Research on Fourier Coefficient-Based Energy Capture for Direct-Drive Wave Energy Generation System Based on Position Sensorless Disturbance Suppression

Abstract

In order to improve the energy capture efficiency of direct-drive wave power generation (DDWEG) systems and enhance the robustness of the reference power tracking control, a Fourier coefficient-based energy capture (FCBEC) and a position sensorless disturbance suppression (PSDS) control strategy are proposed. For energy capture, FCBEC is proposed to construct the objective function by maximizing the average power over a period of time and expanding the variables in the Fourier basis when the maximum power is captured, which is used as the basis for obtaining the reference trajectory. To address the limitations of the mechanical encoder, the position sensorless technique, based on a sliding mode observer (SMO), is used in the power tracking control, and the position information is obtained through an inverse tangent function. The perturbation caused by the inverse electromotive force error in the system is theoretically analyzed. A full-order terminal sliding mode approach is employed to design a current controller that suppresses the perturbation and ensures accurate tracking of the reference current. Simulation results show that the ocean-wave energy capture strategy proposed in this paper can make the energy captured by the PTO reach the optimal value under the impedance matching condition, and that the response speed and robustness of the full-order terminal sliding mode are better than the traditional PI control.

1. Introduction

Direct-drive wave energy generation (DDWEG) systems boast advantages such as low loss and high conversion efficiency, making them a highly viable option for ocean-wave energy generation [1]. The inherent randomness of ocean waves significantly affects the efficiency of energy capture, necessitating comprehensive research on effective energy capture control strategies [2]. By leveraging the reference power obtained from these strategies, the system regulates the electromagnetic force of the permanent magnet linear synchronous generator (PMLSG) to facilitate energy transfer. Kinematic position information is a crucial control factor in this process [3]. Typically, PMLSGs are anchored to the seafloor, where mechanical sensors are vulnerable to environmental factors such as temperature fluctuations, humidity, and mechanical jitter, thus necessitating the use of position sensorless techniques to obtain positional information [4]. Nonetheless, inaccuracies in the position information obtained through sensorless methods can disrupt the system, highlighting the need for robust control schemes designed to mitigate such errors [5].

The energy capture control strategy determines the efficiency of wave energy capture, directly impacting the economic viability and sustainability of the system. Current ocean-wave energy capture control strategies are broadly categorized into causal control [6] and non-causal control [7], with each strategy utilizing real-time sea state and predicted sea state information, respectively, to control the electromagnetic force. In causal control, robustness of the hydroelectric unit to the model uncertainty is achieved; however, the core principle involves obtaining the dominant frequency in irregular waves, resulting in sub-optimal energy capture [8]. Conversely, non-causal control optimizes the predicted future sea state over a limited period, thereby maximizing ocean-wave energy capture and making it the most widely used strategy currently [9]. Model Predictive Control (MPC) algorithms are extensively applied in non-causal energy capture control [10], where the optimal trajectory is determined through quadratic programming (QP) [11,12]. While the algorithm is capable of accurately calculating the power reference value, its high computational demand and dependence on the accuracy of the Wave Energy Converter (WEC) mathematical model present certain limitations [13].

After the reference power is calculated, actuator position information must be obtained to control the electromagnetic force for power transfer. The PMLSG commonly employs position sensorless techniques to acquire this information. The position sensorless technique relies on sophisticated algorithms, which require substantial computational resources. However, it eliminates the need for physical sensors, thereby reducing system complexity and cost. By removing components that require maintenance, this approach enhances reliability, making it suitable for marine environments. Existing position sensorless methods are broadly categorized into two types based on the convex pole effect of the generator [14] and the fundamental wave model [15]. The former is suitable for the zero/low-speed phase, with position information extracted by injecting high-frequency voltage to excite the convex pole effect of the PMLSG [16]. However, when wave energy is fully exploited, linear generators operate in a nonlinear state, and actuator position information is obtained from the reversed electromotive force. Common algorithms for this include the SMO [17] and the extended Kalman filter [18]. Position estimation accuracy is a crucial index for position sensorless technology. For instance, a high-frequency rotating voltage injection method was used within 10% of the motor’s rated speed, resulting in maximum estimation errors of ±5° and ±10° during steady-state and dynamic processes, respectively [14]. Factors such as controller frequency and DC bus voltage affect position estimation, leading to perturbations [19]. These position estimation errors introduce perturbations in the system and reduce its robustness. Consequently, a new current tracking strategy can be designed to suppress these perturbations.

To suppress disturbances caused by errors in position sensorless systems, the controller strategy needs to be improved. Traditional vector control combined with a PI controller cannot adapt to complex nonlinear systems [20]. In medium- and high-speed motion, the system inertia and dynamic characteristics often result in large overshoots and an inability to effectively suppress system disturbances [21]. Sliding mode control (SMC), on the other hand, can handle system uncertainties more effectively, exhibiting strong anti-disturbance capabilities and robustness. The control variables of this method converge rapidly, making it suitable for complex marine environments [22]. However, the inherent switching function in SMC can induce high-frequency jitter and singularities in the control rate. These issues can be mitigated by designing the sliding mode surface parameters carefully, employing a switching mechanism to avoid singularities, and replacing the switching function with a broadened boundary layer and a continuous function to reduce jitter. Additionally, high-order SMC and filters can further suppress high-frequency vibrations.

In this paper, a position sensorless disturbance suppression control strategy for DDWEG based on Fourier coefficient-based energy capture is proposed and addressed under nonlinear conditions. Firstly, the method is developed by projecting the solution function onto a Fourier basis, with maximum power as the solution objective. Secondly, the perturbations generated by the position sensorless system are analyzed. Subsequently, a full-order terminal sliding mode controller is designed to suppress these perturbations, enhancing the stability and robustness of the system while ensuring rapid response to changes in reference power. Finally, the proposed strategy is verified through simulation.

2. Generation-Side Dynamic Modeling

2.1. Hydrodynamic Model

The DDWEG device discussed in this paper primarily consists of cylindrical floats, connecting rods, a permanent magnet linear generator, and other components. The schematic diagram of the device is shown in Figure 1. It is assumed that the device operates as a point absorber, where the float and the PMLSG move synchronously in a vertical direction, exhibiting single-degree-of-freedom motion under the influence of ocean waves.

The nonlinear term in the ocean wave is a quadratic viscous force, which is combined with Newton’s second law and Cummings’ equation to create the equations of motion:

$$M\ddot{z}(t)=F_e(t)+F_r(t)+F_h(t)+F_{pto}(t)+F_v\left(t\right)$$

(1)

where $F_e$ represents the excitation force, and it is assumed that the excitation n force is known in this paper. $F_{pto}$ represents the power take-off (PTO) damping force; because the PTO device is a linear generator, it can be considered that the PTO force is an electromagnetic force. $F_r$ and $F_h$ represent the wave radiation and the hydrostatic restoring force, which are linear, i.e., proportional to the WEC’s displacement, velocity, and acceleration. $F_v=C_D\dot{z}\left(t\right)\left|\dot{z}\left(t\right)\right|$ represents the secondary viscous damping force and is nonlinear, where $z\left(t\right)$, $\dot{z}\left(t\right)$, and $\ddot{z}\left(t\right)$ denote the displacement, velocity, and acceleration of the device, respectively. $C_D$ represents a constant.

The expression for the radiative force includes convolution terms that are related to future variables. Consequently, the classical power capture strategy can be analyzed using frequency expansion expressions:

$$F_e\left(\omega \right)+F_{pto}\left(\omega \right)+F_v\left(\omega \right)=j\omega \left[M+A\left(\omega \right)\right]\dot{Z}\left(\omega \right)+B\left(\omega \right)\dot{Z}\left(\omega \right)+K_h\dot{Z}\left(\omega \right)/j\omega$$

(2)

where $A(\omega )$ and $B(\omega )$ represent frequency-dependent radiation-added mass and damping. $A(\omega )$ and $B(\omega )$ can be calculated using Nemoh software (v3.0.1). The right-hand side of the equation can be equated to the inductance, resistance, and capacitance in the circuit, representing the intrinsic parameters of the wave energy generator. The velocity can be equated to the current in the circuit, and the equivalent circuit diagram is shown in Figure 2. When the intrinsic parameters are coupled with the PTO parameters, maximum power point tracking (MPPT) can be achieved. In practical applications, ocean waves are modeled as a superposition of multiple regular waves with varying frequencies and phases, necessitating the incorporation of nonlinear force considerations.

2.2. Hydrodynamic Model

The rotary motion of a permanent magnet synchronous rotary generator can be transformed into linear motion by a PMLSG, effectively aligning both types of motion. The stator voltage equation of the PMLSG in the d-q coordinate system can then be derived:

$$\begin{array}{l}{u}_d=R_s{i}_d+L_d{\displaystyle \frac{d{i}_d}{dt}}-{\omega }_e{L}_q{i}_q\\ u_q=R_s{i}_q+L_q{\displaystyle \frac{d{i}_q}{dt}}+{\omega }_e{\psi }_f+{\omega }_e{L}_d{i}_d\end{array}$$

(3)

where ${\omega }_e=\pi v/\tau$, ${\omega }_e$ represents the electrical angular frequency, $v$ represents the speed of the linear motion, and $\tau$ is the pole distance. For a surface-mounted PMLSG, the electromagnetic force can be expressed as

$$F_{em}={\displaystyle \frac{3}{2}}\left[{\displaystyle \frac{\pi }{\tau }}{\psi }_f{i}_q+{\displaystyle \frac{\pi v}{\tau }}\left(L_d-L_q\right)i_d{i}_q\right]={\displaystyle \frac{3\pi }{2\tau }}{\psi }_f{i}_q$$

(4)

3. Energy Capture Strategy for DDWEG

Once an accurate mathematical model is obtained, the optimal trajectory must be computed. In this paper, it is assumed that the future sea state is known. The FCBEC control strategy proposed here addresses challenges such as irregular waves and nonlinearities. FCBEC comprises three main components: the objective function, trajectory computation, and interval optimization.

3.1. Construction of the Objective Function

The objective function is formulated as a minimization problem over the interval:

$$\min \begin{array}{cc} & P\left(z,f_{PTO}\right)={\displaystyle \frac{1}{T}}\end{array}{\displaystyle {\int }_0^T\dot{Z}\left(t\right)F_{PTO}\left(t\right)}dt$$

(5)

where the PTO force is regarded as electromagnetic force, i.e., $F_{PTO}=F_{em}$. Signals with time-dependent velocities and forces can be expanded into the following Fourier series expansion form:

$$X\left(t\right)\approx x_1+{\displaystyle \sum _{k=1}^N\left[x_{2k}\cos \left({\omega }_{k}t\right)+x_{2k+1}\sin \left({\omega }_{k}t\right)\right]}$$

(6)

where ${\omega }_k=k{\omega }_0$ and ${\omega }_0=2\pi /T$ represent the frequency step, T represents the selected period of signal, and N represents the number of harmonics. ${\omega }_c=2N\pi /T$ represents the cutoff frequency. $X\left(t\right)$ stands for parameters such as $Z\left(t\right)$, $F_e\left(t\right)$, $F_{PTO}\left(t\right)$, $F_v\left(t\right)$, etc.

Vectors $Z$, $F_e$, $F_{PTO}$, and $F_v$ are defined as consisting of the Fourier coefficients of $Z$, $F_e$, $F_{PTO}$, and $F_v$, respectively. They are expressed as $X={\left[\begin{array}{cccccc}{x}_1& x_2& x_3& \dots & x_{2N}& x_{\left(2N+1\right)}\end{array}\right]}^T$.

Based on the calculus relationship between displacement and velocity, the relational equation for the Fourier coefficient matrix of displacement and velocity can be expressed as

$$\dot{Z}=\left[\begin{array}{c}{\dot{z}}_1\\ {\dot{z}}_2\\ {\dot{z}}_3\\ \dots \\ {\dot{z}}_{2N}\\ {\dot{z}}_{2N+1}\end{array}\right]=\left[\begin{array}{cccccc}0& & & & & \\ & 0& {\omega }_1& & & \\ & -{\omega }_1& 0& & & \\ & & & \dots & & \\ & & & & 0& {\omega }_N\\ & & & & -{\omega }_N& 0\end{array}\right]\left[\begin{array}{c}{z}_1\\ z_2\\ z_3\\ \dots \\ z_{2N}\\ z_{2N+1}\end{array}\right]=\Omega Z$$

(7)

Based on the orthogonality of the trigonometric basis, along with the hydrodynamic model and Fourier expansion mentioned above, the objective function can be formulated as follows:

$$P\left(Z,F_{PTO}\right)={\displaystyle \frac{1}{2}}{Z}^{\mathrm{T}}{\Omega }^{\mathrm{T}}{F}_{PTO}$$

(8)

3.2. Optimization of the Objective Function

An exciting force can be decomposed into different frequency components, each acting as an AC source. According to the superposition theorem, the response of a branch circuit containing multiple independent sources is equal to the algebraic sum of the responses of each independent source acting alone. In an AC circuit, the overall current response is therefore the sum of the responses from multiple single-frequency sources. Solving for an AC circuit with frequency ${\omega }_k$,

$$F_{ek}=\sqrt{f_{e2k}^2+f_{e\left(2k+1\right)}^2}\sin \left({\omega }_{k}t+{\alpha }_k\right)\Rightarrow {\dot{F}}_{ek}=\sqrt{f_{e2k}^2+f_{e\left(2k+1\right)}^2}\angle {\alpha }_k=f_{e\left(2k+1\right)}+j{f}_{e2k}$$

(9)

$${\dot{Z}}_k={\omega }_k\sqrt{z_{2k}^2+z_{2k+1}^2}\sin \left({\omega }_{k}t+{\beta }_k\right)\Rightarrow {\dot{\dot{Z}}}_k={\omega }_k\sqrt{z_{2k}^2+z_{2k+1}^2}\angle {\beta }_k=-{\omega }_k{z}_{2k}+j{\omega }_k{z}_{2k+1}$$

(10)

The solution is derived by analyzing the AC circuit and separately solving for its real and imaginary parts. From this, the Fourier coefficient matrix of $F_{PTO}$ under irregular wave excitation, derived in the Fourier basis according to the superposition theorem, can be formulated as follows:

$$\begin{array}{c}{F}_{PTO}=HZ-F_e-F_v\\ H=\left[\begin{array}{cccc}0& & & \\ & H_1& & \\ & & \dots & \\ & & & H_N\end{array}\right]\\ H_k=\left[\begin{array}{cc}-(m+A({\omega }_k)){\omega }_k^2+k_h& {\omega }_{k}B({\omega }_k)\\ -{\omega }_{k}B({\omega }_k)& -(m+A({\omega }_k)){\omega }_k^2+k_h\end{array}\right]\end{array}$$

(11)

where $H_k$ is a matrix representing the characteristics of eigenparameters at different frequencies. According to (11), the PTO force is expressed as a function related to $Z$. Minimizing this objective function can be achieved through gradient-based optimization techniques, ultimately reducing it to a quadratic programming problem.

3.3. Optimal Trajectory Solution

By substituting (11) into (8), maximizing power involves solving for the optimal speed of motion, thereby enabling effective power tracking control. The expression for power is differentiated into linear and nonlinear components:

$$P\left(Z\right)=P_l\left(Z\right)+P_{nl}\left(Z\right)={\displaystyle \frac{1}{2}}\left(Z^{\mathrm{T}}{\Omega }^{\mathrm{T}}HZ-Z^{\mathrm{T}}{\Omega }^{\mathrm{T}}{F}_e\right)-{\displaystyle \frac{1}{2}}\left(Z^{\mathrm{T}}{\Omega }^{\mathrm{T}}{F}_v\right)$$

(12)

where $P_l$ represents the power generated by linear forces, and $P_{nl}$ represents the power generated by nonlinear forces. From (12), it is evident that power varies with $Z$. To determine the maximum power value, the first-order optimality condition is utilized, and the matrix expression combining the Fourier coefficients is formulated as follows:

$$\begin{array}{c}{\nabla P|}_Z={\nabla P_l|}_Z+{\nabla P_{nl}|}_Z==\left(DZ-{\displaystyle \frac{1}{2}}{\Omega }^{\mathrm{T}}{F}_e\right)+\left(-{\displaystyle \frac{1}{2}}{Z}^{\mathrm{T}}{\Omega }^{\mathrm{T}}{F}_v\right){|}_Z=\mathit{0}\\ D=\left[\begin{array}{cccccc}0& & & & & \\ & {\omega }_1{H}_{1\left(1,2\right)}& & & 0& \\ & & {\omega }_1{H}_{1\left(1,2\right)}& & & \\ & & & \dots & & \\ & 0& & & {\omega }_n{H}_{n\left(1,2\right)}& \\ & & & & & {\omega }_n{H}_{n\left(1,2\right)}\end{array}\right]\end{array}$$

(13)

where $D$ represents the transition matrix generated by gradient calculation, and $H_{k\left(1,2\right)}$ represents the element in the first column and second row of the $H_k$ matrix. The secondary viscous drag term is velocity-dependent and can therefore be derived as follows:

$${\displaystyle \frac{\partial P_{nl}}{\partial Z}}={\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^T{\left(\dot{Z}\right)}^{\prime }}\left(\dot{Z}{f}_v^{\prime }\left(\dot{Z}\right)+f_v\left(\dot{Z}\right)\right)dt$$

(14)

The orthogonality formula holds true for three sets of trigonometric functions. The gradient solution of the quadratic viscous force, decomposed into a Fourier series of N frequencies, is expressed as follows:

$$\begin{array}{cc}{\displaystyle \frac{\partial P_{nl}}{\partial z_{2k}}}& ={\displaystyle \frac{1}{T}}{\displaystyle {\int }_0^T{\omega }_k\sin \left({\omega }_{k}t\right)}\left(\dot{z}{f}_v^{\prime }\left(\dot{z}\right)+f_v\left(\dot{z}\right)\right)dt\hfill \\ & =-{\displaystyle \frac{3}{8}}{\omega }_k({\omega }_m{z}_{2m+1}{\omega }_{\left|k-m\right|}{z}_{2\left|k-m\right|}+{\omega }_m{z}_{2m+1}{\omega }_{k+m}{z}_{2\left(k+m\right)}+{\omega }_m{z}_{2m}{\omega }_{\left|k-m\right|}{z}_{2\left|k-m\right|+1}\hfill \\ & +{\omega }_m{z}_{2m}{\omega }_{k+m}{z}_{2\left(k+m\right)+1})-{\displaystyle \sum _{m=n-k+1}^N\left({\omega }_m{z}_{2m+1}{\omega }_{\left|k-m\right|}{z}_{2\left|k-m\right|}+{\omega }_m{z}_{2m}{\omega }_{\left|k-m\right|}{z}_{2\left|k-m\right|+1}\right)}\hfill \end{array}$$

(15)

$$\begin{array}{cc}{\displaystyle \frac{\partial P_{nl}}{\partial z_{2k+1}}}& =-{\displaystyle \frac{1}{T}}{\displaystyle {\int }_0^T{\omega }_k\cos \left({\omega }_{k}t\right)}\left(\dot{z}{f}_v^{\prime }\left(\dot{z}\right)+f_v\left(\dot{z}\right)\right)dt\hfill \\ & =-{\displaystyle \frac{3}{8}}{\omega }_k{\displaystyle \sum _{m=1}^{N-k}({\omega }_m{z}_{2m+1}{\omega }_{\left|k-m\right|}{z}_{2\left|k-m\right|+1}+{\omega }_m{z}_{2m+1}{\omega }_{k+m}{z}_{2\left(k+m\right)+1}+{\omega }_m{z}_{2m}{\omega }_{\left|k-m\right|}{z}_{2\left|k-m\right|}}\hfill \\ & +{\omega }_m{z}_{2m}{\omega }_{k+m}{z}_{2\left(k+m\right)})-{\displaystyle \sum _{m=n-k+1}^N\left({\omega }_m{z}_{2m+1}{\omega }_{\left|k-m\right|}{z}_{2\left|k-m\right|+1}+{\omega }_m{z}_{2m}{\omega }_{\left|k-m\right|}{z}_{2\left|k-m\right|}\right)}\hfill \end{array}$$

(16)

When $z>0$, the expression for the gradient optimal solution is presented above. Conversely, the equation reverses under the condition $z<0$. The optimal trajectory equation can be derived by incorporating this equation into (13). Additionally, iterative methods can be employed to obtain the solution. Starting with the displacement under linear conditions as the initial guess, the secondary viscous force is handled with the same gradient as the excitation force. Iterating within an acceptable error range allows for determination of the motion velocity.

3.4. Interval Time Domain Optimization

During practical analysis, the spectrum is limited to a finite time period, necessitating truncation of the signal. Due to the non-periodicity of the excitation force signal, it is necessary to truncate the signal using a window function and extend it periodically. Non-periodic truncation of the signal can lead to severe sidelobe effects, causing frequency leakage. The window function improves the periodic handling of the signal in the time domain, reducing leakage. A Hanning window $\omega \left(n\right)=0.5-0.5\cos \left(2n\pi /N\right),n=0~N-1$ is employed for this purpose, as depicted in Figure 3.

After window function processing, the reference trajectory is determined using the FCBEC control strategy, and the reference power is calculated. Once the reference trajectory for the current interval is obtained, the above operations are repeated for the next interval until the entire solving range is completed. Based on the above analysis, the structure of the proposed FCBEC control strategy is illustrated in Figure 4.

4. Perturbation Analysis of Position Sensorless Theory

After obtaining the reference power through FCBEC, power transmission is achieved by controlling the electromagnetic force. In Field-Oriented Control (FOC), coordinate transformation is used to equivalently represent an AC motor as a DC motor, making position information critically important. Errors between the actuator position obtained through sensorless techniques and the actual actuator position can disrupt the system. Therefore, this paper proposes the PSDS control strategy and designs corresponding controllers to suppress disturbances.

4.1. Position Sensorless Strategy

In this paper, it is mentioned that DDWEG operates in a nonlinear state, characterized by high wave frequencies and amplitudes. During such conditions, the PMLSG moves at high speeds, generating significant reactive forces within the system, from which position information can be estimated. A Sliding Mode Observer (SMO) does not require high accuracy in the system model and is insensitive to parameter variations and external disturbances, making it a robust control method. In the control system of a PMLSG, this method designs an SMO based on the error between the desired current and feedback current. The extended back electromotive force (EMF) encapsulates all positions and velocities of the generator actuator; hence, accurate estimation of the extended back EMF is essential for determining the generator’s position information. Based on the equation of state of the generator, the stator current error is designed as follows:

$${\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{\widehat{i}}_{\alpha }-i_{\alpha }\\ {\widehat{i}}_{\beta }-i_{\beta }\end{array}\right]={\displaystyle \frac{1}{L_d}}\left[\begin{array}{cc}-R_s& -{\omega }_e\left(L_d-L_q\right)\\ {\omega }_e\left(L_d-L_q\right)& -R_s\end{array}\right]\left[\begin{array}{c}{\widehat{i}}_{\alpha }-i_{\alpha }\\ {\widehat{i}}_{\beta }-i_{\beta }\end{array}\right]-{\displaystyle \frac{1}{L_d}}\left[\begin{array}{c}{\widehat{E}}_{\alpha }-E_{\alpha }\\ {\widehat{E}}_{\beta }-E_{\beta }\end{array}\right]$$

(17)

where ${\widehat{E}}_{\alpha }$ and ${\widehat{E}}_{\beta }$ represent the observed values of the back electromotive force. ${\widehat{i}}_{\alpha }$ and ${\widehat{i}}_{\beta }$ represent the observed values of the current and the design slip mold surface: $s_{\alpha }={\widehat{i}}_{\alpha }-i_{\alpha }$, $s_{\beta }={\widehat{i}}_{\beta }-i_{\beta }$. We define the Lyapunov function as $V=0.5\left(s_{\alpha }^2+s_{\beta }^2\right)$. To ensure that $s_{\alpha }d{s}_{\alpha }/dt+s_{\beta }d{s}_{\beta }/dt<0$, we design the SMC law as

$$\left[\begin{array}{c}{\widehat{E}}_{\alpha }\\ {\widehat{E}}_{\beta }\end{array}\right]=\left[\begin{array}{c}k\mathrm{sgn}\left(s_{\alpha }\right)\\ k\mathrm{sgn}\left(s_{\beta }\right)\end{array}\right]$$

(18)

When k satisfies $k>\max \left\{-R_s\left|{\widehat{i}}_{\alpha }-i_{\alpha }\right|+E_{\alpha }\mathrm{sgn}\left({\widehat{i}}_{\alpha }-i_{\alpha }\right),-R_s\left|{\widehat{i}}_{\beta }-i_{\beta }\right|+E_{\beta }\mathrm{sgn}\left({\widehat{i}}_{\beta }-i_{\beta }\right)\right\}$, $\dot{V}\le 0$ is constant, when $\dot{V}\equiv 0$ and $s_{\alpha }\equiv s_{\beta }\equiv 0$, the system is gradually stabilized according to the LaSalle invariance principle, and when $t\to 0$, $s\to 0$. The rate of convergence depends on k.

Discontinuous high-frequency switching signals are usually processed using a low-pass filter. Position information can usually be obtained using the inverse tangent function method, with compensation for the delay effect of the low-pass filter:

$${\widehat{\theta }}_e=-\arctan \left({\widehat{E}}_{\alpha }/{\widehat{E}}_{\beta }\right)+\arctan \left({\widehat{\omega }}_e/{\omega }_c\right)$$

(19)

where ${\omega }_c$ represents the cutoff frequency of the low-pass filter $G\left(s\right)={\omega }_c/\left(s+{\omega }_c\right)$, and ${\widehat{\omega }}_e$ represents the observed value of the motor angular velocity. The block diagram of the SMO algorithm is shown in Figure 5.

4.2. Perturbation Analysis Based on Angular Error

Due to generator parameter asymmetry and other factors, inaccurate decoupling of the alternating and direct axis components may occur, leading to current position estimation errors and subsequent disturbances in the system. To mitigate the impact of position information errors, it is necessary to analyze the disturbances caused by these errors. Assuming the actual position is ${\theta }_e$, the relationship between the estimated position and the actual position is given by ${\widehat{\theta }}_e={\theta }_e+\Delta \theta$. The relationship between the estimated and actual values of the d-axis and q-axis currents obtained through coordinate transformation satisfies

$$\begin{array}{l}{i}_d={\widehat{i}}_d\cos \left(\Delta \theta \right)-{\widehat{i}}_q\sin \left(\Delta \theta \right)\\ i_q={\widehat{i}}_d\sin \left(\Delta \theta \right)+{\widehat{i}}_q\cos \left(\Delta \theta \right)\end{array}$$

(20)

where ${\widehat{i}}_d$ and ${\widehat{i}}_q$ are the d- and q-axis current values calculated from the estimated position information, $i_d$ and $i_q$ are the actual d- and q-axis current values, and $\Delta \theta$ is the error of the position information. When $\Delta \theta$ is small, it satisfies $\sin \left(\Delta \theta \right)\approx \Delta \theta$, $\cos \left(\Delta \theta \right)\approx 1$. By neglecting higher-order terms and rewriting the equation, the current equations in the d-axis and q-axis coordinate system are given as follows:

$$\begin{array}{l}{\displaystyle \frac{d{\widehat{i}}_d}{dt}}-\Delta \theta {\displaystyle \frac{d{\widehat{i}}_q}{dt}}=-{\displaystyle \frac{1}{L_d}}\left(R_s-{\omega }_e{L}_q\Delta \theta \right){\widehat{i}}_d+{\displaystyle \frac{1}{L_d}}\left(R_s\Delta \theta +{\omega }_e{L}_q\right){\widehat{i}}_q+{\displaystyle \frac{1}{L_d}}{u}_d\\ {\displaystyle \frac{d{\widehat{i}}_q}{dt}}+\Delta \theta {\displaystyle \frac{d{\widehat{i}}_d}{dt}}=-{\displaystyle \frac{1}{L_q}}\left(R_s-{\omega }_e{L}_d\Delta \theta \right){\widehat{i}}_q-{\displaystyle \frac{1}{L_q}}\left(R_s\Delta \theta +{\omega }_e{L}_d\right){\widehat{i}}_d-{\displaystyle \frac{1}{L_q}}{\omega }_e{\psi }_f+{\displaystyle \frac{1}{L_q}}{u}_q\end{array}$$

(21)

Based on the above analysis, the expression for the disturbance caused by the position information error in the system can be derived. The disturbance reduces the robustness of the system and can be suppressed by designing an appropriate controller.

4.3. Full-Order Terminal SMC Theory

SMC is an effective disturbance suppression solution. Compared to Linear Sliding Mode (LSM), terminal sliding mode (TSM) enables system states to converge to zero within a finite time, offering better dynamic performance. By meticulously designing the sliding mode surface, TSM minimizes the influence of switching terms, effectively mitigating chattering. This study proposes a comprehensive full-order terminal SMC strategy that adeptly addresses both chattering and singularity issues, thereby enhancing overall control performance. Taking a second-order nonlinear system as an example, its state equation is given by

$$\{\begin{array}{c}{\dot{x}}_1=x_2\\ {\dot{x}}_2=f\left(x,t\right)+d\left(x,t\right)+bu\end{array}$$

(22)

where $x={\left[\begin{array}{cc}{x}_1& x_2\end{array}\right]}^T$, $f\left(x,t\right)$, and $d\left(x,t\right)$ represent functions with respect to $x$, $d\left(x,t\right)$ represents a systematic perturbation term satisfying $\left|d^{\prime }\left(x,t\right)\right|\le k_d$, and $u$ represents the sliding mode control rate. The following sliding mode surface is selected:

$$\begin{array}{c}{s}_1={\ddot{e}}_1+c_2{\left|{\dot{e}}_1\right|}^{a_1}\mathrm{sgn}\left({\dot{e}}_1\right)+c_1{\left|e_1\right|}^{a_2}\mathrm{sgn}\left(e_1\right)\\ e_1=x_1^{*}-x_1\end{array}$$

(23)

where $x_1^{*}$ represents the given reference value; $c_n>0$ and $a_n>0$ represent the design parameters; and the polynomial $p^2+c_2{p}^{a_1}+c_1{p}^{a_2}$ satisfies the Hurwitz stability condition. The full-order terminal SMC achieves smoother control inputs by introducing high-order sliding mode surface design, effectively reducing or eliminating the chattering issues associated with traditional SMC methods. The full-order terminal control rate consists of the equivalent control rate $u_{eq}$ and the switching robust control $u_{sw}$, as shown in the following equation:

$$u=u_{eq}+u_{sw}$$

(24)

Disregarding the disturbance and uncertainty terms, the equivalent controller can be designed as

$$u_{eq}={\displaystyle \frac{1}{b}}\left({\ddot{x}}_1^{*}-f\left(x,t\right)+c_2{\left|{\dot{e}}_1\right|}^{a_2}\mathrm{sgn}\left({\dot{e}}_1\right)+c_1{\left|e_1\right|}^{a_1}\mathrm{sgn}\left(e_1\right)\right)$$

(25)

We define the Lyapunov function as

$$V=0.5{s_1}^2$$

(26)

$$s_1{\dot{s}}_1=s_1\left(-b{\dot{u}}_{sw}-d^{\prime }\left(x,t\right)\right)$$

(27)

To ensure system stability and establish the sliding mode arrival condition, the switching control is designed as follows:

$${\dot{u}}_{sw}=k\mathrm{sgn}\left(s_1\right)/b$$

(28)

Bringing (28) into (27) gives

$$\dot{V}=s_1{\dot{s}}_1=s_1\left(-b\cdot {\displaystyle \frac{k}{b}}\mathrm{sgn}\left(s_1\right)-d^{\prime }\left(x,t\right)\right)=-k\left|s_1\right|-s_1{d}^{\prime }\left(x,t\right)\le 0$$

(29)

To satisfy the Lyapunov condition, ensuring $k\ge k_d$, the control rate of the equivalent SMC strategy can be expressed as follows:

$$\begin{array}{c}u=u_{eq}+u_{sw}\\ u_{eq}={\displaystyle \frac{1}{b}}\left({\ddot{x}}_1^{*}-f\left(x,t\right)+c_2{\left|{\dot{e}}_1\right|}^{a_2}\mathrm{sgn}\left({\dot{e}}_1\right)+c_1{\left|e_1\right|}^{a_1}\mathrm{sgn}\left(e_1\right)\right)\\ {\dot{u}}_{sw}={\displaystyle \frac{k}{b}}\mathrm{sgn}\left(s_1\right)\end{array}$$

(30)

The design of the full-order terminal sliding mode control law excludes the negative exponential term of the state variables, thereby avoiding singularities and enabling precise trajectory tracking and control objectives.

4.4. Design of Current Loop Controller

The full-order terminal sliding mode control is introduced into the mathematical model of the PMLSG, which includes position disturbance, for current tracking. Based on the optimal speed, the given reference value of q-axis current $i_q$ is defined, and the current error variable is

$$e_q=i_q^{*}-i_q\approx i_q^{*}-\left({\widehat{i}}_q+{\widehat{i}}_d\Delta \theta \right)$$

(31)

$${\dot{e}}_q={\displaystyle \frac{d}{dt}}{i}_q^{*}-{\displaystyle \frac{d}{dt}}\left({\widehat{i}}_q+{\widehat{i}}_d\Delta \theta \right)={\displaystyle \frac{d}{dt}}{i}_q^{*}+{\displaystyle \frac{1}{L_q}}\left(R_s-{\omega }_e{L}_d\Delta \theta \right){\widehat{i}}_q+{\displaystyle \frac{1}{L_q}}\left(R_s\Delta \theta +{\omega }_e{L}_d\right){\widehat{i}}_d+{\displaystyle \frac{1}{L_q}}{\omega }_e{\psi }_f-{\displaystyle \frac{1}{L_q}}{u}_q$$

(32)

The full-order terminal slip mode surface is designed based on the q-axis current equation of state:

$$s_q={\dot{e}}_q+c_q{\left|e_q\right|}^{a_q}\mathrm{sgn}\left(e_q\right)$$

(33)

According to the current error system shown in (32), the control rate expression can be obtained by choosing the full-order terminal slip mode surface as in (33) as follows:

$$\begin{array}{c}{u}_q=u_{eq\_q}+u_{sw\_q}\\ u_{eq\_q}=L_q{\dot{i}}_q^{*}+R_s{\widehat{i}}_q+{\omega }_e{L}_d{\widehat{i}}_d+{\omega }_e{\psi }_f+L_q{c}_q{\left|e_q\right|}^{a_q}\mathrm{sgn}\left(e_q\right)\\ {\dot{u}}_{sw\_q}=L_q{k}_q\mathrm{sgn}\left(s_q\right)\end{array}$$

(34)

The Lyapunov function is chosen for the proof:

$${\dot{V}}_q=s_q{\dot{s}}_q=s_q\left[{\left(-{\displaystyle \frac{1}{L_q}}{\omega }_e{L}_d\Delta \theta {\widehat{i}}_q+{\displaystyle \frac{1}{L_q}}{R}_s\Delta \theta {\widehat{i}}_d\right)}^{\prime }-k_q\mathrm{sgn}\left(s_q\right)\right]\le 0$$

(35)

Similarly, the sliding mode surface of the d-axis current can be designed as

$$s_d={\dot{e}}_d+c_d{\left|e_d\right|}^{a_d}\mathrm{sgn}\left(e_d\right)$$

(36)

The control rate expression is

$$\begin{array}{c}{u}_d=u_{eq\_d}+u_{sw\_d}\\ u_{eq\_d}=L_d{\dot{i}}_d^{*}+R_s{\widehat{i}}_d-{\omega }_e{L}_q{\widehat{i}}_q+L_d{c}_d{\left|e_d\right|}^{a_d}\mathrm{sgn}\left(e_d\right)\\ {\dot{u}}_{sw\_d}=L_d{k}_d\mathrm{sgn}\left(s_d\right)\end{array}$$

(37)

Based on the analysis above, the DDWEG system mentioned in this paper can be divided into wave energy capture, a sensorless module, and a power tracking module. The overall control diagram is shown in Figure 6.

5. Simulation Verification

5.1. FCBEC Simulation and Discussion

To validate the superiority of the proposed FCBEC control strategy, a comparison is made with the amplitude control strategy. This paper utilizes Matlab/Simulink (2023a) as a simulation platform to analyze the proposed control strategy, with a cylindrical buoy shape, where the cylindrical float has a base radius of 5 m and a height of 10 m. The JS spectrum is used to analyze the ocean waves. It is assumed that the sea state for the next 40 s is known, simulating a JS spectrum for 40 s, with a significant wave height of 4 m and a peak period of 2.4 s. A 40 s interval is chosen as the window length for analysis, serving as the period for Fourier analysis. Under the excitation force, according to a gradient-based solution scheme, the calculated PTO force is shown in Figure 7, and the optimal trajectory’s optimal velocity is determined with the objective of maximizing average power, as shown in Figure 8.

Figure 9 displays the energy and power of the excitation force alongside the energy and power generated by the PMLSG after obtaining the optimal velocity through solving. From Figure 9, it is evident that the power of the excitation force remains predominantly positive, indicating that in the equivalent circuit diagram, the excitation force and velocity are in phase, and the overall load is resistive. For the PMLSG, its power fluctuates around zero, implying bidirectional power flow under conditions that satisfy impedance matching in the equivalent circuit. It is noteworthy that the power of the excitation force occasionally falls below zero, which may be attributed to inaccuracies introduced by the Fourier decomposition in approximating the original function. In terms of cumulative energy, the energy of the excitation force continues to rise. Numerically, the energy output from the PTO accounts for approximately 50%, aligning with the optimal energy distribution under impedance matching conditions in the circuit, thus validating the feasibility of the FCBEC control strategy. Furthermore, it is essential to equip the PTO with energy storage devices to ensure stable energy flow and system stability.

Figure 10 illustrates the power and energy captured by the PTO under amplitude control conditions. Since it employs single-degree-of-freedom control, it lacks the characteristic of bidirectional energy flow. It is evident that the captured energy is significantly less than that achieved by the proposed FCBEC control strategy. From the perspective of Fourier decomposition, the excitation force used in this study contains substantial high-frequency components, distributed uniformly in the high-frequency range. The amplitude control has limited efficiency in absorbing wave energy from these high-frequency components, thus demonstrating the superiority of the FCBEC strategy.

5.2. PSDS Simulation and Discussion

The simulation model was developed based on the positionless sensing technique using an SMO and the inverse tangent function. Figure 11 illustrates the actual electrical angle, as well as the electrical angle and angular error obtained through the positionless sensing technique.

The reference value of the q-axis current can be calculated according to (4). From Figure 10, it can be observed that after removing the error singularities caused by the residual 2π, the estimation error of the actuator position ranges between 0 and 5°. The disturbance range of the system is considered to be ±0.05, thereby enabling the design of a sliding mode controller. Using the control method with id = 0, the reference trajectories for iq and id are obtained. The waveforms of id and iq obtained through this method are shown in Figure 12, alongside those obtained from PI control.

From Figure 12, it can be observed that the full-order terminal sliding mode control strategy tracks the reference value more rapidly during sudden current changes and exhibits smaller overshoot. At the wave peaks and troughs, the tracking error of the full-order terminal sliding mode control strategy is significantly smaller than that of the PI control strategy. Furthermore, the full-order terminal SMC demonstrates a quicker settling time and reduced overshoot relative to the PI control method, indicating that it achieves steady-state operation more rapidly and exhibits superior performance compared to the conventional PI controller.

6. Conclusions

• Addressing the challenge of wave energy capture, this paper introduces an FCBEC control strategy that is applicable in nonlinear marine environments by optimizing the Fourier coefficients of the objective function through gradient-based methods. Simulation results demonstrate that within a defined timeframe, the FCBEC strategy ensures that the excitation force remains in phase with velocity. Moreover, the PTO’s equivalent impedance meets impedance matching conditions, enabling the capture of approximately 50% of the excitation force’s total energy. The superiority of the FCBEC strategy is clearly demonstrated through comparison with amplitude control. However, this approach requires a high-capacity energy storage unit for the power take-off (PTO) device, which leads to increased costs. This represents a potential avenue for future research aimed at optimizing the cost-effectiveness of the system.
• The position error of the actuator is determined by comparing the SMO-based back electromotive force (BEMF) method with encoder data. To mitigate perturbation effects on the system, an analysis of the disturbances caused by position errors is conducted, and a full-order terminal SMC strategy is developed to precisely track the current. Simulation results confirm that the full-order sliding mode control strategy achieves faster tracking speeds and enhances system robustness compared to traditional PI controllers.