An EMD-Based Power Allocation Approach for Hybrid Energy Storage Systems to Smooth PMLG Output Power

Abstract

Direct-drive wave power generation systems based on permanent magnet linear generators (PMLGs) produce fluctuating electromagnetic power under irregular wave excitation, which may affect DC-bus voltage stability and load-side power quality. To smooth the fluctuating output power, this paper develops an empirical mode decomposition (EMD)-based power allocation strategy for a battery–supercapacitor hybrid energy storage system (HESS). In the proposed strategy, EMD is used to decompose the fluctuating electromagnetic power into low-frequency and high-frequency components according to their time-scale characteristics. The low-frequency component is assigned to the battery for energy buffering, while the high-frequency component is assigned to the supercapacitor for transient power compensation. Finite-control-set model predictive current control (FCS-MPCC) is adopted on the generator side to improve the current response of the PMLG, and an MPC-based HESS controller is designed to track the assigned power commands and regulate the DC-bus voltage. Simulation results show a battery power-tracking error of 3.93 W and a DC-bus voltage standard deviation of 0.108 V; compared with LPF, EMD reduced the load-step voltage deviation by 11.94%. Experiments confirm that the PMLG back-EMF follows the translator velocity, the storage currents track their references, and the DC-bus voltage remains within ±2 V of its reference.

1. Introduction

Wave energy has attracted increasing attention as a promising ocean renewable energy source for coastal and island power supply due to its wide availability, relatively high energy density, and predictable characteristics [1,2,3]. Among the various wave energy conversion technologies [4,5], direct-drive wave energy converters (DDWECs) based on permanent magnet linear generators (PMLGs) [6,7,8,9] have attracted considerable attention because they eliminate complex mechanical transmission stages and enable direct energy conversion from buoy motion to electrical power [10,11,12,13]. In addition, linear-rotary generators [14] and magnetic lead screw systems [15,16,17] have also been investigated to improve the energy conversion performance of DDWECs. However, wave excitation forces in realistic sea states are inherently stochastic and non-stationary [18]. As a result, the electromagnetic power generated by the PMLG usually contains multi-frequency [19] fluctuations. If these fluctuations are directly transferred to the DC bus or the load side, they may affect voltage stability and power quality. Therefore, power smoothing is an important issue in direct-drive wave power generation systems.

In a direct-drive wave power system, the generator-side converter regulates the PMLG stator currents to control the electromechanical conversion process [20,21]. Conventional dual-loop PI control is simple and widely used, but its dynamic response may be limited under rapidly varying operating conditions [22,23]. This may lead to current tracking errors during transient conditions. Finite-control-set model predictive current control (FCS-MPCC) is a suitable option because it directly evaluates candidate switching states and provides a fast current response [23,24,25,26].

However, current control alone cannot remove the power fluctuations caused by irregular wave excitation. The electrical power of the PMLG is closely related to the reciprocating motion of the translator. Therefore, both slow power variations and fast transient components may appear in the DC-link power. These components may cause DC-bus voltage variations and increase the burden on the energy storage system. Therefore, an additional energy storage and power smoothing stage is required between the generator-side converter and the load side [27].

A battery–supercapacitor hybrid energy storage system (HESS) is commonly used to smooth the fluctuating power of wave energy converters by exploiting the complementary characteristics of the two storage devices. The battery is suitable for low-frequency energy buffering due to its relatively high energy density, whereas the supercapacitor is more suitable for high-frequency power compensation because of its high power density and fast response [28]. Therefore, the smoothing performance of the HESS largely depends on how the fluctuating power is allocated between the battery and supercapacitor branches. If the allocation is not appropriate, high-frequency power components may enter the battery branch, thereby increasing battery current fluctuations.

Low-pass filtering (LPF) is a simple and commonly used method for power allocation in HESSs. It separates the power signal into low-frequency and high-frequency components according to a fixed cutoff frequency. However, wave power is usually non-stationary, and its dominant frequency may vary with time. In this case, a fixed cutoff frequency may not separate the low-frequency and high-frequency components effectively. In addition, LPF may introduce delay in the generated power reference, which can affect the coordination between the battery and the supercapacitor. Empirical mode decomposition (EMD) [29] provides an alternative way to process non-stationary power signals. It decomposes a signal into several intrinsic mode functions according to its local time-scale characteristics. Unlike LPF, EMD does not require a predefined basis function or a fixed cutoff frequency. Therefore, EMD can be used to divide the fluctuating electromagnetic power into components with different time scales. The slow-varying component can be assigned to the battery, while the fast-varying component can be assigned to the supercapacitor. This allocation helps reduce high-frequency power stress on the battery and improves the power smoothing function of the HESS.

Recent studies on direct-drive wave energy systems can be grouped into three main topics. For front-end control, Lin et al. included electrical losses of the PTO in the control objective and studied loss-aware wave-to-wire optimization [30]. Liao and Li developed a nonlinear MPC framework for a wave-to-wire WEC system with nonlinear dynamics and constraints [31]. Huang et al. combined loss-aware MPC with variable DC-voltage regulation to improve the wave-to-wire efficiency of direct-drive WECs [32]. These studies improved front-end control, but they mainly focused on conversion efficiency and constraint handling rather than storage-side power smoothing. For HESS energy management, Zhu et al. proposed a fast nonlinear MPC strategy to reduce the computational burden in battery–supercapacitor systems for wave energy converters [33]. They also developed a learning-augmented MPC method for online weight tuning and battery degradation mitigation [34]. Tan et al. [35] combined an IDA-PBC outer voltage loop with MPC inner current loops to coordinate the battery and supercapacitor. In their subsequent study [36], MFPC–HOSMO and MPC were employed to improve system robustness against parameter variations, but the power allocation still relied on a predefined cutoff frequency. Although these studies improved HESS control performance, adaptive allocation of non-stationary power using an automatically determined frequency boundary has not been fully addressed. In addition, studies on linear generators have mainly focused on topology optimization and electromagnetic performance [37,38,39,40,41,42]. Zhu et al. designed and experimentally validated a direct-driven linear-rotary wave generator [43], while Wang et al. enhanced the back-EMF and thrust density of a linear permanent-magnet Vernier machine through structural optimization [44]. However, less attention has been paid to a coordinated control framework that links generator-side current control, adaptive power decomposition, and HESS-side predictive control.

To address this problem, this paper proposes an EMD-based power allocation strategy for a battery–supercapacitor HESS in a PMLG-based direct-drive wave power generation system. The fluctuating electromagnetic power is decomposed by EMD into low-frequency and high-frequency components according to its time-scale characteristics. The low-frequency component is assigned to the battery for energy buffering, while the high-frequency component is assigned to the supercapacitor for fast transient power compensation. FCS-MPCC is adopted on the generator side to improve the current response of the PMLG, and an MPC-based HESS controller is designed to track the assigned power commands and regulate the DC-bus voltage. The proposed strategy was evaluated through simulations under irregular-wave excitation and compared with a conventional causal first-order LPF under the same 80 s input and control settings. Supplementary experiments were conducted to verify the operating characteristics of the PMLG, the current-tracking performance of the storage branches, and the DC-bus voltage regulation.

2. Mathematical Modeling of the Direct-Drive Wave Power Generation System with Hybrid Energy Storage

Mathematical models of the buoy, PMLG, and HESS are required for controller design and simulation analysis. This section first describes the heave-motion model of the point-absorber buoy. The PMLG model is then derived in the synchronous dq reference frame. Finally, the battery–supercapacitor HESS and its bidirectional DC–DC converters are modeled to support the subsequent control design.

2.1. Heave-Motion Model of the Point-Absorber Buoy

Considering the vertical heave motion of the device, the buoy is mechanically coupled to the PMLG translator through a stiff and non-slack connection. In this direct-drive configuration, the heave motion of the buoy is transmitted directly to the translator. By neglecting the compliance and slack-line motion of the connection, the buoy and translator have the same vertical displacement and velocity and are therefore represented as a single-degree-of-freedom coupled mechanical system [45,46]. Based on Newton’s second law, the heave dynamics of the coupled mechanical system can be described as:

$$M{x}^{″}(t)=F_{\mathrm{hyd}}(t)+F_{\mathrm{pto}}(t)-M_{\mathrm{g}}$$

(1)

where x(t) is the vertical displacement of the translator, M is the total structural mass of the moving components, F_hyd(t) is the hydrodynamic force acting on the buoy, F_pto(t) is the reaction force generated by the power take-off (PTO) system, and g is the gravitational acceleration. Specifically, under the synchronous-motion assumption adopted in this study, M is determined as the sum of the dry mass of the buoy, the mass of the PMLG translator, and the mass of any rigid connecting components moving with them. Therefore, M includes both the buoy and the PMLG translator, whereas the fixed stator and supporting structure are excluded [47].

Under linear wave theory and the small-amplitude motion assumption, the hydrodynamic force can be expressed as the sum of the excitation force, radiation force, and hydrostatic restoring force:

$$F_{\mathrm{hyd}}(t)=F_{\mathrm{ex}}(t)+F_{\mathrm{rad}}(t)+F_{\mathrm{hs}}(t)$$

(2)

where F_ex(t) is the wave excitation force induced by incident waves, F_rad(t) is the radiation force associated with the buoy motion, and F_hs(t) is the hydrostatic restoring force.

When the static equilibrium position is taken as the reference point, the hydrostatic restoring force caused by vertical displacement can be approximated by a linear stiffness term:

$$F_{\mathrm{hs}}\left(t\right)=-Kx(t)$$

(3)

where K is the hydrostatic restoring coefficient determined by the buoy geometry and fluid properties.

In the simplified linear model used in this study, the radiation force accounts for the inertial and damping effects resulting from fluid–structure interaction and can be expressed as:

$$F_{\mathrm{rad}}(t)=-M_a{x}^{″}(t)-C{x}^{\prime }(t)$$

(4)

where M_a is the added mass and C is the radiation damping coefficient. The hydrodynamic added mass M_a is not included in the structural moving mass M but is considered separately in the radiation-force model. Consequently, the effective inertia in the final heave-motion equation is represented by M + M_a [48].

The excitation force is related to the incident wave motion and can be equivalently expressed as:

$$F_{\mathrm{ex}}(t)=M_a{x}_w^{″}(t)+C{x}_w^{\prime }(t)+K{x}_w(t)$$

(5)

where x_w(t) denotes the equivalent displacement associated with the incident wave.

By substituting Equations (2)–(5) into Equation (1), the heave motion equation of the wave energy converter can be obtained as:

$$(M+M_a)x^{″}(t)+C{x}^{\prime }(t)+Kx(t)-F_{\mathrm{pto}}(t)-M_g=F_{\mathrm{ex}}(t).$$

(6)

Because the gravitational term determines only the static equilibrium position, the displacement is redefined relative to this equilibrium position. The gravitational term is then removed, and the simplified linear equation is obtained as:

$$(M+M_a)x^{″}(t)+C{x}^{\prime }(t)+Kx(t)-F_{\mathrm{pto}}(t)=F_{\mathrm{ex}}(t).$$

(7)

Equation (7) gives the linear time-domain model used for electromechanical coupling analysis and subsequent controller design.

2.2. Mathematical Model of the PMLG in the Synchronous dq Reference Frame

For a control-oriented PMLG model in the synchronous dq reference frame, the following assumptions are made. Magnetic saturation, core losses, cogging force, and higher-order harmonics are neglected. The three-phase armature windings are assumed to be symmetrical, the phase inductances are treated as constants, and the air gap between the stator and translator is considered uniform.

These assumptions establish a control-oriented model that retains the principal relationships among the linear velocity, dq-axis currents, electromagnetic thrust, and electromagnetic power required for the generator-side current control and subsequent HESS power allocation. Under practical operating conditions, magnetic saturation may change the equivalent excitation flux linkage and phase inductances, while cogging force and higher-order harmonics may introduce additional electromagnetic-thrust and current ripple. Core losses contribute an additional loss component. Incorporating these non-ideal effects would further improve the detailed electromagnetic and loss predictions of the PMLG model over a wider operating range.

The stator voltage and flux linkage equations of the PMLG are first written in the three-phase stationary abc reference frame as:

$${\Psi }_{abc}=L_{abc}{I}_{abc}+{\Psi }_{f,abc}$$

(8)

$$U_{abc}=R_s{I}_{abc}+{\displaystyle \frac{d{\Psi }_{abc}}{dt}}$$

(9)

where U_abc, I_abc, and ψ_abc are the three-phase voltage, current, and flux linkage vectors; R_s is the winding phase resistance; and ψ_f,abc is the permanent magnet excitation flux vector.

To obtain a model suitable for current control, the three-phase variables are transformed into the synchronous dq reference frame. In this reference frame, the stator voltage and flux linkage equations can be written as:

$$\left\{\begin{array}{l}{\Psi }_d=L_d{i}_d+{\Psi }_f\\ {\Psi }_q=L_q{i}_q\end{array}\right.$$

(10)

$$\left\{\begin{array}{l}{u}_d=R_s{i}_d+{\displaystyle \frac{d{\Psi }_d}{dt}}-{\displaystyle \frac{\pi v}{\tau }}{\Psi }_q\\ u_q=R_s{i}_q+{\displaystyle \frac{d{\Psi }_q}{dt}}+{\displaystyle \frac{\pi v}{\tau }}{\Psi }_d\end{array}\right.$$

(11)

where τ is the pole pitch, v is the linear velocity, and ψ_f is the equivalent excitation flux linkage.

For a surface-mounted PMLG, the d-axis and q-axis inductances are assumed to be equal, i.e., L_d = L_q. Under this condition, the reluctance thrust term vanishes, and the electromagnetic thrust is proportional to the q-axis current:

$$F_e={\displaystyle \frac{3\pi }{2\tau }}{\Psi }_f{i}_q=K_e{i}_q$$

(12)

where K_e is the electromagnetic thrust coefficient.

The electromagnetic power of the PMLG is then calculated as:

$$P_e=F_e·v={\displaystyle \frac{3\pi }{2\tau }}v{\Psi }_f{i}_q.$$

(13)

In Equation (13), P_e denotes the electromagnetic power calculated from the electromagnetic thrust and linear velocity. In the present control-oriented PMLG model, core losses are not introduced as a separate loss term. The dq-axis model and the electromagnetic power expression are used in the generator-side current control and the subsequent HESS power allocation.

2.3. Mathematical Modeling of the HESS

To smooth the fluctuating output power and maintain DC-bus stability, a battery–supercapacitor HESS is used in the direct-drive wave power generation system. Based on the complementary characteristics of the two storage devices, the battery branch is used to handle the low-frequency power component, while the supercapacitor branch is used to compensate for the high-frequency transient component. Bidirectional DC–DC converters are adopted as the power electronic interfaces between the storage branches and the DC bus. The overall topology of the HESS is shown in Figure 1.

2.3.1. Battery Model

The PNGV equivalent circuit of the battery is shown in Figure 2. The model includes an open-circuit voltage source, an ohmic resistance, and polarization branches to describe the steady-state and transient electrical behavior of the battery.

The model equations are written as:

$$I_{bat}(t)={\displaystyle \frac{U_n(t)}{R_n}}+C_n{\displaystyle \frac{d{U}_n(t)}{dt}}$$

(14)

$$U_{bat}(t)=U_{oc}(t)+U_p(t)-I_{bat}(t)R_b-U_{bat}(t)$$

(15)

$$I_{bat}(t)=-C_p{\displaystyle \frac{d{U}_p(t)}{dt}}$$

(16)

where U_oc is the open-circuit voltage, R_b is the ohmic internal resistance, R_n is the polarization resistance, C_n is the polarization capacitance, C_p is the capacitance used to describe the dynamic behavior of the battery, U_n is the voltage across the polarization branch, U_p is the voltage across C_p, I_b is the battery current, and U_b is the battery terminal voltage.

With the discharge current defined as positive, the battery power is calculated as:

$$P_{bat}(t)=U_{bat}(t)I_{bat}(t)$$

(17)

where P_b > 0 denotes battery discharge and P_b < 0 denotes battery charge.

2.3.2. Supercapacitor Model

The equivalent circuit of the supercapacitor is shown in Figure 3. In the HESS, the supercapacitor is used to compensate for the high-frequency transient power component.

The model equations are written as:

$$U_{sc}(t)=U_c(t)+I_{sc}(t)R_{\mathrm{ESR}}$$

(18)

$$I_{sc}(t)={\displaystyle \frac{U_c(t)}{R_{\mathrm{EPR}}}}+C_c{\displaystyle \frac{d{U}_c(t)}{dt}}$$

(19)

where R_ESR is the equivalent series resistance, R_EPR is the equivalent parallel resistance, C_c is the supercapacitor capacitance, U_c is the capacitor voltage, I_sc is the supercapacitor current, and U_sc is the terminal voltage of the supercapacitor.

Since R_EPR is large, the leakage current is small over the simulation time scale considered in this study. Therefore, the leakage branch is neglected, and the simplified model is written as:

$$I_{sc}(t)=C_c{\displaystyle \frac{d{U}_c(t)}{dt}}.$$

(20)

The stored energy of the supercapacitor is expressed as:

$$W_{sc}(t)={\displaystyle \frac{1}{2}}{C}_c{U}_c^2(t).$$

(21)

3. EMD-Based Power Allocation and Control Strategy of the DDWEC with HESS

3.1. Generator-Side FCS-MPCC of the PMLG

FCS-MPCC is used as the generator-side current control method to improve the current response of the PMLG under fluctuating wave excitation. The overall configuration of the generator-side FCS-MPCC is shown in Figure 4.

According to the dq-axis voltage equations of the PMLG, the continuous-time current dynamics used for prediction can be written as:

$$\left\{\begin{array}{l}{e}_d=R_s{i}_d(t)+L_d{\displaystyle \frac{d{i}_d(t)}{dt}}-{\displaystyle \frac{\pi v(t)}{\tau }}{L}_q{i}_q(t)+u_d(t)\\ e_q=R_s{i}_q(t)+L_q{\displaystyle \frac{d{i}_q(t)}{dt}}-{\displaystyle \frac{\pi v(t)}{\tau }}{L}_d{i}_d(t)+u_q(t)\end{array}\right.$$

(22)

where e_d and e_q are the d-axis and q-axis back electromotive forces, i_d and i_q are the stator currents, u_d and u_q are the dq-axis converter output voltages, R_s is the stator resistance, L_d and L_q are the d-axis and q-axis inductances, v(t) is the translator velocity, and τ is the pole pitch.

The current derivatives are discretized using the first-order Euler approximation:

$${\displaystyle \frac{di}{dt}}\approx {\displaystyle \frac{i(k+1)-i(k)}{T_s}}$$

(23)

where T_s is the sampling period. The one-step-ahead prediction model of the PMLG currents is then obtained as:

$$\left\{\begin{array}{l}{i}_d(k+1)=(1-T_s{\displaystyle \frac{R_s}{L_d}})i_d(k)+T_s{\displaystyle \frac{\pi v(k)}{\tau }}{\displaystyle \frac{L_q}{L_d}}{i}_q(k)-T_s{\displaystyle \frac{u_d(k)}{L_d}}\\ i_q(k+1)=T_s{\displaystyle \frac{\pi v(k)}{\tau L_q}}{\Psi }_f+(1-T_s{\displaystyle \frac{R_s}{L_q}})i_q(k)-T_s{\displaystyle \frac{\pi v(k)}{\tau }}{\displaystyle \frac{L_d}{L_q}}{i}_q(k)-T_s{\displaystyle \frac{u_d(k)}{L_d}}\end{array}\right..$$

(24)

For the surface-mounted PMLG considered in this study, the d-axis current reference was set to zero to reduce unnecessary d-axis current. Thus,

$$i_d^{*}(k+1)=0.$$

(25)

Based on the predicted currents, the generator-side FCS-MPCC cost function is defined as:

$$J=\left|i_{\mathrm{d}}^{*}(k+1)-i_{\mathrm{d}}(k+1)\right|+\left|i_{\mathrm{q}}^{*}(k+1)-i_{\mathrm{q}}(k+1)\right|.$$

(26)

To ensure safe operation, the current constraint is written as

$$\left\{\begin{array}{l}-i_{\max }\le i_d\le i_{\max }\\ -i_{\max }\le i_q\le i_{\max }\end{array}\right..$$

(27)

Although the generator-side MPCC improves the current response of the PMLG, the electromagnetic power produced under irregular wave excitation still requires a power allocation stage before HESS-side control.

3.2. EMD-Based Power Allocation of the HESS

EMD was therefore used in this study as the power allocation method. It decomposes the HESS power command into components with different local time scales. The physical basis of this allocation lies in the complementary power and energy characteristics of the battery and supercapacitor. The relatively high energy density of the battery makes it suitable for the low-frequency component associated with slowly varying and longer-duration energy exchange. In contrast, the high power density, rapid charge–discharge response, and high cycle endurance of the supercapacitor make it suitable for the high-frequency component associated with short-duration transient power fluctuations [49,50]. Accordingly, the reconstructed low-frequency component is used as the battery power reference, whereas the high-frequency component is used as the supercapacitor power reference. This allocation reduces the exposure of the battery to rapid current variations while utilizing the supercapacitor for fast transient power compensation.

3.2.1. EMD-Based Decomposition of Fluctuating Power

In the proposed system, the electrical power generated by the PMLG is primarily delivered to the load through the common DC bus. The HESS does not absorb the entire generated power, but only compensates for the instantaneous power imbalance between the generated power and the load demand. Neglecting line loss and converter loss, the HESS power command is defined as:

$$P_{\mathrm{HESS}}(t)=P_{\mathrm{load}}(t)-P_{\mathrm{wave}}(t)$$

(28)

where P_load(t) is the load-side power demand and P_wave(t) is the electrical power generated by the direct-drive wave power generation system. When the generated wave power exceeds the load demand, the HESS absorbs the surplus power. When the generated power is lower than the load demand, the HESS supplies the power deficit.

The fluctuating power signal to be decomposed is defined as:

$$P^{\prime }(t)=P_{\mathrm{HESS}}(t).$$

(29)

The local maxima and local minima of P′(t) are first identified. The upper envelope P_max(t) and lower envelope P_min(t) are then constructed by cubic spline interpolation. Their mean value is expressed as:

$$m(t)={\displaystyle \frac{{P^{\prime }}_{\max }(t)+{P^{\prime }}_{\min }(t)}{2}}.$$

(30)

By subtracting the envelope mean from the input signal, the local detail component is then obtained as:

$$r(t)=P^{\prime }(t)-m(t).$$

(31)

If r(t) satisfies the intrinsic mode function (IMF) criterion, it is extracted as an IMF component, denoted by imf_i(t). Otherwise, the sifting process is repeated until the IMF criterion is satisfied. After the i-th IMF component is extracted, the residual signal is updated according to:

$$P^{\prime }(t)=P^{\prime }(t)-{\mathrm{imf}}_i(t).$$

(32)

This procedure is repeated until the residual becomes monotonic or contains no further oscillatory component. The fluctuating power signal is finally expressed as the sum of all IMF components and a residual term.

3.2.2. Frequency Boundary Determination and Power Reconstruction

After EMD, the frequency characteristics of each IMF component are estimated using the Hilbert transform. For the $i$-th IMF component imf_i, the Hilbert transform is defined as:

$$H\left[{\mathrm{imf}}_i(t)\right]={\displaystyle \frac{1}{\pi }}\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{-\infty }^{+\infty }\frac{{\mathrm{imf}}_i(\tau )}{t-\tau }d\tau }$$

(33)

where P.V. is the Cauchy principal value.

The corresponding analytic signal is expressed as:

$$z_i(t)={\mathrm{imf}}_i(t)+j{\mathrm{imf}}_i(t)=a_i(t)e^{j{\theta }_i(t)}$$

(34)

where a_i(t) is the instantaneous amplitude and θ_i(t) is the instantaneous phase.

The instantaneous frequency of the i-th IMF component is obtained as:

$$f_i(t)={\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{d{\theta }_i(t)}{dt}}.$$

(35)

Based on the instantaneous-frequency distributions of the IMF components, each possible division between adjacent IMF components is evaluated as a candidate frequency boundary. For each candidate, a frequency-aliasing index is calculated according to the instantaneous-frequency crossings between neighboring IMF components. The candidate corresponding to the minimum nonzero index is automatically selected as the frequency boundary. A smaller index indicates weaker frequency aliasing and a clearer separation between the fast and slow power fluctuations. Based on the determined boundary, the low-frequency and high-frequency powers are reconstructed as follows:

$$P_{\mathrm{low}}(t)={\displaystyle \sum _{i=n_b}^N{\mathrm{imf}}_i(t)+\mathrm{res}\left(\mathrm{t}\right)},$$

(36)

$$P_{\mathrm{high}}(t)={\displaystyle \sum _{i=1}^{n_{b=1}}{\mathrm{imf}}_i(t)}.$$

(37)

Accordingly, the reference powers of the battery branch and the supercapacitor branch are defined as:

$$P_{bat,\mathrm{ref}}(t)=P_{\mathrm{low}}(t),$$

(38)

$$P_{sc,\mathrm{ref}}(t)=P_{\mathrm{high}}(t).$$

(39)

where the low-frequency component is assigned to the battery branch and the high-frequency component is assigned to the supercapacitor branch.

The frequency-aliasing index is used to select the boundary with the weakest instantaneous-frequency overlap between adjacent IMF components. This criterion reduces the influence of mode mixing on the power-allocation boundary by avoiding the separation of strongly overlapping neighboring modes. The low-frequency and high-frequency power references are reconstructed by directly summing the selected IMF components at their original sampling instants. Therefore, the reconstructed power components retain the original time coordinates without introducing an additional fixed group delay. EEMD and CEEMDAN are noise-assisted extensions that can further improve modal separation and reconstruction completeness [51,52]. Endpoint-extension approaches based on mirror expansion and signal prediction have also been developed to improve the boundary treatment of EMD [53]. Considering the computational efficiency required by the power-allocation framework, classical EMD combined with the automatic frequency-aliasing-based boundary-selection criterion was adopted in this study.

3.3. MPC-Based Control Strategy of the HESS

Based on the EMD-based power allocation, the HESS-side controllers are designed to track the assigned power commands and regulate the DC-bus voltage. The battery branch tracks the low-frequency power reference through a power outer loop and an MPC current inner loop. For the supercapacitor branch, the current reference is generated by the DC-bus voltage regulation term, and the resulting current command is tracked by an MPC current inner loop. The overall control structure is shown in Figure 5.

3.3.1. Battery-Branch Control

The battery branch is used to track the low-frequency power reference P_bat,ref obtained from the EMD-based allocation. The power error is processed by a PI regulator to generate the battery current reference:

$$I_{bat,\mathrm{ref}}(k)=G_{PI,bat}(P_{bat,\mathrm{ref}}(k)-P_{bat}(k))$$

(40)

where P_b(k) is the actual battery power and G_PI,bat is the battery-side PI regulator.

Based on the battery-branch prediction model, the battery current at the next sampling instant is predicted as:

$$I_{bat}(k+1)=I_{bat}(k)+{\displaystyle \frac{T_s}{L_1}}(V_{bat}(k)-V_{dc}(k)S_{w1}(k))$$

(41)

where I_bat(k) is the battery current, V_bat(k) is the battery terminal voltage, V_dc(k) is the DC-bus voltage, L₁ is the battery-branch inductance, T_S is the sampling period, and S_w1(k) is the candidate switching state.

The battery-side cost function is defined as:

$$J_1={[I_{bat,\mathrm{ref}}(k)-I_{bat}(k+1)]}^2.$$

(42)

The switching state that minimizes J₁ is selected and applied in the next control interval.

The battery-side MPC prioritizes the tracking of the current reference generated from the allocated battery power command, while the influence of rapid power variations on the battery is mitigated primarily through the preceding EMD-based power-allocation stage. EMD adaptively decomposes the fluctuating HESS power command into IMF components with different local time scales. Based on the automatically determined boundary, the slowly varying low-frequency IMFs and residual component are reconstructed as the battery power reference, whereas the rapidly fluctuating high-frequency IMFs are reconstructed as the supercapacitor power reference. Because the battery current reference is generated from the reconstructed low-frequency power component, high-frequency fluctuations are prevented from entering the battery current-control loop. This produces a smoother battery current command and reduces rapid current variations and frequent charge–discharge transitions.

3.3.2. Supercapacitor-Branch Control

The supercapacitor branch is used to compensate for fast power fluctuations and support DC-bus voltage regulation. Owing to its fast dynamic response, the outer-loop control objective of the supercapacitor branch is selected as DC-bus voltage regulation. The voltage error is defined as:

$$e_{dc}(k)=V_{dc,\mathrm{ref}}(k)-V_{dc}(k).$$

(43)

The supercapacitor current reference is generated by the voltage outer-loop PI controller as:

$$I_{sc,\mathrm{ref}}(k)=G_{PI,sc}(V_{dc,\mathrm{ref}}(k)-V_{dc}(k))$$

(44)

where G_PI,sc is the supercapacitor-side PI regulator.

Based on the supercapacitor-branch prediction model, the supercapacitor current at the next sampling instant is predicted as:

$$I_{sc}(k+1)=I_{sc}(k)+{\displaystyle \frac{T_s}{L_2}}(V_{sc}(k)-V_{dc}(k)S_{w4}(k))$$

(45)

where I_sc(k) is the supercapacitor current, V_sc(k) is the supercapacitor terminal voltage, L₂ is the supercapacitor-branch inductance, and S_w4(k) is the candidate switching state.

The supercapacitor-side cost function is defined as:

$$J_2={[I_{sc,\mathrm{ref}}(k)-I_{sc}(k+1)]}^2.$$

(46)

The switching state that minimizes J₂ is selected and applied in the next control interval.

4. Results

4.1. Simulation Setup

To verify the proposed coordinated control strategy, a Simulink model of DDWEC with battery–supercapacitor HESS was established. The model includes the PMLG-based WEC unit, the generator-side FCS-MPCC, the common DC bus, and the battery–supercapacitor HESS. The main simulation parameters are listed in

Table 1. Main simulation parameters of the proposed system.

Parameter	Value
Simulation time (s)	80
Fixed-step size (s)	0.00001
DC-bus voltage reference (V)	330
d-axis inductance (H)	0.031
q-axis inductance (H)	0.031
Stator resistance (Ω)	2.8
Flux linkage (Wb)	0.38
Pole pitch (m)	0.1
d-axis current reference (A)	0
Battery nominal voltage (V)	30
Battery nominal capacity (Ah)	30
Initial battery SOC (%)	59.95
Battery internal resistance (Ω)	0.01
Battery-branch inductance (H)	0.1
Supercapacitor capacitance (F)	100
Supercapacitor ESR (Ω)	8.9 × 10−3
Rated supercapacitor voltage (V)	600
Initial supercapacitor voltage (V)	181

.

4.2. Generator-Side MPCC Performance

The generator-side control performance was evaluated by the d-axis and q-axis current responses of the PMLG, as shown in Figure 6.

As shown in Figure 6a, the d-axis current remained close to zero during the simulation, with an RMS tracking error of 0.174 A. Figure 6b shows that the q-axis current followed its reference under irregular wave excitation, with an RMS tracking error of 0.184 A. These results indicate that the generator-side FCS-MPCC provides stable current regulation for the PMLG.

Mechanical and electromagnetic power characteristics of the PMLG are shown in Figure 7.

As shown in Figure 7a, the mechanical captured power increases during the initial transient stage and then enters a stable oscillatory state. Figure 7b shows that the electromagnetic power varies with the reciprocating translator velocity, reflecting the pulsating power characteristic of the direct-drive PMLG. The resulting electromagnetic power signal is subsequently used as the input for the EMD-based HESS power allocation.

4.3. EMD-Based Power Allocation Results and LPF Comparison

The EMD decomposition result of the fluctuating power signal is presented in Figure 8a. Each colored curve represents the temporal evolution of one IMF component, and the three axes denote time, IMF order, and power amplitude, respectively. IMF1–IMF6 exhibit densely distributed and rapidly varying oscillations over the 0–80 s interval, indicating that these components mainly contain short-time-scale power fluctuations. From IMF7 to IMF10, the oscillations become progressively more widely spaced, and the corresponding components represent medium- to long-time-scale power variations. The IMF components exhibit different power-amplitude ranges, and IMF order primarily reflects the variation in their oscillation time scales.

To determine the reconstruction boundary quantitatively, the frequency-aliasing indices of the valid candidate boundaries are compared in Figure 8b. The minimum positive index was 0.187 at the IMF6/IMF7 boundary, compared with 1.871 at IMF5/IMF6 and 0.474 at IMF7/IMF8. According to the criterion described in Section 3.2.2, a smaller index indicates weaker frequency aliasing between neighboring IMF components. Therefore, the boundary between IMF6 and IMF7 is selected automatically. Accordingly, IMF1–IMF6 are combined to form P_high, with IMF7–IMF10 together with the residual form P_low.

Based on the IMF6/IMF7 boundary automatically selected in Figure 8b, the reconstructed P_high and P_low are shown in Figure 9. The low-frequency component presents a smooth and slowly varying waveform and contains the main energy variation of the fluctuating power. In contrast, the high-frequency component oscillates around zero and represents short-time power fluctuations. These reconstructed components are consistent with the time-scale hierarchy presented in Figure 8a and the quantitative boundary selection shown in Figure 8b.

Figure 10 compares the EMD- and LPF-based power-allocation results over the same 80 s record. A causal first-order LPF was adopted as the conventional comparison method. The LPF output was assigned to P_low, while the difference between the original power and the LPF output was assigned to P_high. For a fair comparison, all system and controller settings were kept unchanged. The cutoff frequency was determined from the characteristic frequencies of the two adjacent IMFs on either side of the automatically selected EMD boundary, giving f_c = 0.4457 Hz. Low-frequency leakage was calculated as the proportion of the spectral energy below f_c in P_high relative to its total spectral energy. A lower leakage value indicates that less slowly varying power remains in the supercapacitor power command.

As shown in

Table 2. Quantitative comparison of the EMD- and LPF-based power-allocation methods.

Index	EMD	LPF	Relative Change
Standard deviation of Plow (W)	579.2	542.8	+6.70%
Standard deviation of Phigh (W)	56.4	204.9	−72.47%
Low-frequency leakage in Phigh (%)	6.07	93.14	−93.48%
Additional causal-filter lag (s)	-	0.314	-
Maximum Udc deviation after the 45 s load step (V)	0.426	0.484	−11.94%

, the proposed EMD method reduced the standard deviation of P_high by 72.47% and the low-frequency leakage in P_high by 93.48% compared with LPF. The low leakage value of 6.07% shows that the EMD-based high-frequency component mainly contains rapidly varying power, whereas the value of 93.14% for LPF indicates that a large amount of slowly varying power remains in its high-frequency component. In addition, the causal LPF produced an additional lag of 0.314 s. After the load change at 45 s, EMD reduced the maximum DC-bus voltage deviation by 11.94%. These results show that EMD provides clearer power separation and improves the DC-bus response during the load change.

4.4. Battery and Supercapacitor Control Performance

The HESS-side control performance was evaluated from the power and current responses of the battery and supercapacitor branches. The power tracking results are shown in Figure 11.

As shown in Figure 11a, the battery power follows the low-frequency power reference generated by the EMD-based allocation, indicating that the battery branch executes the energy-buffering command. Figure 11b shows that the supercapacitor power varies more rapidly around zero, which is consistent with its role in compensating fast power fluctuations. The actual power follows the main fluctuation trend of the reference signal, indicating that the supercapacitor branch provides the required fast compensation.

The current tracking results of the storage branches are shown in Figure 12.

As shown in Figure 12a, the battery current follows its reference with a relatively smooth waveform. This behavior is consistent with the role of the battery branch in handling the low-frequency power component generated by the EMD-based allocation. Figure 12b shows the current tracking response of the supercapacitor branch. Compared with the battery current, the supercapacitor current contains faster variations, which is consistent with its function of compensating rapid power fluctuations and supporting DC-bus voltage regulation. The actual current follows the main variation trend of the reference current after the initial transient.

4.5. Load Power Condition and DC-Bus Voltage Stabilization

The load power profile used in the simulation is shown in Figure 13. As shown in Figure 13, the load power was approximately 330 W from 0 s to 45 s and then increased to approximately 990 W after 45 s.

As shown in Figure 14, the DC-bus voltage rose during the startup stage and then converged to the 330 V reference. A short transient overshoot appeared at the beginning of the simulation. The maximum DC-bus voltage deviation during this transient was 39.72 V, and the voltage recovered to within ±1% of its reference value within approximately 0.011 s. During the load transition at 45 s, the maximum voltage deviation was only 0.43 V, and the DC-bus voltage remained within the ±1% band. The enlarged view further shows that the voltage fluctuation remained small under the fluctuating power condition.

4.6. Quantitative Summary of Simulation Results

To further evaluate the proposed coordinated control strategy, the key quantitative indices obtained from the simulation are summarized in Table 3.

Table 3. Key performance indices of the proposed control strategy.

Item	Index	Value
Generator-side current control	RMS error of iq tracking	0.184 A
	RMS value of id	0.174 A
Electromechanical conversion	Mean electromagnetic power	682.0 W
EMD-based power allocation	Standard deviation of Plow/Phigh	579.2/56.4 W
Battery power tracking	RMS error of Pbat	3.93 W
Supercapacitor power tracking	RMS error of Psc	14.63 W
HESS current tracking	RMS error of ibat/isc	0.099/0.081 A
DC-bus voltage regulation	Std. of Vdc after transient	0.108 V

Table 3. Key performance indices of the proposed control strategy.

Item	Index	Value
Generator-side current control	RMS error of iq tracking	0.184 A
	RMS value of id	0.174 A
Electromechanical conversion	Mean electromagnetic power	682.0 W
EMD-based power allocation	Standard deviation of Plow/Phigh	579.2/56.4 W
Battery power tracking	RMS error of Pbat	3.93 W
Supercapacitor power tracking	RMS error of Psc	14.63 W
HESS current tracking	RMS error of ibat/isc	0.099/0.081 A
DC-bus voltage regulation	Std. of Vdc after transient	0.108 V

The quantitative indices in Table 3 provide an overall evaluation of the proposed coordinated control strategy. The RMS errors of the q-axis and d-axis currents were 0.184 A and 0.174 A, respectively, indicating stable generator-side current regulation. The mean electromagnetic power of 682.0 W characterizes the average electromechanical output under the considered irregular-wave condition. The standard deviations of P_low and P_high were 579.2 W and 56.4 W, respectively, quantifying the different variation levels of the reconstructed slow and fast power components. The RMS power-tracking errors of the battery and supercapacitor branches were 3.93 W and 14.63 W, while their RMS current-tracking errors were 0.099 A and 0.081 A, respectively. These indices quantify the ability of the two storage branches to follow their assigned power and current references. After the initial transient, the standard deviation of the DC-bus voltage was 0.108 V, corresponding to approximately 0.033% of the 330 V reference, which indicates stable DC-bus voltage regulation under fluctuating power conditions.

4.7. Experimental Validation

To supplement the simulation results, a laboratory-scale experimental platform was established for the DDWEC with a battery–supercapacitor HESS, as shown in Figure 15. The hardware mainly consisted of a wave power generation emulator, a permanent magnet linear generator (PMLG), a three-phase half-bridge rectifier, a battery, a supercapacitor, two bidirectional DC–DC converters, and a DC load. The control system was implemented using a DSP controller and a rapid-control-prototyping controller. The wave power generation emulator consisted of an induction motor, a frequency converter, a gearbox, a crank–connecting-rod mechanism, and the PMLG. The output of the PMLG was connected to the common DC bus through the three-phase half-bridge rectifier. The battery and supercapacitor were interfaced with the DC bus through their respective bidirectional DC–DC converters. The DSP controller was used to implement the generator-side control algorithm, whereas a rapid-control-prototyping controller was employed for the control of the hybrid energy storage system.

The PMLG was a TLM160-B0954SP-1-0300-E110 (Dongguan Aipusi Automation Technology Co., Ltd., Dongguan, China). The mechanical wave emulator employs a gearbox with a reduction ratio of 25 and a crank–connecting-rod mechanism with a connecting-rod length of 360 mm. The DC-bus voltage was measured using an LV25-P voltage transducer (LEM International SA, Meyrin, Switzerland) with a measurement range of 10–500 V and an accuracy of ±0.9%, while the electrical signals were acquired using a 16-bit AD7656 (Nanjing Yanxu Electrical Technology Co., Ltd., Nanjing, China) data-acquisition module with a bipolar input range of ±10 V. The generator-side control algorithm was implemented using a YXDSP-F28335 DSP (Nanjing Yanxu Electrical Technology Co., Ltd., Nanjing, China) controller, whereas the HESS-side MPC was implemented using a YXSPACE-SP2000 rapid-control-prototyping controller (Nanjing Yanxu Electrical Technology Co., Ltd., Nanjing, China) based on a TMS320F28377 processor (Texas Instruments Incorporated, Dallas, TX, USA). The three-phase half-bridge rectifier employs FF50R12RT4 IGBT modules (Infineon Technologies AG, Neubiberg, Germany) rated at 1200 V/50 A and a TX-DA102D6 six-channel PWM driver (Beijing LMY Electronics Co., Ltd., Beijing, China).

The main parameters of the laboratory-scale PMLG and the wave power generation emulator are summarized in

Table 4. Main parameters of the laboratory-scale PMLG.

Parameter	Value	Parameter	Value
Permanent-magnet flux linkage	0.278 Wb	Pole pitch	16 mm
Stator resistance	4.4 Ω	Gearbox reduction ratio	25
Synchronous inductance	21 mH	Connecting-rod length	360 mm

. The simulation model and the laboratory prototype represent PMLGs with different physical dimensions and rated parameters. Therefore, the simulation controller used the parameters listed in Table 1, whereas the experimental controller was parameterized using the actual prototype parameters listed in Table 4. Although the parameter values are different, the mathematical model, control algorithm, and control objectives remain unchanged.

The laboratory PMLG used sintered Nd–Fe–B permanent magnets of grade N42SH. At 20 °C, the nominal remanent magnetic flux density B_r was 1.27–1.33 T, the normal coercivity H_cB was 0.923–1.035 MA/m, the intrinsic coercivity H_cJ was at least 1.671 MA/m, and the maximum energy product (BH)_max was 310–342 kJ/m³. The second-quadrant B–H curve decreased approximately linearly from B ≈ 1.30 T at H = 0 toward B = 0 as the reverse magnetic field approached H_cB.

The experimental results were used to verify the main control functions investigated in the simulations. Figure 16 verifies the electromechanical response of the PMLG through the measured relationship between the translator velocity and the phase-A back-EMF. Figure 17 evaluates the tracking of the battery and supercapacitor current references generated from the EMD-based power commands. Figure 18 further evaluates the coordinated operation of the two storage branches and the regulation of the DC-bus voltage.

Figure 16 shows the measured phase-A back-EMF and translator velocity of the PMLG. As the magnitude of the translator velocity increased, both the amplitude and electrical frequency of the phase-A back-EMF increased. Near the reversal points of the reciprocating motion, the translator velocity approached zero and the back-EMF correspondingly decreased. This relationship is consistent with the electromagnetic induction characteristics of the PMLG and confirms the electromechanical response of the experimental platform under reciprocating motion.

Figure 17 compares the reference and measured currents of the battery and supercapacitor branches. The peak-to-peak variations of the battery reference and measured currents were approximately 0.04 A and 0.02 A, respectively. Excluding the two narrow downward pulses, the peak-to-peak variations of the supercapacitor reference and measured currents were approximately 0.55 A and 0.44 A, respectively. The substantially smaller current variation in the battery branch indicates that the slowly varying power component is assigned to the battery, whereas the larger variation in the supercapacitor branch reflects its response to the rapidly fluctuating power component. The low-frequency and high-frequency power components generated by the EMD-based allocation were separately supplied as the reference power commands for the battery and supercapacitor control branches. The branch controllers converted the two power commands into their corresponding current references. The measured battery and supercapacitor currents followed their respective references, demonstrating that the two HESS branches can effectively execute the EMD-based allocated power commands under MPC control.

Figure 18 shows the battery current, supercapacitor current, and DC-bus voltage measured simultaneously under coordinated HESS operation. According to the oscilloscope vertical scales, the peak-to-peak variations of the battery and supercapacitor currents were approximately 0.4 A and 1.4 A, respectively. The substantially smaller variation in the battery current is consistent with its response to the slowly varying power component, whereas the larger supercapacitor current variation reflects its response to the rapidly fluctuating power component. Meanwhile, the DC-bus voltage was maintained around its reference value with a fluctuation of approximately ±2 V. Together with the reference-current tracking results in Figure 17, these results demonstrate that the two storage branches perform their respective power-allocation functions while maintaining the DC-side power balance and voltage regulation.

4.8. Limitations

The present study has several limitations. The PMLG was represented by a control-oriented model in which magnetic saturation, core losses, cogging force, and higher-order harmonics are neglected, while the experimental validation was conducted under the operating conditions achievable with the laboratory wave power generation emulator and focused on the back-EMF characteristics, the current-tracking performance of the battery and supercapacitor branches, and DC-bus voltage regulation. Future work will extend the experimental operating conditions, refine the PMLG model, and experimentally compare classical EMD with EEMD and CEEMDAN in terms of mode separation, endpoint behavior, phase response, and computation time.

5. Conclusions

This study proposed a control strategy that combines generator-side FCS-MPCC, EMD-based power allocation, and HESS-side MPC for a PMLG-based direct-drive wave power generation system. The EMD reconstruction boundary is selected automatically using the frequency-aliasing index, without requiring a fixed cutoff frequency.

Under the same 80 s input and control settings, the proposed EMD method reduced the standard deviation of the high-frequency power component from 204.9 W with LPF to 56.4 W and reduced its low-frequency leakage from 93.14% to 6.07%. The LPF introduced an additional lag of 0.314 s. After the load change at 45 s, the maximum DC-bus voltage deviation decreased from 0.484 V with LPF to 0.426 V with EMD. These results show that EMD provides clearer power separation and better DC-bus response. The rapidly changing power is assigned to the supercapacitor, while the battery receives a smoother power command, which reduces rapid battery-current changes and frequent charging and discharging.

The experimental results show that the battery and supercapacitor currents followed their references. Their peak-to-peak variations were approximately 0.4 A and 1.4 A, respectively, while the DC-bus voltage fluctuation remained within approximately ±2 V. These results confirm that the two storage branches can carry out the assigned power commands while maintaining the DC-bus voltage. Future work will consider battery SOC, current constraints, and battery degradation in the HESS-side MPC and conduct system-level experiments under irregular-wave conditions.