Power-setpoint extremum seeking control for maximizing wind power capture of turbine and farm operation

Abstract

In this paper, we propose a power-setpoint based Extremum Seeking Control (ESC) framework for model-free Region-2 controls for maximizing the power capture for turbine and farm operation, without dependency on wind measurement. As a major obstacle for retrofitting wind turbine/farm controls is that only the power setpoint is accessible, the power-setpoint based ESC framework is proposed with a back-calculation anti-windup structure. If increasing the power demand cannot further increase actual power output, the anti-windup structure automatically holds the power demand setpoint. For farm operation, the proposed method is integrated into the Delay-compensated Nested-loop ESC. The proposed method is evaluated by simulations on the SimWindFarm platform for both single-turbine and farm operation scenarios. The results demonstrate the capability of tracking the achievable optimum power for turbine and farm operation, with only reasonable increase of some loads. The proposed method promises an easy-to-implement model-free retrofitting control strategy for enhancing wind energy capture.

Keywords

Region-2 wind turbine control extremum seeking control power-setpoint tracking wind farm control nested-loop extremum seeking control energy capture

Introduction

As a primary sector of renewable energy, wind power has its global capacity exceeding 650GW in 2020 (Lee and Zhao, 2019). With the ever-increasing scale of wind power generation, the levelized cost of energy (LCOE) can be significantly reduced by improving the efficiency of power capture, especially for the below-rated-wind-speed (i.e. Region-2) operation. For individual turbine operation, successful Region-2 control maintains suitable tip speed ratio (TSR) and blade pitch angle for a given wind condition to maximize the energy capture. In order to achieve this goal, various region-2 control strategies have been developed, mainly with model-based approaches. A typical scheme is to utilize a pre-determined map of power coefficient C_P (also known as the “power curve”) in terms of TSR and blade pitch (Manobel et al., 2018; Sohoni et al., 2016; Spudic et al., 2018). The turbine C_P map has been obtained using analytical modelling, numerical simulation and/or empirical model fitting. As TSR is defined as the ratio of rotor tip speed to the incoming wind speed, the turbine rotor speed (and/or blade pitch angle) is regulated to the setpoint corresponding to the maximum C_P, provided that some wind measurement or estimate is available. Different model-based control methods have been studied. Mirzaei et al. (2016) design a proportional-integral (PI) Region-2 controller via the pole–placement technique, based on a reduced-order model derived from HAWCStab2 (Larsen and Hansen, 2015) simulation, and the PI parameters are tuned with constrained optimization. Bianchi et al. (2012) propose a gain-scheduled $H_{\infty}$ technique based on high fidelity local wind turbine models. Henriksen et al. (2011) study a nonlinear model predictive control scheme with a simplified turbine model, in which the preview information of incoming wind speed can enhance power capture. Ostergaard et al. (2007) design a gain-scheduled linear quadratic controller combined with state and disturbance estimation, where the wind speed is estimated by inversion of the static aerodynamic model. Besides, linear parameter-varying control of wind turbines have been explored in (Bobanac et al., 2010; Ostergaard et al., 2009).

Although the model-based region-2 control strategies are straightforward in the sense of regulation control, the achievable performance of energy capture depends on two major aspects: (i) aerodynamic characteristics of turbine rotor (due to their impact of the actual C_P map), and (ii) measurement/estimate of the effective incoming wind speed. Such dependencies lead to inherent limitations under the challenge of field operation. Due to the complex nature of aerodynamics, the turbine aerodynamic characteristics can vary significantly, due to wind shear and variations in ambient condition, and especially blade surface wear (Ashuri et al., 2016), accumulations of ice, snow, dirt and bugs (Jasinski et al., 1998; Ren and Ou, 2009). Single-point wind measurement with nacelle-top anemometer can seldom reflect well the effective wind speed, in addition to strong distortion by near-field wake effect. Good wind estimation relies on quality blade aerodynamic model, which is in turn highly affected by the aforementioned uncertain factors. Due to all these factors, model-free control strategies are highly desirable, which are intended to achieve near-optimum performance without detailed knowledge of turbine model or accurate wind measurement. Johnson et al. (2006) propose a model-reference adaptive control approach to the torque-gain control, with the feedback of normalized power output based on met-tower wind measurement. Recently, the Extremum Seeking Control (ESC) has emerged as a promising solution of model-free real-time optimization for region-2 control. With the gradient estimate realized with dither-demodulation scheme, the optimum search process can be more robust against the wind fluctuation. Creaby et al. (2009) study ESC based region-2 control with power feedback only, with rotor torque, blade pitch and yaw input. Xiao et al. (2019) have conducted field tests of ESC based region-2 controllers on the NREL CART3 turbine, and significant improvement in energy capture is reported without negligible sacrifice in fatigue loads.

The benefits of ESC based region-2 control have also extended to wind farm operation. Although model-based region-2 wind farm control have been investigated using induction and/or wake steering (Knudsen et al., 2015), using model predictive control (MPC) (Boersma et al., 2018), linear quadratic regulator (LQR) (Soleimanzadeh et al., 2012, 2014), sequential convex optimization (Hovgaard et al., 2012), and dynamic programming (Rotea, 2014), effectiveness of these strategies depend primarily on, besides quality turbine models, the wake models. In addition to the characteristic uncertainties for individual turbines in field operation as described above, wake phenomenon presents a more complex situation in terms of wind farm modelling, due to its inherent temporal-spatial variation, wake meandering, and other effects. Therefore, model-free wind farm control strategies have been investigated, for example game theory (Marden et al., 2013), gradient descent (Gebraad and van Wingerden, 2015), Bayesian ascent (Park and Law, 2016), and random search (Ahmad et al., 2016) methods. ESC has been extended to wind farm control as well (Rotea, 2000). Seem and Li (2012) proposed a Nested-loop ESC (NLESC) for wind farm power maximization in which the maximum power output of a wind turbine array can be achieved by controlling each turbine with ESC to maximize a utility that combines its own power with the power of the downstream turbines. Yang et al. (2015) demonstrated the benefits of the NLESC wind farm control strategy via SimWindFarm simulation. Later studies with LES simulations further validated the effectiveness (Ciri et al., 2016, 2017, 2019).

As for the existing work on ESC based region-2 control for wind turbines and wind farms, the axial-induction-factor oriented turbine or farm controls have been based on control authorities like generator torque and blade pitch. Access to these manipulated inputs is straightforward for control system developers. Therefore, ESC development can be carried out relatively easily. However, when ESC is used to retrofit supervisory control for existing wind turbines or wind farms, the control framework that requires access to these manipulated inputs cannot be implemented. As for the control system interface that wind farm operators can operate, only the Power Setpoints for individual turbines are effectively manipulatable inputs while the torque and blade pitch are controlled in the inner-loop regulators for power, rotor speed and load controls. Therefore, for control retrofitting of existing wind turbine/farms, it is desirable to develop an ESC control framework based on power-setpoint manipulation.

The major contributions of this paper are two-fold: i) development of a power setpoint based ESC for region-2 operation of individual turbines, and ii) development a power setpoint based delay-compensated nested-loop ESC (that maximizes power demand of the wind farm as a whole) for each individual turbine in a wind farm. Particularly, for free-stream wind speed slighter higher than the rated wind speed, we show that the proposed farm controller realizes easy and automatic switching between region-2 and region-3 operation of a given turbine without violating maximization of farm total power output. These strategies can be easily implemented in existing wind power plants and could send the reference power directly as the optimal available power in the wind. Such control strategies can significantly mitigate the possible energy loss due to errors in wind/wake measurement/estimation and turbine aerodynamic characteristics which are currently used for wind power control system developers to determine the power-setpoint for individual turbines or turbines in farms. The proposed control strategies are evaluated with a simulation study. In spite of the availability of several high-fidelity wind farm control simulation platforms such as SOWFA, we have chosen the SimWindFarm platform. This platform has implemented a comprehensive wind plant control strategy based on power-setpoint scheduling and regulation, which best resembles the control system implementation of commercial wind farms.

The remainder of the paper is structured as follows. Section II describes the control framework of SimWindFarm, which is a power setpoint based wind farm control scheme. Section III presents the idea of power setpoint based ESC. Section IV describes the design and simulation of power setpoint ESC for turbine control, while the work on wind farm control is given in Section V. The paper is concluded in Section VI.

SimWindFarm: A power-setpoint based wind farm control platform

This study adopts SimWindFarm as the simulation platform (Grunnet et al., 2010), as its farm-level and turbine-level control framework is established with power setpoint as reference input. The wake model used in SimWindFarm is based on Larsen et al. (2008). The turbine controller of the NREL 5 MW turbine is based on the description in (Jonkman et al., 2009). The network operator in SimWindFarm is designed with different operating modes:

(1) Absolute Power Mode: the wind farm power output is explicitly specified.

(2) Delta Mode: a specific amount of reserve power could be maintained.

(3) Balance Mode: load balancing or load matching for grid integration.

(4) Rate Limit Mode: a specific rate of increase for power demand is imposed on the wind farm operation.

(5) Frequency Mode: the wind farm power output is controlled to maintain the specified grid frequency.

All these modes require knowledge of wind speed to determine total available power at the site as input to decide on the total farm power demand from the turbines. With the knowledge of wind speed in front of each turbine, it divides the total farm power demand in proportion to the available wind power in front of each turbine (Grunnet et al., 2010). The available wind power at turbine i is

P_{a, i} = \frac{1}{2} ρ {AV}_{meas, i}^{3} C_{p, max}

(1)

where $V_{meas, i}$ is the freestream wind velocity in front of turbine i, $ρ$ is the air density and A is the cross-sectional area of the turbine. The total available wind farm power P_a is just the sum of available powers in front of each turbine, that is

P_{a} = \sum P_{a, i}

(2)

The power demand for turbine i, P_d,i, is determined as the portion of the total farm power demand P_d, (set by the network operator) according to the share of it available power in the total available power, that is

P_{d, i} = P_{d} \times \frac{P_{a, i}}{P_{a}}

(3)

To calculate the available power in front of the turbines, the maximum power coefficient (C_p,max) of 0.47 is adopted in equation (1) for the NREL 5 MW turbine model. For wind farm operation, the available power of the turbines in the leading row (with respect to the prevailing wind direction) is calculated based on the freestream wind speed as specified in equation (1), while that of a downstream turbine is calculated based on the wake corrected wind velocity at the nacelle as described in (Grunnet et al., 2010). With dependence on the available power thus calculated, the default wind farm control strategy in SimWindFarm can be considered as a model-free approach with high-fidelity wind measurement.

The controller for the NREL 5 MW used in SimWindFarm is described fully in (Jonkman et al., 2009) and a Simulink implementation can be found in SimWindFarm toolbox as discussed in (Grunnet et al., 2010). A conventional variable-speed, variable blade-pitch-to-feather configuration has been chosen for designing this controller. The power production depends on a generator-torque controller and a full-span rotor-collective blade-pitch controller. The main goal of the generator-torque controller is to maximize power capture below the rated value while the purpose of blade-pitch controller is to regulate generator speed above the rated point. Both torque and pitch controllers use the generator speed measurement as the feedback inputs.

For the 5 MW turbine model, NREL’s baseline control strategy can be depicted by the torque-speed curve as shown in Figure 1. The generator torque is determined by the filtered generator speed, with five control regions defined: 1, 1.5, 2, 2.5, and 3. Table 1 gives the corresponding look-up table. Region 1.5 is a start-up region, which is designed as a linear transition between Regions 1 and 2. Region 2.5 is a linear transition between Regions 2 and 3, with a torque slope corresponding to the slope of an induction machine.

Figure 1.

Torque-speed curve NREL control strategy of NREL 5 MW turbine model.

Table 1.

Definitions of torque control regions for NREL 5 MW turbine model.

Generator speed	Torque control regions
0–670 rpm	Region 1
670–871 rpm	Region 1.5
871–1161.9 rpm	Region 2
1161.9–1173.7 rpm	Region 2.5
1173.7 rpm–above	Region 3

The generator torque control is realized by the so-called “torque-gain controller,” that is the generator torque command is given by

τ_{g} = k_{t} ω^{2},

(4a)

where ω denotes the generator speed and k_t is known as the “torque gain.” The theoretical optimal value of k_t can be found as

k_{opt} = \frac{1}{2} \frac{ρ A R^{3}}{N^{3}} \frac{C_{p, max}}{λ_{opt}^{3}}

(4b)

N is the gear ratio, R is the rotor radius and λ_opt is the optimum tip speed ratio. The optimum torque gain (k_opt) is obtained as 2.2 Nm/(rad/s)² with the values $ρ$ = 1.2231 kg/m³, R = 63m, N = 97, λ_opt = 7.55 and $C_{p, max}$ = 0.47. The generator torque in region 1 ( $τ_{1}$ ) is set to 0, in region 2 ( $τ_{2}$ ) follows the optimal torque curve, and in region 3 $τ_{3} = P_{rated} / ω$ since the generated power is kept constant at the rated value. For region 1.5, the torque command follows:

τ_{1.5} = T_{2} \times \frac{ω_{1.5} - W_{1.5}}{W_{2} - W_{1.5}}

(5)

where W_1.5 and W₂ are the generator speeds at the beginning of region 1.5 and region 2, respectively, which are 670 rpm and 871 rpm for the 5 MW turbine model. T₂ is the optimal torque achieved at generator speed W₂ based on equation (4a). For region 2.5, the torque command follows:

τ_{2.5} = T_{2.5} + (\frac{P_{rated}}{W_{3}} - T_{2.5}) \times \frac{ω_{2.5} - W_{2.5}}{W_{3} - W_{2.5}}

(6)

where W_2.5 and W₃ are the generator speeds at the beginning of region 2.5 and region 3, respectively, which are 1100 rpm and 1161.9 rpm. T2.5 is the optimal torque at generator speed W_2.5, again based on equation (4a).

In region 3, the blade pitch is used to regulate the rotor speed to its rated value, via a gain scheduled PI controller, that is

β_{ref} = K_{P} (β) ω_{err} + K_{I} (β) \int_{0}^{t} ω_{err}

(7a)

K_{P / I} (β) = K_{P / I, 0} \frac{β_{2}}{β_{2} + β}

(7b)

K_P and K_I are based on linearization of the power production sensitivity to blade pitch angle. ω_err denotes the regulation error for the generator speed, K_P_/I,0 denotes the base values of K_P and K_I at β = 0°, and β₂ is the pitch angle where the sensitivity is doubled.

Control of power setpoint using ESC

ESC principle and design guidelines

ESC is a model-free real-time optimization strategy that can search for unknown input parameters for optimizing a selected performance index of a nonlinear dynamic system. For the single-input ESC scheme, Figure 2 shows the Wiener-Hammerstein approximation which is typically considered for ESC design (Rotea, 2000). F_in(s) and F_out(s) are unity-gain linear time-invariant (LTI) approximation of the input and output dynamics, respectively, and y = f(t,u) denotes a possibly time-varying static map of the nonlinear plant. The online gradient estimation of ESC operation is based on applying periodic dither d₁(t) = asin $(ω t)$ and demodulation signal d₂(t) = $\frac{2}{a}$ sin(ωt+θ), where a is the dither amplitude, $ω$ is the dither frequency and θ is the phase compensation. The high-pass filter F_HP(s) is applied to the performance index y that is aimed to optimize (maximize/minimize) by tuning the input u. The low-pass filter F_LP(s) with an integrator and gain k completes the ESC loop.

Figure 2.

Block diagram of conventional ESC.

The gradient of output with respect to input u, ∂f/∂u, is estimated from the measurements of the objective function y with dither signal d₁(t) = asin $(ω t)$ added to the input, that is

u = \hat{u} + a \sin ω t

(8)

The corresponding plant output is then approximated as

y = f [\hat{u} + a \sin (ω t + φ_{in})]

(9)

where ϕ_in represents the phase shift caused by the input dynamics at the dither frequency. The Taylor series expansion of equation (9) is

y = f (\hat{u}) + {\frac{\partial f}{\partial u} |}_{u = \hat{u}} [a \sin (ω t + φ_{in})] + H . O . T .

(10)

The high-pass filter F_HP (s) is designed to suppress the DC term in equation (10) while passing the AC terms. Its output is approximated as

{\frac{\partial f}{\partial u} |}_{u = \hat{u}} a [\sin (ω t + + φ_{in} + φ_{out} + φ_{HP})] + H . O . T .

(11)

where $φ_{out}$ and $φ_{HP}$ denote the phase shifts caused by the F_out(s) and F_HP(s) at the dither frequency, respectively. The signal in equation (11) is then multiplied by the demodulation signal d₂(t) = $\frac{2}{a}$ sin(ωt+θ) to obtain

{\frac{\partial f}{\partial u} |}_{u = \hat{u}} [\cos (θ - φ_{in} - φ_{out} - φ_{HP}) - \cos (2 ω t + φ_{in} + φ_{out} + φ_{HP})] + H . O . T .

(12)

Using the low-pass filter F_LP(s), DC term in equation (13) can be retained which is proportional to the gradient ∂f/∂u:

\frac{\partial f}{\partial u} \cos (θ - φ_{in} - φ_{out} - φ_{HP})

(13)

Finally, closing the loop with an integrator drives the gradient to zero in steady state, provided that the closed-loop system is asymptotically stable; thus, resulting in an optimum control input.

In order to maximize the gradient information extracted, the phase of the demodulation signal θ is chosen to satisfy $θ = φ_{in} + φ_{out} + φ_{HP}$ .

The standard ESC design guidelines (Rotea, 2000) have been adopted in the study:

Estimate the input dynamics based on open-loop tests.

Choose the dither frequency within the bandwidth of the input dynamics.

Choose the dither amplitude so that the dithered output has appropriate SNR (signal-to-noise ratio) at the dither frequency with respect to the section of output due to measurement noise, external disturbance and process variation.

Design the high-pass filter with highest possible cut-off frequency while the dither frequency remains in the pass band.

Design the low-pass filter with highest possible bandwidth while providing appropriate suppression on dither related harmonics.

Determine the phase shift angle between the additive dither d₁(t) and the demodulating signal d₂(t) to compensate for the phase change caused by plant dynamics and the high-pass filter that is, $∠ θ = ∠ F_{in} (j ω) + ∠ F_{HP} (j ω)$ , where $∠ F_{in} (j ω)$ and $∠ F_{HP} (j ω)$ are the phase response of the plant input dynamics and the high-pass filter, respectively, evaluated at the dither frequency.

Choose an integrator gain to improve the transient performance.

As mentioned in Section 1, all the previous studies on ESC based wind turbine region-2 control have adopted the generator torque (or equivalently torque gain), blade pitch, and/or yaw as manipulated inputs. Access to these actuations is inconvenient for control system retrofitting by wind farm operators, for which only the power setpoint can be accessed as potential manipulated input. Therefore, we propose a power setpoint based ESC as described in next subsection.

Power setpoint based ESC for region-2 wind turbine control

As reviewed in Section 1, the primary interest of this paper is on how to utilize the power demand sent to turbine operation for realizing the ESC based region-2 control, first at the turbine level, and later for the farm-level operation. Figure 3 shows a basic overview of where power setpoint based ESC would come into wind turbine control picture. Baseline torque and pitch controllers are as described in Section 2.

Figure 3.

Block diagram showing overall wind turbine control structure.

For maximizing power output with tuning the power demand, the relevant ESC design will be based on the following rationale:

1) The power demand can be satisfied when the available wind power is greater than the power demand. Under such circumstance, the power demand can be further increased. From the ESC standpoint, the use of integral control will naturally fit the need for increasing the power demand.

2) When the available power is not greater than the power demand, the difference will be reflected as the steady-state error for the power demand regulation loop. If such power tracking error can be fed back to the input side of the integrator, the power demand will automatically stay at (the neighborhood of) the available wind power.

Conventionally, ESC controllers have been implemented on wind turbines by modifying the baseline torque and pitch controllers as can be seen in Figure 4(a). But, based on above considerations, the power demand driven ESC for wind turbine region-2 control is proposed as shown in Figure 4(b). The difference between the actual power output and the power demand is fed back into the input of the integrator through an anti-windup gain K_aw, which is like the design of anti-windup ESC proposed by Li and Seem (2008). As for power setpoint based ESC, the input dynamics F_IN(s) refers to the closed loop control dynamics for power setpoint regulation. equation (13) can now be modified as below

\frac{\partial f}{\partial u} \cos (θ - φ_{in} - φ_{out} - φ_{HP}) + K_{aw} (P - P_{dem})

(14)

Figure 4.

Conventional and power setpoint based ESC structure for wind turbine control: (a) conventional ESC for torque gain and blade pitch control, and (b) power setpoint based ESC for region-2 wind turbine control.

Together with the dither signal, the integral action can extract the gradient of the unknown static mapping f(t,u). The input variable “u” must live within a particular domain, as restricted by the operational limits of the plant. Given that $\hat{u}$ is driven via an integrator it is quite feasible that this input variable will be driven outside this acceptable domain, and hence the actual input may get saturated on the boundary of the domain as a consequence of actuator limits and will differ from the presumed input. Due to saturation, the objective function is no longer smooth, and the ESC will estimate the gradient of unknown objective function. When the objective function is piece-wise continuous, at some points, the gradients are not well-defined. Anti-windup ESC is shown to find the extremum for a modified objective function which is the sum of the original objective function and a penalty term. The penalty term is related to the saturation, which ensures that the adapted parameter returns to the desired domain where the original objective function can be optimized. When saturation occurs, the anti-windup block ensures that the input $\hat{u}$ will move to saturation bounds. On the other hand, when no saturation occurs, the ESC block will dominate, which serves to find the optimum u*.

Power setpoint based NLESC for wind farm control with delay compensation

The power setpoint based ESC described in the previous subsection is then extended to the region-2 wind farm control, being integrated to the Nested-loop Extremum Seeking Control (NLESC) proposed by Yang et al. (2015). The NLESC maximizes the total power output of the wind turbine array by formulating an ESC loop for every turbine, in which the turbine operation is optimized by using the feedback of the summed power of this very turbine and its downwind units. The objective function can be given as

P_{i}^{*} = max_{P_{dem, i}} (P_{i} e^{- D_{i, 1} s} + \sum_{j = i + 1}^{n} P_{j} e^{- D_{j, 1} s}), i = 1, . . . ., n - 1

(15)

D_i,1 denotes the time delay of wake propagation from turbine i to turbine 1 in the downstream. It is noteworthy that dynamic control of wind farm operation is always challenged by the significant transport delay of wake propagation, that is . there is a significant dead time between the changes in control input to the change of wind input to the downstream turbines. Yang et al. (2015) estimated the delay by using the cross-correlation between the power outputs of the adjacent turbines. As for dither frequency selection for upstream turbines, such delay is included into the input dynamics with respect to the total power output. Such treatment thus yields dither frequencies of upstream turbines much lower than what can be chosen for wind turbine ESC, which in turn significantly slows down the convergence speed. Later, by adopting a predictor-based ESC scheme (Oliveira et al., 2017), Wu and Li (2017) modified the NLESC based wind farm control into the Delay Compensated NLESC (DCNLESC). By using delay compensation, the dither frequency selection would not be required to be lowered to compensate for the delay and thus faster convergence can be obtained. In this study, we integrate the idea of power setpoint ESC into the DCNLESC framework, as shown in Figure 5. Like the previous studies, the wind farm considered is a cascaded turbine array consisting of turbines T1, T2, …, Tn, with turbine T1 being the most downstream unit, and turbine Tn being the leading unit hit by the wind first.

Figure 5.

Schematic of delay compensated nested-loop ESC based wind farm control.

Figure 6 shows the detailed implementation of the power setpoint based DCNLESC, illustrated with the case of turbine i. The dither signal S_i(t), demodulation signals M_i(t) and N_i(t) are designed based on (Oliveira et al., 2017) and (Wu and Li, 2017). The delay compensated algorithm as in Figure 6 requires an estimate of the time delay. D_i_-1,i, the propagation delay from turbine i to turbine i−1 is estimated by cross-correlation between P_i and P_i_-1 (i.e. the power outputs of two adjacent turbines) [40].

{\hat{D}}_{i - 1, i} = \underset{P_{dem}}{\arg} [max {\frac{1}{N} \sum_{k = 1}^{N} {(P_{i} (kT) - {\bar{P}}_{i}) (P_{i - 1} (kT + τ) - {\bar{P}}_{i - 1})}}]

(16)

Figure 6.

Block diagram of delay compensated power setpoint NLESC for turbine i.

Figure 7 shows the block diagram for the correlation-based delay estimation (Wu and Li, 2017). Firstly, the estimation errors are limited using a saturation block. The upper and lower bounds for delay estimate are set at 150 and 0 seconds, respectively. A median filter F_MF(.) with sampling period of 1 second, buffer size and overlap set at 100 and 50 is used to reduce the negative impact of turbulent/gusty wind, followed by a first order low-pass filter F_LP,D(s) with time constant of 100 seconds to smoothen the obtained estimate. The delayed power thus obtained is denoted as P_i,del. Now, the cost function for each ESC is the summation of this delayed power obtained from all downstream turbines.

Figure 7.

Cross-correlation based estimation of wake propagation delay.

Design and simulation for turbine control

The proposed power setpoint based ESC for wind turbine region-2 control is first designed and simulated with a single NREL 5 MW turbine model in SimWindFarm. The main parameters of the turbine model are listed in Table 2.

Table 2.

Main parameters of NREL 5 MW turbine model.

Rated power	5 MW
Cut-in, rated, cut-out wind speed	3 m/s, 11.4 m/s, 25 m/s
Cut-In, rated rotor speed	6.9 rpm, 12.1 rpm
Rotor diameter	126 m

Design of power setpoint based ESC wind turbine control

As described in Section 2.1, the dither frequency of ESC should be selected within the bandwidth of the plant dynamics. The input dynamics essentially reflects the actuator dynamics and rotor inertia. Regarding the feedback of power output, the output dynamic corresponds to sensory dynamics and/or signal conditioning. In this case, the output dynamic is dictated by the averaging operation or low-pass filter used to smooth the power measurement, which is typically needed under turbulent wind. As a simplified treatment, the input and output dynamics are combined as for the parameter estimation from open-loop step test responses under constant wind input. The step response of power output under power demand indicates a first-order dynamics, as shown in Figure 8(a). The estimated time constant from the power demand to the output power ranges from 0.6 to 0.8 s. The largest time constant was adopted for ESC design, that is, 0.8 s. The corresponding bandwidth of the combined input-output dynamic for the power demand-based control is 1.25 rad/s. Based on the estimated bandwidth, the dither frequency is selected as 0.125 rad/s. The Bode plots for the estimated plant dynamic and the filters in ESC design are shown in Figure 8(b).

Figure 8.

Estimation of plant dynamics and ESC design: (a) step responses and fitted responses, and (b) bode plots of plant dynamic and filters in ESC design.

The ESC parameters are listed in Table 3. Note that the input to the ESC was normalized power with respect to the rated power (i.e. 5 MW in this study) for each turbine. The low-pass and high-pass filters were designed with the 2nd-order Butterworth filters. In particular, a first-order low-pass filter with time constant of rotor inertia is placed at the ESC output that is, on power demand signal to avoid sudden change in the power demand.

Table 3.

Design parameters for single-turbine power setpoint ESC.

Parameter	Value
Dither frequency (rad/s)	0.125
Dither amplitude	0.05
Phase compensator (rad)	0.785
Integrator gain	0.1
Anti-windup gain	8
Cut-off frequency for LPF (rad/s)	0.125
Cut-off frequency for HPF (rad/s)	0.125

Simulation of power setpoint ESC based wind turbine control

The ESC controller designed in Section 4.1 is then evaluated with SimWindFarm simulations, with the hub-height mean wind speed of 8m/s, no wind shear, under turbulence intensity (TI) of 2% and 10%, respectively. For all the cases, the ESC is turned on at t = 120 s, initial power demand is set at 0.5 MW and simulated data between 1000s and 3000s are used for evaluations of steady-state performance.

The ESC simulation results for 8 m/s wind with 2% TI are shown in Figure 9. It reveals that the power demand converges towards the available power in wind without any wind speed measurement or estimation. Plot (a) portrays the wind profile, generator torque and rotor speed, while plot (b) depicts the power trajectories as well as the FFT plots of the output and available power.

Figure 9.

Simulation results of power setpoint based ESC for single-turbine operation under 8 m/s wind with 2% TI: (a) wind speed, generator torque and rotor speed, and (b) power trajectories and FFT plots.

The generator torque changes from 4.32 kN·m to a mean value of 19.51 kN·m, and the rotor speed changes from 12.1 rpm to a mean value of 9.75 rpm. The output power has a mean value of 1.82 MW, and the available power (plotted using equation (1)) averages 1.80 MW for the same period. The FFT plot of the output power shows a peak at 0.02 Hz (i.e. the dither frequency) due to the ESC operation. It also reveals that the output power tracks the available wind power quite well for lower frequencies, while not for those above the dither.

The ESC simulation results for 8 m/s wind with 10% TI are shown in Figure 10. In plot(a), the generator torque changes from 4.33 kN·m to a mean value of 18.06 kN·m, and the rotor speed changes from 12.1 rpm to a mean value of 10.28 rpm. The average output power found by the ESC is about 1.79 MW, while the average available wind power is 1.90 MW in the same interval as shown in plot (b). It is not surprising that the power tracking capability becomes degraded under higher turbulence intensity. Again, the FFT plot shows a peak at the 0.02 Hz dither frequency. Compared to the FFT power plot under the 2% TI, the capability of tracking low-frequency available wind power is degraded under the 10% TI.

Figure 10.

Simulation results of power setpoint based ESC for single-turbine operation under 8 m/s wind with 10% TI: (a) wind speed, generator torque and rotor speed, and (b) power trajectories and FFT plots.

For any control strategies developed for enhancing energy capture capability, the impact on fatigue loads must be evaluated. The damage equivalent load (DEL) for the tower-base fore-aft bending moment and shaft torsional moment are also evaluated with data for [1000, 3000] s for all simulations, and the results are summarized in Table 4. The DEL results of ESC simulations are compared with those with the SimWindFarm default controller, with the relative changes calculated as

R e l a t i v e C h a n g e (%) = \frac{D E L w i t h E S C - D E L w i t h S i m W i n d F a r m C o n t r o l l e r}{D E L w i t h S i m W i n d F a r m C o n t r o l l e r} \times 100 %

(17)

Table 4.

Comparison of DEL results of ESC and SimWindFarm controller for single-turbine operation.

	Turbulence intensity %	DEL with SimWindFarm controller (kN-m)	DEL with ESC (kN-m)	Relative change (%)
Shaft torsional moment	2	4.93 × 10⁵	5.21 × 10⁵	+5.83
Shaft torsional moment	10	3.28 × 10⁶	2.25 × 10⁶	−31.31
Tower fore-aft bending moment	2	8.36 × 10⁶	9.32 × 10⁶	+11.45
Tower fore-aft bending moment	10	4.71 × 10⁷	4.98 × 10⁷	+5.73

For 2% TI wind, the DELs of the shaft torsional moment and tower fore-aft bending moment show respective increase of 5.83% and 11.45%. For 10% TI wind, the DEL of shaft torsional moment decreases by 31.31%, while that of tower fore-aft bending moment increases by 5.73%. Apparently, the SimWindFarm controller demonstrates more aggressive control authorities under higher turbulence. The capability of real-time search for the optimum power setpoint by the proposed control strategy does not induce to substantial increase in the structural loads.

Design and simulation of power setpoint DCNLESC wind farm control

The power setpoint based DCNLESC wind farm control is first evaluated for a two-turbine array, and then for a three-turbine array. For both scenarios, all the turbines adopt the NREL 5 MW turbine model, the turbines are aligned along the wind direction (i.e. no yaw error), and the row spacing is 5D (D: turbine diameter). For all the cases, the ESC is turned on at t = 120 s and the initial power demand is set at 0.5 MW for all the turbines. Simulated data between 1000s and 3000s are used for evaluations of the output power and available power. As described in Figure 6, a single-input power setpoint ESC is implemented for turbine T1, while for turbines T2 through Tn, the single-input power setpoint DCNLESC is implemented and a first-order low-pass filter with time constant of rotor inertia was placed at the power demand signal of each turbine to avoid sudden change in the power demand.

Design of power setpoint based NLESC wind farm control with delay compensation

For the NLESC design, the plant dynamic that was estimated for the single turbine operation is used. With delay compensation, the dither frequencies for upstream turbines can be selected by addressing the plant dynamics only, that is no need to compensate for the wake propagation delay. Figure 11 shows the dither frequencies selected as well as the Bode plots of plant dynamics and the relevant filters. The ESC design parameters are listed in Tables 5 and 6 for the two-turbine and three-turbine cases, respectively.

Figure 11.

Dither frequencies and bode plots of plant dynamics and filters for DCNLESC design.

Table 5.

Design parameters of power setpoint based DCNLESC wind farm control for two-turbine case.

Parameter	Values
Parameter	T2	T1
Dither frequency (rad/s)	0.08	0.125
Dither amplitude	0.025	0.005
Phase compensation (rad)	1.0	0.785
Integrator gain	0.07	0.2
Anti-windup gain	100	100
Cut-off frequency for LPF (rad/s)	0.08	0.125
Cut-off frequency for HPF (rad/s)	0.08	0.125

Table 6.

Design parameters for power setpoint based DCNLESC wind farm control for three-turbine case.

Parameter	Values
Parameter	T3	T2	T1
Dither frequency (rad/s)	0.06	0.08	0.125
Dither amplitude	0.025	0.02	0.005
Phase compensation (rad)	1.12	1.0	0.785
Integrator gain	0.05	0.08	0.2
Anti-windup gain	100	100	100
Cut-off frequency for LPF (rad/s)	0.06	0.08	0.125
Cut-off frequency for HPF (rad/s)	0.06	0.08	0.125

Again, the low-pass and high-pass filters are 2nd-order Butterworth filters. The filters are designed with the highest possible bandwidths while still providing satisfactory attenuation of the corresponding dither frequencies and the unwanted harmonics. The input to the DCNLESC is shown in Figure 12. P_d,i refers to the power demand for turbine i, P_r denotes the rated power (i.e., 5 MW for the simulation example in this study), P_iⁿ denotes the normalized output power for DCNLESC at turbine i, P_d,iⁿ denotes the normalized demand power at turbine i, and P_j,del denotes the delayed power output from the downstream turbines. Therefore, the input to the DCNLESC is the sum of power output of turbine i and the delayed output powers of all the downwind turbines with normalization to the rated power.

Figure 12.

Normalized power input for DCNLESC implementation at turbine i.

Simulation results for two-turbine array

With the controller designed in Section 5.1, the two-turbine array is first simulated in SimWindFarm, with the hub-height mean wind speed of 8 m/s, no wind shear, under TI of 2% and 10%. The wind profiles for the two turbines are shown in Figure 13. The wake effect of turbine T2 is clearly revealed, with the decreased wind speed in front of turbine T1. The power trajectories of ESC simulations under 2% and 10% TI are shown in Figure 14(a) and (b), respectively.

Figure 13.

Wind profiles of DCNLESC simulations for the two-turbine case with 8 m/s mean wind speed: (a) 2% TI and, (b) 10% TI.

Figure 14.

Power trajectories of DCNLESC simulations for the two-turbine case with 8 m/s mean wind speed: (a) 2% TI and, (b) 10% TI.

For the 2% TI case, the mean power outputs for T2 and T1 are 1.78 MW and 1.25 MW, respectively, by averaging over the interval of [1000, 3000] s. During the same interval, the available wind powers for T2 and T1 have respective mean values of 1.76 MW and 1.23 MW. For the 10% TI case, the mean power outputs for T2 and T1 are 1.72 MW and 1.18 MW respectively, while the available wind powers for T2 and T1 have respective mean values of 1.84 MW and 1.23 MW. Thus, the results confirm that the power demand by the proposed control strategy can automatically track the available power in the wind. In other words, the DCNLESC can greatly reduce the loss of power without turbine and farm models or wind speed measurements.

The DELs of shaft torsional moment and tower fore-aft bending moment are evaluated for both SimWindFarm controller and the proposed DCNLESC, with the results tabulated in Table 7. Again, these results are obtained with the data of [1000, 3000] s. For the 2% TI case, the DCNLESC leads to the DEL of shaft torsional moment increasing by 23% for T2 while decreasing by 4.36% for T1, and the DEL of tower fore-aft bending moment reducing by 7.06% for T2 and 3.25% for T1. For the 10% TI case, the DCNLESC leads to the DEL of shaft torsional moment increasing by 0.39% for T2 and 10.94% for T1, and the DEL of tower fore-aft bending moment increasing by 13.05% for T2 and 11.91% for T1. The smaller changes in turbine load can be perceived since the dither signal is much slower than the primary modes of tower and shaft dynamics.

Table 7.

DEL for two-turbine array simulations.

	Turbulence intensity %	Turbine no.	DEL with SimWindFarm controller (kNm)	DEL with ESC (kNm)	Relative change (%)
Shaft torsional moment	2	T2	5.22 × 10⁵	6.45 × 10⁵	+23.62
	2	T1	4.68 × 10⁵	4.48 × 10⁵	−4.36
	10	T2	2.69 × 10⁶	2.7 × 10⁶	+0.39
	10	T1	2 × 10⁶	2.21 × 10⁶	+10.94
Tower fore-aft bending moment	2	T2	8.45 × 10⁶	7.85 × 10⁶	−7.06
	2	T1	6.16 × 10⁶	5.95 × 10⁶	−3.25
	10	T2	4.14 × 10⁷	4.67 × 10⁷	+13.05
	10	T1	3.15 × 10⁷	3.52 × 10⁷	+11.91

Simulation results for three-turbine array operation

Then, the power-setpoint DCNLESC wind farm controller is simulated for the three-turbine array, with the same wind profiles as the previous subsection. The wind profiles before individual turbines are shown in Figure 15. The reduction in wind speed is clearly revealed for T2 and T1 due to the wake effect of T3 and T2. The power trajectories of DCNLESC simulations under 2% TI and 10% TI wind conditions are as shown in Figures 16 and 17, respectively. For the 2% TI case, the mean power outputs for T3, T2, and T1 are 1.80 MW, 1.17 MW, and 0.90 MW, respectively, by averaging over the interval of [1000, 3000] s. The available wind powers during the same interval have mean values of 1.77 MW for T3, 1.17 MW for T2 and 0.90 MW for T1, respectively. For the 10% TI case, the mean power outputs for T3, T2, and T1 are 2.05 MW, 1.38 MW, and 1.05 MW, respectively. The available wind powers during the same interval have mean values of 2.1 MW for T3, 1.40 MW for T2 and 1.05 MW for T1, respectively. Therefore, the proposed power setpoint DCNLESC works well in tracking the actual optimum of available power in model-free fashion without wind measurement.

Figure 15.

Wind profiles of DCNLESC simulations for the three-turbine case with 8 m/s mean wind speed: (a) 2% TI and, (b) 10% TI.

Figure 16.

Power trajectories of DCNLESC simulation for three-turbine array: 8 m/s mean wind speed and 2% TI: (a) output power for each turbine, and (b) total power trajectories.

Figure 17.

Power trajectories of DCNLESC simulation for three-turbine array: 8 m/s mean wind speed and 10% TI: (a) output power for each turbine, and (b) total power trajectory.

For the three-turbine case, the DELs of shaft torsional moment and tower fore-aft bending moment are evaluated for both SimWindFarm controller and the proposed DCNLESC, with the results tabulated in Table 8. These results are obtained with the data of [1000, 3000] s. For the 2% TI case, the DCNLESC leads to the DEL of shaft torsional moment increasing by 25.01% for T3 and 3.34% for T2 while decreasing by 0.36% for T1, implying an average 9.33% increase. The DEL of tower fore-aft bending moment increases by 31.29% for T3 and 1.77% for T2, while decreases by 0.86% for T1, thus resulting in an average increase of 10.73%. For the 10% TI case, the DCNLESC leads to the DEL of shaft torsional moment reducing by 5.17% for T3, increasing by 7.14% for T2 and 2.68% for T1 respectively. The DEL of tower fore-aft bending moment increases by 18.14% for T3, 24.08% for T2 and 21.83% for T1. Thus, only reasonable changes in some loads are observed as a trade-off of optimum power tracking by the proposed controller.

Table 8.

DEL for three-turbine array.

	Turbulence intensity %	Turbine no.	DEL with SimWindFarm controller (kNm)	DEL with ESC (kNm)	Relative change (%)
Shaft torsional moment	2	T3	6.16 × 10⁵	7.7 × 10⁵	+25.01
		T2	5.35 × 10⁵	5.52 × 10⁵	+3.34
		T1	4.53 × 10⁵	4.5 × 10⁵	–0.63
	10	T3	2.68 × 10⁶	2.54 × 10⁶	–5.17
		T2	2.01 × 10⁶	2.15 × 10⁶	+7.14
		T1	1.88 × 10⁶	1.93 × 10⁶	+2.68
Tower fore-aft bending moment	2	T3	1.00 × 10⁷	1.31 × 10⁷	+31.29
		T2	7.43 × 10⁶	7.56 × 10⁶	+1.77
		T1	6.42 × 10⁶	6.37 × 10⁶	–0.86
	10	T3	3.93 × 10⁷	4.64 × 10⁷	+18.14
		T2	3.04 × 10⁷	3.78 × 10⁷	+24.08
		T1	2.75 × 10⁷	3.35 × 10⁷	+21.83

Automatic switching between region-2 and region-3 operation for wind farm control

When the lower end of Region-3 operation, that is the mean wind speed is a bit higher than the rated speed, the wind farm control operation is complicated in that some turbine(s) in front rows may work in Region 3, while other row(s) may operate in Region 2. A major concern for operating a downwind turbine is whether the ESC controller can automatically switch between region-2 and region-3 or vice versa. The desirable scenario is that the controller region switching of a downwind turbine can be realized without violating maximization of the total power output for a given free-stream wind. In large wind farms, the wind speed in front of the first turbine in a row would be much higher and less turbulent than the wind speed in front of further downwind turbines due to wake effect. There would surely be situations when the front turbine experiences a wind speed that is in region-3 and downwind turbines experiences wind speed in region 2. Therefore, to evaluate the effectiveness of the proposed strategy under such situation, a simulation is performed with a wind profile with 12 m/s mean speed, for 2% and 10% TI, respectively, as shown in Figure 18. The ESC parameters used in both cases are same as given in Table 6. The wind profiles in front of individual turbines are shown in Figure 18. The power trajectories are shown in Figures 19 and 20 for 2% and 10 % TI, respectively.

Figure 18.

Wind profiles of DCNLESC simulations for the three-turbine array: 12 m/s mean wind speed: (a) 2% TI, and (b) 10% TI.

Figure 19.

Power trajectories of DCNLESC simulation for three-turbine array: 12 m/s mean wind speed and 2% TI: (a) power trajectories for individual turbines, and (b) trajectories of wind farm total power.

Figure 20.

Power trajectories of DCNLESC simulation for three-turbine array: 12 m/s mean wind speed and 10% TI: (a) output power for each turbine, and (b) total power trajectories.

For the 2% TI case, the mean power outputs for T3, T2, and T1 are 4.87 MW, 4.57 MW, and 3.75 MW, respectively, by averaging over the interval of [1000, 3000] s. The available wind powers during the same interval have mean values of 5 MW for T3, 4.72 MW for T2 and 3.83 MW for T1, respectively. The T3 operation is shown to be brought to Region 3 automatically by the power-setpoint ESC, and the output power is kept at the rated power of 5 MW for the rest of simulation period. The T2 operation switches between Region 3 and Region 2 occasionally, while the operation of T1 essentially stays in Region 2.

The proposed controller does not show significant deviation from the SimWindFarm default controller without knowledge of plant model or wind measurement. For the 10% TI case, the mean power outputs for T3, T2, and T1 are 4.73 MW, 4.20 MW, and 3.82 MW, respectively. The available wind powers during the same interval have mean values of 4.74 MW for T3, 4.28 MW for T2 and 3.96 MW for T1, respectively. Figure 18 shows that, for all three turbines, switching between Region 2 and Region 3 occurs frequently, due to the higher fluctuation of wind speed. Nevertheless, the proposed controller realizes smooth inter-region switching like the operation enabled by the SimWindFarm controller. For this case also, the DELs of shaft torsional moment and tower fore-aft bending moment are evaluated for both the SimWindFarm controller and the proposed DCNLESC, with the results tabulated in Table 9. These results are obtained with the data segments within [1000, 3000] seconds.

Table 9.

DEL for three-turbine array with automatic inter-region switching.

	Turbulence intensity %	Turbine no.	DEL with SimWindFarm controller (kNm)	DEL withESC (kNm)	Relative change (%)
Shaft torsional moment	2	T3	8.03 × 10⁵	8.02 × 10⁵	–0.03
		T2	9.18 × 10⁶	9.17 × 10⁶	–0.15
		T1	1.37 × 10⁶	1.29 × 10⁶	–5.9
	10	T3	2.01 × 10⁶	2.01 × 10⁶	+0.11
		T2	2.59 × 10⁶	2.74 × 10⁶	+5.89
		T1	2.99 × 10⁶	2.95 × 10⁶	–1.4
Tower fore-aft bending moment	2	T3	2.55 × 10⁷	2.55 × 10⁷	–0.1
		T2	2.55 × 10⁷	3.03 × 10⁷	+18.53
		T1	1.62 × 10⁷	2.29 × 10⁷	+41.65
	10	T3	7.49 × 10⁷	7.67 × 10⁷	+2.47
		T2	7.17 × 10⁷	7.54 × 10⁷	+5.25
		T1	6.88 × 10⁷	7.08 × 10⁷	+2.84

Conclusion

This paper presents a power-setpoint based ESC controller strategy for Region-2 operation of wind turbine and wind farm operation, which is like the back-calculation framework in anti-windup integral control. For wind farm operation, the power-setpoint ESC strategy is integrated with an existing DCNLESC algorithm. The proposed control strategy is evaluated with SimWindFarm under different wind inputs. Simulation results show that the output power with the proposed controller can track the achievable maximum power output in a model-free fashion without the need for wind measurement, while only reasonable load increase is observed for some situations. In addition, the proposed strategy can realize desirable switching between Region-2 and Region-3 operation for wind farm control without sacrificing real-time optimization of power yield.

A major significance of the proposed control strategy is to realize the benefit of ESC based model-free Region-2 control using the power setpoint as the manipulated input instead of the typical physical actuations such as torque (gain) and/or blade pitch. Therefore, the proposed strategy can be easily applied for retrofitting wind farm control in terms of enhancing energy capture.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Devesh Kumar

Yaoyu Li

Zhongyou Wu

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