values, found via trial and error, allows for converging values within a low amount of iterations. s represents the location of the particles, which is the set of variables that are being optimised. Using a weighted average from Gbest
and Pbest, the locations of the particles for the next iteration are determined. w is the inertial
Table 2.2: Values of the constants and inertial factors used in equation 2.27 - 2.29.
c1,P SO 2
c2,P SO 2
r1 Random number between 0 - 1
r2 Random number between 0 - 1
wmax 0.9
wmin 0.4
Chapter 2. Literature review 20
2.9
CFD simulations using OpenFOAM
In order to identify the wake expansion parametric values, a CFD study will be conducted. The CFD software chosen was OpenFOAM, an open-source CFD program that is based on the C++computing language. OpenFOAM was written with C++ as it allows for object oriented
programming, which in turns allow for physical or non physical parameters to be grouped into classes. This allows for easier management of the programming code, as traits can be inherited between classes. An example will be how a vectorfield class can be generated from a vector and field class [OpenFOAM, 2016].
The OpenFOAM program consist of two general applications: solvers and utilities. Solvers are applications that are designed to solve a specific problem that is based on a set of equations describing the initial conditions and the problem. As for utilities, they are a set of application that deals with data manipulation depending on the requirement of the problem. Similar to other CFD softwares, the CFD process in OpenFOAM consist of three major operation, namely: pre-processing, solving and post processing. The following flow chart shows the overview of OpenFOAM structure.
Figure 2.10: Diagram showing the overview of OpenFOAM programming structure. Source: OpenFOAM, 2016.
The first major step of the OpenFOAM process is the pre-processing stage of the program. In this stage, the initial conditions, such as the desired wind direction and velocity, and the model definitions, such as the time step and duration of simulation, will have to be entered. These values are stored in a database, from which subsequent calculations will be conducted. It is also possible to control the simulation conditions, such as the frequency for autosaves. This data entry process is handled by the utility applications in OpenFOAM.
are acceptable, while ensuring that the run-time of the simulation is reasonable [Skaperdas and Iordanidis, 2012]. However, a mesh study will not be conducted for this thesis, as the mesh was already verified from a previous project within the company.
Figure 2.11: Some of the commonly used elements for the mesh structure. Source: Open-FOAM, 2016.
The second major step of the simulation process involves solving the problem defined during the pre-processing stage. This solution will rely on a set of mathematical equations which describe the physical characteristics of the air flow through and around the wind turbines. Depending on the purpose of the simulation, different equations can be chosen to give a better prediction of the actual fluid behaviour.
As OpenFOAM is an open source software, custom solvers can be written easily to suit most simulation purposes. To do so, the solver should be written with the structure and program-ming language defined in the OpenFOAM user manual. However, OpenFOAM also comes with various pre-installed solvers that are commonly used in commercial application. These pre-set solvers include: the Direct Numerical Simulation (DNS), the Large Eddy Simulation (LES) and the Lagrangian solver.
Chapter 3
Methodology
3.1
Outline of pitch-based optimisation model
Using the equations and concepts discussed in chapter 2, it is possible to develop a framework for the pitch-based power optimisation model. Unfortunately, combining both the yaw and pitch based optimisation strategy is not a straight forward task. This is largely due to both the CT and CP values being dependent on the values of ϕ and β (see equation 2.23). Therefore,
it is necessary to develop both a pitch and yaw based optimisation model separately before combining both models.
As the end goal of the thesis is to develop a pitch optimisation model that can be integrated its yaw based counterpart (outlined in section 2.2 and 2.4), the pitch optimisation model should fulfil the following conditions:
• The input conditions required for the pitch-based optimisation model should be similar to that of the yaw based model.
• The wake model used for the velocity calculations should be similar for both optimisation models. Due to this condition, the wake model should account for three different wake zones to allow easy integration.
• The parametric values used for the modified Jensen wake model should be consistent between both optimisation models.
• Although the pitch-based power model will not be able to capture the effects of the yaw angle on farm power production, the effects of rotor rotation on the wake center-line (Equation 2.8 - 2.9) should be considered. This is required as it will reduce the discrepancy between the power estimations from both optimisation models.
Using the concepts covered in the literature review, the following pitch-based power optimisa-tion model was proposed.
Figure 3.1: Flowchart showing the major steps in the pitch-based power optimisation model. The subroutines used in step 4 and 5 are outlined separately for clarity.
The flowchart shown in figure 3.1 outlines the major steps that will be included in the pitch-based power optimisation model. As step 4 and 5 of the optimisation model contain important subroutines, the subroutines are also listed for clarity.
From figure 3.1, the process flow of the pitch optimisation process can be seen. Step 0 - step 3 are not iterative in nature for a single flow condition, as the wake coverage will is not a function of the turbine control setting. This is an important distinction, as step 0-3 will not result in an increase of computational time when the sampled solutions are increased during optimisation.
Step 4 of the optimisation model consist of three major steps: determining the CT and CP
Chapter 3. Methodology 24
provided by step 0 - 3, step 4 of the model will be able to give the power production of the wind farm as an output.
Step 5 of the process is the PSO optimisation model that is outlined in section 2.8. Step 5.1 will evaluate the total farm power production from step 4 to determine the Gbest and Pbest
values using the conditions outlined in figure 2.9. After the Gbest and Pbest are determined,
the next set of β and ϕ will be generated randomly using a random number generator and equation 2.27 - 2.29. However, these sets of solutions will have to fulfil a condition, Kopt, due
to the generator requirements of the wind turbines, which will be covered in the next section.
Figure 3.2: Flowchart showing how the proposed pitch-based power optimisation model will integrate with a yaw-based optimisation model. The process steps highlighted in yellow are based on yaw optimisation and are not included in the pitch-based optimisation model.
Figure 3.2 shows how the pitch-based power optimisation model shown in figure 3.1 can inte-grate with a yaw-baw optimisation model. Due both CT and CP being coupled with ϕ and β,
it is not possible to optimised for both ϕ and β in tandem. Therefore, the integrated model will require an additional optimisation model that will locate the optimal value of ϕ and β separately for a given wind speed and direction.
3.2
Verification of parametric values
The modified Jensen wake model proposed by Gebraad P.M.O, et. al. requires a set of parametric constants in order to describe the wake characteristics of a wind farm. These values were obtained using a high fidelity CFD software, known as the Simulator for Onshore/Offshore Wind Farm Application (SOWFA), which is based on Large Eddy Simulation (LES). The constants they found were based on two NREL 5 MW reference wind turbines and is not representative of the wind turbines examined in the thesis [Gebraad et al., 2014].
Table 3.1: Table showing the parametric values derived by Gebraad P.M.O, et. al. Source: Gebraad et al., 2014.
Wake parametric model parameters
Deflection Expansion Velocity kd 0.15 ke 0.065 MU,1 0.5
ad -4.5 me,1 -0.5 MU,2 1
bd -0.01 me,2 0.22 MU,3 5.5
me,3 1 aU 5
bU 1.66
These parametric values can be be broken down into three categories: wake deflection (kd,
ad and bd), wake expansion (ke and me,q) and wake velocity deficit parameters (aU, bU and
MU,q). These parameters describe the deflection, expansion and velocity characteristics of
wake respectively.
While this set of parametric values were obtained from a high fidelity CFD program and are verified to be highly accurate for the NREL reference wind turbines, they might not apply for the wind farm that will be examined in the thesis. Due to the difference in terrain and meteorological characteristics, the turbulence intensity of the examined wind farm will be different. As wake characteristics are affected by the turbulence intensity [Wang et al., 2004], these values will have to be verified before it is applied to the wake model.
Ideally, all of the parameters should be confirmed by a high fidelity CFD program before they are used in the wake model. This will be done through OpenFOAM and the methodology for the CFD process will be covered in chapter 3.5.
After verifying that both the wake expansion and deflection coefficients are accurate, the parametric values describing the wake velocity deficit can be corrected by comparing the CFD results with the modified Jensen wake model. This additional step will allow the wake model to better simulate the actual power production of the wind farm.
Chapter 3. Methodology 26
of values for MU,q can be obtained. The new set of parametric values should allow for the
wake model to better model the actual wind velocity and power production of the wind farm. This method of correcting the parametric values is in line with the methodology suggested by Gebraad P.M.O, et. al. [Gebraad et al., 2014].
3.3
Wake deflection in pitch-based optimisation model
As the optimisation model aims to maximise the power production of the wind farm by only optimising the pitch angle of each turbine, the yaw angle of the wind turbine will be set to zero for the purpose of analysis. However, while the wake deflection due to turbine yaw, δyw,yaw,
is zero, there will be a wake deflection component that occurs from rotation induced wake deflection, δyw,rotation(see chapter 2.4.1).
Therefore, the model will have to account for wake deflection, even though the deflection contribution from turbine yaw can be neglected. This contribution is described in equation 2.8 and will be included in the model under step 2 in figure 3.1.
This wake deflection is a geometrical variable that determines the wakes’ center-line after deflection. As such, it will affect the wake area coverage ratios and the wakes’ contributions by the different wake zones. The deflections will have an impact on the power calculations, as the wake area coverage ratios and their contributions will affect the velocity deficit for the downwind turbines. Moreover, the addition of the wake deflection will prevent a discontinuity from occurring when the pitch-based optimisation model is integrated with the yaw-based optimisation model (i.e: discrepancy in power prediction when turbine yaw angle = 0o).
3.4
Determination of wake area coverage ratios and wake decay
constants
With the wind farm rotated with respect to the incoming wind direction, the wake dependency and area ratios for the three wake zones can be determined. The modified Jensen wake model will require two inputs for it to function, namely:
• A matrix containing the wake coverage area ratios between a downwind turbine and any upwind wakes that it bisects. This matrix will have to contain the information for the three wake zones.
