Power maximization of a point absorber wave energy converter using improved model predictive control

This paper considers controlling and maximizing the absorbed power of wave energy converters for irregular waves. With respect to physical constraints of the system, a model predictive control is applied. Irregular waves’ behavior is predicted by Kalman filter method. Owing to the great influence of controller parameters on the absorbed power, these parameters are optimized by imperialist competitive algorithm. The results illustrate the method’s efficiency in maximizing the extracted power in the presence of unknown excitation force which should be predicted by Kalman filter.


Introduction
Wave energy is produced by wind energy and the main source of wind is solar energy. This energy conversion makes wave power be great energy source with enormous potential (Ringwood et al., 2014). Energy harvesting technologies have significant impact on taking advantage of wave power. Wave energy converters (WECs) are designed to harvest the maximum energy from ocean waves. The best way for decreasing the consumption of fossil fuel is using this absorbed power. The big challenge, in order to use the extracted power, is controlling a WEC, while guaranteeing the best performance with reducing the risk of damage.
This work proposes the control aspect of WEC by focusing on model predictive control. Also with the aim of maximizing the absorbed power, controller parameters are optimized by revolutionary algorithm. Wave prediction is done by Kalman filter.
There are several control strategies which are applied to WEC aiming at improving its efficiency, such as complex conjugate control (Fusco and Ringwood, 2013), latching control (Kara, 2010;Babarit and Clément, 2006) and declutching control (Babarit et al., 2009). Schoen et al. (2011) compared fuzzy and robust controllers and showed that the result of applying robust controller can minimize the error created by unknown parameters more effectively and extract more power rather than fuzzy controller. Model Predictive Control (MPC) is another controller with specific characteristics that make it really useful. MPC is an online constraint optimization technique, which is implemented in this paper. MPC has been applied on WEC by Cretel et al. (2011), Brekken (2011), and Li and Belmont (2014), and it was also used for an array of WECs by Li and Belmont (2014b). In addition to the necessity of using an appropriate controller, the wave excitation force prediction is required. Multiple techniques have been explored to predict the excitation force. Ringwood (2009a, 2009b) illustrated that how Autoregressive and Autoregressive Moving Average models allow the access to prediction into the future. The comparison between autoregressive method and neural network has been done (Fusco and Ringwood, 2009a). Recently Kalman filter has been employed to make prediction for linear and nonlinear systems due to its simplicity, robustness and suitability for real time implementations as shown in Ringwood et al. (2014); wave prediction for two body self-reacting devices has been done by Kalman filter. This method was tried by Hals et al. (2011) and Fusco and Ringwood (2013) for a simple WEC. Prediction horizon, control horizon and sampling time are basic parameters in MPC, which affect the controller performance dramatically. Finding their optimized values is a new method to improve the controller performance. In this paper the optimization is done by revolutionary algorithm and then the improved MPC is applied to a WEC, which has been done for the first time in this paper.
The structure of this paper is as follows. WEC model is addressed in Section 2. Wave prediction method using Kal-man filter and the model of incoming wave are illustrated in Sections 3 and 4, respectively. Section 5 focuses on the MPC performance on WEC and illustrates the necessity of optimization of MPC parameters. Imperialist competitive algorithm as an optimization method is explained in Section 6. Finally simulation results are shown in Section 7.

Wave energy converter model
In this part, the motion equation and forces acting on WEC are illustrated. Details of the state space model of WEC are shown. Intended WEC is shown in Fig. 1, which has a floating body moves in heave. Power take off (PTO) converts the body motion into electricity. The system motion equation is expressed as follows: where M m is the body mass; x(t) and v(t) are its related position and movement velocity, respectively; z(t) is the Kernel function; k f is the friction constant of a modeling and k s is the stiffness coefficient; is the force of the PTO part, which is applied to the WEC. Excitation force , which is known as disturbance, is the result of the incident wave elevation . Its transfer function in the frequency domain is defined in Eqs.
(2) and (3) which is a non-casual transfer function. In these relations, and are the Fourier transform of the wave elevation and excitation force , and the transfer function is represented by . The output is the excitation force in frequency domain (Fusco and Ringwood, 2013): (2) To cope with the problem of non-causality, we need to predict the motion of the body that leads to predict the wave elevation. The WEC model and the relation between the velocity and forces are defined below: where Z i denotes the mechanical impedance and V is the movement velocity of a WEC in frequency domain. Based on Falnes (1980), the total force f t acting on the WEC is where Z is a square matrix which includes real part R, the radiation resistance element, and imaginary part X, the radiation reactance element. And V is the movement velocity. The absorbed power by a WEC is V opt where * denotes complex conjugate. The maximum absorbed power for the optimum velocity is (Falnes, 2002) In the general case, the absorbed power is (Li and Belmont, 2014a) PTO power is in the reverse direction of the buoy movement and applied to the buoy as shown in Fig. 1, so this relation is given minus. The extracted energy over a period [0, T] is The state space model of a WEC is In this relation PTO force is the input control and is the time derivative of the wave excitation force considered as disturbance. System outputs are the buoy displacement y [k] and buoy velocity z [k]. The details of state space model, shown from Eq. (13a) to Eq. (13d), are based on WEC parameters which are listed in Table 1. Where and is equivalent to the stiffness coefficient k s in the system motion equation in Eq. (1) (Yu and Falnes, 1995).
In some cases, and can be different, for example when the PTO force works through some mechanism or when the model includes several modes of motion but power is only extracted from one of them .
In order to increase the longevity of WEC, two physical constraints are considered as mentioned in Eqs. (14) and (15). Farideh MILANI, Reihaneh Kardehi MOGHADDAM China Ocean Eng., 2017, Vol. 31, No. 4, P. 510-516 511 φ max The first one is defined to keep WEC floating on the sea surface, so that neither sink nor rise above the water. In this constraint, z w is the water level and z v is the middle height of WEC's float. This difference should be smaller than . The implementation of the second constraint guarantees the safety of WEC Belmont, 2014a, 2014b).

Wave prediction using Kalman filter
The procedure of wave prediction using Kalman filter with its framework is demonstrated in this part. Using Kalman filter is an efficient and accurate way to predict the incoming wave. Kalman filter was introduced in 1960 by Rudolf E. Kalman, but before it was given in a simple algorithm by Swerling. This estimator-predictor filter includes a set of mathematics equations with optimal performance; it means that, if all noises are considered as Gaussian noise, Kalman filter minimizes the mean square error of all estimated parameters. To implement Kalman filter, discrete state space model of incoming wave as mentioned in Eq. (17a) and Eq. (17b) is considered. w(k) is the process noise or wave elevation and v(k) is the measurement noise with the following assumptions (Li and Belmont, 2014a): .
(17b) The state is estimated as follows: (20) And the predicted output is defined as:ŷ where L is the Kalman filter gain and P is the solution of Recatti equation with relations as follows: The Kalman filter framework is shown in Fig. 2. The control input, u, and the process noise, w, are applied to the plant. The output signal, y, is generated by impressing the measurement noise v. u and y are considered as the Kalman filter inputs and its outputs are estimated output and estimated state .

State space model of incoming wave ω λ
In order to make prediction of wave, a certain model of wave is needed which is addressed in this part. In overall model of regular wave, displacement and velocity are considered as states and other characteristics of system have been assumed constant. But in irregular waves, the natural frequency and damping coefficient are added to states. State space model of an irregular wave is shown below ): The input is white Gaussian noise and the excitation force is output. c F is the matrix of the PTO force limitations.
is the measurment noise. The state transition matrix , with sampling time , is given by Eq. (25) This wave model will be applied to Kalman filter to predict the future excitation force.

Model predictive control
Model predictive control (MPC) optimizes problems by considering the physical constraints, and it has been used in many fields such as chemical engineering, process control, and control systems. Fig. 3 shows the implementation of MPC for a plant. Unmeasured disturbance d is an unknown signal that always exists in the system and is not dependent on the controller or plant. Controller adjusts the manipulated variable u such that the output tracks the set point r and this is the purpose of control. Noise, z, represents the electrical noise, sampling error or any factor that has a destructive effect on the accuracy of the output. MPC can consider the constraints on . This ability is useful when the system has several outputs with different constraints. As shown in Fig. 4, the prediction horizon P and control horizon M are two important parameters in MPC. At the current time t, the output process from the next moment (t+1) to the next P steps (t+P) is predicted by the model. By solving an optimization problem, the future inputs, from the time t to time (t+M-1), are obtained. Then only the current control signal u(t/t) is sent to the process and the procedure is repeated at later stages. In order to obtain the control law, the optimized cost function is used. Here, a criterion of the cost function is that the output can be predicted by the model as close as possible to determine the optimal path which results in the minimization of control vector variations (Abedi et al., 2013;Holkar and Waghmare, 2010). (26) Sampling time is an important parameter which has great influence on MPC performance. With low , the ac-T s curacy of the system to follow the set point is enhanced but the calculation time will increase. High P slows down the performance of the system. Selecting low M increases the performance velocity of the system, on the other hand, high M decreases the system error. So in order to enhance the performance of MPC, certain and optimized values for , P and M have to be selected. In this paper imperialist competitive algorithm is chosen to optimize these three parameters of MPC applied on WEC. This algorithm is a quick search technique that can obtain exact or approximate local optimum responses in the desired space.
6 Imperialist competitive algorithm P n C n C i Imperialist competitive algorithm (ICA) is a revolutionary algorithm inspired by social and political progress of human life. This algorithm has shown a good performance in convergence rate and global optimal achievement. Its flowchart is introduced in Fig. 5. Optimization procedure starts with initial populations which are countries. These countries are classified into imperialists and colonies. Depending on the imperialist power, colonies are divided among imperialists. The power of an imperialist is shown in Eq. (28), where is the imperialist cost and is the cost of each imperialist.
Each imperialist has initial colonies in Eq. (29). is the total number of colonies. After deviding colonies, they move toward their imperialist. During the moving, a colony can be placed in the imperialist position because of its low cost. Competition among imperliasts continues and   Farideh MILANI, Reihaneh Kardehi MOGHADDAM China Ocean Eng., 2017, Vol. 31, No. 4, P. 510-516 some of them will be eliminated if they lose their colonies (Atashpaz-Gargari and Lucas, 2007).
7 Numerical results As illustrated previously, Kalman filter is used in order to predict the excitation force. Fig. 6 shows the overall procedure of implementing Kalman filter and MPC on WEC. As illustrated in the wave state space model in Eq. (24), white Guassian noise w [k] or incoming wave is the single input of the wave plant and the real excitation force is the output. By implementing Kalman filter using "Kalman" function in Matlab, the predicted output will be the excitation force . In Fig. 7 a comparison has been done between the real excitation force and the predicted excitation force . As it is shown, the predicted signal is close to the real one, so Kalman filter has proper performance. The process noise covariance (Q) and the sensor noise covariance (R) are two parameters which affect the input w[k] and measurement noise v[k], respectively. In simulation process, these parameters are considered as 1×10 6 . By means of the system, identification toolbox state space model of the extracted is considered as unmeasured disturbance and is applied to WEC plant. Constraints of the system are the PTO force limit [-30, 30] and heave motion limit [-0.7, 0.7]. MPC parameters are selected as P=50, M=30 and T s =0.02. Fig. 8 shows the velocity and displacement trajectory during 30 s. Two inputs are shown in Fig. 9. The relative constraints are satisfied over the whole period for the PTO force and displacement.
The extracted power and extracted energy by WEC controlled with MPC during 30 s are calculated by Eqs. (10) and (11) and shown in Figs. 10 and 11, respectively. High absorbed energy shows that MPC has acceptable efficiency to control WEC. To increase the absorbed energy of a WEC under the control of MPC, it is necessary to optimize the basic parameters of MPC. By imperialist competitive algorithm, three optimized values will be obtained. The cost function, which should be minimized, is integral absolute      error (IAE). The error in this system is the difference between control inputs produced by MPC and the maximum PTO force which is supposed to be 30. With the maximum PTO force, the maximum energy will be transferred to the network.
The optimized values are shown in Table 2. Constraints of P and M have to be considered such that P is larger than M, otherwise the controller will not work properly. The more initial population causes accurate values, in exchange operation time of algorithm increases.
As described, these new values will change the performance of the system. As the optimized values of P and M are larger than their non-optimized values, the speed of controller decreases but as shown in Figs. 12 and 13, the absorbed power increases effectively. So it can be claimed that the optimization procedure has positive effect on the model predictive control of WEC, with the aim of increasing the absorbed energy.

Conclusion
In this paper, model predictive control is applied to WEC in irregular waves. Wave prediction is done by means of Kalman filter and the predicted excitation force is similar to the real excitation force, proving that Kalman filter has proper performance. The performance of controller is improved by optimizing the prediction horizon, control horizon and sampling time, using imperialist competitive algorithm. As shown, the absorbed power and absorbed energy increase dramatically after the optimized values are applied. So using MPC with new optimized parameters makes us closer to achieve the purpose of control. More study of other optimization methods is still needed to achieve better result in the future work. Also it will enhance investigations to consider an array of WECs with optimized values.