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Nonlinear Model Predictive Control of a Point Absorber Wave Energy Converte

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118 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 4, NO. 1, JANUARY 2013 Nonlinear Model Predictive Control of a Point Absorber Wave Energy Converter Markus Richter, Mario E. Magaña, Senior Member, IEEE, Oliver Sawodny, and Ted K. A. Brekken, Member, IEEE Abstract—This paper presents the application of nonlinear model predictive control (NMPC) to a point absorber wave energy converter (WEC). Model predictive control (MPC) is generally a promising approach for WECs, since system constraints and actuator limits can be taken into account. Moreover, it provides a framework for defining optimal energy capture and it can benefit from predictions. Due to possible nonlinear effects, such as the mooring forces, an NMPC is proposed in this paper, whose performance is compared to that of a linear MPC. Both controllers are supposed to control a nonlinear point absorber model. Computer simulations show that the proposed NMPC is able to optimize the energy capture while satisfying system limits. Index Terms—Nonlinear model predictive control, point absorber, wave energy converter. I. INTRODUCTION O CEAN wave energy is a promising renewable energy source that can be converted into useful electrical energy using WECs. In general, it is difficult to estimate the amount of exploitable wave energy in the world’s ocean. According to [1], the ocean holds approximately 8000–80 000 TWh/year or 1–10 TW, whereas Falnes in [2] quantifies the world’s exploitable wave power resource to be of the order of 1 TW. Comparing this to the world’s annual energy consumption of approximately 148 000 TWh in 2008, [3] shows that wave energy could play an important role in the world’s energy portfolio. Furthermore, most renewable energy resources are variable and nondispatchable and thus have a significant impact on the utility reserve requirements. Halamay et al. showed in [4] that this impact can be reduced by a diversified energy mix, where WECs could play an important role. In order for WECs to become a commercially viable alternative, operating the WEC in an optimal fashion is a key task. Much work on optimizing the energy generation of WECs has been done which leads to control laws such as latching control and phase and amplitude control [5]–[7]. Manuscript received November 08, 2011; revised May 11, 2012; accepted May 21, 2012. Date of publication July 31, 2012; date of current version December 12, 2012. M. Richter and O. Sawodny are with the Institute for System Dynamics, University of Stuttgart, Stuttgart 70049, Germany (e-mail: markus-c.richter@arcor.de; sawodny@isys.uni-stuttgart.de). M. E. Magaña and T. K. A. Brekken are with Oregon State University, Corvallis, OR 97331 USA (e-mail: magana@eecs.oregonstate.edu; brekken@eecs. oregonstate.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSTE.2012.2202929 Furthermore, Schoen et al. [8] have proposed a hybrid control strategy in order to increase the energy conversion while being robust to modeling errors. The short-term tuning of the converter is done by a fuzzy logic controller, while the robust controller attempts to minimize the modeling errors. Recently, researchers increasingly deal with MPC algorithms in order to control point absorbers. They have shown that MPC is a very promising control method, since it is able to exploit the entire power potential of a WEC on the one hand, while respecting the constraints on motions and forces on the other hand. All the above-mentioned control approaches require prediction data of the wave’s motion or at least they can benefit from predictions. The problem of short-term wave forecasting has already been studied by many researchers. Fusco and Ringwood [9], for example, focused on wave forecasting and prediction requirements for unconstrained phase and amplitude control. In that work, the required forecasting horizon and the achievable performance of wave predictions for different properties of the floating system has been quantified. The same authors presented several wave prediction algorithms in [10]. They implemented cyclical models, autoregressive models, and neural networks in order to predict the wave’s motion and validated these models against real observations from data buoys. It was shown that an accurate prediction for up to two typical wave periods into the future can be calculated. Different prediction models have been presented in [8]. There, a predictive Kautz model and a combination of a Kautz and an autoregressive model have been proposed, where both filters have reduced orders compared to conventional prediction filters. Hence, the proposed filters can lower the computational effort in real-time applications. This paper deals with model predictive control, where a prediction of the wave’s motion is required. However, it does not focus on wave prediction and an ideal prediction is assumed. However, the above-mentioned work on wave prediction could be used to estimate the wave horizon for our model predictive controller as well. Current research into MPC for wave energy applications exclusively focuses on linear MPC. Brekken in [11] and Hals et al. in [12] successfully applied linear MPC to one-body WEC models. Moreover, the application to a linear two-body model with mooring was demonstrated in [13]. In order to deal with possible nonlinear effects, such as the mooring forces, a nonlinear model predictive controller is proposed in this paper. Its performance is compared to that of a linear MPC, also controlling the nonlinear system. The proposed controllers are validated and compared through simulation for irregular sea states. 1949-3029/$31.00 © 2012 IEEE RICHTER et al.: NONLINEAR MODEL PREDICTIVE CONTROL OF A POINT ABSORBER WAVE ENERGY CONVERTER 119 culate the radiation forces. Using impulse response functions yields convolution terms in the expressions for the radiation forces. Furthermore, a highly nonlinear mooring force law [17] is used. The motion of the buoy and the spar is denoted by and , respectively. The equation of motions (EOMs) can be derived using Newton’s second law. The formulation is based on linear wave theory (LWT) and the frequency-dependent parameters of the L10 are assumed to be constant. The EOMs for the two bodies are (1) (2) where the radiation forces , the hydrodynamic force , and the mooring force are described by (3) (4) (5) (6) (7) Fig. 1. L10 Wave Energy Coverter [14] (top) and schematic diagram (bottom). (8) The paper is organized as follows. In Section II, a nonlinear two-body model in the time-domain is presented, where the discretization of the nonlinear model is also discussed. The implementation of the NMPC is described in Section III. By means of computer simulation, the performance of the proposed NMPC is validated and compared to that of the linear MPC from the work in [13]. II. POINT ABSORBER MODEL This paper exclusively focuses on one subclass of WECs, namely, point absorbers. The point absorber L10, developed at Oregon State University is considered in this work and is shown on the left side of Fig. 1. It consists of a float, also called buoy, floating on the ocean surface and a second body consisting of a spar and a ballast tank (in what follows, just spar), where the spar’s motion is damped through mooring. The relative motion of the two bodies can be converted into usable energy through a power takeoff (PTO) system. A schematic diagram of the L10 is shown at the top of Fig. 1, where and denote the positions of the spar and the float, respectively. The readers are referred to [15] for detailed information about the L10. A. Equations of Motion The proposed nonlinear two-body model follows the work in [16]. Thus, impulse response functions are used in order to cal- where the impulse response functions of the different radiation forces are denoted by . The hydrodynamic parameters are the added masses, are the viscous damping factors, and are the hydrostatic stiffnesses. Also, there are coupled ra- diation forces resulting from the interaction of the spar and the buoy. denotes the PTO force which is the control input for the system, whereas the excitation forces and are the system disturbances. The mooring system is based on the experimental mooring configuration in [17]. A top view schematic diagram is shown in Fig. 2. There, the buoy is moored to a static reference around the buoy. There are two layers of cables, where each layer consists of four cables as can be seen in Fig. 2. Here, no mooring to the sea floor is assumed. is the stiffness of one cable, and is the horizontal length from the reference to the buoy. Table I lists the parameters used for modeling, where a cable stiffness of N m and a cable length to the reference of m is assumed for the mooring force calculation. In what follows, we neglect the convolution terms because the inertia and the damping term are the dominating terms for the ra- diation forces. The convolution terms are fairly small compared to other external forces. Simulation results confirm that assump- tion. In case there was a larger influence of these terms, the 120 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 4, NO. 1, JANUARY 2013 can be formulated. Thereby, it is assumed that the entire state is measurable. The model can be written as (13) where Fig. 2. Mooring configuration (top view). TABLE I SYSTEM PARAMETER OF THE L10 WEC (14) (15) (16) (17) convolution terms can be approximated by a linear state space model by methods described in [18] and [19]. However, this is not necessary here. For model predictive control in general, using a model in state space form is convenient. Equations (1)–(8) can be reformulated as (9) (10) These equations can be transformed in state space form by sub- stituting (9) into (10) and vice versa to get rid of the coupling terms and . The equations can be restated as where the PTO force is now denoted by and the excitation forces and by and , respectively. In our work, we consider three different models. The first model (1)–(8) with convolution terms is called the extended model. The plant model in the simulations is always this model. The second model (13)–(17) without convolution terms is called the reduced model. The nonlinear MPC approach is based on this one. The only nonlinearity in the reduced model is the mooring force. Replacing the nonlinear mooring force by the linear law (18) yields the linear model, which the linear MPC is based on in this paper. B. Discretization A finite parameterization of the controls and constraints is (11) used to find a direct solution of the optimization problem. There- fore, the system must be described as a discrete-time nonlinear state space model in the form (19) (12) Now, the nonlinear state space model with the state vector and the initial conditions where the function maps the current state , the control- lable input , and the uncontrollable input to the next state . According to [20], a nonlinear system (20) RICHTER et al.: NONLINEAR MODEL PREDICTIVE CONTROL OF A POINT ABSORBER WAVE ENERGY CONVERTER 121 with real analytic vector fields and can be dis- cretized by an approximate sampled-data representation under zero-order hold assumption by (21) where (22) with . Here denotes the order of the discretization, and is the sampling interval. Simulation shows that a discretization of order 1 (comparable to Euler forward method) is not appropriate for the proposed NMPC approach. In fact, it yields unreasonable results. Due to this fact, the discretization order is chosen to be in the ensuing work. With the nonlinear reduced system (13)–(17), it follows that (23) (24) Using (21), after some manipulations the discrete-time system can be described by (25) where (26) (27) (28) with the Jacobian matrix, shown in (29) at the bottom of the page. III. PROBLEM FORMULATION AND IMPLEMENTATION OF NMPC Linear MPCs are well known and have been applied since the 1970s, whereas NMPC’s have been used since the 1990s [21]. Linear MPC theory is quite mature today and system theoretic attributes such as stability and optimality are well addressed [22]. Also, many different industrial MPC applications can be found. The case is different with NMPC. While theoretical characteristics are well discussed, industrial applications are difficult to find [23]. Linear MPCs and NMPCs have basically the same concepts. In general, the NMPC problem is formulated as solving a finite horizon optimal control problem which is subject to constraints and to system dynamics [21]. The readers are referred to [21] and [24] regarding general information about MPC. A fundamental problem germane to NMPC schemes is that the constrained optimization problem needs to be solved within a specified time limit. In the case of linear MPC, the problem is convex and for the class of LQP, proven optimization algorithms exist to solve the problem efficiently. NMPC requires the solution of a nonlinear problem, though. In general, these problems are nonconvex, thus it cannot be assured to find the global optimum. Additionally, the solution can be computationally expensive. Therefore, it is important to exploit the special structure of each problem to obtain a real-time feasible optimization problem. This work does not focus on real-time applicability. The focus is on the qualitative performance of NMPC regarding a nonlinear WEC model whose nonlinearity results from a nonlinear mooring force. So, this work attempts to establish if NMPC is advantageous to use for controlling the selected nonlinear WEC compared to the linear MPC. The NMPC and the linear MPC are implemented using Matlab/SIMULINK. The linear MPC is based on the work in [13] and uses the reduced system with the linear mooring law (18). The NMPC uses the solver “fmincon” of Matlab which can handle nonlinear problems with nonlinear constraints as well. Here, the interior-point method is used as the optimization algorithm of “fmincon.” In the following, the solution of the nonlinear optimization problem using Matlab is outlined. The objective function is formulated to include one term expressing the generated power and another term presenting the energy use. In general, the generated power for point absorbers is the product of relative velocity and PTO force. Furthermore, the optimization problem includes slack variables and as (29) 122 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 4, NO. 1, JANUARY 2013 in [13] to avoid infeasibilities. The optimization problem can be stated as TABLE II NMPC/MPC PARAMETER VALUES (30) where (31) subject to (32) (33) (34) (35) (36) where , , and are the relative position, velocity, and the generator constraints, respectively. and are the weighting factors that penalize the slack variables and and are the constraints for them. The problem is implemented using the following optimization vector: (37) with variables. It is straightforward to express the ob- jective function by means of . The solver “fmincon” can handle box constraints, linear, and nonlinear inequality and equality constraints. The slack vari- ables (36) and the input constraints (35) are considered as box constraints which yield . The position (33) and velocity (34) limits are considered as linear inequality constraints which yield . Additionally, the system dynamics (32) need to be included as constraints, here as nonlinear equation constraints with equations. According to [10], the wave’s motion can be accurately pre- dicted for up to two typical wave periods into the future. Thus, a prediction time of at least 10 s is realistic. However, it should be noted that a large horizon time normally improves the perfor- mance, but at the same time, the computational effort increases since the optimization problem increases. Due to this fact, is chosen through simulation results. Since the results with a larger horizon time are not significantly better, a horizon time of 3 s is used which normally contains an half wave period and thus, the dominant dynamics. In the following, s and s, and thus the horizon length is 30 steps. In summary, the optimization problem consists of 152 opti- mization variables and 304 equations for the constraints. This is a large problem for online solving of the problem within a step time of s. However, as stated above, the goal of this work is to determine the qualitive behavior of the system con- trolled by the NMPC in comparison with a linear MPC and not to discuss real-time applicability. Fig. 3. Nonlinear and linear mooring forces for