Maximizing the Absorbed Power of A Point Absorber Using An FA-Based Optimized Model Predictive Control

This paper presents an extended model predictive controller for maximizing the absorbed power of a point absorber wave energy converter. Owing to the great influence of controller parameters upon the absorbed power, the optimization of these parameters is carried out for the first time by a firefly algorithm (FA). Error, the difference between output velocity of buoy and input wave speed which leads to power maximization in the optimized MPC is compared with the classical MPC. Simulation results indicate that given the high accuracy and acceptable speed of the algorithm, it can adjust the parameters of the controller to the point where system error decreased effectively and the absorbed energy increased about 4 MW.


Introduction
Wave energy is one of the renewable and predictive energies with high-energy potentials and low environmental effects. In most countries of the world, electricity production by wave energy is considered as a main and promising source. On the other hand, the majority of remote areas, such as islands, are far from electricity networks. Therefore, providing powers to such areas is often costly and incompatible with the environment. In such cases, using a renewable energy, such as wave energy, is the best selection (Fadaeenejad et al., 2014). Among other benefits of the wave energy we can refer to: • The extraction of electricity from wave energy overnight can be continuously performed and waves can travel long distances with very low energy losses (Fadaeenejad et al., 2014).
• Wave energy is an indirect form of solar energy that in addition to its purity and renewable state, has more energy flux (about 2-3 kW/m), compared with solar (about 0.1-0.3 kW/m 2 ) and wind (~0.5 kW/m 2 ) energies .
• The most important advantage of the wave energy, as a renewable energy, is its ability to reduce the distribution of pollutants. Hence, the production of 300 kg of carbon dioxide per megawatt-hour can be prevented by using the wave energy (Fadaeenejad et al., 2014).
Over the past few years, the extraction of wave energy has been one of the most challenging issues in designing and controlling wave converters. A wave energy converter is the device used for the extraction of mechanical energy of sea waves and turning them into electrical wave (Fadaeenejad et al., 2014;Jama et al., 2013). According to Fig. 1, there is a type of converter called point absorber, which has attracted the attention of business centers. Some of the benefits of this type of converter are of low costs and complications, compared with other converters (Falnes, 2002). Compared to the length of waves, these converters have significantly low sizes (López et al., 2013). In this study, wave energy converter of point absorber is applied. One of the most important issues of wave energy producers is the control of wave converters to obtain the maximum power. Therefore, the design of controllers, which can control the fluctuations of these converters during wave collision is very important (Fusco and Ringwood, 2013). Various methods have been proposed by different research groups to maximize the energy extracted from wave China Ocean Eng., 2018, Vol. 32, No. 6, P. 696-705 DOI: https://doi.org/10.1007/s13344-018-0071-4, ISSN 0890-5487 http://www.chinaoceanengin.cn/ E-mail: coe@nhri.cn energy converters using different controllers. In Jama et al. (2012), an inactive control method with the use of self-tunable fuzzy logic controller along with particle swarm optimization algorithm was proposed to maximize the absorbed energy for controlling a wave energy converter. The study showed that the self-tunable fuzzy logic controller along with particle swarm optimization algorithm has a better performance, compared with the simple fuzzy controller. According to Schoen et al. (2008), the fuzzy logic controller (FLC) has been emphasized in the industry due to its simplicity and ability to be used in nonlinear systems. This type of controller is independent of the mathematical model of the system. However, one of the defects of fuzzy controllers is their creation based on human experiences and application for great uncertainties, which is suitable for systems with chaotic dynamics and variable with time.
During the early 1980s, Budal and Falnes showed that one of the conditions to achieve maximum power absorption in a wave energy converter was keeping the speed and wave excitation force in the same phase, which leads to the introduction of a specific type of phase control, called latching control (Babarit and Clément, 2006;Kara, 2010). Henriques et al. (2016) applied a receding horizon of latching control on a wave energy converter of oscillating water column (OWC) type, validated by hardware-in-the-loop setup. One of the limits of this type of controller is implementing the actual time for irregular waves since a long duration is required to predict wave behaviors. On the other hand, there is a great load on latching controller. Babarit et al. (2009) proposed two methods of pseudo-continuous and declutching controls to control the energy imposed by hydraulic power take-off (PTO) system. Pseudo-continuous control methods can cause the complexity of PTO by a series of discrete values. Thus, the declutching method can be used as a simple technique to control this hydraulic system. The main benefit of the declutching method is the need for only one bypass valve, which reduces the complexity of the process. Garrido et al. (2012) applied a sliding mode controller to adjust the generator's slide, providing an appropriate high-frequency switching control method for nonlinear systems in the presence of uncertainty.  used a sliding mode controller to control the wave energy converter of point absorber type. Although this controller is resistant against noise, disturbance, and uncertainty, it has a stable and linear sliding surface and chattering issues. Castro et al. (2014) applied artificial neural networks (ANN) to predict the power obtained from nearshore waves. The results of this study indicated that the ANN model had the ability to accurately predict wave behaviors. However, one of the disadvantages of this network is the high volume of information processing, which leads to the use of this method in systems with more complicated dynamics.
Today, the control of industrial processes and the selec-tion of proper methods in this field are of paramount importance. Despite the conventional use of controllers, such as proportional integral derivative (PID), in the industry, it should be noted that complicated industrial processes include a wide range of complex behaviors, which limits the application of such controllers. Therefore, attention has been paid to the use of effective control methods, including model predictive control, to reduce the costs and to prevent damage to the device. Model predictive control (MPC) has been assessed by universities and industries since the early 1980s, leading to its recognition as one of the most advanced technologies, which predicts the future amounts of a system and promotes its performance in industrial applications using the dynamic process modeling (Yakub and Mori, 2014). Other benefits of the predictive controller are the easy implementation of control law, easy deployment of multivariate systems, natural compensation for the effect of disturbance and measurability (Li and Belmont, 2014). In this article, the model predictive control has been used as a control strategy for a wave energy converter of point absorber. While this controller is suitable for most of the cases, it faces some problems in fast dynamics due to a high volume of calculations. In the present research, it is aimed to use the MPC for fast dynamics and to improve its performance for obtaining optimal parameters (i.e., prediction and control horizon), which has been performed for the first time by the firefly algorithm (FA) in this article and helps to reduce error, increase the absorbed energy, and enhance WEC performance. The general control diagram of the proposed method has been presented in Fig. 2. The hydrodynamic structure and equations of state space of a wave energy converter of point absorber are analyzed in the second section. The operation method of MPC on point absorber is assessed in the third section of this article, and firefly algorithm is introduced in the fourth section. Also, simulation results and recommendations for future improvement are provided in the fifth section.

Point absorber
Generally, point absorbers are defined as active wave oscillators with extremely small sizes, compared with the input wavelength. These absorbers are limited to a degree of freedom to move up and down. The general structure of a point absorber is shown in Fig. 3, which contains a buoy, tether, permanent magnet linear generator (PMLG) (power transfer section PTO) and restoring spring units. The vertical movement of the buoy leads to the movement of PTO power transfer system and stimulating voltage in stator circuit of the device leads to power production. After that, the power obtained from waves is changed into an electrical power by PTO. In the final stage, electrical power converter applies this power on the network. End-stop spring has been installed to prevent the extreme movements of the buoy and damage to the device (Wahyudie et al., 2015).

Hydrodynamic model for a point absorber
The mathematical model of a point absorber is complex and nonlinear. In most cases, a simple and linear dynamic model is obtained under specific conditions and assumptions. Forces applied to the converter are generally divided into two categories: (1) Forces applied to the buoy by fluids surrounding the buoy, known as hydraulic forces, and (2) Forces applied to the buoy by other components of the converter, such as PTO and spring forces. The governing equation of motion of the WEC's buoy can be constructed using Newton's second law as (Guo et al., 2018): where, is the total physical mass of the buoy. Variable of is the vertical acceleration of the moving buoy, is the wave excitation force, is the wave radiation force, is the hydrostatic buoyancy force, is the mechanical and hydrodynamic force losses, is the restoring force and is the mechanical control force applied by PMLG generator (PTO section) on the buoy. Eq. (1) can be replaced by a linear equation, which accurately presents the system model (Guo et al., 2018). All of the forces in Eq. (1) are expressed concerning the linear functions of buoy movement (displacement, velocity and acceleration of buoy), with the exception of and . Therefore, wave excitation force and the force applied to the buoy by PTO are regarded as system inputs. Radiation force of is a causal force and modeled as follows: where is the mass added in infinite frequency, meaning that a hydrodynamic constant mass is added to the physical mass of the buoy. In Eq. (2), convolution statement of is estimated by a fourth-order invariant linear model, and is defined as the velocity of buoy movement.
where is the derivative of the approximate radiation state, and , , and are the matrices of radiation state, input, and output. It should be noted that these matrices are determined by solving the radiation issue for the buoy using a numerical hydrodynamic tool known as WAMIT (Zhan and Li, 2018). The Excitation force is a noncausal hydrodynamic force regarded as the main stimulating force of wave converter system and defined as a force that shows the relationship between random force and the buoy and obtained by a sudden collision of the buoy by waves. The linear approximation of wave excitation force is expressed as follows: In Eq. (4), is the convolution of the kernel of excitation or impulse response function, and is the height of the estimated wave (Pastor and Liu, 2014). According to Eq. (5), the buoyancy force of is the force originated from the spring-like property of liquids surrounding the buoy and reaction force applied to the buoy, in which is a constant and has been approximated. The linear buoyancy force is regarded as a function of the vertical displacement of buoy . The mathematical model of buoyancy force is defined as follows: where is the water density, is the gravity acceleration of earth, is the buoyancy stiffness factor, and is the displacement of the buoy. Another force is the saving force of , which is applied to the converter by the restoring spring unit under the generator and its simulative parts are known as the translators. This is a linear force and is modeled as follows: where is the spring constant coefficient. There are other forces leading to power loss.
is non-linear and can even be variable with time. Nevertheless, it is considered as the linear function of buoy velocity due to its simplicity (Wahyudie et al., 2015).
R loss where is the loss resistance. Model of a WEC can be expressed in the frequency domain as Eq. (8): where , and represent the Fourier transform of the velocity , excitation force , control force , respectively and is the frequency domain. In a way that all the properties of the system are incorporated in the intrinsic mechanical impedance of defined as: where is the radiation resistance (real and even) and is the frequency-dependent added mass, often replaced by its high-frequency asymptote (Ringwood et al., 2014). f The Excitation force , which is known as a disturbance, is the result of the incident wave elevation . Its transfer function which is represented by in the frequency domain is defined in Eq. (10), and it is a non-casual transfer function. In Eq. (11), is also the Fourier transform of the wave elevation. The excitation force in frequency domain is formulated as follows (Milani and Moghaddam, 2017): The force reaching from wave to PTO is as follows: F where is the amount of impedance of power extraction system and is regarded as the velocity. Given Eq. (8), the relationship between velocity and load is expressed below: (12) Using the control law Eq. and considering Eq. (13), the PTO force can be written as follows: F The average power over the time is the product of the control force and the system velocity P Eq. (15) can be expressed in the frequency domain using Parseval's theorem. The average power absorbed by the PTO system in the frequency domain is as follows: P indicates the complex-conjugate operation. Note that Parseval's theorem can only be applied if the force and velocity signals are supposed to be zero outside the time interval. The integrand of Eq. (16) is proportional to the average power absorbed at each frequency P Therefore, regarding Eqs. (13), (14) and (17), the ab-sorbed power can be estimated through Eq. (17): P where is the intrinsic mechanical impedance. If is maximized, in Fig. 4, optimized will be equal to the complex-conjugate of the intrinsic impedance and the maximum power will transfer from waves to the load. This is independent of in the wave energy, optimal control law in Eq. (19) is called 'complexconjugate' as the load impedance is required to produce the optimal force (Fusco and Ringwood, 2012). P 2.2 State space model of a point absorber After the combination of all linear forces and turning them into state space form, the general form of state space equation for a wave converter is as follows: ; y =ż(t) = CX(t). (20) According to Eq. (20), is the vector of state variables, is the buoy displacement, is the velocity of buoy movement, and . State matrices dimensions are in the form of , , and . Matrices of A, B and C are defined as follows: And radiation matrices are estimated as follows: In Eq. (20), the system has two inputs; the first, defined by is the control force applied by PTO section and can be designed and changed and has an effect on obtaining the maximum power. The second input is the wave excitation force of , which is expressed in the form of disturbance and cannot be changed or manipulated. These two forces are combined for simplicity and turning the system into a single-input and single-output (SISO) system (Wahyudie et al., 2015).

Model predictive control
Model predictive control refers to a set of computer control algorithms, which directly use the dynamic model of the system to predict its future behavior (Scattolini, 2009). The base structure of MPC has been presented in Fig. 5. Due to Fig. 5, the performance of the MPC is in a way that a model of the process is used to predict its behavior in the future. The predicted output is compared with the reference input, and the obtained error is minimized by a cost function and its constraints. Moreover, control signal changes must be minimized in this controller. MPCs have some features in common, such as the explicit use of process model to predict the output of the system in the future, the estimation of the control signal based on the minimization of a cost function, and the application of the first sample of the calculated control signal (based on a horizon in the target function) (Rossiter, 2003).
3.1 Discrete-time model predictive control using state space model With the ability of MPC to predict state variables and future behaviors of a system, this controller has been used to control WEC performance and maximize absorbed energy. The procedure of prediction state variables and future output variables using Discrete-Time Model Predictive Control with its framework is demonstrated in this part.
With the discrete-time state space equations of an SISO system, we will have: u y According to Eqs. (22) and (23), is the input variable which is consisted of PTO force and excitation force, is the output variable of the process that is the buoy velocity and is the vector of state variables, the dimension's size of which is . Also according to Eqs. (21), (22) and (23), , and are considered. Given the need for plant input information to predict and control, it has been assumed based on the principle of reducing the control horizon and for simplicity that the input of has no effect on output (at the same time of k). is considered (Wang, 2009).

Prediction of states and output variables
Assuming that is the time of the current sampling and is always a positive number, it can be concluded that the state variable of contains the current system information. The future control vector is considered as follows: where is the control horizon and indicates the number of parameters used to obtain the future control path. With the current information of the system, future state variables for samples are predicted. is the prediction horizon length. Future state variables are defined as follows: where, denotes the predicted state variables at the time of considering the current information of . It should be noted that the amount of control horizon of must be less than or equal to the prediction horizon of . According to the state space model of the system and matrices of , , and , future state variables can be calculated according to a series of control parameters, as follows: And with regard to the predicted state variables, the future output variables are estimated, as follows: And the output vector can be defined as Eq. (28): With respect to the mentioned equations, it is observed that the predicted variables are estimated based on the information of current variables of and and the future control path of in a way that , (Wang, 2009). Eventually, can be written as: In a way that According to Eq. (29), , , and are the state variables matrices, and and are the prediction and control horizon, respectively.

Optimization in discrete-time model predictive control
With the reference signal of , the aim is to approach the predicted output as much as possible to the reference signal. As it is mentioned in Eq. (30), the goal of the first statement in the cost function is to minimize the error between the predicted output and the reference signal, whereas the aim of the second statement of the cost function is to find the control vector of which minimizes the cost function.
In Eq. (30), is defined. is a diagonal matrix in the form of , in a way that can be an adjustable parameter. It should be noted that when is selected, the aim is to minimize the first statement, meaning error, and when is big as selected, the effect of the first statement is reduced. In order to find the optimal control path, the cost function in Eq. (30) must be minimized. By replacing Eq. (29) in Eq. (30), the cost function is rewritten in the form of Eq. (31).
Now, if the first derivative of cost function is estimated relative to ∆u and is considered equal to zero and assuming that the reverse matrix of exists, the optimal control signal is obtained in the form of Eq. (32): After obtaining the optimal control vector, the first vector is applied to WEC and the considered output is obtained.
In the next step, all of these stages are repeated (Wang, 2009). To improve the MPC performance, two important parameters of control horizon and prediction horizon are optimized using the firefly algorithm, which has great performance in terms of convergence rate and global optima achievement (Milani and Moghaddam, 2015).

I r
Firefly algorithm is a metaheuristic and inspired by nature, which was first introduced by Yang in 2008. This algorithm is designed based on the behavior of fireflies based on their reflected light. The cost function of this algorithm can be easily fit to the brightness level of fireflies. It is always assumed that the attractiveness of a firefly is determined by its level of brightness, which is related to the cost function. Moreover, the intensity of light has a reverse relationship with inverse distance law. In the simplest mode, the intensity of light of is changed continuously and exponentially with the distance of (Yang, 2010). The mathematical expression of changes in the intensity of light is presented in Eq. (33): Given Eq. (33), the primary intensity of light is in the distance of , and is the light absorption coefficient. Currently, the attractiveness level of fireflies can be defined using the mentioned concepts. Hence, the attractiveness of a firefly is obtained through Eq. (34): where is the primary attractiveness in , is the light absorption coefficient and is the distance between two fireflies. The distance between two fireflies of and is defined in the form of the equation below: r i j Nevertheless, the estimation of distance can be carried out using other criteria. The movement of firefly , which is the absorption of firefly , is calculated by Eq. (36): i In Eq. (36), the first statement is the current location of firefly, the second statement is related to the low attraction of firefly , which regarding the evaluation of intensity of light, the firefly with low light moves toward the firefly with more brightness. The third statement is a random movement of firefly (in cases where there is no two homogenous fireflies in terms of light attraction) (Yang, 2014). In this article, the cost function, which must be minimized, is the integral of absolute magnitude of the error. Thus, the error is the difference between the output of buoy and buoy velocity and is calculated with Eq. (37): Flowchart of firefly algorithm is defined with regard to Negar RAHIMI, Reihaneh Kardehi MOGHADDAM China Ocean Eng., 2018, Vol. 32, No. 6, P. 696-705 (N c ) (N P ) α β γ i j i j Ali (2015) and the expressed optimization issue in this article in Fig. 6. As can be seen in Fig. 6, the optimization process starts with random production of the particles (fireflies), called control horizon and prediction horizon , within the specified boundaries and setting parameters of , , and . Then, the produced new particles are evaluated by updating the MPC and checking the WEC system response through computing a cost function in Eq. (37). Afterwards, the fireflies with higher IAE move toward the fireflies with lower IAE so as to determine the attraction between the fireflies, which is caused by the cost function for each firefly and . The position of each particle is updated when firefly is attracted to another more attractive firefly , which is determined based on Eq. (36). The procedure continues until reaching the maximum iteration and obtaining an optimum set of MPC parameters (Jama et al., 2014).

Results
In this section, the proposed control system is analyzed in MATLAB software using JONSWAP wave spectrum (for irregular waves) based on nominal period , determined wave height and frequency . Dynamic equations of wave energy converter of point absorber type used in Section 2.2, which is explained before, are regarded as a system model for MPC. Furthermore, the required parameters in MATLAB simulation for design of WEC according to the reference (Wahyudie et al., 2015) are expressed in Table 1. The height of JONSWAP wave in the time domain is estimated according to Eq. (38): According to Eq. (38), is the desired wave spectrum in the frequency area of , and is con- The desired wave spectrum is defined in the form of Eq. (39) (Veritas, 2011):  After the estimation of input wave height, wave excitation force must be obtained. With regard to Eq. (4), it is observed that wave excitation force can be estimated using convolution of wave height and impulse response function . The important point is that the value of can be obtained through second-to five-order approximation of an exponential function using a digitizer software and cftool in MATLAB. Impulse response function with five-order approximation is shown in Fig. 8.
The excitation force f ex of the input wave is shown in Fig. 9. Reference signal used in this article is obtained applying the reference (Fusco and Ringwood, 2012) approximately according to Eq. (40):  (ż) excitation force obtained in the previous stage. The main goal is that the point absorber is controlled by the predictive controller in a way that first, it can extract maximum output energy and second, the speed of wave energy converter, which is in fact the desired system output , can follow the reference speed (input wave speed).

5.1
Simulation results without using model predictive control In this section, the results obtained by point absorber without the presence of controller are expressed. As mentioned earlier, control inputs are wave excitation force and control force . When there is no controller applied to the system, the only point absorber input is wave excitation force. In this state, as shown in Fig. 10, output velocity of buoy and input wave speed are not in phase, and the output does not follow the reference signal. Therefore, with regard to Fig. 11, mean force value, which can be extracted from buoy, is about 3 kW.

Simulation results in the presence of model predictive control
In this section, the results of point absorber in the pres-ence of MPC are expressed. If the control signal obtained from the predictive controller, which is in fact the input force by power take-off (PTO), is in the form of Fig. 12 and power obtained by buoy is estimated according to Eq. (41), buoy output, which is its speed, can properly follow the input speed after the use of controller, which itself leads to the maximum power transfer, as seen in Fig. 13. In Fig. 14, medium power obtained without a controller is compared with the power obtained after the use of con-       troller, observing that the system can receive a power of about 4.376 MW in the presence of controller.
5.3 Optimization results obtained by using firefly algorithm Given the need for parameters optimization to increase the performance of MPC, this process is carried out by firefly algorithm for the first time. The ranges of effective parameters in this algorithm including light absorption coefficient (γ), random vector coefficient (α), and primary attractiveness (β) are defined according to Table 2. Because of the effectiveness of two important parameters of control horizon and prediction horizon in determining the appropriate performance of controller, determining their optimal value can lead to their improved performance and decreased error.
(IAE = 23.14) N There is a difference between the optimal values of each effective parameter in MPC and their non-optimal amount. In addition, any change in each of these parameters has an impact on the performance of controller. Generally, elevated prediction horizon leads to decreased speed of the system and increased volume of calculations. Moreover, the selection of low control horizon leads to solving a smaller number of variables in each control range of controlling equations, causing an increase in the speed of calculations. Optimal values of prediction and control horizons are shown in Figs. 15 and 16. It is observed that we have reached the minimum cost function value as and . At the end and with regard to Fig. 17, error obtained by optimal values of parameters of the predictive controller with the use of firefly algorithm is compared with error obtained by classical MPC without using this algorithm. This comparison indicates that error obtained by optimal values of parameters along with the use of firefly algorithm is about 0.4, which decreased significantly compared with the error obtained by the classical MPC without the use of firefly algorithm (about one).
The extracted power in this article was also compared with references (Li and Belmont, 2014;Cretel et al., 2011). It is observed that the amount of manufactured power in this paper is equal to 4.376 MW, which shows an increase compared with the manufactured powers of two other articles, which were reported to be about 3.5 MW and 4 MW, respectively. Despite the system model of this paper is very similar to the provided references, but since the firefly algorithm is used in this study to optimize the MPC paramet-ers, we have an increased power value.

Conclusion
In this article, model predictive control has been presented to control a type of wave converter called point absorber. The goal is to maximize the power extracted from point absorber by tracking the reference speed. In addition, effective parameters of MPC have been optimized by firefly algorithm to improve the performance of the controller. Two parameters of prediction horizon and control horizon play a significant role in the increase of calculation volume and system velocity. Therefore, firefly algorithm can optimize these parameters, causing an improvement in the performance of the controller and a decrease in optimization error. At the end and by seeing the simulation results of this article, it was shown that MPC can easily maximize the power attracted by point absorber. It is recommended that Laguerre functions and intelligent combined algorithms can be used to decrease the volume of calculations in the predictive controller and increase the speed and accuracy of the   system, respectively. It is suggested that an array of wave energy converters with various intervals should be evaluated to increase the energy obtained from the converter in future studies.