Bi-Layer Model Predictive Control with Extended Horizons for Multi-Axis Underactuated Wave Energy Converters

Abstract

In the field of wave energy, multi-axis wave energy converters (WECs) have emerged as a research priority owing to their enhanced energy absorption, leading to increased computational complexity. Conventional model predictive control (MPC) approaches have demonstrated limitations in the trade-off between real-time requirements and control performance. This paper proposes a bi-layer MPC strategy, including a long-term energy maximization layer and a short-term trajectory-tracking layer. First, a multi-axis underactuated WEC (MU-WEC) is proposed, which incorporates an inertial cable-driven parallel mechanism to absorb energy from multiple directions. In addition, a control-oriented dynamic model of a MU-WEC is established. Then, a bi-layer MPC strategy is proposed, which decouples computationally intensive optimization processes from time-sensitive real-time control, alleviating the computational burden significantly. Therefore, the upper layer achieves enhanced control performance by enabling extended prediction horizons, whereas the lower layer serves to ensure real-time requirements. Moreover, numerical simulations under irregular wave conditions demonstrate the performance of the proposed bi-layer MPC: under different waves, bi-layer MPC improves energy absorption by 127–311% over conventional MPC. This performance enhancement stems from the 5 times extension of the prediction horizon enabled by the reduced computational burden.

1. Introduction

In recent years, the field of wave energy conversion has witnessed a shift toward multi-axis wave energy converters (WECs) owing to their demonstrated superiority in terms of power capture capacity through expanded absorption axes [1,2,3,4]. Ali et al. [5] demonstrated that the power absorption upper limit of multi-axis WECs substantially exceeds that of single-axis devices. Owing to this theoretical advantage, several studies have investigated multi-axis WECs and empirically validated their superior performance. Chen et al. [6] proposed a three-DOF (degree of freedom) WEC that uses a linkage mechanism to drive a hydraulic system, enabling multi-axis energy absorption. Ma et al. [7] studied the hydrodynamic performance of a multi-DOF point absorber under different wave periods and mechanical parameters. Yu et al. [8] developed a multi-axis WEC based on a multi-DOF combined mechanism and established a hydrodynamic model for wave–buoy interaction. Some multi-axis WECs achieve multi-axis energy absorption through parallel mechanisms. Michailides et al. [9] conducted a preliminary evaluation of the response and power absorption ability of a fully enclosed WEC with an internal parallel mechanism, demonstrating that additional absorption axes can increase power absorption. Shadmani et al. [10] performed shape optimization on a fully enclosed multi-axis WEC. Yao et al. [11] introduced a multi-axis WEC based on a three-UPU parallel mechanism and built a laboratory prototype.

However, the marine environment presents complex physical, chemical, and biological challenges for WECs. With an increased number of moving components exposed to seawater, multi-axis WECs exhibit particular vulnerability to extreme wave conditions, corrosion, and biofouling. To address these challenges, researchers have developed fully enclosed WECs with multiple axes, where all moving components are hermetically sealed within protective shells, effectively isolating them from seawater and marine organisms. Pendulum-type WECs are a category of fully enclosed WECs which absorb energy by an internal pendulum with an inertial mass placed inside a sealed outer floater [12]. Carapellese et al. [13] implemented MPC for a pendulum-type WEC. Antoniadis et al. [14] designed a fully enclosed WEC that uses an internal four-bar mechanism for energy absorption. TALOS is another type of fully enclosed WEC based on parallel mechanisms [9,15]. Using a multi-DOF parallel mechanism formed by internal hydraulic cylinders, TALOS achieves multi-axis energy absorption.

Due it being fully enclosed, WEC only allows for the control of the relative position between the outer floater and internal moving components, this constituting an underactuated system. These compact fully enclosed structures offer additional advantages, including simplified deployment procedures, reduced maintenance requirements, and enhanced reliability. In this paper, a fully enclosed underactuated wave energy converter (MU-WEC) is proposed, featuring a fully enclosed two-body structure comprising an inertial mass and an outer floater. These components are interconnected with the power take-off (PTO) system through tensioned cables, where relative motion induces changes in the cable length to generate energy. Compared with rigid components, the cable-based system provides distinct advantages of low inertia and an extended workspace. Additionally, the cable length changes can be efficiently transformed into rotational motion, enabling optimal spatial utilization for PTO integration. Internally, the cable arrangement forms a cable-driven parallel mechanism (CDPM) with three spatial degrees of freedom (DOFs), facilitating multi-axis energy absorption. However, the two-body structure of the MU-WEC introduces high DOFs and underactuated characteristics, presenting significant challenges for energy-maximizing control.

The primary objective of WECs is to maximize energy absorption from ocean waves under irregular and time-varying wave conditions. Advanced control strategies are essential for maintaining high energy capture efficiency across dynamic sea states. In fact, efficient control strategies have been proven critical for reducing the levelized cost of energy (LCOE) in wave energy systems, representing a key factor in achieving commercial viability [16]. To date, various control methodologies have been proposed for WECs, including impedance matching control, latching/declutching control, and phase control [17,18,19,20]. These control methods are all based on the impedance-matching principle, which defines the WEC’s velocity as a scaled version of the wave excitation force, generating dual optimal prerequisites—the optimal phase condition and amplitude condition for energy absorption [16]. While theoretically sound, these methods encounter significant difficulties in practical applications. The control of WECs requires maximizing energy capture under constraints (e.g., control force and displacement limits), which these impedance-based methods struggle to handle effectively. Moreover, these control methods are non-causal, as their implementation depends on information about future wave excitation. To address this limitation, researchers have developed various causal control approaches. Zou et al. [21] proposed a time-varying linear quadratic control strategy for a three-DOF WEC, which employs an extended Kalman filter to estimate the instantaneous wave excitation force, leveraging the WEC’s historical and current states to maximize energy absorption over a receding horizon. Similarly, a causal MPC strategy for a stochastically excited WEC has been researched [22], which does not require explicit forecasting of the incident wave force. Nevertheless, because these causal control strategies do not incorporate future wave inputs, they remain inherently suboptimal.

Model predictive control (MPC) has emerged as a promising solution for WEC control, offering both system constraint handling capabilities and optimal energy absorption potential [23]. Hall et al. [24] applied the MPC method to TALOS and compared the effects of two different models on power output. The results show that the model incorporating both hydrodynamics and PTO dynamics can increase power output. Sergiienko et al. [25] investigated the influence of MPC parameters on the performance of MPC controller. However, existing MPC strategies still face significant challenges in real-time implementation for WECs, where the requirement for real-time optimization creates substantial computational burdens [26,27]. MPC faces a fundamental trade-off in practical implementation: conventional MPC approaches struggle to achieve real-time control performance. While reducing the prediction horizon can alleviate computational costs, the control performance can be degraded. This implementation paradox becomes particularly pronounced in multi-axis WECs, where increased system DOFs exacerbate the computational complexity, making real-time implementation even more challenging. Existing literature has explored various approaches to reduce the computational burden of MPC. The nonlinearity of WEC systems leads to high computational costs in the optimization process of MPC. The pseudo-spectral method has been proposed to decrease computational cost by parameterizing system states to approximate the original dynamics of WECs [28,29]. Guerrero-Fernández et al. [30] introduced an MPC approach that integrates a moving window blocking technique for a point absorber, which reduces the number of decision variables by input-parameterized solutions, thereby significantly alleviating the computational burden of the optimization process. Zhan et al. [31] proposed computationally efficient MPC to reduce the computational cost caused by complex linear hydrodynamic models of multi-axis WECs, where the linear time-variant control law is determined offline by the analytic solution of the optimal control problem. Neural networks offer another approach to reduce the computational cost of MPC for WECs. Studies have shown that modeling nonlinear dynamics with neural networks, or directly approximating computationally intensive processes like optimization, can effectively lower computational costs. These methods have been successfully applied in WEC research [32,33,34].

These computational-efficient methods effectively reduce computational cost of MPC. However, fundamentally speaking, the objective of reducing MPC’s computational cost is to enhance its real-time performance, thereby meeting the requirements for real-time MPC implementation. From this perspective, an alternative approach is valuable to research: By decoupling the most computational-intensive optimization task from real-time requirements, enabling parallel execution between the optimization layer and the control layer, real-time MPC implementation can likewise be guaranteed. This approach does not inherently reduce the total computational cost; rather, it strategically distributes the most computationally intensive task and the real-time execution task into different layers, and sustains its computation through parallel computing to ensure real-time performance. Such parallel computation can be readily implemented on multi-threaded/multi-core hardware. Currently, there exist some bi-layer MPC methods that distribute different tasks across two layers to accomplish additional control goals. Bu et al. [35] developed two-level MPC to address the long-term nonlinear optimization challenges in gas pipeline systems. The upper layer employs a simplified nonlinear MPC formulation for long-term optimization, while the lower layer provides an operating scheme to control the actual gas pressure using a high-fidelity model. However, gas pipeline systems are slow-changing systems whose control strategies execute periodically on a time scale of hours. The computation time for a single execution loop is negligible compared to this execution period. For the electronics area, some bi-layer model predictive controllers are proposed to separate the computational processes of different control purposes [36,37]. Consequently, the bi-layer MPC architecture can distribute different tasks across two layers, which demonstrates potential for decoupling the computationally intensive receding horizon optimization from real-time requirements. However, existing research on bi-layer MPC still lacks dedicated investigations addressing real-time MPC implementation in rapidly changing environments.

In the field of wave energy, some research has also focused on bi-layer MPC. Luan and Wang [38] developed a double-layer MPC method for WEC that incorporates an additional compensation layer to generate compensation control forces, thereby decreasing the effects of model mismatch. The results show that the additional compensation layer effectively reduces the adverse effects of model mismatch and increases energy output. Zhan et al. [39] proposed a bi-layer MPC method, in which the upper layer identifies time-varying system dynamics online, while the lower layer maximizes energy output. This method effectively improves energy absorption. Overall, the abovementioned bi-layer MPC methods aim to achieve multiple control objectives through bi-layer structure, such as compensating for model mismatch and maximizing energy absorption. Furthermore, research on bi-layer MPC in rapidly changing environments remains underexplored. Inspired by these bi-layer MPC methods, this paper chooses a different bi-layer structure to further improve energy absorption. The proposed method decomposes the MPC tasks across two layers. This achieves decoupling between the optimization task and real-time requirements, thereby ensuring the real-time performance of the proposed MPC. In practice, such real-time performance enables the use of an extended prediction horizon, which significantly enhances energy absorption capacity [25].

This paper presents a bi-layer model predictive control (bi-layer MPC) strategy featuring two parallel control layers. An upper layer performs long-term energy maximization through computing and updating the energy-maximized trajectory while considering system constraints. Moreover, a lower layer functions as a short-term trajectory-tracking layer, aiming to track the energy-maximized trajectory in real time. The proposed bi-layer MPC decouples computationally intensive optimization from real-time control requirements, with the lower layer handling the control force calculation, whereas the upper layer focuses solely on trajectory generation. This bi-layer structure significantly relaxes the stringent computational requirements of conventional MPC implementations. The computational tasks of the upper layer and lower layer can be executed concurrently on separate hardware platforms or computed in parallel on a single hardware device. Inevitably, the actually energy-maximized trajectory exhibits a latency relative to the optimal trajectory (though this effect can be mitigated by reducing modeling errors). However, it simultaneously enables the MPC to achieve a longer prediction horizon under relaxed computational requirements, with the generated trajectory possessing a higher upper limit of energy absorption. Moreover, the proposed bi-layer framework maintains compatibility with other MPC approaches and can be easily integrated with other existing methods to achieve better performance.

This paper is organized as follows. In Section 2, a fully enclosed underactuated wave energy converter is proposed, where the energy conversion principle of the MU-WEC is presented. Additionally, a control-oriented dynamic model is established, which serves as the control plant for subsequent investigations. In Section 3, a bi-layer model predictive control (MPC) strategy is presented. The two layers of the bi-layer model predictive controller are constructed separately, and its real-time computational principle is illustrated. In Section 4, numerical simulations are conducted to validate the bi-layer MPC, with a comparative performance analysis between three type of MPC.

2. Energy Conversion Principle and Control-Oriented Dynamic Modeling of MU-WEC

2.1. Inertia-Driven Mechanism and Energy Conversion Principle of MU-WEC

A fully enclosed underactuated WEC (MU-WEC) is proposed. Figure 1 presents a schematic of MU-WEC, with the corresponding parameter values provided in

Table 1. Dimension parameters of MU-WEC.

Parameters	Values	Parameters	Values
$D_{of}$	5 m	$H_{id}$	3.6 m
$D_{in}$	4 m	$H_d$	2.5 m
$H_{of}$	9 m	$D_d$	2.5 m
$H_{in}$	3 m	$H_{1/2}$	1.5 m

. As shown in Figure 1, the MU-WEC consists of a fully enclosed outer floater, an inertial mass contained within the outer floater, and a cable-based PTO system connecting them. The inertial mass contains cable attachment points, where each cable routes through a pulley system before winding onto a reel that shares a shaft with the PTO. The cables maintain positive tension by PTO stiffness. Specifically, the PTO spring is preloaded to maintain positive tension and avoid slack. Therefore, the cable length changes can be transformed into rotation of the PTOs, while the PTOs can actively regulate the cable tension through the torque output. Four such modules form a cable-driven parallel mechanism with the floater as the base platform and the inertial mass as the moving platform, allowing multi-axis control and energy absorption through cables and PTOs. Wave-induced motion drives relative displacement between the inertial mass and the outer floater, altering the cable length to rotate the reels and actuate the PTOs. The fully enclosed design eliminates seawater corrosion and biofouling risks for moving components, resulting in simplified deployment procedures while reducing maintenance and improving reliability. The internal cable-driven parallel mechanism enables the MU-WEC to absorb wave energy across multiple axes (all translational axes), enhancing the energy absorption performance.

2.2. Control-Oriented Modeling of MU-WEC with Two-Body Coupling Dynamics

This section presents the development of a linear mathematical model to describe the dynamics of the MU-WEC system. Although a nonlinear dynamic model could describe the system behavior more comprehensively, this aspect falls beyond the scope of this paper. The present study focuses on establishing a linear model at the remaining position, which subsequently serves as the plant for the control strategy developed in Section 3.

The MU-WEC dynamics are modeled as two parts: the float dynamics and the dynamics of the internal CDPM. The dynamics of the MU-WEC in this paper consists of two parts: one describes the dynamics of the outer floater [40], and the other describes the internal CDPM [41]. Therefore, the dynamics of the MU-WEC system can be represented by the following equations:

$$\left\{\begin{array}{l}{M}_{in}{\ddot{X}}_{in}=-F_k+F_u+F_p-G\hfill \\ M_{out}{\ddot{X}}_{out}=F_h+F_{exc}+F_{rad}+F_k-F_u\hfill \end{array}\right.$$

(1)

where $M_{in}$ and $M_{out}$ represent the mass matrix of the inertial mass and outer floater, respectively; $X_{in}$ and $X_{out}$ indicate the displacement of the inertial mass and outer floater, while the relative displacement between the inertial mass and outer floater is defined as $X_r=X_{in}-X_{out}$ ; $F_k=K_{pto}{X}_r$ is the PTO stiffness force acting on the inertial mass, where $K_{pto}=J^T\mathrm{diag}\left(\begin{array}{cccc}{k}_1& k_2& k_3& k_4\end{array}\right)J$ , $J$ is the Jacobian matrix (see Appendix A), $k_i$ is the stiffness of the PTO connected to the i-th cable; $F_u$ represents the control force acting on the inertial mass; $F_p$ is the preloaded force, which is utilized to establish equilibrium against the gravity G at the rest position; $F_h=K_h{X}_{out}$ indicates the hydrostatic force; $F_{exc}$ is the wave excitation force; and $F_{rad}$ represents the radiation force, which can be expressed as [40]:

$$F_{rad}=M_{\infty }{\ddot{X}}_{out}+{\displaystyle {\int }_0^t\mathbb{K}\left(t-\tau \right){\dot{X}}_{out}\left(\tau \right)d\tau }$$

(2)

where $M_{\infty }$ indicates the added mass at infinite frequency and where $\mathbb{K}$ is the radiation impulse response function. These hydrodynamic terms are computed by boundary element methods. For the subsequent control strategy, the convolution term in (2) is approximated in state-space form [23]:

$$\left\{\begin{array}{l}{\dot{X}}_{rad}=A_{rad}{X}_{rad}+B_{rad}{\dot{X}}_{out}\hfill \\ {\displaystyle {\int }_0^t\mathbb{K}\left(t-\tau \right){\dot{X}}_{out}\left(\tau \right)d\tau }\approx C_{rad}{X}_{rad}\hfill \end{array}\right.$$

(3)

Thus, (1) can be expressed as a state-space model:

$$\left\{\begin{array}{l}\dot{X}=A_{c}X+B_{uc}U+B_{wc}W\hfill \\ Y_r=C_{yrc}X\hfill \\ Z_r=C_{zrc}X\hfill \\ Z_{out}=C_{zoc}X\hfill \end{array}\right.$$

(4)

where $X={\left[\begin{array}{cc}{X}_{wec}& X_{rad}\end{array}\right]}^T$ is the state vector, and $X_{wec}={\left[\begin{array}{cccc}{X}_r& X_{out}& {\dot{X}}_r& {\dot{X}}_{out}\end{array}\right]}^T$ ; $A_c$ is the state matrix; $U=F_u$ is the control input vector; $B_{uc}$ is the input matrix; $W=F_{exc}$ is the disturbance vector (wave excitation); $B_{wc}$ is the disturbance matrix; $Y_r={\dot{X}}_r$ , $Z_r=X_r$ and $Z_{out}=X_{out}$ are the output vectors; $C_{yrc}$ , $C_{zrc}$ and $C_{zoc}$ are the output matrix.

3. Bi-Layer Model Predictive Control Strategy

3.1. Framework of Bi-Layer MPC Strategy

This section proposes a bi-layer MPC strategy and outlines its underlying framework. As demonstrated in (1), the MU-WEC is a complicated system with underactuated and high-dimensional characteristics, imposing a substantial computational burden on the conventional MPC strategy. This situation introduces a fundamental trade-off—reducing the control horizon alleviates the computational burden but leads to degradation of the control performance. To achieve real-time control with control performance retention, this paper proposes a bi-layer MPC strategy.

The framework of the bi-layer MPC strategy is depicted in Figure 2. The upper layer functions as a long-term energy maximization layer, acquiring wave information and computing constrained energy-maximized trajectories for the MU-WEC system. The lower layer is a short-term trajectory-tracking layer designed to achieve real-time tracking of the energy-maximized trajectories. The distinctive feature of this bi-layer MPC strategy is that each layer contains an independent model predictive controller, while the computation processes of the two model predictive controllers are parallel. Specifically, the upper layer model predictive controller adopts a long prediction horizon to compute energy-maximized trajectories over future time intervals, subsequently transmitting these trajectories to the lower layer upon computation completion. Concurrently, during the computation cycles of the upper layer, the lower-layer model predictive controller continuously tracks the most recent trajectory until the upper layer inputs an updated trajectory, at which point tracking recommences from the current time instant of the updated trajectory. For example, as shown in Figure 2b, at time $t_1$ , the upper-layer controller initiates computation of a future energy-maximized trajectory (red trajectory) on the basis of the system state at $t_1$ . The computation is complete at $t_2$ , after which the trajectory is transmitted to the lower-layer controller. Although the computed trajectory spans from $t_1$ onward, the lower-layer controller tracks only the trajectory starting from $t_2$ (the current time instant). Upon completing the current computation, the upper-layer controller immediately acquires the system state at $t_2$ and proceeds to the next cycle of energy-maximized trajectory generation. Therefore, the computational tasks of the upper layer and lower layer can be executed concurrently on separate hardware platforms or computed in parallel on a single hardware device. This bi-layer structure significantly relaxes the stringent computational requirements of conventional MPC implementations. The pseudocode has been provided in

Table 2. Pseudocode of the bi-layer MPC strategy.

Initialization //upper layer while ture   Predict wave forces   Update the current system state $X$ and time step $t_i$   Calculate energy-maximized trajectory   Transfer the energy-maximized trajectory to the lower layer end //lower layer While ture   if the calculation of upper layer is completed     Update energy-maximized trajectory     Predict wave forces     Update the current system state $X$ and time step $t_j$     Track the updated energy maximization trajectory from the current time step $t_j$   else     Predict wave forces     Update the current system state $X$ and time step $t_i$     Track the energy maximization trajectory   end end

.

This framework of the bi-layer MPC strategy has several significant advantages: (1) the parallel computation framework enables a long control horizon in the upper layer while maintaining computational isolation from real-time control command execution, thereby preserving control performance on power absorption; (2) the lower-layer model predictive controller employs shortened prediction horizons for energy-maximized trajectory tracking, significantly reducing computational burdens to satisfy the requirements of real-time control.

3.2. Long-Term Energy Maximizing Layer

This section presents the long-term energy maximizing layer (i.e., the upper layer) of the proposed bi-layer MPC strategy. The primary objective of the upper layer is to generate energy-maximized trajectories while satisfying system constraints. The energy-maximized trajectories are subsequently tracked by the lower layer. Notably, the energy absorption performance of a controlled plant is fundamentally determined by the trajectories generated by the upper layer.

To develop the bi-layer MPC strategy, the continuous model described in (4) must be transformed into a discrete model [42]. In this paper, the standard Zero-Order Hold (ZOH) method is used to discrete the continuous model, which is widely used in the field of wave energy [43]. The discrete model is expressed as follow:

$$\left\{\begin{array}{l}\dot{X}\left(k+1\right)=AX\left(k\right)+B_{u}U\left(k\right)+B_{w}W\left(k\right)\hfill \\ Y_r\left(k\right)=C_{yr}X\left(k\right)\hfill \\ Z_r\left(k\right)=C_{zr}X\left(k\right)\hfill \\ Z_{out}\left(k\right)=C_{zout}X\left(k\right)\hfill \end{array}\right.$$

(5)

The energy absorbed by the discrete model can be expressed as follows:

$$\mathbb{E}=-{\displaystyle \sum T_s{Y}_r{\left(k\right)}^{T}U\left(k\right)}$$

(6)

where $T_s$ is the sample time of the discrete model.

It is well known that achieving computational efficiency in MPC fundamentally requires formulating the optimization problem as a convex quadratic programming (QP) problem. However, WEC control differs from conventional trajectory tracking problems, as its primary objective involves maximizing absorbed energy. This energy maximization objective leads to a non-convex risk, which significantly restricts the applicability of efficient optimization algorithms. To address this challenge, this paper uses the following objective function to ensure the convexity of the QP problem [43]:

$$\mathcal{J}={\displaystyle \sum _{k=0}^N\left[Y_r{\left(k\right)}^{T}U\left(k\right)+U{\left(k\right)}^{T}RU\left(k\right)+Z_r{\left(k\right)}^{T}Q{Z}_r\left(k\right)\right]}$$

(7)

where N is the control horizon; $R=\mathrm{diag}\left(r_1,\dots ,r_{n_u}\right)$ is the weight matrix of the control input; $n_u$ represents the dimension of the control input; and $Q=\mathrm{diag}\left(q_1,\dots ,q_{n_r}\right)$ indicates the weight matrix of the relative displacement between the inertial mass and the outer floater. In (7), $Y_r{\left(k\right)}^{T}U\left(k\right)$ represents the energy absorption term; $U{\left(k\right)}^{T}RU\left(k\right)$ denotes the control force penalty term, which constrains the magnitude of the control force; and $Z_r{\left(k\right)}^{T}Q{Z}_r\left(k\right)$ serves as a penalty term for the relative displacement between the inertial mass and the outer floater, acting as a soft constraint on the motion of the inertial mass. It is important to note that $\mathcal{J}$ is not monotonically correlated with the absorbed energy, because both $Y_r{\left(k\right)}^{T}U\left(k\right)$ and $U{\left(k\right)}^{T}RU\left(k\right)$ are influenced by $U\left(k\right)$ . Consequently, a small weight $R$ should be selected to maximize the impact of the energy absorption term $Y_r{\left(k\right)}^{T}U\left(k\right)$ on $\mathcal{J}$ . The inclusion of penalty terms for both the control force and the relative displacement in the objective function may reduce the absorbed energy, but it simultaneously relaxes the requirements of the actuators and decreases the collision risk between the inertial mass and the outer floater. Moreover, through appropriate selection of weight matrices, the objective function enables flexible adjustment among three competing objectives: energy absorption maximization, control force minimization, and relative displacement limitation.

Rewriting (7) in matrix form, the objective function can be expressed in matrix form as follows [43]:

$$\mathcal{J}={\mathbb{Y}}_r^T\mathbb{U}+{\mathbb{U}}^Tℝ\mathbb{U}+{\mathbb{Z}}_r^T\mathbb{Q}{\mathbb{Z}}_r$$

(8)

where $ℝ=\mathrm{diag}\underset{N+1}{\underbrace{\left(R,\dots ,R\right)}}$ ; $\mathbb{Q}=\mathrm{diag}\underset{N+1}{\underbrace{\left(Q,\dots ,Q\right)}}$ ; $\mathbb{U}$ is the control sequence; ${\mathbb{Y}}_r$ and ${\mathbb{Z}}_r$ are the future output sequences:

$$\left\{\begin{array}{l}{\mathbb{Y}}_r={\mathbb{G}}_{y}X\left(k\right)+{\mathbb{F}}_{y\_u}\mathbb{U}+{\mathbb{F}}_{y\_w}\mathbb{W}\hfill \\ {\mathbb{Z}}_r={\mathbb{G}}_{zr}X\left(k\right)+{\mathbb{F}}_{zr\_u}\mathbb{U}+{\mathbb{F}}_{zr\_w}\mathbb{W}\hfill \end{array}\right.$$

(9)

where $\mathbb{W}$ is the future wave information, and

$${\mathbb{G}}_y={\left[\begin{array}{ccccc}{\left(C_{yr}\right)}^T& {\left(C_{yr}A\right)}^T& {\left(C_{yr}{A}^2\right)}^T& \dots & {\left(C_{yr}{A}^N\right)}^T\end{array}\right]}^T$$

(10)

$${\mathbb{F}}_{y\_u}=\left[\begin{array}{c}0\\ C_{yr}{B}_u& 0\\ C_{yr}A{B}_u& C_{yr}{B}_u& \ddots \\ ⋮& ⋮& & 0\\ C_{yr}{A}^{N-1}{B}_u& C_{yr}{A}^{N-2}{B}_u& \dots & C_{yr}{B}_u& 0\end{array}\right]$$

(11)

$${\mathbb{F}}_{y\_w}=\left[\begin{array}{c}0\\ C_{yr}{B}_w& 0\\ C_{yr}A{B}_w& C_{yr}{B}_w& \ddots \\ ⋮& ⋮& & 0\\ C_{yr}{A}^{N-1}{B}_w& C_{yr}{A}^{N-2}{B}_w& \dots & C_{yr}{B}_w& 0\end{array}\right]$$

(12)

$${\mathbb{G}}_{zr}={\left[\begin{array}{ccccc}{\left(C_{zr}\right)}^T& {\left(C_{zr}A\right)}^T& {\left(C_{zr}{A}^2\right)}^T& \dots & {\left(C_{zr}{A}^N\right)}^T\end{array}\right]}^T$$

(13)

$${\mathbb{F}}_{zr\_u}=\left[\begin{array}{c}0\\ C_{zr}{B}_u& 0\\ C_{zr}A{B}_u& C_{zr}{B}_u& \ddots \\ ⋮& ⋮& & 0\\ C_{zr}{A}^{N-1}{B}_u& C_{zr}{A}^{N-2}{B}_u& \dots & C_{zr}{B}_u& 0\end{array}\right]$$

(14)

$${\mathbb{F}}_{zr\_w}=\left[\begin{array}{c}0\\ C_{zr}{B}_w& 0\\ C_{zr}A{B}_w& C_{zr}{B}_w& \ddots \\ ⋮& ⋮& & 0\\ C_{zr}{A}^{N-1}{B}_w& C_{zr}{A}^{N-2}{B}_w& \dots & C_{zr}{B}_w& 0\end{array}\right]$$

(15)

In practical applications, the change rate of the control force usually needs to be limited. The limits of control force and motion are also needed. Therefore, by incorporating both control force and motion constraints, (8) can be reformulated into the following quadratic form:

$$\begin{array}{rl}\underset{\mathrm{U}}{min}\mathcal{J}& =\frac{1}{2}\mathrm{\Delta }{\mathbb{U}}^T\mathcal{H}\mathrm{\Delta }\mathbb{U}+{\mathcal{F}}^T\mathrm{\Delta }\mathbb{U}\\ s.t.\mathcal{T}\mathrm{\Delta }\mathbb{U}& ⩽\mathcal{E}\end{array}$$

(16)

where $\Delta \mathbb{U}$ is input slew rate form of the control sequence; $\mathcal{H}$ is the Hessian matrix; $\mathcal{F}$ is the linear term; the constraints are defined by $\mathcal{T}$ and $\mathcal{E}$ , including maximum allowable control force, maximum change rate of the control force and limits of the relative displacement between the inertial mass and the outer floater. The mathematical expression of each term is detailed below:

$$\mathcal{H}={\mathbb{I}}_{\Delta u}^T\left({\mathbb{F}}_{y\_u}+{\mathbb{F}}_{y\_u}^T+2ℝ+2{\mathbb{F}}_{z\_u}^T\mathbb{Q}{\mathbb{F}}_{z\_u}\right){\mathbb{I}}_{\Delta u}$$

(17)

$$\begin{array}{cc}\mathcal{F}& ={\mathbb{I}}_{\Delta u}\left[\left({\mathbb{G}}_y+2{\mathbb{F}}_{z\_u}^T\mathbb{Q}{\mathbb{G}}_z\right)X\left(k\right)\right.\hfill \\ & +\left({\mathbb{F}}_{y\_u}+{\mathbb{F}}_{y\_u}^T+2ℝ+2{\mathbb{F}}_{z\_u}^T\mathbb{Q}{\mathbb{F}}_{z\_u}\right){\mathbb{I}}_{u}U\left(k-1\right)\hfill \\ & \left.+\left({\mathbb{F}}_{y\_w}+2{\mathbb{F}}_{z\_u}^T\mathbb{Q}{\mathbb{F}}_{z\_w}\right)\mathbb{W}\right]\hfill \end{array}$$

(18)

$$\mathbb{U}={\mathbb{I}}_{u}U\left(k-1\right)+{\mathbb{I}}_{\Delta u}\Delta \mathbb{U}$$

(19)

$${\mathbb{I}}_u={\left[\begin{array}{cccc}I& I& \dots & I\end{array}\right]}^T$$

(20)

$${\mathbb{I}}_{\Delta u}={\left[\begin{array}{cccccc}I& I& \dots & I& \dots & I\\ 0& I& \dots & I& \dots & I\\ ⋮& ⋮& \ddots & ⋮& \ddots & ⋮\\ 0& 0& \dots & I& \dots & I\end{array}\right]}^T$$

(21)

$$\mathcal{T}={\left[I-I\hspace{0.17em}{\mathbb{I}}_{\Delta u}^T-{\mathbb{I}}_{\Delta u}^T{\left({\mathbb{F}}_{z\_u}{\mathbb{I}}_{\Delta u}\right)}^T-{\left({\mathbb{F}}_{z\_u}{\mathbb{I}}_{\Delta u}\right)}^T\right]}^T$$

(22)

$$\mathcal{E}=\left[\begin{array}{c}\Delta {\mathbb{U}}_{\max }\\ \Delta {\mathbb{U}}_{\max }\\ {\mathbb{U}}_{\max }-{\mathbb{I}}_{u}U\left(k-1\right)\\ {\mathbb{U}}_{\max }+{\mathbb{I}}_{u}U\left(k-1\right)\\ {\mathbb{Z}}_{\max }-{\mathbb{G}}_{z}X\left(k\right)-{\mathbb{F}}_{z\_w}\mathbb{W}-{\mathbb{F}}_{z\_u}{\mathbb{I}}_{u}U\left(k-1\right)\\ {\mathbb{Z}}_{\max }+{\mathbb{G}}_{z}X\left(k\right)+{\mathbb{F}}_{z\_w}\mathbb{W}+{\mathbb{F}}_{z\_u}{\mathbb{I}}_{u}U\left(k-1\right)\end{array}\right]$$

(23)

$${\mathbb{U}}_{\max }={\underset{N+1}{\underbrace{\left[U_{\max },\cdots ,U_{\max }\right]}}}^T$$

(24)

$$\Delta {\mathbb{U}}_{\max }={\underset{N+1}{\underbrace{\left[\Delta U_{\max },\cdots ,\Delta U_{\max }\right]}}}^T$$

(25)

$${\mathbb{Z}}_{\max }={\underset{N+1}{\underbrace{\left[Z_{\max },\cdots ,Z_{\max }\right]}}}^T$$

(26)

where $n_u$ is the dimension of the control force; $U_{\max }=\left[{\mathfrak{u}}_1,\dots ,{\mathfrak{u}}_{n_u}\right]$ represents the maximum allowable control force in each dimension; $\Delta U_{\max }=\left[\Delta {\mathfrak{u}}_1,\dots ,\Delta {\mathfrak{u}}_{n_u}\right]$ indicates the maximum change rate of the control force; and $Z_{\max }=\left[{\mathfrak{z}}_1,\dots ,{\mathfrak{z}}_{n_r}\right]$ indicates the limits of the relative displacement between the inertial mass and the outer floater.

At time step k, the upper-layer model predictive controller starts to solve the optimization problem described by (31), with the solution available at time step $k+i$ . The resulting energy-maximized displacement trajectories of the inertial mass and outer floater are as follows:

$$\begin{array}{cc}\hfill {\mathbb{Z}}_{ref}& ={\left[\begin{array}{ll}{\mathbb{Z}}_r^T& {\mathbb{Z}}_{\mathrm{out}}^T\end{array}\right]}^T\hfill \\ \hfill & ={[Z_{\mathrm{ref}}(k\mid k)\cdots Z_{\mathrm{ref}}(k+i\mid k)\cdots Z_{\mathrm{ref}}(k+N\mid k)]}^T\hfill \end{array}$$

(27)

where ${\mathbb{Z}}_r$ is calculated by (9), and ${\mathbb{Z}}_{out}$ can be calculated in a similar way:

$${\mathbb{Z}}_{out}={\mathbb{G}}_{zout}X\left(k\right)+{\mathbb{F}}_{zout\_u}\mathbb{U}+{\mathbb{F}}_{zout\_w}\mathbb{W}$$

(28)

$${\mathbb{G}}_{zout}={\left[\begin{array}{cccc}{C}_{zout}^T& {\left(C_{zout}A\right)}^T& \dots & {\left(C_{zout}{A}^N\right)}^T\end{array}\right]}^T$$

(29)

$${\mathbb{F}}_{zout\_u}=\left[\begin{array}{c}0\\ C_{zout}{B}_u& 0\\ C_{zout}A{B}_u& C_{zout}{B}_u& \ddots \\ ⋮& ⋮& & 0\\ C_{zout}{A}^{N-1}{B}_u& C_{zout}{A}^{N-2}{B}_u& \dots & C_{zout}{B}_u& 0\end{array}\right]$$

(30)

$${\mathbb{F}}_{zout\_w}=\left[\begin{array}{c}0\\ C_{zout}{B}_w& 0\\ C_{zout}A{B}_w& C_{zout}{B}_w& \ddots \\ ⋮& ⋮& & 0\\ C_{zout}{A}^{N-1}{B}_w& C_{zout}{A}^{N-2}{B}_w& \dots & C_{zout}{B}_w& 0\end{array}\right]$$

(31)

Upon completion of the upper-layer computation at time step $k+i$ , the receding horizon is updated to initiate the next round of optimization. The reference trajectories for the lower-layer controller start from the current time step $k+i$ . If the computation time of the upper layer remains constant as i, the condition $N>2i$ must be satisfied. Otherwise, the remaining trajectory length after each optimization round would be shorter than the computation time, compromising real-time control feasibility. In practice, the computation time fluctuates, so the control horizon length N must be selected with a sufficient margin for these temporal fluctuations. However, for the linear model of the MU-WEC and convex optimizations, this constraint is virtually impossible to reach. Moreover, the computation time also directly affects the trajectory update latency. Given that the control horizon length affects the computation time, a balance must be established between the trajectory update latency and the control horizon length to enhance the energy absorption performance. Notably, in contrast to the conventional MPC, the bi-layer MPC permits the upper layer to employ an extended prediction horizon, resulting in generated trajectories with theoretically superior energy capture potential.

3.3. Short-Term Trajectory-Tracking Layer

In this section, the short-term trajectory-tracking layer (i.e., the lower layer) is constructed. The lower layer is designed to receive and track the energy-maximized trajectories generated by the upper-layer optimization. The implementation of a short prediction horizon in the lower-layer model predictive controller ensures compliance with real-time control requirements.

As formulated in (1), the control force $F_u$ simultaneously influences both $X_{in}$ and $X_{out}$ . Because $X_r=X_{in}-X_{out}$ , $F_u$ concurrently drives $X_r$ and $X_{out}$, the dimension of $F_u$ is less than the DOFs of the MU-WEC system. Consequently, the MU-WEC is an underactuated system. For the MU-WEC, independent tracking of either $X_r$ or $X_{out}$ can be readily achieved without considering constraints. However, simultaneous tracking of both $X_r$ and $X_{out}$ presents a significant challenge. Furthermore, the control of the MU-WEC must also account for actuator saturations and motion constraints. This complexity renders most constraint-handling closed-loop control methods unsuitable (as they are either designed for fully actuated systems or require complex gain conditions and ignore actuator saturation), whereas the MPC method is uniquely suited to address these problems [44]. Consequently, a model predictive controller is implemented in the lower layer to minimize tracking errors. Theoretically, the underactuated nature of the MU-WEC system implies that a prolonged tracking duration for a trajectory may lead to error divergence. However, the trajectory updates provided by the upper layer periodically reset the tracking error in the lower layer, which enables the lower-layer model predictive controller to maintain tracking errors within acceptable bounds through optimization.

The lower and upper layers employ the same sampling time and discrete model, as shown in (5). The objective of the lower-layer model predictive controller is to minimize the tracking error between the actual trajectories and the energy-maximized trajectories under constraints. Accordingly, the optimization problem of the lower layer is formulated as follows:

$$\begin{array}{cc}\hfill \underset{U_L}{min}{\mathcal{J}}_L& ={\displaystyle \sum _{k=0}^{N_L}}{\parallel R_{Lr}[Z_r(k)-Z_{r\_\mathrm{ref}}(k)]\parallel }^2\hfill \\ & +{\parallel R_{\mathrm{Lout}}[Z_{\mathrm{out}}(k)-Z_{\mathrm{out}\_\mathrm{ref}}(k)]\parallel }^2\hfill \\ s.t.|Z_r(k)|& ⩽Z_{max},\hspace{1em}k=0,1,\cdots ,N_L\hfill \\ |U_L(k)|& ⩽U_{\max },\hspace{1em}k=0,1,\cdots ,N_L\hfill \end{array}$$

(32)

where $U_L$ is the output control force; $N_L$ is the control horizon; and $R_{Lr}$ and $R_{Lout}$ are the weight vectors of $X_r$ and $X_{out}$ .

Similarly to the upper layer, the optimization problem can be reformulated in quadratic form as follows:

$$\begin{array}{rl}\underset{U_L}{min}{\mathcal{J}}_L& =\frac{1}{2}{\mathbb{U}}_L^T{\mathcal{H}}_L{\mathbb{U}}_L+{\mathcal{F}}_L^T{\mathbb{U}}_L\\ s.t.{\mathcal{T}}_L{\mathbb{U}}_L& ⩽{\mathcal{E}}_L\end{array}$$

(33)

where ${\mathbb{U}}_L$ is the output control sequence; the constraints are defined by ${\mathcal{T}}_L$ and ${\mathcal{E}}_L$ ; and

$${\mathcal{H}}_L={\mathbb{F}}_{zr\_u}^T{ℝ}_{Lr}{\mathbb{F}}_{zr\_u}+{\mathbb{F}}_{zout\_u}^T{ℝ}_{Lout}{\mathbb{F}}_{zout\_u}$$

(34)

$${\mathcal{F}}_L^T={\mathcal{F}}_r+{\mathcal{F}}_{out}$$

(35)

where

$$\{\begin{array}{l}{\mathbb{R}}_{Lr}=\mathrm{diag}([\underset{N_L+1}{\underbrace{R_{Lr},R_{Lr},\cdots ,R_{Lr}}}])\\ {\mathbb{R}}_{\mathit{Lout}}=\mathrm{diag}([\underset{N_L+1}{\underbrace{R_{\mathit{Lout}},R_{\mathit{Lout}},\cdots ,R_{\mathit{Lout}}}}])\\ {\mathcal{F}}_r=(X^T(k){\mathbb{G}}_{zr}^T+{\mathbb{W}}^T{\mathbb{F}}_{zr\mathrm{\_}w}^T-{\mathbb{Z}}_{r\mathrm{\_}\mathit{ref}}){\mathbb{R}}_{Lr}r{\mathbb{F}}_{zr\mathrm{\_}u}\\ {\mathcal{F}}_{\mathit{out}}=(X^T(k){\mathbb{G}}_{\mathit{zout}}^T+{\mathbb{W}}^T{\mathbb{F}}_{\mathit{zout}\_\mathit{w}}^T-{\mathbb{Z}}_{\mathit{out}\_\mathit{ref}}){\mathbb{R}}_{\mathit{Lout}}{\mathbb{F}}_{\mathit{zout}\_\mathit{u}}\end{array}$$

(36)

3.4. Cable Tension Distribution of the Control Force

The control force $F_u$ is distributed to the cables by solving the following equation [45]:

$$J^T{T}_u-F_u=0$$

(37)

where $T_u={\left[t_{u1},t_{u2},t_{u3},t_{u4}\right]}^T$ is the cable tension vector; $t_{u1}$ , $t_{u2}$ , $t_{u3}$ and $t_{u4}$ represent the cable tensions for the respective cables. To avoid cable slack, it is necessary to ensure that the tension in each cable is positive. Therefore, $T_u$ is the positive solution of (37). If a positive solution exists for any $F_u$ in (37), it means that the cables can always maintain positive tension under. According to Figure 1 and Table 1, the Jacobian matrix $J$ of the internal CDPM can be calculated, and its representation in the 6-D space is as follows:

$$J=\left[\begin{array}{cccccc}0.80& 0& 0.60& 0& 0& 0\\ -0.40& -0.69& 0.60& 0& 0& 0\\ -0.40& 0.69& 0.60& 0& 0& 0\\ 0& 0& -1.0& 0& 0& 0\end{array}\right]$$

(38)

where the row vectors of the Jacobian matrix $J$ represent the unit wrenches of the tension in each cable, respectively. As shown in (38), the rank of $J$ is 3, and the 3-D subspace spanned by its row vectors represents all translational DOFs (surge, sway, and heave axes). This means that the wrenches of all cables can exert control forces on the inertial mass only in the 3 DOFs (surge, sway, and heave) through linear combination. In other words, MU-WEC can only generate control forces in 3 translational DOFs. Therefore, the internal CDPM is a 3-DOF CDPM with 4 cables [45].

Next, it will be proved that a positive solution of (37) always exists within the 3-D subspace spanned by the row vectors of $J$ . This problem is equivalent to prove that the convex hull formed by the row vectors of $J$ contains the origin in the 3-D subspace, where the criterion is given as follows [46]:

$$ℂJ_4^T<0$$

(39)

where $\mathbb{C}={\left[\begin{array}{lll}{\mathcal{C}}_1^T& {\mathcal{C}}_2^T& {\mathcal{C}}_3^T\end{array}\right]}^T$ represents the convex cone formed by the first 3 rows of $J$ , ${\mathcal{C}}_i^T$ is the corresponding normal vector of the convex cone planes; $J_4$ is the fourth row vector of $J$ . Calculation shows that (38) holds. Therefore, for any control force $F_u$ generated by the MU-WEC, a positive solution $T_u$ always exists. In other words, for any control force produced by the proposed bi-layer MPC method, the tension distributed to each cable remains positive. In this case, the solution space of (37) is a 1-D subspace of the cable tension space [47], where the solution can be obtained through linear programming (LP). To reduce the burden on the actuators, $T_u$ needs to be as small as possible. Thus, the solving of $T_u$ can be formulated as the following LP problem:

$$\begin{array}{ll}{\displaystyle \underset{\mathbb{U}}{min}}& {\mathcal{J}}_u=E_u{T}_u\\ s.t.& J^T{T}_u-F_u=0\\ & 0<T_u<T_{umax}\end{array}$$

(40)

where $E_u=\left[\begin{array}{cccc}1& 1& 1& 1\end{array}\right]$ , $T_{u\max }$ is the upper-limit of cable tensions. Consequently, it is proved that cable tensions corresponding to any control force can be kept positive, and the method of calculating the positive cable tensions is also provided, thereby preventing cable slackness. On the basis of ensuring $T_u>0$ , it is also necessary to ensure that the cable tension has a solution without exceeding the upper limit ($T_u<T_{u\max }$). This problem can be transformed into whether the convex hull formed by all possible $J^T{T}_u$ contains all permitted $F_u$ . If the convex hull contains all permitted $F_u$ , then (40) must have a solution that satisfies the constraints.

4. Simulation and Results

4.1. Simulation Settings

This section details the fundamental parameters of the MU-WEC implemented in numerical simulations. The MU-WEC’s dimensional configuration and parameter values are shown in Figure 1 and Table 1. The mooring system incorporates four mooring cables characterized by a mass density of 150.92 kg/m and a horizontal tension of 10 kN. Water depth is set as $h_d=30$ m. The mooring forces are linearized about the equilibrium position through application of the catenary method [48]. The remaining hydrodynamic parameters are computed by AQWA software (integrated in Ansys 2020 R2), which is a widely used commercial software package in the area of ocean engineering [49,50,51]. For the MU-WEC, pulley friction probably induces mechanical energy losses. With an assumption of a pulley diameter of 300 mm which us a cylindrical roller bearings with a bore of 120 mm, the frictional force can be calculated by the SKF bearing friction model [52]. Under conditions of 30 °C, 60 rpm, and radial loads ranging from 10 to 150 kN, the frictional force of the bearing remains below 0.1% of the inertial mass gravity. Therefore, this paper considers the effects of pulley friction can be ignored.

The MU-WEC model is discretized with a sampling interval of 0.05 s. The optimization problem is solved using MATLAB’s quadprog solver with the interior-point-convex algorithm. The solver is configured with a constraint tolerance of 0.001 and an optimality tolerance of 0.001. For the upper layer (long-term energy maximization control), the weighting coefficients for relative displacements are uniformly set to $q_i=5100$ , whereas those for control forces in all directions are set to $r_i=1\times {10}^{-5}$ . In the upper layer, the control forces of all directions are constrained to a maximum of 50 kN, with a rate limit of 10 kN; the allowable relative displacement is limited to 2.5 m in the surge direction and 1.5 m in the heave direction. For the lower layer (short-term trajectory tracking control), the prediction horizon is fixed at 0.1 s to ensure rapid tracking performance. To ensure solvability, all constraints in the lower layer are limited to 95% of their upper-layer counterparts. All computations are performed on a device with an Intel Core i7-10700 CPU and MATLAB R2021a. The maximum tension of a single cable actuator is 70 kN. The green area in Figure 3 represents the convex hull of all feasible resultant forces (i.e., $J^T{T}_u$ ) exerted by the cable tension on the inertial mass, while the orange area represents the range of control forces permitted by the bi-layer model predictive controller. Clearly, the convex hull formed by all possible $J^T{T}_u$ contains all permitted $F_u$ . According to the conclusion in Section 3.4, for all control forces permitted by the bi-layer model predictive controller, it always obtains cable tensions $T_u$ within the tension bound using (40). To prevent collisions caused by excessive displacement, a contingency strategy is introduced. When the displacement exceeds the constraint, the lower layer no longer tracks the trajectory generated by the upper layer. Instead, its reference trajectory switches to the rest position. That is, the current actual output control force drives the inertial mass back to its rest position until the displacement constraint is satisfied and a feasible trajectory can be generated again by the upper layer.

4.2. Generation of Wave Excitation

The modeling of irregular waves and wave excitation are detailed in this section. The irregular waves are generated using the JONSWAP wave spectrum, mathematically expressed as follows:

$$S\left(\omega \right)=319.34{\displaystyle \frac{H_s^2}{T_p^4{\omega }^5}}\exp \left[-{\displaystyle \frac{1948}{{\left(T_p\omega \right)}^4}}\right]{3.3}^{\exp \left[-{\displaystyle \frac{{\left(0.159\omega T_p-1\right)}^2}{2{\sigma }^2}}\right]}$$

(41)

where $T_p$ is the peak period; $H_s$ is the significant wave height; and $\sigma =0.07$ for $\omega \le {\omega }_p$ and $\sigma =0.09$ for $\omega >{\omega }_p$ , where ${\omega }_p$ is the peak frequency. The generated wave excitation force is calculated using the following formula:

$$f_{exc}\left(t\right)={\displaystyle \sum _{j=1}^{N}A\left({\omega }_j\right)}\mathrm{Re}\left\{B_{wave}\left({\omega }_j\right)e^{i\left({\omega }_{j}t+{\theta }_j\right)}\right\}$$

(42)

where $A\left({\omega }_j\right)=\sqrt{2S\left(\omega \right)\Delta \omega }$ is the amplitude of the jth regular wave component; $\Delta \omega$ is the interval of consecutive angular frequencies; $B_{wave}\left({\omega }_j\right)$ is the frequency-dependent wave excitation force per unit wave amplitude; and ${\theta }_j$ is the random phase of the jth regular wave component.

4.3. Optimization of Mechanical Parameters

The mechanical parameters of the MU-WEC exert a significant influence on energy absorption. Appropriate mechanical parameters can not only enhance energy absorption without active control, but also provide a superior plant for subsequent control strategies. This section determines the mechanical parameters of the MU-WEC through an optimization in the frequency-domain. It should be noted that, consistent with the majority of the literature on active control for WECs, the PTO damping force is considered a component of the control force. The energy absorbed by PTOs is treated as the negative work component of its active control force. Consequently, the PTO damping optimized in this section is used to determine other mechanical parameters.

The four mechanical parameters of the MU-WEC are selected as optimization variables: the PTO damping $c$ , PTO stiffness $k$ , mass ratio between the inertial mass and outer floater $r_m$ , and total mass $m_{total}$ . Considering practical constraints, both PTO damping $c$ and stiffness $k$ are bounded to resist gravity and prevent excessive cost; the mass ratio $r_m$ and total mass $m_{total}$ are also limited; the motion of the inertial mass is constrained to avoid collision. Furthermore, the annual mean power of the MU-WEC is used as the objective function. Therefore, the optimization problem can be formulated as

$$\begin{array}{ll}\max & {\mathit{P}}_{\mathit{a}}\left(\mathit{c},\mathit{k},{\mathit{r}}_{\mathit{m}},{\mathit{m}}_{\mathit{total}}\right)={\displaystyle \sum _{\mathit{j}}{\mathit{Y}}_{\mathit{j}}{\mathit{P}}_{\mathit{j}}}\\ s.t.& {\widehat{X}}_{rS}\in \left[-1.6,1.6\right]\\ & {\widehat{X}}_{rH}\in \left[-0.6,0.6\right]\hfill \\ & c\in \left[0,30000\right]\hfill \\ & k\in \left[10000,100000\right]\hfill \\ & r_m\in \left[0,1\right]\hfill \\ & m_{total}\in \left[20000,39270\right]\hfill \end{array}$$

(43)

where ${\widehat{X}}_{rS}$ is the surge displacement, ${\widehat{X}}_{rH}$ is the heave displacement; ${\mathit{Y}}_j$ is the percentage of time in a year for the j-th wave condition; $P_j={\displaystyle {\int }_0^{\infty }2S\left(\omega \right)}{P}_M\left(\omega \right)d\omega$ is the mean absorbed power of the j-th wave condition, $S\left(\omega \right)$ is the JONSWAP wave spectrum, $P_M\left(\omega \right)$ is the mean power of PTO damping which is calculated by the dynamics under the irregular wave component at wave frequency $\omega$ . The MU-WEC is designed to work near Chudao Island of China, where the wave climate matrix is shown in Figure 4a [53]. The particle swarm optimization (PSO) method is employed to perform the optimization [54]. The optimization process is shown in Figure 4b, and the optimal mechanical parameters and the optimal annual mean power are presented in

Table 3. Mechanical Parameters of MU-WEC.

Parameters	Values
PTO damping	15.1 kN·s/m
PTO stiffness	26.4 kN/m
Mass ratio	1
Total mass $m_{total}$	33,697 kg
Optimal annual mean power	4201 W

.

4.4. Comparison Between the BI-Layer MPC and Conventional MPC

This section presents a comparative performance analysis between the proposed bi-layer MPC and conventional MPC. Conventional MPC employs the same objective function, constraints, sampling interval and weighting coefficients as those used in the upper layer of the bi-layer MPC. Notably, as demonstrated in Section 3, the optimization process of the bi-layer MPC’s upper layer is inherently decoupled from real-time control requirements. Specifically, while the prediction horizon length of the upper layer governs power absorption, it remains independent of real-time computational constraints; conversely, the single-step computation time of the lower layer dictates real-time feasibility, with its prediction horizon fixed at 0.1 s to guarantee rapid tracking capability. Consequently, throughout the subsequent discussions in this section, the term “prediction horizon” exclusively refers to the upper layer’s prediction horizon, whereas “computation time” specifically denotes the lower layer’s single-step computation time.

Under wave conditions with a significant wave amplitude of 0.35 m and a peak period of 5 s, the influences of the prediction horizon on the single-step computation time and average power are investigated for both the conventional MPC and the proposed bi-layer MPC, as illustrated in Figure 5. Energy absorption has a positive correlation with increasing prediction horizon length. In Figure 5a, the curves represent the average computation time, whereas the error bars indicate the fluctuation range of the computation time. To ensure real-time control, the single-step solution must be completed within one sampling period. When the prediction horizon exceeds 1.75 s, the maximum single-step computation time of the conventional MPC surpasses the sampling interval, which means that real-time control cannot be guaranteed. Beyond 2.25 s, the average computation time of the conventional MPC exceeds the sampling interval, rendering real-time control theoretically unattainable. In contrast, the proposed bi-layer MPC ensures real-time control capability because of its decoupled prediction horizon and real-time control requirements. Consequently, although the conventional MPC demonstrates similar performance to the bi-layer MPC at identical prediction horizons (as shown in Figure 5b), the conventional MPC cannot reach a long prediction horizon because of the computation time and real-time requirements. Moreover, the bi-layer MPC can achieve superior performance over the conventional MPC by enabling longer prediction horizons under the same hardware, overcoming the real-time limitations inherent in conventional MPC implementations.

Figure 6 presents the computation time and average power of the conventional MPC and bi-layer MPC under irregular waves with varying peak periods. The significant wave amplitude is set as 0.35 m. According to Figure 5a, the conventional MPC adopts a conservative prediction horizon of 1.5 s to guarantee control stability. Figure 6 demonstrate that the wave period has a negligible effect on the computation time, indicating that the prediction horizon becomes the dominant factor influencing energy absorption under the same wave conditions. Figure 6b shows that when the bi-layer MPC adopts a 5 s prediction horizon, its average power increases by 127–311% compared to the conventional MPC. By employing a longer prediction horizon, the bi-layer MPC can achieve higher average power than the conventional MPC under different wave conditions. Under the same hardware constraints, the most computationally intensive part of the bi-layer MPC is decoupled from real-time control. This bi-layer framework allows the use of a larger prediction horizon compared to conventional MPC, which leads to a better performance. Although a conventional MPC can achieve comparable performance improvements by setting a same prediction horizon, it would fail to meet real-time control requirements.

As depicted in Figure 7, the motion response and control force output of the MU-WEC under the bi-layer MPC are presented, as calculated by (5). The red dotted lines are the maximum limits of control force. The wave significant wave amplitude is set as 0.35 m and the peak period is set as 5 s. The results demonstrate that for all prediction horizons, both the relative displacements and control forces remain within their maximum limits, ensuring safety under all constraint conditions. Notably, there is a trend where longer prediction horizons lead to more pronounced relative displacements. This systematic trend indicates that the bi-layer MPC intentionally increases relative motion to optimize power absorption. The proposed control strategy successfully achieves a balance between two competing objectives: maximizing energy absorption while maintaining all the constraints.

The influence of varying the weights within the objective function on the absorbed power is explored in Figure 8. It can be observed that a higher weight assigned to either the control force or the relative displacement term decreases the absorbed power. This effect is attributed to the increased weight on these terms, which in turn reduces the relative weight on the energy absorption term, leading to diminished energy absorption. Therefore, reducing the weights of the control force and the relative displacement term helps to make the energy absorption term dominate the objective function.

4.5. Effects of the Trajectory Update Latency

As previously discussed, the trajectory update latency in the bi-layer MPC results in reduced energy absorption compared with that of the conventional MPC under the same prediction horizons. This section investigates the influence of trajectory update latency on control performance. The simulation employs an upper-layer prediction horizon of 5 s and a lower-layer prediction horizon of 0.15 s, with computational analysis revealing that each upper-layer cycle requires approximately 0.48 s to complete. Consequently, a fundamental lower bound of 0.5 s for achievable trajectory update latency is established. For the length of the trajectory received by the lower-layer controller equal to the upper-layer prediction horizon, the maximum effective tracking duration for the lower layer is determined by the difference between the trajectory length and the trajectory update latency, which is 4.5 s. The irregular wave is characterized by a significant wave amplitude of 0.35 m and a peak period of 5 s.

Figure 9 shows the energy output of the MU-WEC under varying trajectory update latencies. Overall, the results demonstrate an inverse relationship between energy absorption and trajectory update latency. This phenomenon occurs because the trajectories generated by the upper layer are computed on the basis of the system states of one trajectory update latency earlier. As this latency increases, the resulting hysteresis effect becomes more pronounced, causing progressively greater deviations between the implemented control forces and the theoretically optimal control sequence. Notably, the performance degradation remains relatively limited, which can be attributed to the long motion period of WEC systems compared with the trajectory update latency. Consequently, it is feasible to extend the prediction horizon in the upper layer, as the improved energy absorption achieved through an extended prediction horizon can effectively compensate for the negative impacts of increased trajectory update latency. Notably, only white noise in the control input is considered. Although modeling errors and external disturbances could exacerbate the effect of update latency, the proposed method provide a higher theoretical upper bound of performance while the real-time requirement is guaranteed, which the authors believe is of significant value.

4.6. Performance Evaluation Under Extreme Wave Conditions

As shown in Figure 4, for approximately 0.05% of a year, Chudao Island is subjected to extreme wave conditions of 3.5–4 m significant wave height and 6–8 s peak period. Therefore, this section employs irregular waves with a significant wave height of 4 m and a peak period of 6 s to evaluate the performance of the proposed bi-layer MPC under such extreme wave conditions.

From Figure 10a, it can be observed that under extreme wave conditions, the energy output remains positively correlated with the prediction horizon. Furthermore, Figure 10b–d demonstrate that the proposed bi-layer MPC continues to satisfy all constraints even in extreme wave environments. The larger the prediction horizon, the more the actuator tends to saturate.

4.7. Performance Evaluation Under Wave Prediction Errors and Extreme Wave Conditions

In the previous sections, future wave excitation is perfectly obtained. However, in practice, wave excitation is impossible to predict perfectly, which leads to model mismatch and leads to a deterioration in control performance. Therefore, this section incorporates wave prediction errors and model mismatch to study their effect on the proposed bi-layer MPC. The performance of the proposed bi-layer MPC is compared against both the conventional MPC and a double-layer MPC for model mismatch. For detailed information regarding the double-layer MPC for model mismatch (MM-MPC), please refer to [38]. The same parameters are used for both the MM-MPC and the proposed bi-layer MPC. To better simulate practical conditions, all outputs of the state-space model are superimposed with white noise (variance = 0.01) to simulate sensor noise, and the model states are estimated using a Kalman filter. Additionally, performance under extreme wave conditions is simulated to investigate the effects of actuator saturation.

The prediction error of wave excitation is modeled as follows [55]:

$$w_{er}\left(k+i+1|k\right)={\lambda }_{er}{w}_{er}\left(k+i|k\right)+{\displaystyle \frac{H_s}{2}}{\xi }_{er}$$

where $w_{er}\left(k+i|k\right)$ is the prediction error of at step $k$ for time step $k+i$ , $w_{er}\left(k|k\right)~\mathcal{N}\left(0,3000\right)$ is the initial error; ${\lambda }_{er}=1.01$ is a constant, which indicates the growth of the prediction error over prediction time; ${\xi }_{er}~\mathcal{N}\left(0,3000\right)$ is the Gaussian white noise. The wave excitation with prediction error is shown in Figure 11.

Under wave conditions with a significant wave amplitude of 0.35 m and a peak period of 5 s, Figure 12 shows the computation time and average power of the three MPC methods under different prediction horizons. When the prediction horizon is identical, the average power of the three methods is close. However, both conventional MPC and MM-MPC are constrained by their single-step computation time and cannot realize long prediction horizons. As shown in Figure 12a, under real-time constraints, conventional MPC and MM-MPC can only achieve a prediction horizon of 1.75 s. Consistent with the conclusions in Section 4.3, the proposed bi-layer MPC benefits from the computational advantage of its bi-layer framework, allowing it to absorb more energy by employing longer prediction horizons. Figure 12b shows the average power of the three methods under identical prediction horizons. The right area of the orange dash dot line in Figure 12b corresponds to prediction horizons where conventional MPC and MM-MPC fail to achieve real-time control. Nevertheless, this region allows us to isolate the advantages attributable to different prediction horizons and draw some conclusions. When the prediction horizon is short, the average power of all three methods is very similar. As the prediction horizon increases, the average power of the proposed bi-layer MPC becomes slightly lower than that of conventional MPC and MM-MPC. This is due to the negative effect of the increasing trajectory update latency, as discussed in Section 4.5.

To better evaluate the performance of the three MPC methods under real-time constraints, a series of simulations is performed, and statistical data across all iterations are analyzed under wave conditions with a significant wave amplitude of 0.35 m and a peak period of 5 s. Every simulation have duration of 100 s.

Table 4. Statistical data of the bi-layer MPC.

	Prediction Horizon	Iteration	± 95% CI	P95	P99	Max.	Min.	Ave.	Deadline-Miss Rate	Constraint Violation Rate
U	0.5	2001	[3.5 × 10−3, 3.6 × 10−3]	4.9 × 10−3	5.9 × 10−3	0.011	2.7 × 10−3	3.5 × 10−3	0	0
L		2000	[7.5 × 10−4, 7.7 × 10−4]	1.3 × 10−3	1.8 × 10−3	2.5 × 10−3	4.7 × 10−4	7.6 × 10−4	0	0
U	0.75	2001	[3.5 × 10−3, 3.6 × 10−3]	4.3 × 10−3	4.9 × 10−3	6.7 × 10−3	2.9 × 10−3	3.5 × 10−3	0	0
L		2000	[6.0 × 10−4, 6.1 × 10−4]	9.5 × 10−4	1.1 × 10−3	2.3 × 10−3	4.2 × 10−4	6.1 × 10−4	0	0
U	1	2001	[5.5 × 10−3, 5.6 × 10−3]	6.6 × 10−3	7.2 × 10−3	8.1 × 10−3	4.1 × 10−3	5.5 × 10−3	0	0
L		2000	[5.9 × 10−4, 5.9 × 10−4]	8.1 × 10−4	9.4 × 10−4	1.4 × 10−3	4.5 × 10−4	5.9 × 10−4	0	0
U	1.25	2001	[9.3 × 10−3, 9.4 × 10−3]	0.011	0.012	0.014	6.6 × 10−3	9.3 × 10−3	0	0
L		2000	[6.7 × 10−4, 6.8 × 10−4]	8.8 × 10−4	9.9 × 10−4	1.5 × 10−3	4.8 × 10−4	6.7 × 10−4	0	0
U	1.5	2001	[0.015, 0.015]	0.019	0.021	0.023	0.010	0.015	0	0
L		2000	[7.1 × 10−4, 7.3 × 10−4]	9.9 × 10−4	1.3 × 10−3	2.5 × 10−3	5.7 × 10−4	7.2 × 10−4	0	0
U	1.75	2001	[0.023, 0.024]	0.029	0.033	0.039	0.016	0.023	0	0
L		2000	[7.2 × 10−4, 7.4 × 10−4]	1.0 × 10−3	1.3 × 10−3	2.4 × 10−3	5.9 × 10−4	7.3 × 10−4	0	0
U	2	1999	[0.033, 0.033]	0.041	0.045	0.050	0.022	0.033	0	0
L		2000	[7.1 × 10−4, 7.3 × 10−4]	9.8 × 10−4	1.1 × 10−3	2.5 × 10−3	4.5 × 10−4	7.2 × 10−4	0	0
U	2.25	1691	[0.044, 0.044]	0.055	0.060	0.066	0.030	0.044	0	0
L		2000	[6.8 × 10−4, 6.9 × 10−4]	9.8 × 10−4	1.1 × 10−3	1.4 × 10−3	4.3 × 10−4	6.8 × 10−4	0	0
U	2.5	1096	[0.058, 0.059]	0.075	0.081	0.089	0.039	0.058	0	0
L		2000	[6.1 × 10−4, 6.2 × 10−4]	9.1 × 10−4	1.1 × 10−3	1.2 × 10−3	4.2 × 10−4	6.1 × 10−4	0	0
U	2.75	979	[0.075, 0.076]	0.099	0.11	0.12	0.050	0.076	0	0
L		2000	[5.9 × 10−4, 6.0 × 10−4]	9.1 × 10−4	1.1 × 10−3	1.5 × 10−3	4.0 × 10−4	6.0 × 10−4	0	0
U	3	850	[0.096, 0.098]	0.13	0.13	0.16	0.067	0.097	0	0
L		2000	[5.7 × 10−4, 5.8 × 10−4]	8.6 × 10−4	1.0 × 10−3	1.5 × 10−3	3.9 × 10−4	5.8 × 10−4	0	0
U	3.25	677	[0.12, 0.12]	0.16	0.18	0.20	0.078	0.12	0	0
L		2000	[5.5 × 10−4, 5.6 × 10−4]	8.4 × 10−4	1.0 × 10−3	1.6 × 10−3	3.9 × 10−4	5.6 × 10−4	0	0
U	3.5	578	[0.15, 0.15]	0.20	0.22	0.24	0.097	0.15	0	0
L		2000	[5.7 × 10−4, 5.9 × 10−4]	9.7 × 10−4	1.2 × 10−3	2.5 × 10−3	3.7 × 10−4	5.8 × 10−4	0	0
U	3.75	457	[0.19, 0.20]	0.28	0.39	0.45	0.11	0.20	0	0
L		2000	[5.6 × 10−4, 5.8 × 10−4]	9.7 × 10−4	1.4 × 10−3	3.9 × 10−3	3.7 × 10−4	5.7 × 10−4	0	0
U	4	378	[0.23, 0.24]	0.32	0.36	0.38	0.15	0.24	0	0
L		2000	[5.7 × 10−4, 5.9 × 10−4]	9.8 × 10−4	1.6 × 10−3	2.8 × 10−3	3.8 × 10−4	5.8 × 10−4	0	0
U	4.25	336	[0.27, 0.28]	0.35	0.40	0.41	0.19	0.28	0	0
L		2000	[5.3 × 10−4, 5.4 × 10−4]	8.3 × 10−4	1.2 × 10−3	1.9 × 10−3	3.8 × 10−4	5.4 × 10−4	0	0
U	4.5	279	[0.33, 0.34]	0.44	0.48	0.58	0.21	0.34	0	0
L		2000	[5.1 × 10−4, 5.3 × 10−4]	8.5 × 10−4	1.3 × 10−3	2.9 × 10−3	3.7 × 10−4	5.2 × 10−4	0	0
U	4.75	243	[0.38, 0.40]	0.50	0.56	0.60	0.24	0.39	0	0
L		2000	[4.8 × 10−4, 4.9 × 10−4]	7.3 × 10−4	1.1 × 10−3	2.2 × 10−3	3.6 × 10−4	4.9 × 10−4	0	0
U	5	212	[0.44, 0.46]	0.62	0.70	0.79	0.30	0.45	0	0
L		2000	[4.8 × 10−4, 4.9 × 10−4]	7.5 × 10−4	1.1 × 10−3	1.8 × 10−3	3.7 × 10−4	4.9 × 10−4	0	0

summarizes the ± 95% confidence interval, the 95th and 99th percentiles, maximum, minimum, and average of the computation time for the proposed bi-layer MPC, along with the deadline miss rate and constraint violation rate. In Table 4, U represents the upper layer; L represents the lower layer; ± 95% CI is ± 95% confidence interval of computation time; P95 and P99 are the 95th and 99th percentile of computation time; Max., Min. and Ave. are the maximum, minimum and average computation time. As can be seen from Table 4, throughout thousands of iterations, the computation time of the lower layer consistently remained below 0.01 s, ensuring a deadline miss rate of zero. The computation time of the upper layer increases with the prediction horizon. However, since the upper layer is decoupled from real-time requirements, its computation time does not affect real-time performance. Therefore, the proposed bi-layer MPC demonstrates excellent real-time capability over thousands of iterations. Meanwhile, the constraint violation rate remains zero.

Table 5. Statistical data of the conventional MPC and MM-MPC.

	Prediction Horizon	Iteration	± 95% CI	P95	P99	Max.	Min.	Ave.	Deadline -Miss Rate	Constraint Violation Rate
C	0.5	2000	[3.6 × 10−3, 3.7 × 10−3]	5.0 × 10−3	5.9 × 10−3	7.4 × 10−3	2.7 × 10−3	3.6 × 10−3	0.00%	0
MM		2000	[3.7 × 10−3, 3.7 × 10−3]	4.3 × 10−3	4.8 × 10−3	0.010	3.0 × 10−3	3.7 × 10−3	0.00%	0
C	0.75	2000	[3.4 × 10−3, 3.5 × 10−3]	4.1 × 10−3	4.8 × 10−3	7.0 × 10−3	2.8 × 10−3	3.4 × 10−3	0.00%	0
MM		2000	[4.0 × 10−3, 4.0 × 10−3]	4.4 × 10−3	4.7 × 10−3	6.2 × 10−3	3.3 × 10−3	4.0 × 10−3	0.00%	0
C	1	2000	[5.5 × 10−3, 5.6 × 10−3]	6.7 × 10−3	7.5 × 10−3	8.9 × 10−3	4.1 × 10−3	5.6 × 10−3	0.00%	0
MM		2000	[5.8 × 10−3, 5.8 × 10−3]	6.7 × 10−3	7.1 × 10−3	9.9 × 10−3	4.7 × 10−3	5.8 × 10−3	0.00%	0
C	1.25	2000	[8.7 × 10−3, 8.8 × 10−3]	0.011	0.012	0.014	6.4 × 10−3	8.8 × 10−3	0.00%	0
MM		2000	[8.9 × 10−3, 9.0 × 10−3]	0.011	0.012	0.014	6.7 × 10−3	8.9 × 10−3	0.00%	0
C	1.5	2000	[0.014, 0.014]	0.017	0.019	0.022	9.2 × 10−3	0.014	0.00%	0
MM		2000	[0.014, 0.014]	0.017	0.019	0.022	9.9 × 10−3	0.014	0.00%	0
C	1.75	2000	[0.022, 0.022]	0.027	0.029	0.035	0.015	0.022	0.00%	0
MM		2000	[0.021, 0.022]	0.026	0.029	0.033	0.014	0.021	0.00%	0
C	2	2000	[0.031, 0.031]	0.039	0.042	0.054	0.019	0.031	0.05%	0
MM		2000	[0.030, 0.030]	0.038	0.041	0.053	0.019	0.030	0.15%	0
C	2.25	2000	[0.042, 0.043]	0.053	0.058	0.091	0.028	0.042	11.75%	0
MM		2000	[0.041, 0.041]	0.051	0.057	0.061	0.028	0.041	7.25%	0
C	2.5	2000	[0.055, 0.056]	0.071	0.078	0.090	0.037	0.056	73.40%	0
MM		2000	[0.053, 0.054]	0.066	0.074	0.082	0.034	0.053	64.35%	0
C	2.75	2000	[0.072, 0.073]	0.093	0.099	0.12	0.046	0.073	99.50%	0
MM		2000	[0.068, 0.069]	0.087	0.096	0.13	0.045	0.069	98.30%	0
C	3	2000	[0.11, 0.12]	0.17	0.20	0.28	0.057	0.11	100.00%	0
MM		2000	[0.088, 0.089]	0.11	0.12	0.13	0.053	0.088	100.00%	0
C	3.25	2000	[0.16, 0.16]	0.22	0.26	0.43	0.088	0.16	100.00%	0
MM		2000	[0.11, 0.11]	0.15	0.17	0.23	0.072	0.11	100.00%	0
C	3.5	2000	[0.18, 0.18]	0.26	0.30	0.41	0.095	0.18	100.00%	0
MM		2000	[0.15, 0.16]	0.21	0.23	0.41	0.085	0.16	100.00%	0
C	3.75	2000	[0.25, 0.25]	0.33	0.36	0.42	0.12	0.25	100.00%	0
MM		2000	[0.19, 0.19]	0.27	0.34	0.48	0.11	0.19	100.00%	0
C	4	2000	[0.33, 0.34]	0.60	0.72	0.81	0.15	0.33	100.00%	0
MM		2000	[0.25, 0.26]	0.34	0.39	0.52	0.15	0.26	100.00%	0
C	4.25	2000	[0.35, 0.37]	0.60	0.83	0.98	0.17	0.36	100.00%	0
MM		2000	[0.29, 0.30]	0.38	0.44	0.60	0.18	0.29	100.00%	0
C	4.5	2000	[0.36, 0.37]	0.61	0.88	1.1	0.20	0.36	100.00%	0
MM		2000	[0.36, 0.36]	0.48	0.59	0.84	0.22	0.36	100.00%	0
C	4.75	2000	[0.37, 0.37]	0.48	0.53	0.59	0.24	0.37	100.00%	0
MM		2000	[0.43, 0.44]	0.61	0.71	1.0	0.24	0.43	100.00%	0
C	5	2000	[0.44, 0.45]	0.58	0.74	1.1	0.28	0.44	100.00%	0
MM		2000	[0.51, 0.53]	0.73	0.89	1.2	0.28	0.52	100.00%	0

shows the data of the conventional MPC and the MM-MPC, where C represents the conventional MPC and MM represents the MM-MPC. It can be observed that both methods begin to miss deadlines when the prediction horizon is 2 s. When the prediction horizon reaches 2.25 s, both the 95th and 99th percentiles of the computation time exceed the sample interval. At a prediction horizon of 2.5 s, the average computation time surpasses the sample interval. Furthermore, the deadline miss rate reaches 100% when the prediction horizon is extended to 3 s. Consequently, the real-time performance of both the conventional MPC and the MM-MPC severely decreases as the prediction horizon increases.

Under various wave peak periods and a significant wave amplitude of 0.35 m, the computation time and average power of the three MPC methods are shown in Figure 13. As indicated in Figure 12a, to ensure real-time control, the prediction horizon in both MM-MPC and conventional MPC is set to 1.75 s. Comparing Figure 6b and Figure 12b, it can be found that under wave prediction errors, the performance of the proposed bi-layer MPC deteriorates by up to 15.5%, 24.9%, and 38.7% for prediction horizons of 1 s, 3 s, and 5 s, respectively. From Figure 13a, it can be observed that in most situation, the computation time of MM-MPC is slightly higher than that of conventional MPC, due to the additional computational module for handling model mismatch. As shown in Figure 13b, the average power of MM-MPC is higher than that of conventional MPC in most cases, yet lower than that of the proposed bi-layer MPC with prediction horizons of 3 s and 5 s. These results demonstrate that the performance improvement achieved by increasing the prediction horizon is more significant than the benefit provided by the model-mismatch compensation in MM-MPC.

5. Conclusions

This paper proposes a bi-layer model predictive control (MPC) strategy that decouples computationally intensive optimization from real-time requirements, resulting in performance enhancement through an extended prediction horizon. First, a fully enclosed underactuated wave energy converter (MU-WEC) is proposed, providing a detailed explanation of its energy absorption mechanism and establishing a control-oriented dynamic model. Next, a bi-layer MPC strategy is proposed, including a long-term energy maximizing layer and a short-term trajectory-tracking layer. The long-term energy maximizing layer periodically generates and updates energy-maximized trajectories, whereas the short-term trajectory-tracking layer ensures real-time tracking of the updated reference. This bi-layer design separates the computationally intensive part of MPC from real-time requirements, simultaneously satisfying real-time constraints while improving control performance through an extended prediction horizon. Numerical simulations validate the superiority of the proposed bi-layer MPC over conventional MPC in terms of both real-time feasibility and control performance because of its extended prediction horizon, where the absorbed power under the proposed bi-layer MPC is 127–311% greater than that of the conventional MPC.

The computational efficiency of the proposed bi-layer MPC framework creates opportunities for integrating additional high-computational control methods (such as nonlinear MPC, robust MPC) to further enhance system performance. However, this paper employs conventional model predictive controllers in both layers of the bi-layer MPC and does not extend the framework to other advanced model predictive controllers. Furthermore, the use of linear dynamics may ignore the effects caused by nonlinearities.