Control of Direct-Drive Wave Energy Conversion Considering Displacement Constraints and an Improved Sensorless Strategy

Abstract

An integrated control strategy is proposed for direct-drive wave energy conversion (DDWEC) systems to address displacement safety constraints and improve the robustness of sensorless position estimation. Under strong wave excitation, buoy displacement may exceed its stroke limit due to conventional amplitude control, leading to mechanical risks. To mitigate this, a displacement-constrained damping regulation law is introduced, incorporating a displacement-dependent correction factor that retains optimal damping within a safe region and increases additional damping smoothly as the displacement approaches its limit. For sensorless operation, a dual-time-scale adaptive amplitude modulation strategy is developed, based on high-frequency square-wave voltage injection. By decoupling the fast position-estimation loop from the slow injection-amplitude adjustment, the demodulated high-frequency current remains within an optimal band, ensuring a high signal-to-noise ratio (SNR) under disturbances and parameter variations. Simulation results show that displacement boundary violations are eliminated, with a 25.7% reduction in peak displacement and only a 7.65% reduction in average captured power. The injection amplitude is adaptively regulated to maintain the demodulated current within the measurement band, enhancing position-estimation stability and accuracy. A fail-safe boundary for extreme sea states (H_s ≈ 2.2 m) is also identified, ensuring robust operation under varying conditions.

1. Introduction

Ocean wave energy is widely regarded as a clean renewable resource with abundant potential and relatively high predictability, and it has been receiving increasing attention in the context of the global energy transition and the “dual-carbon” goals [1,2]. In DDWEC systems, a permanent magnet linear synchronous generator (PMLSG) is directly coupled to a buoy, and the reciprocating wave-induced motion is converted into electrical power without intermediate mechanical transmission stages. Owing to the resulting simple structure, high energy-conversion efficiency, and high reliability, DDWEC architectures have been considered a promising development direction for wave energy technologies [3,4].

Maximizing wave energy capture is widely recognized as the core objective of DDWEC systems. Within linear hydrodynamic theory and equivalent-circuit representations, classical complex-conjugate control (also known as impedance-matching control) is derived by matching the intrinsic system impedance to the power-take-off (PTO) impedance under regular waves, and maximum power capture can be achieved in theory [5,6]. In realistic ocean environments, however, the stochastic nature of wave excitation and the finite mechanical stroke of the device introduce an inherent trade-off between energy capture and safe operation. Under energetic sea states, continued tracking of the theoretical optimum damping can drive the buoy displacement to, or beyond, its physical stroke limit, potentially causing end-stop impacts, device damage, or safety incidents [7,8]. Although maximum power point tracking under irregular waves has been widely studied, including wave-prediction-based model predictive control [9,10] and model-free intelligent control [11], further investigation is still needed into strategies that can incorporate buoy-displacement constraints directly and smoothly into the real-time power-capture loop while preserving dynamic energy-capture optimization. In particular, achieving a smooth and adaptive transition from maximum-power-capture operation to safety-first operation as the stroke limit is approached remains a key problem of both theoretical and engineering significance. Recent stroke-constrained control studies for wave energy converters include causal/constrained MPC formulations and displacement-constrained reactive control schemes [12,13]. Although these approaches differ in implementation, they consistently indicate a displacement–power trade-off: stricter motion suppression improves stroke safety, but usually reduces absorbed power when the constraint is enforced more conservatively near the stroke boundary [13].

Accurate mover position information is required for space-vector modulation and high-performance control. In harsh marine environments, however, mechanical position sensors are difficult to install and maintain, are costly, and often exhibit reduced reliability [14]; therefore, position-sensorless techniques are typically preferred. Because the permanent-magnet linear machine in DDWEC systems operates for extended periods in the low-speed region, back-EMF-based observers generally suffer from insufficient SNR, whereas HF signal injection provides effective observability [15,16]. Among HF-injection approaches, square-wave voltage injection is attractive because it can provide fast dynamic response without complex filters [17]. Nevertheless, under wave-induced load fluctuations, time-varying machine parameters, and strong noise, a fixed injection amplitude cannot ensure that the demodulated HF current remains within the optimal observation region. If the injected signal is too weak, the SNR of position estimation deteriorates and the phase-locked loop (PLL) may become unstable. If it is too strong, additional losses and current ripple are introduced, thereby degrading thrust smoothness [18,19]. Moreover, when adaptive amplitude modulation is implemented on a single time scale, coupling between estimation errors and amplitude updates may form a positive-feedback path and reduce system stability [20]. Therefore, a dual-time-scale adaptive mechanism, in which amplitude regulation is decoupled from the position-estimation process, is needed to robustly maintain the observation signal strength within an optimal range across varying sea states.

To address these challenges, an integrated control strategy is proposed for DDWEC systems, combining displacement-constrained optimal power capture with improved sensorless position estimation. First, for power capture, a smooth damping regulation strategy is developed from a displacement-constrained model, where a displacement-dependent correction factor is embedded into conventional optimal damping control. Within a prescribed safe operating range, the theoretical optimal damping is retained to maximize energy capture. As the buoy displacement approaches its stroke limit, an additional damping term is increased smoothly to suppress motion amplitude, ensuring mechanical safety. A traditional end-stop spring–damper limiter is included as a baseline for comparison. For sensorless control, a dual-time-scale adaptive amplitude modulation method based on high-frequency square-wave voltage injection is introduced. High-frequency current extraction and position estimation are updated on a fast time scale, while injection amplitude optimization is performed on a slower time scale. By monitoring the demodulated HF current, the injection amplitude is regulated to maintain the observation signal within a predefined optimal interval, ensuring a consistently high signal-to-noise ratio (SNR) under disturbances and parameter variations. Simulation results show that displacement boundary violations are eliminated, with a 25.7% reduction in peak displacement and only a 7.65% decrease in average absorbed power. The proposed method also results in a 3.76% higher average power than the mechanical limiter baseline under the same zero-violation condition. A sea-severity sweep further identifies a fail-safe boundary around H_s ≈ 2.2 m for a 1% violation criterion. The effective measurement ratio is increased from 25.84% to 99%, while position-estimation accuracy is maintained with a position-error metric of 0.06 m.

2. Mathematical Model of the Wave Power Generation System

2.1. Hydrodynamic Model

The point-absorber buoy is used as the interface between the DDWEC system and the incident ocean waves. The reciprocating wave motion is converted into relative linear motion between the permanent magnets and windings, thereby enabling the conversion of mechanical energy into electrical power. In this work, a PMLSG is adopted as the PTO unit, and its mover is mechanically coupled to the point-absorber buoy. A structural schematic of the device is shown in Figure 1. By analyzing the hydrodynamic interaction forces between the buoy and the surrounding fluid, the buoy motion is derived, and a system-level dynamic model of the DDWEC system is established.

When the PMLSG is integrated with the buoy-type wave energy converter, the overall equation of motion of the wave power generation system is formulated by combining the buoy hydrodynamic model with the mechanical dynamics of the permanent-magnet linear machine. According to linear hydrodynamic theory, the net mechanical input force can be expressed as:

$$f_m=f_w+f_{\rho hg}+f_r,$$

(1)

where f_w is the wave excitation force, f_r is the wave radiation force, and f_ρhg is the hydrostatic restoring force of the buoy.

The hydrostatic restoring force is obtained by integrating the hydrostatic pressure (including gravitational effects) over the wetted surface and can be written as:

$$f_{\rho hg}=-K_{\rho hg}x,$$

(2)

where K_ρhg = ρgS_buoy is determined by seawater density and gravitational acceleration, S_buoy is the buoy bottom area, and x denotes the generator mover displacement.

The radiation force f_r represents the fluid reaction induced by buoy motion and accounts for both radiation-damping and added-mass effects. For linear regular waves, f_r can be expressed as:

$$f_r=-m_{add}\ddot{x}-K_z\dot{x},$$

(3)

where m_add is the added mass associated with radiation, and K_z is the radiation-damping coefficient.

2.2. Electromechanical Model of the Permanent-Magnet Linear Machine

To derive the machine model, the following assumptions are adopted: (i) the reluctances of the mover and armature iron are neglected; (ii) end effects are ignored; (iii) hysteresis and eddy-current losses are neglected; and (iv) magnetic saturation is neglected. Using a coordinate transformation, the armature voltage equations of the permanent-magnet linear machine in the three-phase abc reference frame are expressed in the synchronous d–q reference frame as [9]:

$$\left[\begin{array}{c}{u}_d\\ u_q\end{array}\right]=\left[\begin{array}{c}0\\ {\omega }_e{\psi }_f\end{array}\right]-\left[\begin{array}{cc}{R}_s& 0\\ 0& R_s\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]-\left[\begin{array}{cc}{L}_d& 0\\ 0& L_q\end{array}\right]{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}{\omega }_e{L}_q{i}_q\\ -{\omega }_e{L}_d{i}_d\end{array}\right],$$

(4)

where u_d and u_q are the d- and q-axis armature-voltage components, i_d and i_q are the corresponding current components, ω_e represents the electrical angular velocity of the linear generator, ψ_f signifies the permanent magnet excitation flux linkage, and R_s is the armature winding resistance.

The instantaneous electromagnetic power of the permanent-magnet linear machine in the d–q reference frame is obtained as the sum of the d- and q-axis power components:

$$P_{em}={\displaystyle \frac{3}{2}}\left(e_d{i}_d+e_q{i}_q\right),$$

(5)

e_d and e_q represent the d-axis and q-axis induced electromotive forces.

Under steady-state conditions, the integral terms of i_d and i_q can be neglected; therefore, the electromagnetic power of the permanent-magnet linear machine is simplified to:

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

(6)

Accordingly, the electromagnetic thrust generated by the permanent-magnet linear machine is derived as:

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

(7)

τ represents the pole pitch of the linear generator.

For a surface-mounted PMLSG, the direct and quadrature axis inductances are approximately equal (L_d ≈ L_q), rendering the reluctance thrust component negligible. To maximize the thrust-to-ampere ratio and minimize armature copper losses, the zero d-axis current control strategy is implemented. Therefore, the control condition i_dref = 0 is explicitly applied in the outer control loop, making the electromagnetic thrust strictly proportional to the q-axis current i_q.

3. Optimal Power Capture Strategy Based on Damping Control with Displacement Constraints

Wave-energy capture is a key component of DDWEC systems, and the performance of the power-capture strategy directly determines wave-energy utilization efficiency. In single-degree-of-freedom (SDOF) amplitude control, the PTO is modeled as a pure resistor and magnitude matching is required; in dual-degree-of-freedom (2DOF) complex-conjugate control, the PTO is modeled as a series resistor–capacitor network and conjugate matching is required. In both modes, the equivalent stiffness and damping of the PTO are tuned to match the amplitude and phase of the incident waves, thereby maximizing energy capture efficiency under regular-wave conditions.

Under energetic wave excitation or rapidly varying operating conditions, if the fixed optimal damping K_e,opt is maintained, the buoy displacement may be driven to, or beyond, the stroke limit. Therefore, displacement constraints should be embedded into the maximum-power-capture controller so that the motion amplitude is automatically attenuated as the stroke limit is approached, thereby ensuring safe operation.

3.1. Determination of the Optimal Damping Coefficient

For linear regular waves, the heave displacement is denoted by x_w, and the corresponding excitation force f_w is expressed as:

$$f_w=m_{add}{\ddot{x}}_w+K_z{\dot{x}}_w+K_{\rho hg}{x}_w,$$

(8)

Equation (8) indicates that the amplitude of f_w depends on the wave-motion displacement x_w, the radiation-damping coefficient K_z, and the buoyancy coefficient K_ρhg, but is independent of the generator mover displacement x.

By combining the buoy hydrodynamic model with the electromechanical dynamics of the permanent-magnet linear synchronous generator, the equation of motion of the wave power generation system is obtained as:

$$(m+m_{add})\ddot{x}+K_z\dot{x}+K_{\rho hg}x=f_w-f_e,$$

(9)

Under regular-wave excitation, the force acting on the floating body is approximately sinusoidal. For steady operation, the electromagnetic force is commanded to vary at the same frequency, to be phase-aligned with the mover velocity, and to have an adjustable amplitude. With an appropriate force amplitude, wave energy can be absorbed at the maximum average rate under steady operation.

Under this control law, the electromagnetic force f_e can be written as:

$$f_e=K_e\dot{x},$$

(10)

where K_e denotes the electromagnetic damping coefficient. Equation (9) can be mapped to an equivalent electrical circuit, as shown in Figure 2, where f_w is represented by a voltage source and the mover velocity x is analogous to current. The maximum-power-capture problem is thereby converted into the classical maximum power transfer problem for a resistive load (Figure 2). From circuit analysis, the average power delivered to the equivalent resistive load is given by:

$$P_e={\displaystyle \frac{f_w^2{K}_e}{{(K_z+K_e)}^2+{\left[{\omega }_w(m+m_{add})-\frac{K_{\rho hg}}{{\omega }_w}\right]}^2}},$$

(11)

where P_e is the average power captured by the generator and ω_w is the wave angular frequency. For a given wave power generation system, K_e is the only variable in P_e, and the condition for P_e to attain its maximum is:

$$K_{e,opt}=\sqrt{{K_z}^2+{\left[{\omega }_w(m+m_{add})-{\displaystyle \frac{K_{\rho hg}}{{\omega }_w}}\right]}^2},$$

(12)

Although the optimal-damping derivation assumes monochromatic waves, the coefficient was tuned at the spectral peak frequency as an equivalent regular-wave approximation for the considered sea state.

The corresponding maximum average captured power is obtained as:

$$P_{emax}={\displaystyle \frac{f_w^2}{2K_z+2\sqrt{{K_z}^2+{\left[{\omega }_w(m+m_{add})-\frac{K_{\rho hg}}{{\omega }_w}\right]}^2}}},$$

(13)

where P_emax denotes the maximum captured power. This quantity represents the period-averaged captured power over one wave period, rather than the instantaneous power.

Therefore, during steady regular-wave operation, maximum average energy capture is achieved when the electromagnetic force is commanded to be sinusoidal at the wave frequency, phase-aligned with the mover velocity, and generated with K_e set to the optimal value in Equation (12).

3.2. Improved Control Strategy Considering Displacement Constraints

Equations (11)–(13) show that, under steady-state conditions, maximum energy is captured when K_e takes the optimal electromagnetic damping in Equation (12), denoted as K_e,opt. However, the mechanical stroke of direct-drive wave energy converters is typically constrained by structural size and layout, and the buoy displacement is required to satisfy:

$$\left|x\right|\le x_{\max },$$

(14)

To enforce this constraint without altering the baseline control structure, a displacement-dependent correction term is introduced into the electromagnetic damping. First, the displacement correction factor is defined as:

$$\mathsf{\varphi }(x)=\left\{\begin{array}{cc}0& \left|x\right|\le \alpha x_{\max }\\ {(3s^2-2s^3)}^n& \alpha x_{\max }<\left|x\right|<x_{\max }\\ 1& \left|x\right|\ge x_{\max }\end{array}\right.,s={\displaystyle \frac{\left|x\right|-\alpha x_{\max }}{(1-\alpha )x_{\max }}},$$

(15)

where 0 < α < 1 is a buffer coefficient (typically 0.7–0.85) defining the activation threshold, and n ≥ 1 shapes the transition steepness. In the following simulations, the nominal values x_max = 0.5 m, α = 0.8, and n = 2 are adopted, and their influence on the displacement–power trade-off is quantified in Section 5.1 through a one-factor-at-a-time sensitivity study.

Notably, the smoothstep term 3s²−2s³ satisfies g(0) = 0 and g′(0) = 0; thus, ϕ(x) is C¹-continuous at |x| = αx_max (and also at |x| = x_max), avoiding abrupt damping changes and potential dynamic discontinuities. An additional damping gain K_e,add > 0 is then introduced. The resulting equivalent electromagnetic damping under displacement constraints can be written as:

$$K_{e,eff}=K_{e,opt}+K_{e,add}\varphi (x),$$

(16)

When the buoy displacement remains within the safe region, |x| ≤ αx_max, ϕ(x) = 0 and the original optimal damping K_e,opt is retained for maximum energy capture. As the buoy approaches its stroke limit, ϕ(x) increases smoothly with |x|, and the additional damping term K_e,addϕ(x) is activated. As a result, the total equivalent damping is increased, and both buoy velocity and displacement amplitude are reduced progressively, thereby preventing boundary violations.

By substituting Equation (16) into the electromagnetic-force expression, the electromagnetic force under displacement constraints is obtained as:

$$f_e=K_{e,eff}\dot{x}=(K_{e,opt}+K_{e,add}\varphi (x))\dot{x},$$

(17)

Accordingly, the system equation of motion is rewritten as:

$$(m+m_{add})\ddot{x}+(K_z+K_{e,opt}+K_{e,add}\varphi (x))\dot{x}+K_{\rho hg}x=f_w,$$

(18)

The introduced displacement-dependent term does not affect maximum-power capture within the operating region; instead, the equivalent damping is increased only as the buoy approaches its displacement limit, so that the net force acting on the buoy is reduced and further displacement growth is suppressed, enabling smooth enforcement of the stroke constraint.

4. Position-Sensorless Control Strategy Based on High-Frequency Injection and Adaptive Amplitude Modulation

For space-vector modulation (SVM), accurate electrical position information is required. To enhance observability at low speed and improve estimation stability in noisy environments, an adaptive injection-amplitude modulation strategy is developed based on HF square-wave voltage injection. Specifically, the injection-voltage amplitude is adjusted in a closed loop using the real-time demodulated HF-current intensity, so that the HF current response is maintained within a predefined optimal observation band, thereby improving the robustness and stability of electrical position estimation.

4.1. Current Response Under High-Frequency Square-Wave Voltage Injection

For mover position estimation, an estimated synchronous reference frame is established and is denoted as the $\widehat{\mathrm{d}}$-$\widehat{\mathrm{q}}$ frame. The angle between the $\widehat{\mathrm{d}}$-axis and the stationary α-axis is denoted by $\widehat{\theta }$_e, which represents the estimated electrical position of the mover. The relationship between the estimated and actual positions is given by [21]:

$${\widehat{\theta }}_e={\theta }_e+\mathsf{\Delta }{\theta }_e,$$

(19)

where θ_e denotes the actual electrical position and Δθ_e denotes the position-estimation error. The injection direction is determined by $\widehat{\theta }$_e; therefore, the HF voltage is injected along the $\widehat{\mathrm{d}}$-axis. The square-wave voltage injected along the $\widehat{\mathrm{d}}$-axis can be expressed as:

$$\left[\begin{array}{l}{\widehat{u}}_{d\_hf}\\ {\widehat{u}}_{q\_hf}\end{array}\right]=\left[\begin{array}{c}{V}_{inj}{\varphi }_{spr}\left(t\right)\\ 0\end{array}\right],{\varphi }_{spr}\left(t\right)=\left\{\begin{array}{l}+1,0<t<0.5T\\ -1,0.5T<t<T\end{array}\right.,$$

(20)

where $\widehat{u}$_{d_hf} and $\widehat{u}$_{q_hf} denote the injected HF square-wave voltages on the $\widehat{\mathrm{d}}$-axis and $\widehat{\mathrm{q}}$-axis, respectively; V_inj is the injection amplitude; ${\varphi }_{\mathit{spr}}\left(t\right)$ is the unit square-wave function; and T is the period of the injected HF voltage. The injected HF square-wave voltage waveform in the $\widehat{\mathrm{d}}$-$\widehat{\mathrm{q}}$ frame is shown in Figure 3.

To reveal the position-dependent saliency signal more clearly, the response to the injected HF square-wave voltage is analyzed in the stationary α-β reference frame. The injected HF voltage in the α-β frame can be expressed as:

$$\begin{array}{c}\left[\begin{array}{l}{u}_{\alpha \_hf}\\ u_{\beta \_hf}\end{array}\right]=\left[\begin{array}{l}-V_{inj}\cos {\widehat{\theta }}_e\\ -V_{inj}\sin {\widehat{\theta }}_e\end{array}\right],0<t<0.5T\\ \left[\begin{array}{l}{u}_{\alpha \_hf}\\ u_{\beta \_hf}\end{array}\right]=\left[\begin{array}{l}{V}_{inj}\cos {\widehat{\theta }}_e\\ V_{inj}\sin {\widehat{\theta }}_e\end{array}\right],0.5T<t<T\end{array},$$

(21)

where u_{α_hf} and u_{β_hf} are the α- and β-axis components of the injected HF voltage. In the α-β stator-voltage model, the back electromotive force induced by the permanent-magnet flux linkage appears at the fundamental frequency and is associated only with the fundamental components. Because the injection frequency is much higher than the fundamental frequency, the injected square-wave voltage can be treated as a high-frequency perturbation component.

Because the injected square-wave voltage is of high frequency, the differential terms are discretized over the sampling interval. After the Park transformation is applied to map the HF current variation from the α-β frame to the $\widehat{\mathrm{d}}$-$\widehat{\mathrm{q}}$ frame, the forward difference in the HF current can be obtained as:

$$\left[\begin{array}{l}\mathsf{\Delta }{\widehat{i}}_{d\_hf}\\ \mathsf{\Delta }{\widehat{i}}_{q\_hf}\end{array}\right]={\displaystyle \frac{0.5V_{inj}T}{\sum L^2-\mathsf{\Delta }{L}^2}}\left[\begin{array}{l}\sum L-\mathsf{\Delta }L\cos (2{\theta }_e-{\widehat{\theta }}_e)\\ -\mathsf{\Delta }L\sin 2({\theta }_e-{\widehat{\theta }}_e)\end{array}\right],$$

(22)

where $\widehat{i}$_{d_hf} and $\widehat{i}$_{q_hf} denote the forward differences in the HF response current components on the $\widehat{\mathrm{d}}$- and $\widehat{\mathrm{q}}$-axis, respectively. Equation (22) indicates that the $\widehat{\mathrm{q}}$-axis forward difference can serve as a position-dependent observation signal for mover position estimation.

4.2. Extraction of the High-Frequency Response Current

To estimate mover position, the HF component must be extracted from the measured current, and the $\widehat{\mathrm{q}}$-axis forward difference must be computed. A real-time computation method based on a second-order difference equation is adopted, in which the target variable is evaluated directly from current samples at adjacent instants.

Under HF square-wave voltage injection, the linear machine behaves predominantly as an inductive load, and a triangular HF current component is superimposed on the fundamental current. Because the injection frequency is far higher than the fundamental frequency, the fundamental current within one HF period can be approximated as a ramp with an approximately constant slope. A schematic of the extracted HF current within one square-wave period is shown in Figure 4.

The sampling frequency is set to twice the HF injection frequency, i.e., f_s = 2f, so that current responses at both positive and negative transitions are captured within each square-wave period. Although the ADC/current-control sampling runs at 10 kHz, the HF demodulation uses synchronized samples at the positive/negative transitions of the injected square wave, which is equivalent to the f_s = 2f_hf requirement for the second-order difference extraction. Based on the superposition of the fundamental current and the HF response current, the relationship between the sampled current and the $\widehat{\mathrm{d}}$-axis forward difference can be expressed as:

$$\left\{\begin{array}{l}{\widehat{i}}_d(k+1)={\widehat{i}}_d(k)+\mathsf{\Delta }{\widehat{i}}_{d\_f}+\mathsf{\Delta }{\widehat{i}}_{d\_hf}\\ {\widehat{i}}_d(k+2)={\widehat{i}}_d(k+1)+\mathsf{\Delta }{\widehat{i}}_{d\_f}-\mathsf{\Delta }{\widehat{i}}_{d\_hf}\end{array}\right.,$$

(23)

where $\widehat{i}$_d(k) is the sampled $\widehat{\mathrm{d}}$-axis current at instant k, $\widehat{i}$_d(k + 1) is the sampled $\widehat{\mathrm{d}}$-axis current at instant (k + 1), $\widehat{i}$_d(k + 2) is the sampled $\widehat{\mathrm{d}}$-axis current at instant (k + 2), and Δ$\widehat{i}$_{d_f} is the variation in the fundamental current component. Using the second-order difference equation, the forward variation in the $\widehat{\mathrm{d}}$-axis component is obtained as:

$$\mathsf{\Delta }{\widehat{i}}_{d\_hf}={\displaystyle \frac{1}{2}}\left[-{\widehat{i}}_d(k+2)+2{\widehat{i}}_d(k+1)-{\widehat{i}}_d(k)\right],$$

(24)

Similarly, the $\widehat{\mathrm{q}}$-axis forward difference is obtained as:

$$\mathsf{\Delta }{\widehat{i}}_{q\_hf}={\displaystyle \frac{1}{2}}\left[-{\widehat{i}}_q(k+2)+2{\widehat{i}}_q(k+1)-{\widehat{i}}_q(k)\right],$$

(25)

where $\widehat{i}$_q(k), $\widehat{i}$_q(k + 1), and $\widehat{i}$_q(k + 2) are the sampled $\widehat{\mathrm{q}}$-axis currents at instants k, k + 1, and k + 2, respectively. By combining Equations (24) and (25), the mover-position observation signal is obtained as:

$$i_{pos}={\displaystyle \frac{-0.5V_{inj}T}{\left(\sum L^2-\mathsf{\Delta }{L}^2\right)}}\mathsf{\Delta }L\sin 2\left({\theta }_e-{\widehat{\theta }}_e\right)={\displaystyle \frac{1}{2}}\left[-{\widehat{i}}_q(k+2)+2{\widehat{i}}_q(k+1)-{\widehat{i}}_q(k)\right],$$

(26)

The mover electrical position is then estimated using a phase-locked loop (PLL) driven by the observation signal.

The position is obtained using a standard second-order type-II PLL. The normalized phase detector output is treated as the PLL input error, denoted by e_PLL.

K_p and K_i are selected to achieve a desired natural frequency ω_n and damping ratio ζ of the linearized loop. Using the standard mapping K_p = 2ζω_n/K_d and K_i = ω_n²/K_d, where K_d denotes the effective phase-detector gain under the adopted signal normalization, ζ ≈ 0.707 and ω_n ≈ 40–45 rad/s lead to K_p ≈ 60 and K_i ≈ 1200 for the considered operating condition. These values provide a stable compromise between dynamic tracking and noise attenuation in the low-speed reciprocating motion tests.

$$\stackrel{\cdot }{\widehat{{\theta }_e}}=\widehat{{\omega }_e}+K_p{e}_{PLL},\stackrel{\cdot }{\widehat{{\omega }_e}}=K_i{e}_{PLL},$$

(27)

4.3. Adaptive Amplitude Modulation Strategy

From the above derivations, the forward differences in the HF currents, Δ$\widehat{i}$_{d_f} and Δ$\widehat{i}$_{q_f}, are approximately proportional to the injection amplitude V_inj. Moreover, because the injected signal has a frequency much higher than the fundamental electrical frequency, an HF current response remains observable in the low-speed region, including near zero mechanical speed, thereby providing an effective signal for electrical angle estimation.

In conventional adaptive amplitude modulation, the injection amplitude for the next period is updated directly from the current-period HF current intensity I_hf,rms. However, this single-time-scale coupled structure allows position-estimation errors to be fed back into V_inj through I_hf,rms, which can create a cumulative effect in the demodulation loop and potentially form a positive-feedback path, thereby degrading estimation stability.

Therefore, a dual-time-scale adaptive amplitude modulation mechanism is adopted. HF current extraction and position estimation are updated on a fast time scale synchronized with the HF injection, whereas the injection amplitude V_inj is adjusted on a slower time scale with a much lower update rate. On the fast time scale, the sampling frequency is set to twice the injected square-wave frequency, and the HF current forward differences are computed in real time according to Equations (23)–(26) and supplied to the PLL for angle estimation. This loop is kept fast to preserve position-estimation accuracy. On the slow time scale, the injection amplitude is updated with a longer regulation period T_amp rather than once per HF cycle, so that amplitude updates remain much slower than HF current extraction, as described in Equation (28):

$$T_{amp}\gg T_{hf},$$

(28)

where T_hf denotes the HF injection period. In this manner, short-term estimation errors in $\widehat{\theta }$ are not immediately propagated to the amplitude update, thereby mitigating error accumulation and instability.

To enable automatic regulation of the injection amplitude, an indicator that reflects the overall intensity of the HF current component is constructed. The root-mean-square (RMS) value of the HF current component is adopted as the intensity measure:

$$I_{hf,rms}=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=1}^N(i_{d,hf}^2(k)+i_{q,hf}^2(k))}},$$

(29)

where N is the number of HF sampling points within the slow time scale. Because the HF square-wave frequency is high and the response is approximately transient, the RMS-based indicator can suppress noise and provide a stable intensity estimate.

Based on this intensity indicator, a low-bandwidth closed-loop regulator is designed so that the HF current intensity is maintained within a predefined target interval. The desired HF current intensity is denoted by I_ref, and the intensity error is defined as:

$$e(k)=I_{ref}-I_{hf,rms}(k),$$

(30)

On the slow time scale, the injection amplitude is updated once every k_m HF extraction cycles. The regulation law is given by:

$$V_{inj}(m+1)=V_{inj}(m)+K_{p}e(m)+K_i{\displaystyle \sum _{j=1}^{m}e(j)},$$

(31)

where m is the update index on the slow time scale and K_p and K_i are the proportional and integral gains, respectively. The corresponding update frequency satisfies:

$${\displaystyle \frac{T_{hf}}{T_{amp}}}={\displaystyle \frac{1}{k_m}}\ll 1,$$

(32)

To avoid excessive current ripple caused by large-amplitude injection, the updated V_inj(m + 1) is constrained as:

$$V_{inj,\min }\le V_{inj}(m+1)\le V_{inj,\max },$$

(33)

where V_inj,max is jointly determined by the modulation index of space-vector pulse-width modulation (SVPWM), the dc-bus voltage, and the allowable limit of HF ripple in the system.

4.4. Design Rationale of the Optimal Measurement Band and Electrical Cost of HF Injection

The optimal measurement band [I_low,I_high] and the reference value I_ref in the amplitude-regulation loop are selected to achieve a practical balance between (i) a sufficiently high observation SNR for stable PLL operation and (ii) limited electrical ripple and injection-induced losses. Under HF square-wave voltage injection, the HF current response is approximately proportional to the injected voltage amplitude because the machine behaves predominantly as an inductive load at the injection frequency. Neglecting back-EMF and resistive drop at the injection frequency, the incremental HF current change within a half injection period can be approximated by

$$\mathsf{\Delta }{i}_{hf}\approx {\displaystyle \frac{V_{inj}}{L_{hf}}}\cdot {\displaystyle \frac{T_{hf}}{2}},$$

(34)

where L_hf denotes the incremental inductance seen by the injected HF signal and T_hf = 1/f_hf. With the parameters in

Table 1. Parameters used in the time-domain simulation of the DDWEC system.

Category	Parameter	Symbol	Value
Generator (PMLSG)	Stator resistance	Rs	6.35 Ω
	d-axis inductance	Ld	59.07 mH
	q-axis inductance	Lq	70.88 mH
	PM flux linkage	ψf	0.425 Wb
	Pole pitch	τ	0.025 m
Hydrodynamics/structure	Buoy mass	M	342 kg
	Added mass	m∞	84.6 kg
	Radiation damping coefficient	Kz	6000 N·s/m
	Hydrostatic stiffness	K	2.0 × 104 N/m
Stroke constraint	Stroke limit	xmax	0.5 m
	Threshold coefficient	α	0.8
	Correction-factor exponent	n	2
Controllers and simulation	Current control	—	PI
	Base simulation step size	Ts	1 ms
Sensorless	HF injection frequency	fhf	1 kHz
	Current sampling frequency	fs	10 kHz
	Injection amplitude range	Vinj	5–30 V
	Fixed-amplitude baseline	Vinj	15 V
	Optimal measurement band	[Ilow,Ihigh]	[0.11 A,0.15 A]
	Target intensity	Iref	0.13 A
PLL	Proportional gain	Kp	60
	Integral gain	Ki	1200
Amplitude modulation	Update interval (single-time-scale)	km	1
	Update interval (two-time-scale)	km	50

, V_inj = 15 V yields Δi_hf ≈ 0.127 A, which matches the selected target level I_ref = 0.13 A. Accordingly, the band [0.11,0.15] A provides a margin against measurement noise and parameter drift while avoiding excessive injection.

The injection amplitude is bounded in Equation (33) to limit HF ripple and losses. For a square-wave injection, the physical HF current ripple can be approximated as a triangular component with peak value

$$I_{pk}\approx {\displaystyle \frac{V_{inj}}{4L_{hf}{f}_{hf}}},I_{rms}\approx {\displaystyle \frac{I_{pk}}{\sqrt{3}}},$$

(35)

Leading to an additional copper-loss estimate

$$P_{cu,hf}\approx {\displaystyle \frac{3}{2}}{R}_s{I}_{rms}^2,$$

(36)

With R_s = 6.35 Ω, L_hf = 59.07 H, and f_hf = 1 kHz, the estimated HF ripple and copper loss remain small within the adopted injection range. For instance, at V_inj = 15 V, I_pk ≈ 0.064 A and P_cu,hf ≈ 0.01 W; even at V_inj = 30 V, I_pk ≈ 0.13 A and P_cu,hf < 0.05 W. Therefore, the primary motivation for bounding V_inj is to avoid excessive current ripple and potential inverter/iron-loss penalties, rather than copper loss alone. In practice, the band design can be refined by considering the measured noise floor, allowable current ripple, and thermal constraints; however, the above estimates provide a clear and reproducible selection rationale for the adopted parameters.

5. Results and Discussion

To verify the effectiveness of the proposed control strategy, a time-domain simulation model of the DDWEC system was developed in MATLAB/Simulink R2024b. The overall simulation framework is shown in Figure 5 and consists of the following modules: the hydrodynamic subsystem; the PMLSG with machine-side current control; the outer-loop power-capture controller (fixed K_e,opt and displacement-constrained additional damping); and the position-sensorless module implementing high-frequency (HF) square-wave injection, demodulation, a phase-locked loop (PLL), and adaptive amplitude modulation. To rigorously replicate the stochastic dynamics of a deep-ocean environment, the irregular wave excitation was synthesized utilizing a Pierson–Moskowitz (PM-Spectrum) frequency distribution with random phases [22]. This empirical spectral model fundamentally captures the energy distribution characteristic of fully developed wind seas, thereby ensuring that the displacement-constrained control strategy is validated against realistic, chaotic wave forcing. The parameters were set to H_s = 1.6 m, T_p = 8 s, N_w = 200, ω ∈ [0.2,2.5] rad/s; random seed = 2. For the fixed-K_e,opt case, K_e,opt was tuned using the spectral peak frequency ω_p = 2π/T, which serves as an equivalent dominant design frequency for irregular seas. In addition, a traditional mechanical end-stop baseline was implemented using a symmetric spring–damper limiter that engages when |x| > αx_max (α = 0.8), with K_s = 2.5 × 10⁵ N/m and C_s = 2.0 × 10⁴ N·s/m.

The machine-side current loop is implemented in the synchronous d–q frame with standard PI regulators and decoupling feedforward terms. The control laws are

$$\begin{array}{l}{u}_d=K_{p,d}(i_d^{\ast }-i_d)+K_{i,d}{\displaystyle \int (}{i}_d^{\ast }-i_d)dt-{\omega }_e{L}_q{i}_q\\ u_q=K_{p,q}(i_q^{\ast }-i_q)+K_{i,q}{\displaystyle \int (}{i}_q^{\ast }-i_q)dt+{\omega }_e(L_d{i}_d+{\phi }_f)\end{array},$$

(37)

These control laws are followed by a voltage limiter consistent with the DC-bus constraint when applicable. The PI gains are selected using a bandwidth-based tuning rule by targeting a desired current-loop crossover frequency ω_c: K_p = Lω_c and K_i = Rω_c. This yields a predictable first-order closed-loop current dynamics and provides a reproducible parameter-setting procedure for the simulations.

The simulation parameters are summarized in Table 1.

For clarity, the red upper line denotes the positive DC bus, the red/blue blocks in the machine schematic denote opposite permanent-magnet polarities, and abc and dq represent the three-phase stationary frame and synchronous rotating frame, respectively.

5.1. Simulation of Damping-Type Power Capture Considering Displacement Constraints

To compare the proposed displacement-constrained strategy with both the conventional optimal damping strategy and a traditional mechanical stroke-protection method, three schemes were implemented as follows:

• Case A (conventional fixed damping): a constant K_e,opt was applied in the outer loop throughout the simulation. The electromagnetic force reference was generated as F* = −K_e,opt$\dot{x}$ and converted into the q-axis current reference i_q* for tracking by the inner current loop.
• Case B (traditional end-stop spring–damper limiter): the same fixed K_e,opt was used for power capture, while a symmetric mechanical limiter was added to emulate a conventional stroke-protection device. When |x| > αx_max, the limiter generated an additional restoring and damping force to prevent end-stop impacts.
• Case C (proposed smooth additional damping): when |x| ≤ αx_max, K_e,opt was retained; when |x| > αx_max, an additional damping term K_e,addϕ(x) was introduced such that the equivalent damping increased smoothly as |x| approached x_max, thereby suppressing further displacement growth without relying on mechanical limiting forces.

The total simulation time was 200 s with a base step size of 1 ms. The absorbed power was post-processed using a 10 s moving average, and the initial 20 s transient was excluded from the statistics.

Figure 6 shows the irregular-wave excitation force synthesized from the PM spectrum with random phases. The resulting time series is broadband and strongly stochastic, featuring intermittent large peaks superimposed on a continuously varying background. Such nonstationary forcing provides a stringent test for stroke-constrained control, because large excitation bursts can rapidly increase the buoy velocity and drive the displacement toward its stroke limit. Importantly, the same excitation force f_w(t) is applied to all compared control schemes, ensuring a fair assessment of power–safety trade-offs under identical wave realizations.

Figure 7 compares the moving-average absorbed power among the three schemes under the nominal PM-spectrum sea state (H_s = 1.6 m, T_p = 8 s). As expected, the conventional fixed-K_e,opt scheme achieves the highest mean power because optimal damping is pursued continuously without considering stroke safety. However, this comes at the expense of boundary violation risk, as shown later in Figure 8. The two stroke-protection schemes reduce the absorbed power to different extents. With the mechanical end-stop spring–damper limiter, the limiter introduces additional restoring force and dissipation once the motion enters the buffer region, which reduces the buoy velocity and consequently decreases electromagnetic power extraction. The proposed smooth-ϕ(x) strategy preserves a larger fraction of the power because K_e,opt is retained in the safe region and the additional damping is activated only near the stroke boundary with a smooth transition. Quantitatively, the mean moving-average absorbed power is 1508.63 W for Case A, 1342.71 W for Case B, and 1393.17 W for Case C. Compared with Case A, the proposed method incurs only a 7.65% reduction in mean absorbed power, while providing a 3.76% improvement over the traditional limiter baseline under zero-violation operation, demonstrating a favorable power–safety compromise.

Figure 8 presents the buoy displacement responses with the stroke limits (±x_max) and the activation thresholds (±αx_max). The fixed-K_e,opt scheme exhibits a clear risk of boundary violations during energetic intervals, reaching a peak displacement of 0.654 m and a violation fraction of 0.0482. This confirms that pure power-optimal damping is not sufficient to guarantee stroke safety under irregular seas. In contrast, both stroke-protection schemes effectively maintain the displacement within the allowable range. The traditional mechanical limiter achieves strict constraint satisfaction with a peak displacement of 0.453 m and zero violation fraction. However, this is obtained by physically engaging the limiter for approximately 7.6% of the time (limFrac = 0.076), and the peak limiter force reaches 14.17 kN, which indicates potential mechanical loading and wear in practical implementations. The proposed smooth-ϕ(x) method also eliminates boundary violations (violFrac = 0) with a peak displacement of 0.486 m, reducing the peak displacement by 25.7% relative to Case A (0.654 m→0.486 m). Compared with Case B, Case C achieves safety without invoking mechanical end-stop forces, while maintaining a higher absorbed power level (as shown in Figure 7). Overall, the results demonstrate that the proposed smooth damping augmentation can prevent end-stop impacts while preserving energy capture more effectively than a purely mechanical stroke-limiting approach.

Figure 9 evaluates the robustness margin of each scheme by sweeping the significant wave height H_s while keeping the same PM-spectrum realization (identical random phases) and using the violation fraction frac(|x| > x_max) as the safety metric. A practical fail-safe threshold of 1% violation fraction is adopted to identify the operating boundary. The fixed-K_e,opt scheme deteriorates rapidly as sea severity increases and becomes unacceptable at relatively low wave heights, indicating that power-optimal damping alone provides insufficient safety margin. The mechanical limiter baseline exhibits the best robustness, maintaining the lowest violation fractions across the sweep because the additional mechanical restoring and damping forces directly restrict motion; nevertheless, its robustness is achieved at the expense of mechanical engagement and loading. The proposed smooth-ϕ(x) scheme significantly extends the safe operating envelope compared to Case A. Based on the 1% criterion, the fail-safe boundary of the proposed method is identified at approximately H_s* ≈ 2.2 m, beyond which a dedicated fail-safe action should be triggered to guarantee stroke safety. This boundary characterization provides an explicit and practical guideline for deployment: in moderate sea environments where Hs remains below H_s*, the proposed method can realize safe operation without end-stop impacts while preserving a higher power level than the mechanical limiter baseline; for harsher conditions, a fail-safe layer is necessary to ensure survivability.

Under nominal PM-spectrum seas, the proposed smooth-ϕ(x) strategy eliminates stroke violations and reduces peak displacement by 25.7% with only a 7.65% reduction in mean absorbed power, while capturing 3.76% more power than a traditional end-stop spring–damper limiter under the same zero-violation condition. A sea-severity sweep further reveals a clear fail-safe boundary (H_s* ≈ 2.2 m for a 1% violation criterion), providing a practical guideline for safe deployment.

To further clarify the parameter selection in the corrective function ϕ(x), a one factor at a time sensitivity study was carried out around the nominal setting x_max = 0.5 m, α = 0.8, and n = 2 under the same PM-spectrum sea state and the same random phase realization. Only one parameter was varied at a time, while the others were kept unchanged. The reported metrics include the mean absorbed power normalized by Case A (P/P_A), the normalized peak displacement (x_pk/x_max), the safety margin (x_max − x_pk), the violation fraction, and the peak q-axis current as an auxiliary indicator of electrical stress.

As summarized in Figure 10a, increasing x_max from 0.45 m to 0.55 m raises the mean absorbed power from 1324.32 W to 1444.80 W, while the normalized peak displacement decreases from 0.9894 to 0.9502. This behavior is expected because a larger available stroke delays the onset of constraint-dominated operation and allows the baseline optimal damping to remain effective over a wider displacement range. For α, increasing the activation threshold from 0.75 to 0.85 slightly increases the mean absorbed power from 1381.21 W to 1401.95 W, but reduces the safety margin from 23.25 mm to 5.75 mm, with x_pk/x_max increasing from 0.9535 to 0.9885. Therefore, smaller α yields a more conservative intervention, whereas larger α preserves more power but leaves less margin to the stroke boundary. The nominal choice α = 0.8 provides a balanced compromise.

The exponent n mainly affects the spatial distribution of the correction inside the buffer region rather than the activation threshold itself. Under the present irregular-wave realization and the adopted damping-update dynamics, n = 1 gives the highest mean absorbed power and the smallest normalized peak displacement, but it also produces the largest peak q-axis current. By contrast, n = 3 concentrates the correction closer to the stroke boundary and results in a smaller safety margin together with a lower retained-power ratio. The nominal choice n = 2 yields the lowest peak q-axis current (309.4 A) while maintaining zero violation and comparable power retention (92.35%), and was therefore retained as a balanced setting from the joint viewpoint of stroke safety, energy capture, and electrical stress.

Overall, the sensitivity study indicates that the proposed smooth-ϕ(x) law is not overly sensitive around the nominal setting: all tested parameter combinations maintain zero violation under the nominal sea state, while parameter tuning mainly shifts the operating point along the expected displacement–power trade-off. In normalized terms, Case C retains 92.35% of the mean absorbed power of Case A while reducing the peak displacement by 25.7%; compared with the mechanical limiter baseline, it accepts a moderately larger peak displacement in exchange for 3.76% higher mean absorbed power under the same zero-violation condition.

5.2. Position-Sensorless Simulation of the Improved High-Frequency Square-Wave Injection Method

To validate the effectiveness of adaptive amplitude modulation and the two-time-scale mechanism, three high-frequency square-wave injection (HSVI) schemes were compared under a low-speed, wave-like reciprocating motion condition:

• Fixed-amplitude HSVI;
• Single-time-scale adaptive-amplitude HSVI (k_m = 1);
• Two-time-scale adaptive-amplitude HSVI (k_m = 50).

In the simulations, the inductance was assumed to vary slowly with position to emulate parameter drift induced by slotting and/or end effects. In addition, short-term disturbances and sampling anomalies were superimposed around t ≈ 12 s and t ≈ 22 s to mimic abrupt interference in practical operation.

Figure 11 illustrates the time evolution of the HF current RMS I_hf,rms under the three strategies, as well as the fraction of time spent within the optimal measurement band [I_low,I_high]. With fixed-amplitude injection, I_hf,rms frequently deviated from the optimal band due to parameter drift and external disturbances; only 25.84% of the samples remained within the band, which degraded the signal-to-noise ratio of the position-observation signal. With adaptive amplitude modulation, V_inj was adjusted to maintain I_hf,rms within the optimal band for most of the simulation, yielding effective ratios of 99.76% (single-time-scale) and 98.91% (two-time-scale). These results indicate that adaptive amplitude modulation can effectively accommodate parameter variations and external disturbances, thereby maintaining consistently favorable observability for position estimation. In addition to improving observability, the electrical cost of adaptive V_inj remains limited because the regulated HF-current intensity is constrained inside a narrow band. Using the estimates in Equations (35) and (36) and the parameters in Table 1, the corresponding HF ripple and additional copper loss are on the order of 10⁻²–10⁻¹ W within the adopted injection range, indicating that the improved robustness is achieved with a small electrical penalty.

As shown in Figure 12, all three methods tracked the low-frequency, wave-like position trajectory. The estimated trajectories obtained with adaptive amplitude modulation were nearly coincident with the true position. The corresponding position-error metric values were 0.0612 m, 0.0600 m, and 0.0599 m, indicating that observability was improved while estimation accuracy was maintained.

Figure 13 compares the error envelope of the position estimation error. The fixed-amplitude scheme exhibited larger fluctuations in the error envelope, reflecting higher sensitivity to operating-condition variations. By contrast, the error envelopes of both adaptive schemes were significantly narrowed, indicating that a higher demodulation signal-to-noise ratio was achieved within the optimal measurement band and that position-estimation stability was improved.

As shown in Figure 14, during the two short disturbance windows, the single time scale scheme—because its update rate is synchronized with HF extraction—tended to propagate an abnormal measurement to the next cycle through the amplitude-modulation channel, thereby causing a transient enlargement of the error envelope. In contrast, the two-time-scale scheme preserved the slow-variable characteristic of amplitude updates within the disturbance windows, resulting in a more stable error envelope with smaller fluctuations. This behavior demonstrates improved robustness to short-term anomalies.

Square-wave HF voltage injection is widely used to enhance low-speed observability due to its fast transient response and implementation simplicity [13,15]. However, most existing implementations rely on a fixed injection amplitude, which cannot guarantee consistent demodulation SNR under load fluctuation, parameter drift, and noise. Several enhanced injection schemes mainly address voltage errors or audible-noise issues [13,16,17], but they do not explicitly regulate the observation-signal intensity. In contrast, the proposed approach introduces a closed-loop amplitude regulation based on the demodulated HF-current intensity and further decouples the amplitude update from the fast estimation loop via a dual-time-scale mechanism. This decoupling alleviates the positive-feedback risk reported for single-time-scale adaptive updates [18] and improves robustness to short disturbance windows, as evidenced by the narrowed error envelope in Figure 13 and Figure 14.

It should be clarified that the reported “effective measurement ratio” in Figure 11 quantifies the percentage of time that the demodulated HF-current intensity I_hf,rms stays inside the predefined optimal observation band [I_low,I_high], rather than the absolute position-estimation accuracy. The actual estimation performance is reflected by the position-error metric in Figure 12, Figure 13 and Figure 14. Moreover, unlike back-EMF observers that may suffer from drift when position is reconstructed mainly via speed integration at very low speed, the proposed method continuously provides a saliency-based phase reference through HF injection and a PLL. Therefore, the estimation error does not accumulate unboundedly over time; instead, it remains bounded under bounded disturbances, with its envelope determined by the observation SNR and the PLL dynamics.

For extremely energetic sea states or rare interference events, practical deployments typically incorporate supervisory logic to trigger a safe mode or fallback strategy when observability is temporarily insufficient. Such system-level fault management and advanced disturbance-rejection current control can further enhance survivability, but they are beyond the scope of this paper, which focuses on improving the HF-injection sensorless mechanism and its robustness through dual-time-scale amplitude regulation.

6. Experimental Validation on a Laboratory Emulation Platform

6.1. Overview of the DDWEC Emulation Platform

Due to laboratory constraints, it is difficult to reproduce broadband irregular waves and full hydrodynamic interactions in a controlled indoor environment. Therefore, an emulation platform was developed to generate a prescribed reciprocating motion that approximates the regular-wave-induced heave of a DDWEC system. The experimental study in this section focuses on validating the feasibility and robustness of the proposed sensorless strategy, rather than reproducing full ocean-wave power capture with displacement-constrained damping regulation.

Figure 15 illustrates the overall architecture of the laboratory emulation platform. The hardware system consists of a PLC controller, a rotary servo motor and drive, a ball-screw mechanism, voltage and current sensing modules, an IGBT-based converter with gate drivers, a resistive load bank, a regulated DC power supply for the DC bus, an ADC sampling unit, and a TMS320F28335 DSP control board. The DC power supply emulates the DC-link voltage and provides a stable electrical interface for the machine-side converter and the external load. A photograph of the platform is shown in Figure 16; due to space limitations, several auxiliary modules are not fully visible.

6.2. Hardware Configuration

6.2.1. Motion Emulation Unit

The motion emulation unit comprises a PLC, a rotary servo motor with its drive, and a ball-screw transmission. During operation, the PLC generates a sinusoidally varying pulse command; the servo drive converts this pulse train into a speed-controlled rotary motion. The ball screw transforms the rotary motion into a reciprocating linear motion, thereby generating a prescribed sinusoidal velocity profile at the translator of the permanent-magnet linear synchronous generator (PMLSG). This setup provides a repeatable “regular-wave equivalent” mechanical excitation for verifying sensorless position estimation and current control under controlled conditions.

The platform uses an SMG130D-0200-20MA(B)K-4LKG rotary servo motor (rated speed: 2000 rpm) and a Step/Dir servo drive (FD5 series). In pulse mode, the drive resolution is 65,536 pulses per revolution.

6.2.2. Linear Generator and Power Electronics

A Suzhou Zhiwei ZW22P-1100B2-CY5-S440 surface-mounted PMLSG is used as the direct-drive generator. The ball-screw slider and the linear mover are rigidly connected through a coupling rod to ensure synchronous motion. During tests, the mover speed follows an approximately sinusoidal trajectory consistent with the emulated wave motion.

The electrical subsystem includes an IGBT converter with gate drivers, current/voltage sensing, an ADC module, a resistive load bank, and a regulated DC supply.

Table 2. Key parameters of the DDWEC emulation platform.

Item	Value	Item	Value
Translator stroke	440 mm	d-axis inductance Ld	59.07 mH
PM flux linkage ψf	0.425 WB	q-axis inductance Lq	70.88 mH
Pole pitch τ	25 mm	Prime mover rated torque TN	10 N·m
Stator resistance Rs	6.35 Ω	Prime mover rated power P	2 kw
DC bus voltage udc	100 V	Resistive load R	100 Ω

summarizes the key parameters of the emulation setup and the PMLSG.

6.3. Control and Software Implementation

6.3.1. Reference Position Processing for Calibration

Although the proposed sensorless strategy can estimate the electrical position without a mechanical position sensor, an encoder-based reference signal is used in the experiments for initialization and calibration, and as a ground truth for evaluating estimation accuracy. Position processing is implemented using the DSP eQEP module, which decodes the quadrature encoder edges to obtain position increments.

The linear speed is computed using an M-method speed measurement principle. Within a fixed sampling interval T_s, the DSP captures the pulse difference between two adjacent sampling instants via the QPOSLAT register and computes the linear speed. The electrical angle increment is then obtained from the speed–electrical-angle relation (consistent with the pole pitch), and the electrical angle is reconstructed via discrete integration. Before each experiment, an initial phase alignment is performed to reduce the accumulated offset and ensure consistent angle reference.

6.3.2. PWM Generation

The converter switching signals are generated by the DSP ePWM module shown in Figure 17. An up–down counting mode is adopted to generate a symmetric triangular carrier, where the time-base counter TBCTR counts between 0 and TBPRD. A compare register determines the switching instants by toggling the PWM output when TBCTR equals CMPR. Space-vector PWM is implemented at 10 kHz. With a 150 MHz time-base clock, TBPRD is set to 7500 under the up–down counting scheme to achieve the desired switching frequency.

6.3.3. Robustness Evaluation Under Disturbances

The estimated electrical angle obtained from the sensorless strategy is compared against the encoder-based reference. Figure 18 shows that the angle estimation error remains within π/18 under the tested operating conditions.

To evaluate robustness against practical disturbances (e.g., measurement noise, parameter variations, and transient interference), disturbance events are intentionally introduced during operation. Figure 19 and Figure 20 compare the measured d- and q-axis current responses under disturbances when using the proposed adaptive amplitude modulation strategy and a conventional HF square-wave injection scheme. The proposed method yields smoother current transients and smaller tracking errors, without evident chattering. In contrast, the conventional method exhibits larger overshoots, and the disturbance-induced tracking deviations can lead to transient force fluctuations, which may degrade system stability.

6.4. Experimental Results and Discussion

Two sets of experiments were conducted: (i) a no-load test to verify the basic operation of the PMLSG drive and the sensing chain; and (ii) dynamic tests under sinusoidal reciprocating motion to validate the sensorless estimation accuracy and robustness. The results confirm that the proposed adaptive amplitude modulation improves the consistency of the observation signal and enhances disturbance tolerance in comparison with conventional fixed-amplitude injection, supporting the simulation findings in Section 5.2.

It should be emphasized that the platform emulates a regular-wave-equivalent motion and does not reproduce full irregular-wave hydrodynamics; therefore, experimental validation in this paper focuses on the sensorless strategy and its robustness, while full-scale validation of displacement-constrained maximum power capture under realistic sea states is left for future work.

7. Conclusions

An integrated control strategy for DDWEC systems was proposed to address two practical challenges: enforcement of displacement constraints and robust low-speed position-sensorless operation. A displacement-constrained damping regulation scheme was incorporated into the conventional optimal-damping framework through a displacement-dependent correction factor. In parallel, a position-sensorless method based on high-frequency square-wave voltage injection was improved using a two-time-scale adaptive amplitude-modulation mechanism. From the simulation investigations, the following conclusions can be drawn:

• A continuous transition from maximum-power capture to safety-first operation was enabled by the proposed displacement-constrained damping regulation, in which an additional damping term was smoothly increased as the stroke limit was approached while power capture was maintained within the safe operating region.
• Under strong wave excitation, the buoy displacement peak was reduced by approximately 25.7%, with stroke-limit violations prevented, while only a 7.65% decrease in average captured power was observed, indicating a favorable trade-off between safety and energy-capture efficiency. A sensitivity study of x_max, α, and n further showed that x_max and α mainly shift the expected displacement–power trade-off, whereas n primarily affects the smoothness-related electrical stress; the nominal choice x_max = 0.5 m, α = 0.8, and n = 2 was therefore retained as a balanced setting.
• With adaptive amplitude modulation, the demodulated HF current was maintained within the prescribed measurement band, improving position-observation signal conditioning and enhancing disturbance rejection compared to fixed-amplitude injection. The position-error metric remained around 0.06 m, and the two-time-scale scheme reduced error fluctuations, improving estimation robustness.

Future work will focus on extending this approach to more complex irregular wave scenarios and incorporating advanced fault-management strategies for real-world implementation, addressing issues such as power transmission efficiency and operational stability under extreme marine conditions.