Distributed Extremum-Seeking for Wind Farm Power Maximization Using Sliding Mode Control

Abstract

This paper introduces a sliding-mode-based extremum-seeking algorithm aimed at generating optimal set-points of wind turbines in wind farms. A distributed extremum-seeking control is directed to fully utilize the captured wind energy by taking into consideration the wake and aerodynamic properties between wind turbines. The proposed approach is a model-free algorithm. Namely, it is independent of the model selection of the wake interaction between the wind turbines. The proposed distributed scheme consists of two parts. A dynamic consensus algorithm and an extremum-seeking controller based on sliding-mode theory. The distributed consensus algorithm is exploited to estimate the value of the total power produced by a wind farm. Subsequently, sliding-mode extremum-seeking controllers are intended to cooperatively produce optimal set-points for wind turbines within the farm. Scheme performance is tested via extensive simulations under both steady and varying wind speed and directions. The presented distributed scheme is compared with a centralized approach, in which the problem can be seen as a multivariable optimization. The results show that the employed scheme is able to successfully maximize power production in wind farms.

1. Introduction

Grouping wind turbines into wind plants (or farms) is one of the most effective ways to produce energy from wind. It helps to reduce the operational and maintenance cost, usage of land and consequently produce affordable green and renewable energy. However, in a wind farm, the kinetic energy is extracted from the wind flow by wind turbines; therefore, the turbine on the upwind side reduces the wind speed before reaching the wind turbines in the downwind side of the farm. Thus, conventional control and optimization methods that optimized a stand-alone wind turbine system and neglect the aerodynamic wake interaction among wind turbines when maximizing the energy captured by wind farms provide a sub-optimal solution. On the other hand, modeling such an interaction remains a difficult task. This introduces the necessity to employ a model-free optimization method to search for optimal set-points of wind turbines in the wind farm.

Extremum-seeking (ES), also known as self-optimization, control can be categorized as a class of optimal control that concerns the situation when the performance function to be optimized is available for measurement but not exactly known. From another perspective, ES control is an adaptive control method in which the control aim is to drive the system output to track undetermined optimal operating points or settings. The first sliding-mode gradient-free optimization method was proposed by Korovin and Utkin in [1]. The key concept is to design a variable structure control (VSC) so that the system output, i.e., the cost function to be optimized, tracks a monotonic function in time in the direction of the optimal point. Later, the method was generalized as ES control with sliding modes for general nonlinear systems by Özgüner and his coworkers. At first, a periodic search (sinusoidal) function was introduced in [2] and analyzed further in [3]. Then, stability analysis considering different scenarios was studied in [4]. Gradient estimation and a discrete sliding mode ES method were proposed in [5]. In terms of applicability of the method, the sliding-mode-based ES has a wide scope of applications in many areas, for example, automotive [6], robotics [7], and renewable energy systems for both, solar [8] and wind [9]. Recently, different multivariable sliding-mode-based ES control schemes were proposed and analyzed. The first scheme was proposed in [10] where the author proposed a multivariable sliding mode ES with application in Raman optical amplifiers. Another scheme was proposed in [11] for an alternator maximum power point tracking (MPPT). A sequential approach was proposed and analyzed in [12] with application in wind farm power maximization. A common limitation of these schemes is the fact that these multivariable schemes are centralized in nature and the weak coupling assumption. Other than the sliding-mode-based ES control, the literature currently features many design approaches for ES control. A well-known approach is the sinusoidal perturbation method, which was proposed in [13], a nonlinear programming based method [14], Newton-based ES [15], and a Shahshahani gradient ES [16], to name a few.

The literature on the model-free distributed optimization methods for maximizing power extracted from the wind farms shows a multitude of methods. The key idea is to design an algorithm that coordinates and generates the control set-point for an individual wind turbine by considering the wake and aerodynamic properties between the farm’s turbines. Particularly, the goal is to reduce the extracted power from the upwind turbines, allowing the downwind tribunes to extract more energy. To achieve this, the axial induction factor (AIF) is considered the control input for an individual wind turbine. Using recent results in game theory, two learning algorithms were employed for wind farm optimization in [17]. The difference between the two algorithms is with regard to the communication structure and the available information to each decision maker, i.e., controller or agent. In the same context, a cooperative game approach was proposed in [18]. The author used both the yaw offset angles as well as the axial induction factor to optimize the power production. A new data-driven approach was proposed in [19]. The author developed two gradient-based maximum power point tracking (MPPT) schemes that were based on the measurement of the wind speed at the upwind side and the estimation of the average wind speed in the farm [20]. As far as the ES algorithms are concerned, sinusoidal-perturbation-based ES control schemes were proposed in [21,22,23,24] where the authors proposed to introduce a consensus mechanism to exchange information, in real-time, about the value of total performance function among the wind turbines.

In this work, a distributed ES control based on sliding-mode theory is employed to generate the wind turbine axial induction factor and influence the overall performance of the wind farm. The key concept behind this approach is the introduction of a consensus dynamic in order to estimate the average value of the total generated power. Then, the dynamic is suitably interfaced with sliding-mode-based extremum-seeking controllers to obtain the optimal settings. One of the main contributions of this paper is the development of a model-free optimization method via distributed sliding-mode-based ES for wind farm optimization. This paper extends, from the application point of view, the results presented in [25]. Compared with the existing works, the proposed solution is distinguishable in several aspects: First, it is an extension to the centralized multivariable ES control schemes proposed in [12] for wind farm optimization problems. Although the centralized ES approach has proven to be effective in many areas, distributed optimization approaches and decentralized schemes improve the scalability of systems. Second, we compared with the results proposed in [26] to find Nash solution in a non-cooperative game and considering that the Nash solution is not the socially optimal solution, this paper seeks to extend the result using the newly developed cooperative ES control [25] to solve the optimization problem in a cooperative fashion. Third, in contrast to existing works in distributed ES [27,28,29] and wind farm optimization [21,22,24,30] that build on the sinusoidal-perturbation approach, this work employs a sliding-mode-based ES control. The sliding-mode approach has the potential to simplify the implementation and improve the convergence rate. Forth, compared with the work proposed by [9] to optimize a stand-alone wind turbine using a single variable-sliding-mode ES, this work investigates the problem optimizing a wind farm using a multivariable ES. Finally, the implementation section features different simulation scenarios. The proposed scheme was tested under both fixed and varying wind speeds. Then, the proposed distributed scheme is compared with both the Nash-based ES control scheme and the centralized ES approach.

2. Wind Farm Model and Optimization

In this section, we explicate the nature of the problem concerning optimizing a wind farm. In wind farms, the kinetic energy of moving air is extracted by the wind turbines and, consequently, wind is slower after passing through the blades. In other words, the wind speed is reduced in the wake downstream of the wind turbine routers by the wind turbine. This phenomenon is called the wake effect. It reduces the power production of the overall farm due to the decrease in wind speed. By taking into consideration the wake and aerodynamic properties among wind turbines in a farm, a recent result [31] shows that an energy gain of more than ten percent can be achieved, in comparison to the existing National Renewable Energy Laboratory (NREL) controller. This section overviews the wind turbine modeling and control. Then, it previews the wind farm optimization problem.

In this work, a farm consists of an array of n connected wind turbines with the full wake effect considered. As can be seen in Figure 1, the considered configuration is not restrictive, since many wind farms can be grouped to form series-connected turbines.

2.1. Wind Turbine Control

Concerning the power control in wind turbines, three types of control can be found in the literature. Yaw-, pitch-, and torque-based control methods. Yaw-based control, i.e., changing the rotor direction, is a slow method when compared to other control actions and, therefore, it does not influence power production in a wind farm as fast as other actions. On the other hand, pitch and torque control can be related if the problem is formulated in terms of the axial induction factor.

The wind turbine’s axial induction factor is a non-dimensional ratio that expresses the decrease in the wind velocity between the free stream and the rotor plane. It is used as a high-level control input (set-point) to influence the generated power in wind farms (see, for example, ref. [32]). It is a function of two quantities that can be controlled by low-level controllers: The turbine’s blade pitch angle and tip-speed ratio $\lambda$. The tip-speed ratio is a function of the undisturbed wind speed V, rotor’s angular velocity $\omega$, and rotor radius R:

$$\lambda =\frac{\omega R}{V}.$$

(1)

The control strategy of a wind turbine depends on the wind speed. The tip-speed ratio can be adjusted by means of torque control when the wind speed is below the rated value, known as Region-2. On the other hand, blade pitch angle is one of the control inputs when the wind speed is above the rated speed. Details about the wind turbine’s region of operation can be found in [33]. This gives a general idea on how the axial induction factor is manipulated.

Remark 1.

The focus in this work is on the design of a high-level controller that generates set-points that can be tracked by low-level controllers, depending on the turbine specifications.

2.2. Wake Interaction Model

The most challenging part of optimizing a wind farm is modeling the wind velocity profile with the consideration of wake created by multiple wind turbines. A well-studied wake model in the area of wind energy, known as Park model [17,34], is considered in the paper. For each wind turbine i, the effective wind velocity is given by the following:

$$V_i(\theta )=V_{\infty }(1-\delta V_i(\theta ))$$

(2)

where $V_{\infty }$ is the free stream wind speed, $\theta$ is the axial induction factor (to be described later), and $\delta V_i(\theta )$ represents the velocity defects seen by turbine i, which is given by

$$\delta V_i(\theta )=2\sqrt{{\displaystyle \sum _{j<i}}{({\theta }_{j}C\left[j,i\right])}^2}$$

(3)

where $C\in R^{n\times n}$ is a matrix that depends on the layout of the wind farm, (i.e., distance between turbines and the interacting wakes) and given by

$$C\left[j,i\right]={\left(\frac{D_j}{D_j+2k(x_i-x_i)}\right)}^2\frac{A_{j\to i}^{overlap}}{A_i}$$

(4)

where $D_j$ is the diameter of the surface area (disk) produced by the blades of turbine j. $A_i$ is the surface area of disk produced by the blades of the wind turbine i and $A_{j\to i}^{overlap}$ is the region of overlap between the wake produced by the wind turbine j and the surface area produced by the blades of turbine i.

Remark 2.

The proposed ES control scheme is a model-free method, meaning that it is independent of the model selection of the wake interaction. The selected model is used for simulation and results verification.

2.3. Model of Produced Power

For each turbine i, let the axial induction factor, denoted by ${\theta }_i$, be the control parameter, which takes values in $\left[0,\frac{1}{2}\right]$. ${\theta }_i$ is given by

$${\theta }_i=\frac{V_{\infty }-V_{rotor}}{V_{\infty }}$$

(5)

where $V_{rotot}$ is the wind speed at the rotor. The power produced by each turbine can be written in terms of the axial induction factor as

$$J_i({\theta }_1,{\theta }_2,\dots ,{\theta }_i)=\frac{1}{2}{\rho }_{air}{A}_i{C}_p({\theta }_i)V_i{(\theta )}^3,$$

(6)

where:

• ${\rho }_{air}$ is the air’s density.
• $A_i$ is the area swept by blades of turbine i.
• $C_p({\theta }_i)$ is the power efficiency coefficient, which given by

  $$C_p({\theta }_i)=4{\theta }_i{(1-{\theta }_i)}^2,$$

  (7)

From (7), it can be easily verified that (6) is a concave function and attains its maximum value ($C_{p_{max}}=\frac{16}{17}$) when ${\theta }_i=\frac{1}{3}$, which is also known as Betz’s coefficient [32].

2.4. Optimization Problem

The problem concerning optimization of the power generated by a wind turbine can be categorized into two main paths. First, optimizing a stand-alone wind turbine system in which the goal is to maintain the power efficiency coefficient $C_p$ as close to Betz’s coefficient at ${\theta }_i=(\frac{1}{3})$, also known as Maximum Power Point Tracking (MPPT) problem (see, for example, ref. [9]). The other path, which is the subject of this work, is concerning the coordination (cooperation) between individual turbines in order to optimize the total power generated by the Farm and generate a new optimal set values of axial induction factors. Figure 2 presents this fact for the case of two turbines. It is evident that the total power generated by the two turbines is maximized in the event that the upwind turbine operates below its maximum operation point, i.e., ${\theta }_1$ is less than $\frac{1}{3}$.

In this paper, an array of n wind turbines standing in the wake of each other is considered. Each turbine is associated with a controller, and will be refereed to as agents. Each agent i for $i=1,2,\dots ,n$ is able to measure its local cost function $J_i(\theta (t))$ for the turbine i. In addition, agents have the capability to exchange information over a fixed, time-invariant, undirected, and strongly connected graph $G=(V_G,E_G)$. $V_G=\{1,2,\dots ,n\}$ is the vertex set representing the number of agents. $E_G\subset V_G\times V_G$ is the edge set. Considering that it is crucial to accurately model the interaction among the wind turbines in a farm, agent i is assumed to be able to measure the power produced by turbine i rather than estimate the power. The optimizing problem is to maximize the power produced by the farm which is given by

$$P_{Total}({\theta }_1,{\theta }_2,\dots ,{\theta }_n)=\sum _{i=1}^n{J}_i({\theta }_1,{\theta }_2,\dots ,{\theta }_i)$$

(8)

Finally, each cost function $J_i(\theta (t))$ is assumed to have a unique maximizer and to be a concave function. In addition, it is assumed that a unique and finite solution to the problem (8) exists. Note that, from Equations (6) and (7), one can verify that these assumptions are satisfied.

3. Proposed Optimization Strategy

To maximize the energy captured in a wind farm, each agent is required to act in a cooperative manner. In order to achieve cooperation between agents, a consensus protocol is introduced such that all agents can have a good estimate of the total power produced by the farm (8). After that, the goal of each agent is to optimize, in real-time, a multivariable cost function. In the designed coordination scheme, the consensus protocol ensures that each agent receives an estimate of the average of the total cost function via a distributed algorithm. Then, a sliding-mode-based extremum-seeking controller utilizes the received estimate to update the consensus with a new value as a new decision has been taken by the local sliding-mode extremum-seeking controller.

Assumption 1.

Each agent is able to communicate with its neighbors and estimate the average of the total cost function ${\widehat{J}}_i(t)$.

This assumption is reasonable due to the fact that modern wind turbines are equipped with communication equipment to transmit and receive all operational signals. In case of communication loss, the proposed scheme will produce a non-cooperative solution. Under Assumption 1, agent i measures the average of the total cost function, $\frac{1}{n}{\sum }_{i=1}^n{J}_i(\theta (t))$, by running a distributed dynamic consensus algorithm. In particular, the dynamic average consensus protocol proposed by Freeman et al. [35] is utilized in this work. A dynamic average consensus algorithm is used to make sure that the consensus algorithm is, continually, driven by the inputs in real-time such that the average of the changing inputs is tracked. The consensus protocol is given by the following:

$$\begin{array}{cc}\hfill {\dot{\widehat{J}}}_i(t)=-\gamma {\widehat{J}}_i(t)& -\sum _{j\ne i}{a}_{ij}[{\widehat{J}}_i(t)-{\widehat{J}}_j(t)]\hfill \\ \hfill & +\sum _{j\ne i}{b}_{ij}[w_i(t)-w_j(t)]+\gamma u_i(t)\hfill \end{array}$$

(9)

$${\dot{w}}_i(t)=-\sum _{j\ne i}{b}_{ij}[{\widehat{J}}_i(t)-{\widehat{J}}_j(t)]$$

(10)

where ${\widehat{J}}_i(t)$ is the estimated cost function for agent i, $w(t)$ is an auxiliary state, $u_i(t)\in \mathbb{R}$ is the dynamic input, and $\gamma >0$ is a global parameter. The dynamic can be rewritten in compact form as follows:

$$\left[\begin{array}{c}\dot{\widehat{J}}\\ \dot{w}\end{array}\right]=\left[\begin{array}{cc}-\gamma I-L_I& L_P\\ -L_P& 0\end{array}\right]\left[\begin{array}{c}\widehat{J}\\ w\end{array}\right]+\left[\begin{array}{c}\gamma I\\ 0\end{array}\right]u(t)$$

(11)

where $\widehat{J}(t)={[{\widehat{J}}_1(t){\widehat{J}}_2(t)\dots {\widehat{J}}_n(t)]}^T$, and $L_I$ is the integral Laplacian, constructed form the weights $a_{ij}$, and $L_P$ is the proportional Laplacian, constructed form the weights $b_{ij}$. For this approach, $J_i(\theta (t))$ is chosen to be the input to the dynamic $u_i(t)$.

A basic sliding-mode-based extremum-seeking scheme for general nonlinear system is shown in Figure 3. The key concept behind the sliding mode ES is to design a variable structure control input such that the measured performance (cost) function tracks a monotonic function in time towards the optimal points. In the case of maximization problem, the controller follows an increasing function. To design an extremum-seeking controller with sliding mode, a switching function for the $i_{th}$ agent is defined as follows:

$$s_i(t)=J_i(t)-g_i(t).$$

(12)

where $g_i(t)$ is a reference signal that satisfies:

$${\dot{g}}_i(t)=\rho ,g_i(0)=0,\rho >0.$$

(13)

That is, $g_i(t)$ is a monotonic increasing function of time. Next, let the axial induction factor, ${\theta }_i$ be defined as

$$\dot{{\theta }_i}(t)=v_i(t).$$

(14)

where $v_i(t)$ is a variable structure law selected as

$$v_i(t)=-k\mathrm{sgn}\left(sin(\frac{\pi s_i(t)}{{\alpha }_i})\right).$$

(15)

where ${\alpha }_i=2^{i-1}\alpha$ and $k,\alpha$ are positive design parameters. The controller design section is concluded with the following assumption about the controller.

Assumption 2.

The consensus dynamic (11) is much faster than the one of the ${\theta }_i$’s dynamics, that is,

$$|\frac{d}{dt}\widehat{J}(t)|>>|\frac{d}{dt}\theta (t)|.$$

(16)

This assumption ensures a time-scale separation between the consensus dynamic and the ES controller. This assumption is reasonable and can be done by choosing a small k for each ES controller. In other words, the ES controller gain k shall be chosen to be much smaller than the consensus parameters.

4. Controller Analysis

The aim of this section is to show that the output of the proposed scheme converges to a vicinity of points that maximizes the value of $J(\theta (t))$. This will be shown in two main steps. First, the convergence of the consensus algorithm is highlighted. Second, the sliding-mode existence condition need to be studied.

Lemma 1.

Consider the consensus system (11) and let the graph G be a fixed, undirected and strongly connected graph. Then, for any initial conditions, the system states ${[\widehat{J}(t)w(t)]}^T$ remain bounded and each state ${\widehat{J}}_i(t)$ converges exponentially to $\frac{1}{n}{\sum }_{i=1}^n{u}_i(t)$ as $t\to +\infty$.

Proof of Lemma 1.

The proof can be found in [35]. It is worthwhile to mention that this result holds for slowly varying input $u_i(t)$ only. □

Using this lemma and under Assumption 2, we have

$$J_i(t),J_j(t)\to \frac{1}{n}\sum _{i=1}^n{J}_i(\theta (t))\forall i,j\in \{1,2,\dots ,n\}.$$

(17)

Therefore, throughout this section of analyzing the slow dynamics, ${\widehat{J}}_i(\theta (t))=\frac{1}{n}{\sum }_{i=1}^n{J}_i(\theta (t))$ is considered as the measured cost function for each agent i. The condition for the sliding mode to take place is that the deviation from the switching surface $s(t)$ and its derivative $\dot{s}(t)$ should have the opposite sign in the vicinity of the switching surface $s(t)$. In other words, if there exists a constant number $\alpha m,m\in \mathbb{Z}$ such that:

$$(s(t)-\alpha m)\frac{d}{dt}(s(t)-\alpha m)<0,$$

(18)

then, the sliding mode will exist at $s(t)=\alpha m$ (see, for details, [4]). To this end, the following lemma summarizes the sliding-mode existence conditions.

Lemma 2.

For each agent i, with the switching function (12) and ES control input (15). Let the local cost functions $J_i(\theta (t))$ be concave functions and the solution to the problem (8) exist, being unique and finite. Then, the sliding mode-existence condition is ensured and kept on manifold given by $s_i(t)=m\alpha ,m\in \mathbb{Z}$ if the following conditions hold:

• For the given cost function (6), the sum of the absolute values of the partial derivative with respect to ${\theta }_i$ is greater than $\frac{\rho }{k}$, i.e.,

  $$\sum _{i=1}^n|\frac{\partial J(\theta (t))}{\partial {\theta }_i}|>\frac{\rho }{k},\forall i\in \{1,2,\dots ,n\}.$$

  (19)
• The ES controller parameter ${\alpha }_i$ is chosen such that:

  $${\alpha }_i\ne {\alpha }_j\mathrm{and}{\alpha }_i=2^{i-1}\alpha ,\alpha \in {\mathbb{R}}^{+},\forall i,j\in \{1,2,\dots ,n\}$$

  (20)

Proof of Lemma 2.

The proof can be found in [25]. □

Different from the classical sliding mode, the ES controllers are sliding on a number of sliding surfaces while searching for the optimal settings. As long as the sliding mode is active, the switching surface $s(t)$ will be constant; accordingly, the cost function $J(\theta (t))$ will follow the monotonic increasing function towards a vicinity of the optimal solution. After entering the vicinity, which is defined by the region (19), it is possible that either the system output stays inside that vicinity or goes through it. In the latter case, another sliding mode will take place and the system will re-enter the vicinity again on the sliding mode.

5. Implementation and Simulation

The implementation of the proposed scheme is presented for different scenarios of the problem of maximizing power extraction from the wind in a farm. First, the case with fixed wind speed is considered. Then, the case where the wind speed changes over time is implemented. Finally, the decentralized approach is compared with the centralized approach presented in [12]. It is assumed that all turbines are alike and each turbine is equipped with necessary tools for communication and sensing.

More specifically, three wind turbines $n=3$ are considered for simulation, shown in Figure 4. The turbines are assumed to have identical diameter $D=80$ m. Wind turbines are spaced apart by a distance equal to five-times the diameters. That is, the turbines are placed at $\{(0,0),(0,400),(0,800)\}$. The air density ${\rho }_{air}$ is fixed to be equal to 1.225 kg/m³. In addition, each turbine can exchange information with its close neighboring turbines. For this setup, a Laplacian matrix L is given by the following:

$$L=\left[\begin{array}{ccc}1& -1& 0\\ -1& 2& -1\\ 0& -1& 1\end{array}\right]$$

(21)

Scheme gains and parameters are set as $\gamma =2$, $\rho =0.01$, $\alpha =0.01$, $L_I=L_P=10\times L$ and $k=0.02$. In order to satisfy Assumption 2, the ES controller gain k is chosen to be much smaller than the consensus parameters. This will ensure that a consensus is reached in a short time. The implementation scheme is shown in Figure 5. As noted before, the optimal setting of an individual turbine can be determined from (6) and (7). Namely, $\frac{d}{d\theta }{C}_p=0\iff \theta =\frac{1}{3}.$ Since this is an optimal solution for a stand-alone turbine, setting all ${\theta }_i=\frac{1}{3}$ is considered a non-cooperative solution (greedy control strategy). However, if the optimal settings for the total produced power by the farm’s turbines are taken into account, then optimizing the following multivariable function shall be considered:

$$\begin{array}{cc}\hfill P_{Total}& =J_1+J_2+J_3\hfill \\ \hfill & =\frac{1}{2}\rho A(C_p({\theta }_1)V_{\infty }^3+C_p({\theta }_2)V_2{({\theta }_1)}^3+C_p({\theta }_3)V_3{({\theta }_1,{\theta }_2)}^3).\hfill \end{array}$$

(22)

Using a numerical optimizer to maximize the above equation yields an optimal setting given by ${\theta }_1^{*}=0.232,{\theta }_2^{*}=0.208$, and ${\theta }_3^{*}=0.333$. Optimizing the wind farm under a fixed free stream wind speed of $8m/s$ is considered first. Sliding-mode existence is observed in Figure 6. Different from the convergence behavior of a classical first-order sliding mode controller, the ES controllers slide on several sliding surfaces while searching for the optimal settings, i.e., the optimization process, until reaching the proximity of the optimal settings. Once the maximum output is reached, the function gradient will be small enough such that the sliding-mode existence condition (19) will be no longer valid, and hence $s(t)$ will be decreasing. The idea of sliding on several surfaces can also be observed in the convergence behavior of ${\theta }_1,{\theta }_2$ shown in Figure 7a. The initial values were chosen such that individual turbines are optimal, i.e., ${\theta }_i=\frac{1}{3},\forall i=1,2,3$. ${\theta }_3$ converges to $\frac{1}{3}$, as demonstrated in Figure 7b, since the farm is not affected by the last turbine. In selecting the controller parameters, different frequencies are utilized, by selecting different ${\alpha }_i$, in accordance to Lemma 2. Figure 8 shows the power generated by each wind turbine (WT) along with the corresponding axial induction factor. One can observe that the ES controller set the upwind turbine (WT1) to operate at lower capacity, allowing for high wind speed to pass to the downwind turbines. Two control strategies are compared in Figure 9. Namely, the proposed decentralized control is compared with the centralized scheme proposed in [12]. Starting from an optimal setting for individual turbines, the proposed controllers are able to improve the total captured energy by the farm. The power output significantly increased and reached the proximity of the maximum value. Since this is a derivative-free algorithm, the ES controllers can only identify a vicinity of the optimal solution and, therefore, the outputs have noticeable oscillation around the maximum value.

Remark 3.

The observed oscillation is natural, yet it can be lowered by reducing both parameters, ${\alpha }_i$ and k. However, it will result in a slow convergence, see [4] for details.

It should be noted that there is also a trade-off between the scheme’s speed of convergence and the speed of the consensus dynamics. Moreover, it can be observed that the centralized approach provides a more accurate control input as well as a faster convergence, especially for small-scale problems. However, besides the reliability issues associated with the centralized schemes, recall that the decentralized method shows two major advantages in comparison with the centralized architecture. It is computationally less expensive, and in practice, easier to implement. Maximizing the total generated power using the proposed control scheme is compared with the total power generated by applying a non-cooperative control [26] in Figure 10 for a different wind direction. Assuming a fixed yaw angle, the wind direction affects the overlap area between the wake produced by the upwind wind turbine and the (surface area) disk produced by the blades of the downwind wind turbine. A 100 percent overlapping corresponds to the setup in Figure 4, where a zero percent overlapping means that the air direction is perpendicular to the turbines. What is interesting in this comparison is that control strategies only coincide under zero overlapping. Finally, the performance of the designed controller under a time-varying cost function is investigated. In this setup, the wind speed is varied periodically over time to test the robustness of the scheme. Figure 11 shows that the controller is able to track the optimal performance and maintain a maximum power production in the farm.

6. Conclusions

The proposed optimization scheme attempted to develop a model-free approach towards maximizing power captured by a wind turbine farm, namely, through the adoption of a distributed sliding-mode extremum-seeking control scheme. In contrast to conventional control methods that typically optimize individual turbine performance, the proposed scheme maximizes overall captured power in a cooperative fashion, thereby accounting for turbine-to-turbine interactions and ambient conditions in order to reach an optimal set-point for each of the turbines available within the farm. The strategy of introducing a consensus algorithm as a dynamic mechanism to exchange information about the power produced by the farm has been considered. The problem is then reduced to a multivariable extremum-seeking control problem, which is solved by a sliding-mode extremum-seeking controller for each turbine. The stability and convergence of the proposed scheme have been discussed. Performance verification is conducted through a series of experiments and is compared versus several control architecture including centralized as well as non-cooperative schemes under a variety of wind conditions. While this real-time optimization using sliding-mode control highlights the potential of increasing captured power under steady-state conditions, future investigations should aim to further its extension into more adverse conditions in which performance might be disrupted by the presence of measurement noise, time-delay, and poor communication networks.