Platform position control of floating wind turbines using aerodynamic force

Journal Pre-proof Platform position control of floating wind turbines using aerodynamic force Chenlu Han, Ryozo Nagamune PII:

S0960-1481(19)31772-0

DOI:

https://doi.org/10.1016/j.renene.2019.11.079

Reference:

RENE 12627

To appear in:

Renewable Energy

Received Date: 12 February 2019 Revised Date:

13 October 2019

Accepted Date: 15 November 2019

Please cite this article as: Han C, Nagamune R, Platform position control of floating wind turbines using aerodynamic force, Renewable Energy (2019), doi: https://doi.org/10.1016/j.renene.2019.11.079. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

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Platform Position Control of Floating Wind Turbines Using Aerodynamic Force

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Chenlu Han, Ryozo Nagamunea

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a Department

of Mechanical Engineering, University of British Columbia 6250 Applied Science Lane, Vancouver, BC V6T1Z4, Canada Email: [email protected]

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Abstract This paper presents platform position control of utility-scale floating offshore wind turbines in surge and sway directions using aerodynamic thrust force. The position control will be useful in mitigating the wake effect, and thus maximizing the total power capture, of an offshore wind farm. A linear-quadratic-integrator (LQI) controller is designed to achieve the control objectives of platform position transfer to a specified position for the wake loss mitigation, as well as of platform position regulation against disturbances in wind and wave. Power regulation during and after the position transfer is accomplished by the standard constantpower strategy. It is demonstrated, through simulations of a 5-MW wind turbine in the FAST software, that the LQI controller can attain satisfactory platform position transfer and regulation.

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Keywords: Offshore wind energy, Offshore wind farm, Layout optimization, Floating platform, Position control

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1. Introduction

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Wind is a renewable, clean and readily available energy source which has become increasingly popular in the recent years. In the wind energy industry, a fast growing trend lately is to position the wind turbine systems offshore, where the wind condition is stronger and steadier while vast open space is available for wind farm construction [1]. The average offshore wind speed is estimated to be 90% greater than the average onshore [2], giving offshore wind farms the theoretical advantage for more power output. Furthermore, offshore wind fields have less spatial variations in speed due to the flat ocean surface, which reduces excitation of structural fatigue loading as the turbine blades travel through the more uniform wind field in circle. To take advantage of the favorable offshore wind resources, wind turbines have to operate far away from the coastline at deep-water locations. While offshore wind turbines in shallow-water can be attached to the seabed underneath like the onshore ones, deep-water sites require floating platforms to host the

Preprint submitted to Renewable Energy

November 16, 2019

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wind turbines since fixed bottom structures are neither feasible nor economical for water depth larger than 50 meters [3]. The floating platforms add extra degrees of freedom (DOFs) to the wind turbines by allowing the platform to translate and rotate under the excitation of wind and wave, which brings both challenges and opportunities in control. Control challenges include increased complexity in modelling [4], platform vibration suppression and resonance avoidance [5], as well as wave disturbance rejection [6]. On the other hand, the position DOFs of a floating platform provide the opportunity for real-time position control of the wind turbine, which is impossible for onshore or fixed-foundation offshore wind turbines. The benefit of real-time wind turbine position control is clear when a wind farm is considered. When designing the layout of a wind farm, one of the most important considerations, besides the cost of infrastructure and transmission, is to reduce the wake effect. The wake effect is the undesirable aerodynamic interference where the upwind wind turbines in a wind farm block the available wind energy to the downwind ones and also pass on turbulence [1]. To mitigate the wake effect, real-time wind farm control methods have been developed, such as strategies based on the axial induction factor [7, 8, 9], wake redirection [10, 11, 12], and specifically for offshore cases, real-time layout optimization [13, 14, 15]. As for the wind farm layout optimization, the optimized layout is a function of, among other variables, wind speed and direction which may vary during the wind farm operation. Since fixed foundation wind turbines cannot change their positions after installation, they will suffer from reduced power and increase structure vibration unless wind turbines are installed sufficiently far apart each other. In contrast, movability of floating offshore wind turbines (FOWTs) allows the wind farm to track optimal layout with the least amount of wake influence in response to real-time wind condition changes [16]. Such advantage of FOWTs motivates this study on position control problem of a single FOWT in the context of real-time floating offshore wind farm layout optimization. Since the FOWT is typically not designed with position transfer in mind, to generate the necessary actuation force for platform position transfer, either additional actuators should be installed or existing actuators need to be effectively utilized. Our preliminary work [17] explored the novel concept of FOWT position control through only passive utilization of the aerodynamic thrust force from wind. On the other hand, a group of researchers studied offshore wind farm layout optimization by assuming some finite range of movability for each FOWT based on conceptual floating platform design using a winch mechanism [18]. However, any prototyping progress based on this concept has not been reported to date. Another actuator concept is artificial muscle based active mooring line proposed in [19]. It was demonstrated to be effective for stabilization but may not have sufficient range of motion for position control. In comparison, since our proposed aerodynamically-actuated mechanism manipulates the thrust force from the wind by adjusting only commonly-available wind turbine control inputs, it is readily applicable to any FOWT without hardware modification. Furthermore, we proposed in [20] a computational method to determine the movable range of an FOWT under a given wind condition, within which system 2

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constraints (such as rotor speed and generator torque constraints) are met and position transfer is feasible at steady state. We also proposed in [20] open-loop constant control inputs to realize the desired platform position transfer, but such open-loop control has disadvantages in response speed. In this paper, a feedback controller for the FOWT is proposed, which is capable of tracking the platform’s optimal position in a wind farm by the utilization of the aerodynamic thrust force. For obtaining sufficiently large thrust force to move an FOWT, a pitch-to-stall blade control strategy is adopted instead of the standard pitch-to-feather strategy, even though the pitch-to-stall blade control will increase the loads to the blades. The proposed controller has a structure with three sub-controllers, i.e., a power regulator, an inner-loop stabilizing controller, and a position controller. The power regulator based on the standard constant-power strategy guarantees that the FOWT generates the specified power during the platform position transfer and regulation, while the inner-loop stabilizing controller is required for subsequent position controller design. The desired position transfer and regulation are achieved by incorporating a Linear-Quadratic-Integral (LQI) controller in the overall control structure. To the best of our knowledge, there is only very few publications on FOWT platform position controller design. This paper is organized as follows. In Section 2, a platform position control problem of an FOWT is formulated for a semi-submersible wind turbine, and the importance of the position control capability is explained from the viewpoint of wind farm’s total efficiency. The mechanism to adjust the aerodynamic thrust force by using only the standard FOWT control inputs is also briefly discussed in this section. To solve the formulated problem, a feedback control structure which involves a power regulator, an inner-loop stabilizing controller and an LQI position controller is proposed in Section 3. Section 4 examines the performance of the designed controller to track the position command. The performance is quantitatively compared with that of a Proportional-Integral-Derivative (PID) controller, an open-loop controller, and a gain-scheduling PI controller.

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2. Position Control Problem of an FOWT

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In this section, an FOWT to be controlled and its associated signals are introduced in Section 2.1. Then, for this FOWT, a position control problem is formulated in Section 2.2. The importance of this formulated FOWT position control problem in the context of real-time wind farm layout optimization and reconfiguration is discussed in Section 2.3. Finally, in Section 2.4, the mechanism to adjust the aerodynamic force for repositioning the FOWT platform is explained. 2.1. OC4-DeepCwind Semisubmersible To formulate a position control problem and to propose a position control method for FOWTs, a specific virtual FOWT, called OC4-DeepCwind Semisubmersible, is considered throughout this paper. OC4-DeepCwind Semisubmersible,

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illustrated in Fig. 1, is a utility-scale wind turbine with a semi-submersible floating platform published by the National Renewable Energy Laboratory (NREL) [21, 22] in the United States. The simulation software for OC4-DeepCwind Semisubmersible is also provided by NREL, which is named FAST [23]. Due to the limited access to the physical wind turbine by most researchers, the FAST software has been widely adopted as a standard platform for validating control system performance [24, 25, 26]. This paper also considers the FAST simulation results, with all degrees of freedom available in FAST enabled, as closely representing responses of the real plant. The rest of this section briefly reviews the mechanical structure, control inputs, outputs for feedback control, and disturbances of this OC4-DeepCwind Semisubmersible. Remark : Although this paper focuses on position control of only OC4DeepCwind Semisubmersible, the control method proposed in Section 3 will be applicable to FOWTs with platforms other than semi-submersible ones. 2.1.1. Mechanical Structure As shown in Fig. 1, the floating platform consists of three ballasted buoyancy columns spaced in a triangular geometry for improved stability, as well as the center column to which the tower is attached. Heave plates are placed at the bottom of the columns to increase the added mass, and therefore the heave and pitch eigenperiods. Furthermore, to constrain the FOWT’s translational movements, each floating column is attached to a mooring line cable which is anchored to the seabed at the other end. The mooring lines are not pretensioned, but they utilize their heavy weight to provide a gradually-increasing net restoring force as the FOWT deviates from its neutral position. Owing to this increasing restoring force, the FOWT has some limited movability. The rest of the system specifications can be found in extensive details in the original reference on the definition of the wind turbine [21] and the floating platform [22]. In Table 1, we list several main parameter values for interested readers. Table 1: Main parameter values of OC4-DeepCwind Semisubmersible

Parameter Hub-height Blade length Mooring line length Rated rotor speed Gear ratio Generator efficiency Draft Water depth 151

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Value 90 61.5 835.5 12.1 97 94.4 20 200

Unit m m m rpm – % m m

(a) Front View

(b) Top View

Figure 1: Illustration of OC4-DeepCwind Semisubmersible

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2.1.2. Control Inputs The control input vector (u) available to OC4-DeepCwind Semisubmersible is denoted as u := [β, τg , γ]T . (1) Here, collective blade pitch angle (β), indicated in Fig. 1(a), is the rotation of the blades along their longitudinal axes to adjust the magnitude of aerodynamic power and thrust force. Generator torque (τg ) controls power extraction and regulates generator speed. Nacelle yaw angle (γ), indicated in Fig. 1(b), changes the facing angle of the rotor plane which affects the direction and magnitude of thrust force. In this paper, the thrust force caused by the wind is assumed to be perpendicular to the rotor plane in the model [27, 28] for controller design. Thus, the x-direction wind blown into a yawed wind turbine can generate the lateral (y-direction) movement. The validity of this assumption was confirmed in [27, 28], and will be reconfirmed in the controller validation in Section 4 by using FAST simulations having lateral thrust components. All three control inputs in (1) are commonly available in modern horizontalaxis utility-scale wind turbines. In this paper, the position control of FOWTs is realized by using only these three control inputs, i.e., without any additional actuator to existing commercial wind turbines.

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2.1.3. FOWT Outputs The original reference [23] lists a large number of system variables that can be output from the virtual plant. For simple and realistic feedback controller implementation, only fifteen variables are considered as system outputs (y) in this paper, denoted by

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y := [x, y, z, θx , θy , θz , x, ˙ y, ˙ z, ˙ θ˙x , θ˙y , θ˙z , ωr , ωg , ∆θ]T .

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The output vector contains three translational coordinates (x, y, z) as well as three rotational coordinates (θx , θy , θz ), measuring the position of the FOWT’s center of gravity and the FOWT’s orientation. Their derivatives (x, ˙ y, ˙ z, ˙ θ˙x , θ˙y , θ˙z ) are also included in the outputs. In addition, to measure the dynamics of the drivetrain in the nacelle, rotor speed (ωr ), generator speed (ωg ) and elastic deformation of the gearbox transmission (∆θ) are also necessary. In a real physical FOWT, these signals may be measured or estimated by using sensor signals such as global positioning system (GPS), gyroscopes, magnetometers, accelerometers, encoders, and force sensors on mooring lines, and possibly by the fusion of these sensor signals to minimize the measurement inaccuracy1 . 2.1.4. Disturbances System disturbances for OC4-DeepCwind Semisubmersible are in the form of wind and wave. The wind disturbance is the fluctuation of the instantaneous wind velocity (v) around the average velocity (¯ v ). The wave disturbance is the hydrodynamic interaction between the wind turbine platform and the surrounding ocean water, which causes oscillatory motions of the FOWT. 2.2. FOWT Position Control Problem The problem to be considered in this paper is formulated as follow.

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Problem 1. Assume that a target power level Ptar (≤ 5MW) and a target position (xtar , ytar ) are commanded to OC4-DeepCwind Semisubmersible. The position control problem is to move the FOWT to the target position, and maintain its position even in the face of wind and wave disturbances, by adjusting the control inputs u. Constraints in this problem are given as follows.

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C1: Generator power is regulated around the target power level Ptar at all time.

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C2: Control inputs are subjected to actuators’ saturation and rate-limitation.

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C3: The controller moves OC4-DeepCwind Semisubmersible to the target position (xtar , ytar ) within a given time. The constraints C1, C2, and C3 are respectively for generating useful power continuously while the FOWT is changing its position, for meeting the actuators’ physical limitation, and for reasonable layout tracking speed. In Condition C3, ‘within a given time’ means that too-slow movement is not acceptable, because we want the position transition task to be completed before the average wind condition drastically changes. Remark : Note that the target power can be less than or equal to the FOWT’s rated power of 5 megawatts. It was shown in [20] that, for given wind speed and direction, the range of the achievable platform position becomes larger as the target power is lower. Within this enlarged range, the FOWT which generates 1 The selection of measurement hardware and the robustness of the controller proposed in this paper against sensor inaccuracy are beyond the scope of this paper.

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less power can be moved to mitigate the wake effect, thereby enhancing the wind farm’s total efficiency. Remark : For the formulated problem to be feasible, the target position needs to be selected within the movable range of OC4-DeepCwind Semisubmersible corresponding to a specified target power and the average wind speed and direction, calculated by the numerical method in [20]. Selecting a target position within the movable range for the specified target power guarantees the feasibility of the position and power regulations, with safe rotor speeds and generator torques under actuator limit. In this paper, we assume that a target position of the FOWT is specified within the movable range, and leave the question of ‘which specific position to select within the movable range’ as a future work. 2.3. Importance of Position Control for FOWTs Now that the control problem to be solved in this paper has been formulated, the importance of having such position control capability for each individual FOWT will be explained in the context of real-time offshore wind farm layout optimization. In a wind farm, the wake effect is a critical cause of reduced available wind energy and increased fatigue load for downwind FOWTs. To mitigate the wake effect, the floating offshore wind farm has an opportunity to modify its layout for optimizing its total performance. To realize such layout modification, an optimized wind farm layout needs to be sought in real-time in response to wind, wave and demand condition changes, and then the layout update needs to be realized by the platform position controller of each FOWT. Figure 2 illustrates our proposed real-time wind farm layout control structure. The control structure is divided into three levels: wind farm level, turbine level, and actuator level. We will omit the details of the actuator level control mechanisms because they are simple (such as on-off and proportional controllers) and irrelevant to the main contribution of this paper.

Figure 2: Wind farm control structure for tracking optimal layout through position control of individual FOWTs 241

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2.3.1. Wind Farm Level At the wind farm level, the main objective is to search for the optimal layout which maximizes the wind farm performance. In specific, the layout optimization algorithm should measure and estimate real-time wind (by using e.g. anemometers and LIDARs), wave (by using e.g. buoys with inertial sensors and gyros), demand changes to periodically update the optimal wind farm layout, which is then sent to each individual FOWT in the form of target position command and target power command. One attempt for such optimization was presented in [16]. The target platform position needs to be selected within the movable range calculated by the method in [20], which should be large enough for the wake effect mitigation. Most existing layout optimization algorithms focus on the wind farm’s total power as the sole performance measure [13, 14, 15] to generate target layouts, while other considerations such as structural vibration can also be incorporated in the objective function. Although the formulation and solution of this layout optimization are beyond the scope of this paper, it has a critical role in realizing real-time optimal floating offshore wind farm control.

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2.3.2. Turbine Level After receiving target commands of power and position, at the individual FOWT level, each FOWT will encounter the problem formulated in Section 2.2. Thus, it is essential for the FOWT level controller embedded in each FOWT to realize the wind farm layout and operation modification commanded by the wind farm level controller. Design of this FOWT level controller is the main topic of this paper, and will be presented in Section 3.

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2.4. Aerodynamic Force Adjustment Mechanism

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Next, we will explain the mechanism about how the standard wind turbine actuators, i.e., nacelle yaw angle, blade pitch angle and generator torque, will affect the aerodynamic thrust force for the wind turbine repositioning purpose. Nacelle yaw angle γ will affect the thrust force direction. This is because we assume that the thrust force is perpendicular to the rotor plane. For example, as shown in Fig. 1, when the aerodynamic thrust force (Ta ) direction is not aligned with the wind direction (x-direction), the force can be decomposed in the wind direction (Ta cos γ) and its perpendicular direction (Ta sin γ), the latter of which induces the lateral (y-directional) motion of the FOWT. Blade pitch angle β will affect the thrust force magnitude. This is because the thrust coefficient, denoted by CT (β, λ), is a function of β and the tip-speed ratio (TSR) λ. In addition, the TSR λ is also a function β, since the blade pitch angle β will influence the rotor speed ωr , and thus λ. Generator torque τg will also affect the thrust force magnitude. This is because τg affects ωg , which in turn influences λ and thus CT . At the target platform position, when there is no disturbance, the aerodynamic thrust force magnitude and direction need to be balanced with the net force from the mooring lines.

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3. Turbine Level Controller Design 3.1. Feedback Control Structure The proposed structure of the turbine level multi-objective controller is shown in Fig. 3. The controller receives target command in power (Ptar ) and position (xtar , ytar ) from the wind farm level controller, as well as feedback signals (y) from the sensors, and then sends control input commands (u) to the FOWT. As can be seen, the proposed feedback controller consists of three sub-controllers, that is, Power Regulator, Stabilizing Controller, and Position Controller.

Figure 3: Proposed structure of the turbine level multi-objective controller. The matrices Cωg and Cωr are matrices which extract respectively ωg and ωr from y. The red-dashed box indicates the inner-loop feedback system to be used for the design of Position Controller. 293 294 295

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The Power Regulator adopts the standard constant-power strategy (see e.g. [21]), i.e. calculating the generator torque input τg as2 τg =

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where ηg is the generator’s conversion efficiency from mechanical power to electrical power. This equation guarantees that constraint C1 is fulfilled both during and after the position transfer. The role of the Stabilizing Controller is to stabilize the inner-loop system (red-dashed box in Fig. 3), to facilitate the design of the Position Controller that will follow. The proposed Stabilizing Controller is a proportional controller given by ∆β = Kp (˜ ωr − ωr ), (4) where Kp is the proportional gain to be tuned by trial and error, and ω ˜ r is an auxiliary output of the Position Controller to the stabilized inner-loop system 2 Although the constant-power strategy is conventionally used exclusively for above-rated operated conditions, in our proposed platform position control, we uses it for not only aboverated but also below-rated wind speed. The feasibility of the constant-power strategy for a given wind speed is guaranteed by the selection of the target position within the movable range obtained in [20].

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for improving the closed-loop platform position control performance. The generation of the signal ω ˜ will be explained in detail in Section 3.3. The output ∆β of the Stabilizing Controller is to be added to the blade pitch angle command βpos yielded by the Position Controller. This Stabilizing Controller has been introduced in the proposed controller structure for the following reason. In our attempts to design the Position Controller, without the Stabilizing Controller, by using the LQI design technique (to be presented in Section 3.3), it was consistently observed that the design failed numerically. This failure may be due to the weak controllability and the instability of the original plant; in solving the algebraic Riccati equation numerically in finding the LQI controller, the solver did not return a controller which stabilizes the closed-loop system. This is the reason why we decided to stabilize the inner-loop system (red-dashed box in Fig. 3) first by using Stabilizing Controller, and then to design Position Controller which accomplishes both closed-loop stability and position control for the weakly controllable but stable inner-loop system. After stabilizing the inner-loop system, the LQI position controller design has become successful.

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3.2. Simplified Model for Position Controller Design

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In [23], a dynamic model for OC4-DeepCwind Semisubmersible was given and realized in the simulation software FAST. Unfortunately, due to its large number of states (44 states), high complexity of dynamics formulation, and the difficulty of obtaining the linearized model at a user-specified x-y position, the FAST model is inconvenient to be used for model-based position controller design. Therefore, for the purposes of equilibrium calculation and linearization at a given platform position and the given wave condition, another simplified, control-oriented FOWT model [27, 28] was employed, which will be referred to as the Simplified Model in this paper. The dynamics equation of the Simplified Model describing system response to control inputs (u), wind velocity (v), and wave disturbance (w) can be written as x˙ = f (x, u, v, w). (5)

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The state vector (x) in the Simplified Model is the same as the output vector (y) defined in (2). The nonlinear function f was presented in detail in [27, 28]. Despite the significant reduction in DOFs (down to 15 states) of the Simplified Model, it still retains a satisfactory modelling accuracy as compared to FAST [27, 28].

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3.3. LQI Position Controller Design

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Here, a design method is proposed for the Position Controller in Fig. 3. The controller is an LQI controller, which consists of both a state feedback loop and a position error integral feedback loop, as depicted in Fig. 4. The state feedback loop improves the speed of system response and mitigates states fluctuation during transient to satisfy the constraint C3, while the integral loop reduces the steady-state position error for target changes and disturbances. 10

Figure 4: Structure of the LQI position controller (in the dashed box). The matrix Cx,y extracts [x, y]T from y.

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For the LQI controller design, a linearized model of the combination of FOWT, Power regulator, and Stabilizing Controller in Fig. 3 will be utilized. Linearization is taken around an equilibrium point associated with the target position (xtar , ytar ), the target power (Ptar ), and the average wind speed v¯. The numerical method for solving the equilibrium condition f (x0 , u0 , v¯, 0) = 0 was presented in [20]. The input signals to be determined by the LQI position controller is denoted as   ω ˜r ˜ := βpos  . u (6) γ

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Then, the linearized plant for the inner-loop nonlinear system (red-dashed box in Fig. 3) can be written in the form of a linear time-invariant system

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˜ δ x˙ = Aδx + Bδ u,

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˜ are deviations of x and u ˜ from their linearization point x0 where δx and δ u ˜ 0 , respectively. and u Next, this equivalent linearized matrix pair (A, B) is used for computing the feedback gains (K, Ki ) in Fig. 4. The gains are obtained by solving the linear quadratic optimal control problem: Z ∞ ˜˙ T Rδ u ˜˙ )dt, min (eT Qe + δ u (8) ˜ δu

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where e is the position error vector:   xtar − x e := . ytar − y

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(9)

The solution to (8) is given by Z

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˜ = Kδx + Ki δu 11

e dt,

(10)

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where the matrices K and Ki are calculated via the associated algebraic Riccati equation. Weighting matrices Q and R are tuned, by trial and error, for taking the trade-off between platform movement speed and actuation usage, reflecting the constraints C2 and C3 in Section 2.2. Remark : It is worth noting that, although average wind information (¯ v) ˜ 0 in the linearization calculation, as well as K and Ki in conaffects x0 and u troller design, ocean wave and current information (w) is not incorporated in the controller design and implementation due to its difficulty of real-time measurement. These disturbances influence the feedback signals (y), and are rejected by feedback control with an integrator. 3.4. Naive Position Controllers for Comparison For comparison purposes, two naive position controllers are also designed in this paper. The first controller is an open-loop controller sending constant inputs to the FOWT. The second controller is an empirically-tuned PID controller. This PID controller is embedded in Position Controller in Fig. 3, and the same torque controller in (3) and stabilizing controller in (4) are used for fair comparisons between the proposed LQI position controller and the PID controller. 3.4.1. Open-loop Controller The open-loop controller first solves the equilibrium point solution (x0 , u0 ) at the target position (xtar , ytar ) which satisfies the target power (Ptar ). Then, the equilibrium inputs (u0 ) are sent to the FOWT as a constant signal, which, in the absence of modelling error and disturbances, ensures successful position transfer to the target point. This is because control inputs uniquely determine the aerodynamic thrust force at steady-state which, in a monotonically increasing mooring line force field, corresponds to a uniquely determined x-y position. As the name indicates, this open-loop controller does not rely on position feedback signal. Hence, it is expected that its performance is sensitive to modelling errors and disturbances. 3.4.2. PID Controller The PID controller assumes no prior knowledge of the system model, and only applies control inputs in response to error signals for correction. In specific, the auxiliary output of Position Controller (˜ ωr ), blade pitch angle (βpos ), and nacelle yaw angle (γ) are adjusted based on three independent error signals (eP,in , eβ , eγ ) as Z deP ID ˜ = Kp,P ID eP ID + Ki,P ID eP ID dt + Kd,P ID , (11) u dt where Kp,P ID , Ki,P ID , Kd,P ID are 3 × 3 diagonal matrices, and the error signal eP ID is defined as   eP,in eP ID :=  eβ  . (12) eγ 12

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In (12), the first element eP,in is the error between aerodynamic power input and the target power: Ptar eP,in := − τa ωr , (13) ηg where τa is the aerodynamic torque. The other elements eβ and eγ are the position error signals, illustrated in Fig. 5 and defined mathematically by   y xtar −1 , (14) eγ := tan − tan−1 ytar x ˆ eβ := eT d, (15) where e is the position error vector defined in (9) and indicated in Fig. 5, and dˆ is a normalized target position vector given by   1 xtar dˆ := p . (16) 2 ytar x2tar + ytar

Figure 5: Position error signals eγ and eβ 420 421 422 423 424 425 426 427 428 429 430

The logic behind the PID control law (12) is explained next. The position control is decoupled into two tasks, i.e., the angle control task and the displacement control task. The angular error (eγ ) is reduced by using γ for adjusting thrust force direction, whereas the displacement error (eβ ) along the target direction is compensated by using β for adjusting thrust force magnitude. Finally, to establish steady-state operation of generating the target power Ptar , the aerodynamic power entering the drivetrain should be regulated around Ptar ηg . An additional torque sensor is assumed to be installed to measure instantaneous aerodynamic torque (τa ) so that aerodynamic power can be computed. Then the PID controller adjusts the auxiliary signal ω ˜ r to reduce the error in aerodynamic power input (eP,in ) in (13). 13

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4. Simulation Results

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This section examines the performance of the designed controllers on a specified position transfer task in simulations using FAST (Version 7.02) and Matlab/Simulink, with quasi-static mooring line model and first-order wave loading model realized in FAST v7. All the designed controller parameters are provided in Appendix A. Since the controller proposed in Section 3 is responsible for not only position transfer but also power and position regulation against disturbances, controller performances are compared in all these aspects. The GainScheduling Proportional-Integral (GS-PI) controller presented in [21], which has been widely adopted as a baseline controller by researchers in wind turbine control community [5, 29, 30, 31], is also used to demonstrate that power regulation level and platform oscillation level achieved by the proposed controller are comparable to those of the GS-PI controller. Note that the GS-PI controller in [21] is not capable of dealing with the position control requirement, and thus it is not compared in the position control aspect.

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4.1. Position Control Task

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As an example of the FOWT position control problem, a scenario is considered where OC4-DeepCwind Semisubmersible is initially at steady-state operation at position (11, 0) [m], and then commanded to move to a target position (xtar , ytar ) = (8, 8) [m]. This corresponds to about 8 [m] lateral displacement3 . The power is to be regulated at Ptar = 3 [MW]. To mimic realistic offshore environments, both wind and wave profiles were generated by using the widely-accepted software TurbSim [32] and wave theory [33], respectively. The wind field includes vertical wind shear effect, and at each altitude, we assume a fluctuating and spatially uniform wind speed. The time-averaged velocity is v¯ = [18, 0, 0]T [m/s] at the hub height, with fluctuations in each direction, as plotted in Fig. 6. The irregular ocean wave profile was generated using the Pierson-Moskowitz spectrum [33] such that it propagates along the nominal wind direction. The wave height profile at the center column is also plotted in Fig. 6. All analyses and simulations in this section were performed with the wind and wave profiles in Fig. 6. In all the simulations, the states were initially set as follows. x(0)

=

[11 [m], 0 [m], −0.082 [m], 0 [deg], 5.4 [deg], 1.21 [deg]

464

0 [m/s], 0 [m/s], 0 [m/s], 0 [deg/s], 0 [deg/s], 0 [deg/s],

465

12.2 [rpm], 1183.4 [rpm], 0 [deg], ] .

T

466 467 468

Finally, when the control input signals are sent to the FOWT, actuator-level servo loops are required for the physical actuators to track their reference commands. While the commanded signals (u) in (1) can contain any frequencies and 3 Although this movement amount may be too small for wake mitigation, the amount can be increased by modifying mooring line properties such as lengths and anchor points.

14

v¯y + vy [m/s] v¯x + vx [m/s] ¯z + vz [m/s] Wave height [m] v

25 20 15 0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

5 0 −5

4 2 0 −2 −4

4 2 0 −2 −4

Time [min]

Figure 6: Wind and wave disturbance profile

469

any amplitudes, for incorporating physical limitations of actuators, saturations and rate limits are imposed on u as listed in Table 2. Table 2: Saturation and rate-limits for control inputs

Inputs

Saturation

Rate-limit

β τg γ

[−30, 30] [deg] [0, 60.68] [kN·m] [−60, 60] [deg]

[−8, 8] [deg/s] [−30, 30] [kN·m/s] [−0.3, 0.3] [deg/s]

470

471 472 473 474 475 476 477 478 479 480 481

4.2. Simulation Results To examine the controller performances, simulations were conducted for two cases: one case with wind and wave disturbances and the other without. The strong and realistic disturbances resulted in significant fluctuations to the system outputs, making it hard to observe and compare controller performance in convergence speed and position tracking accuracy. Therefore, to have a clear view of the convergence speed and steady-state error of each controller, simulations were first carried out without disturbances, i.e. with constant wind velocity v¯ and no waves, Then, simulations under the aforementioned disturbance profiles are performed to study each controller’s realistic performance of regulating position and power in the face of wind and wave fluctuations.

15

482 483 484

4.2.1. Position Control Performance The x-y position trajectories, as well as the trajectories for other four DOFs (z, θx , θy and θz ), for controllers (except GS-PI controller) are plotted in Fig. 7. In terms of steady-state position, both CLQI and CP ID reached the exact tar-

12 10 8 6

12 10 8 6 0

10

20

30

10

10

5

5

0

0

10

20

30

0

10

20

30

0

10

20

30

0

10

20

30

0

10

20

30

0

10

20

30

0 0

10

20

30

1

1

0

0

-1

-1 0

10

20

30

0 -1 -2 -3

0 -1 -2 -3 0

10

20

30

5

5

0

0 0

10

20

30

2

2

0

0

-2

-2 0

10

20

30

Time [min]

Time [min]

(a) Without Disturbances

(b) With Disturbances

Figure 7: Time response of the FOWT’s x-y position and distance error. LQI controller CLQI : magenta line, open-loop controller COL : blue line, PID controller CP ID : red line 485 486 487 488 489 490 491

get position in the absence of wind and wave disturbances. In contrast, COL converged to a location near the target with a small error due to the modelling discrepancies between the Simplified Model used for controller design and the FAST model used for simulations. As far as the convergence speed (constraint C3) is concerned, CP ID yielded the slowest convergence speed. If we made the PID controller more aggressive, 16

492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510

then the system became unstable with oscillatory responses. Thus, we took a trade-off between speed and oscillation in the controller parameter tuning, and the presented PID control result was the best result that we were able to achieve. The other two controllers (COL and CLQI ) designed based on a mathematical model moved the FOWT platform significantly faster, and both completed the platform position transfer task in less than 5 minutes. Regarding fluctuations during the transient response, the plots in Fig. 7(a) give a clear comparison. Among the three controllers, CP ID has no overshoot, which is mostly due to its slow response. Compared to COL , the proposed controller CLQI was able to reduce overshoot in both x and y directions. This is because COL uses constant control inputs (u0 ) while CLQI can adjust its control inputs to suppress transitional overshoot. When wind and wave disturbances are applied, as plotted in Fig. 7(b), outputs became significantly noisier. However, despite the strong disturbances which caused high frequency fluctuations in the FOWT’s position, all three controllers still successfully moved the FOWT to around the target position. Trajectory plots in the x-y plane are plotted in Fig. 8 for comparison. For interested readers, the trajectory plots for the heave z, the roll θx , the pitch θy and the yaw θz of the platform are also shown in Fig. 7. Additionally, aerodynamic thrust forces are plotted in Fig. 9. 12

12

10

10

8

8

6

6

4

4

2

2

0

0 0

5

10

15

0

(a) Without Disturbances

5

10

15

(b) With Disturbances

Figure 8: Trajectory plots in x-y plane. LQI controller CLQI : magenta line, open-loop controller COL : blue line, PID controller CP ID : red line 511

512 513 514 515 516 517 518

4.2.2. Control Inputs Similar to the comparison in position control result, plots of the control input trajectories in Fig. 10 also demonstrate that CLQI adjusted control inputs much faster than CP ID and was able to use larger actuation to generate a smoother transient response. In the figure, it can be observed that the rate limit of γ in Table 2 was hit until about 2 minutes, with the straight line with the slope 0.3 deg/s = 18 deg/min. 17

800

600

400 0

10

20

30

Aerodynamic Thrust Force [kN]

Aerodynamic Thrust Force [kN]

1000

Time [min]

1000

800

600

400 0

10

20

30

Time [min]

(a) Without Disturbances

(b) With Disturbances

Figure 9: Aerodynamic thrust force. LQI controller CLQI : magenta line, open-loop controller COL : blue line, PID controller CP ID : red line

(a) Without Disturbances

(b) With Disturbances

Figure 10: Time response of control inputs β, τg , γ. LQI controller CLQI : magenta line, open-loop controller COL : blue line, PID controller CP ID : red line

18

519 520 521 522 523 524 525 526

527 528 529 530 531 532 533 534

Remark : Note that the negative blade pitch angle β means that the pitchto-stall strategy [34, 35, 36] is used to generate aerodynamic thrust force appropriate for target platform position. We chose the pitch-to-stall strategy because the larger thrust force, necessary to move OC4-DeepCwind Semisubmersible, can be achieved by this strategy compared to the force attainable with the pitch-to-feather strategy. Another possible benefit of adopting the pitch-to-stall strategy is a potential mitigation of the negative damping effect caused by the right half-plane zero which is inherent to floating offshore wind turbines [37]. 4.2.3. Power Regulation The generator power outputs for tested controllers are shown in Fig. 11. While all controllers were able to regulate power around Ptar = 3 [MW], the power trajectory for COL did not converge to the exact value due to modelling inaccuracies. In the absence of any wind and wave disturbance, the other three controllers were able to regulate generator power around the target value. When disturbances exist, both wind and wave cause fluctuations affect aerodynamic power, making it hard to regulate power output. In comparison, CLQI outperformed CP ID and CGS−P I in power regulation.

Power [MW]

3.1

3.05

3

2.95

2.9 0

10

20

30

Time [min] (a) Without Disturbances

(b) With Disturbances

Figure 11: Time response of generator power output. LQI controller CLQI : magenta line, open-loop controller COL : blue line, PID controller CP ID : red line, GS-PI controller CGS−P I : black line 535

536 537 538 539 540

4.2.4. Time-domain Analyses In the drivetrain, excessive acceleration and deceleration of the rotor (ω˙ r ) and the generator (ω˙ g ) cause fatigue loading to the transmission shafts. Figure 12 plots rotor speed and acceleration (ωr , ω˙ r ). Although CLQI and CP ID appear to involve larger rotational speed variation than COL , their acceleration 19

541

plots indicate that they all generate similar degrees of shaft vibration, which are also comparable to that of the baseline CGS−P I .

(a) Without Disturbances

(b) With Disturbances

Figure 12: Time response of ωr , ω˙ r . LQI controller CLQI : magenta line, open-loop controller COL : blue line, PID controller CP ID : red line, GS-PI controller CGS−P I : black line 542 543 544 545 546 547 548 549 550 551 552 553

On the tower side, fatigue loading is in the form of tower bending due to its interaction with platform and nacelle at the two ends. Its effect can be approximated using the platform’s x-y plane accelerations (¨ x, y¨) and the tower pitch angle acceleration (θ¨y ), which are plotted in Fig. 13. From the plots which are almost overlapping each other, it can be concluded that all four controllers have similar results in tower vibration, and the added task of position control using aerodynamic thrust force did not cause additional platform vibration. Additionally, platform yaw (θz ) and upwind mooring line tension are presented in Fig. 14 for interested readers. Finally, numerical values of the presented performance aspects are summarized in Table 3 to support the observations in the aforementioned figures.

20

(a) Without Disturbances

(b) With Disturbances

Figure 13: Time response of x ¨, y¨, and θ¨y . LQI controller CLQI : magenta line, open-loop controller COL : blue line, PID controller CP ID : red line, GS-PI controller CGS−P I : black line

554

555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571

5. Conclusion This paper proposed a feedback control structure and a linear-quadraticintegrator controller for real-time platform position control of floating offshore wind turbine systems. The platform position of the wind turbine system was transferred to a target position and regulated there by adjusting the aerodynamic thrust force direction and magnitude with the control inputs which are commonly available in modern utility-scale wind turbines. It was demonstrated by simulations that, in the absence of wind and wave disturbances, the proposed controller yielded the best balance of speed and smoothness during position transfer, the best power regulation, and similar degree of drivetrain shaft twisting and tower bending motions compared to other controllers. Even in the face of wind and wave disturbances, the proposed controller structure involving LQI position controller was able to regulate the platform position and the generated electrical power. The designed position controller will be useful in realizing realtime floating offshore wind farm layout reconfiguration, in order to alleviate the wake effect, thereby maximizing the wind farm’s total efficiency. Possible future works are the analysis of the blade loads by the pitch-to-stall blade control and the development of load mitigation methods, frequency-domain controller design 21

Upwind Mooring Line Tension [kN] Platform Yaw [deg]

Upwind Mooring Line Tension [kN] Platform Yaw [deg]

1.5 1 0.5 0 -0.5 -1 0

10

20

30

2000

1.5 1 0.5 0 -0.5 -1 0

10

20

30

0

10

20

30

2000

1500

1500

1000

1000

0

10

20

30

Time [min] (a) Without Disturbances

Time [min] (b) With Disturbances

Figure 14: Time response of platform yaw and upwind mooring line tension. LQI controller CLQI : magenta line, open-loop controller COL : blue line, PID controller CP ID : red line, GS-PI controller CGS−P I : black line

577

such as H∞ control to reduce the platform motions induced by wind and wave disturbances, and real-time wind farm controller design to deal with dynamic change of wind conditions. Especially, the platform position controller proposed in this paper needs to be validated with FAST v8, i.e., with the second-order hydrodynamic loads which are known to excite the natural frequencies of the semisubmersible platform at the difference-frequencies of the incident waves [38].

578

Acknowledgement

572 573 574 575 576

581

This research was financially supported by the Discovery Grant Program of the Natural Sciences and Engineering Research Council in Canada (NSERC) (RGPIN-2017-03753) and the Canada Research Chair Program.

582

Appendix A. Controller Parameters

579 580

584

This appendix collects the parameters of all the controllers designed and used in simulation studies in Section 4 for interested readers.

585

Stabilizing Controller in (4): Kp = 0.5.

583

22

Table 3: Quantitative Comparison of Controller Performances

Without Disturbances

x-y Position Power

CGS−P I

CLQI

COL

CP ID

% Overshoot

n/a

5%

20 %

0%

Rise-time (0-90%) [min]

n/a

2

2.5

14.3

RMS error (regulation) [m]

n/a

0.85

0.83

0.91

RMS error [W]

204 2

Other signals

112 −3

1.49 · 10−2

3.81 · 10

x ¨ RMS [m/s2 ]

4.23 · 10−3

6.78 · 10−3

7.24 · 10−3

6.48 · 10−3

−3

−3

−3

3.63 · 10−3

2.81 · 10−3

2.39 · 10−3

2.02 · 10

y¨ RMS [m/s ] θ¨y RMS [rad/s2 ]

2.16 · 10−3

4.94 · 10

2.43 · 10−3

2.39 · 10

592

−2

ω˙ r RMS [rad/s ] 2

2.71 · 10

41047 −2

3.75 · 10

With Disturbances CGS−P I % Overshoot x-y Position

Other signals

586

CP ID

Due to Signal Fluctuations

RMS error [W]

11368 2

COL

Not Available

Rise-time (0-90%) [min] RMS error (regulation) [m]

Power

CLQI

−1

609

1.01 · 10−1

2.97 · 10

x ¨ RMS [m/s2 ]

1.68 · 10−1

1.67 · 10−1

1.67 · 10−1

1.69 · 10−1

y¨ RMS [m/s2 ] θ¨y RMS [rad/s2 ]

6.53 · 10−2

6.61 · 10−2

6.61 · 10−2

6.58 · 10−2

1.61 · 10−1

1.51 · 10−1

1.51 · 10−1

1.55 · 10−1

−4.296 · 10−4 8.323 · 10−6 5.023 · 10−6 2.0 · 10−3 −1.8 · 10−6 4.4 · 10−3 −6.89 · 10−2 5.28 · 10−2 9.634 · 10−5 6.9 · 10−3 2.7 · 10−2 −1.969 · 10−1 −4.3 · 10−3 −6.229 · 10−6 −3.162 · 10−4 23

9.17 · 10

4256

−2

ω˙ r RMS [rad/s ]

LQI Controller in (10):  3.1 · 10−3  8.857 · 10−4  −2.812 · 10−5   −1.91 · 10−2   1.86 · 10−2   −4.06 · 10−2   2.171 · 10−1  −2 K=  9.04 · 10 −4 −9.305 · 10   −4.14 · 10−2  −1.758 · 10−1   −1.112   4.45 · 10−2   6.407 · 10−5 3.5 · 10−3

1.03 · 10

67583 −1

T 6.176 · 10−4 1.771 · 10−4   −5.624 · 10−6   −3.8 · 10−3   3.7 · 10−3   −8.1 · 10−3   4.34 · 10−2   1.81 · 10−2   . −1.861 · 10−4   −8.3 · 10−3   −3.52 · 10−2   −2.225 · 10−1   8.9 · 10−3   1.281 · 10−5  6.987 · 10−4

 −4.442 · 10−4  Ki = 2.153 · 10−4 −8.885 · 10−5

587

588

 −6.919 · 10−5 −1.257 · 10−4  . −1.384 · 10−5

Open-loop Controller in Section 3.4.1:   −1.455 · 10−1 [rad] u0 =  3.022 · 104 [N · m]  . 6.38 · 10−1 [rad] PID Controller in (11):

589

Kp,P ID

590

Ki,P ID

591

Kd,P ID

592

593 594

595 596

597 598 599

600 601 602

603 604

605 606

607 608

609 610

611 612 613

614 615 616 617

  diag 10−9 , −2 · 10−3 , 0.5 .   = diag 10−9 , −10−4 , 10−3 .   = diag 5.0 · 10−11 , 0, 0.3 . =

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26

Highlights -

Control mechanism for floating offshore wind turbine position control Linear quadratic integrator as a platform position controller Platform position control with only aerodynamic force Position control simulation with realistic wind and wave profiles