Real-Time Nonlinear Model Predictive Controller for Multiple Degrees of Freedom Wave Energy Converters with Non-Ideal Power Take-Off

Abstract

An increase in wave energy converter (WEC) efficiency requires not only consideration of the nonlinear effects in the WEC dynamics and the power take-off (PTO) mechanisms, but also more integrated treatment of the whole system, i.e., the buoy dynamics, the PTO system, and the control strategy. It results in an optimization formulation that has a nonquadratic and nonstandard cost functional. This article presents the application of real-time nonlinear model predictive controller (NMPC) to two degrees of freedom point absorber type WEC with highly nonlinear PTO characteristics. The nonlinear effects, such as the fluid viscous drag, are also included in the plant dynamics. The controller is implemented on a real-time target machine, and the WEC device is emulated in real-time using the WECSIM toolbox. The results for the successful performance of the design are presented for irregular waves under linear and nonlinear hydrodynamic conditions.

1. Introduction

Renewable energy technologies present a viable and sustainable contribution to the world’s growing energy demands, and the ocean provides potential for an enormous untapped energy resource for the world’s energy portfolio [1,2]. The prospect of ocean wave energy has triggered research in optimal power capture techniques for wave energy converters, including non-ideal operating conditions, such as the non-ideal PTO system constraints [3] and nonlinear sea conditions. Achieving optimal power capture by a WEC in practice is a multifaceted objective. It depends on various factors, such as the physical design of the WEC, the design of the PTO system, the ocean conditions, and the control techniques.

Model predictive control (MPC) is a promising control approach for wave energy converters’ relatively slow plant dynamics because it maximizes energy capture while respecting the system’s mechanical limits. MPC is a look-ahead control strategy that predicts future system behavior to solve a constrained optimization problem and determines the best control action to maximize the output power of WEC. MPC and other optimal control schemes, such as pseudospectral methods and MPC-like algorithms, have been comprehensively studied in the literature for a single WEC device and an array of wave energy converters [4,5,6,7,8,9]. An MPC algorithm uses an internal model of the plant to predict the system’s future states [10]. Nonlinear control algorithms can consider the non-ideal operating conditions and nonlinear effects, including but not limited to non-ideal power take-off mechanism [11], nonlinear viscous drag terms [12,13], and nonlinear mooring dynamics [2]. The non-ideality of PTO systems in most literature is limited to the efficiency of the PTO mechanism [13,14,15,16]. One of the motivations for this research is to consider higher-order nonlinear PTO characteristics as an optimization objective for the NMPC problem. The economic MPC techniques consider a general economic cost function directly in real time [17,18,19]. However, we have deployed a real-time iterative (RTI) algorithm [20,21] to optimize a more general class of non-ideal PTO mechanisms using pseudo-quadratic formulations [3]; this method also supports nonlinearities in the plant dynamics, such as mooring and fluid viscous drag. Another motivation for this work is investigating nonlinear multiple degrees of freedom WEC coupled to non-ideal PTO.

Lots of work has been focused on studying multiple degrees of freedom WEC devices that prove a significant improvement of power capture by the WEC device. Multi-resonant feedback control of a three degree of freedom WEC is presented [22], where a linear hydrodynamic model is considered, and multi-resonant proportional-derivative control law is proposed where the focus is linear plant dynamics under unconstrained control. An analysis of a multi-degree-of-freedom point absorber WEC in the surge, heave, and pitch directions is presented in [23], and frequency and time-domain formulations are presented for the linear plant dynamics. A time-domain model for a point absorber WEC in six degrees of freedom is developed in [24] with an optimal resistive loading. The three degrees of a freedom model of a WEC is presented in [25], where the capture performance of various PTO systems is investigated for a linear plant model. An active control strategy based on the optimal velocity trajectory tracking for a multi-DoF submerged point absorber WEC is presented in [26], where a linearized dynamic system model is considered along with an ideal PTO mechanism. A nonlinear MPC design and implementation based on differential flatness parameterization has been proposed in [27]. Given that most of the work focused on linear plant dynamics for multiple degrees of freedom WEC or ideal PTO mechanisms, and the lack of application of NMPC for such class or problems, we have investigated the application of NMPC to nonlinear multiple DoF WEC plant with a non-ideal PTO mechanism, and focus on the real-time implementation of the control algorithm on a real-time target machine.

This research presents the maximization of power extraction by a 2-DoF WEC device, a WECSIM [28] model of the full-scale version of the Dehlsen Associates, LLC multi-pod CENTIPOD [29]. Although the CENTIPOD device is an array device, the cross-coupling between pods is ignored for this study, which is negligible for the sea conditions of interest in this work and will be investigated in the future. The goal is to optimize the power extracted by the heave and pitch PTOs subject to actuation and velocity constraints. The objective function is a nonstandard and nonquadratic functional of PTO force and velocity, resulting from a practical PTO generator power loss characteristic. The WEC model includes nonlinear viscous drag terms; hence, the resulting plant model is a nonlinear dynamic system. We have implemented an NMPC for the problem. To tackle a free-formed objective function subjected to nonlinear system dynamics, we have used the extended version of the NMPC design from [30], based on pseudo-quadratization using an ACADO toolkit [21]. No prior knowledge of wave excitation is assumed. The WEC model is simulated on a real-time emulator machine, while control is deployed on a Speedgoat real-time performance machine [31], which is interfaced with the WEC emulator machine through an ethernet port. The simulation results for real-time NMPC are presented for the linear and nonlinear hydrodynamics conditions simulated in WECSIM.

2. Time Domain Model of a Multiple Degree of Freedom WEC

The WEC device is a full-scale version of the Dehlsen Associates, LLC multi-pod CENTIPOD [29,32]. A 1:35-scale version of the device is shown in Figure 1. This CENTIPOD device has three floating pods and three spars fixed to a backbone structure. The backbone is anchored using mooring lines, as shown in Figure 2. In its 2-DoF version, each pod is attached to a PTO mechanism in the heave and pitch degrees of freedom. All pods in Figure 2 are assumed identical, and since the CENTIPOD device is an array device, the array effect [33] could become prominent as the significant height of the waves increases and the incident angle of the waves is not parallel to the x-axis in Figure 2. For this study, incident waves are assumed parallel to the x-axis, and for the sea state of interest in this work, the cross-coupling between the pods is very small and is neglected, although it will be investigated in future work.

We will follow the subscript notation of the WEC-Sim toolbox [28] for the degrees of freedom for WEC, in which the integers from 1, 2, … 6 correspond to surge, sway, heave, roll, pitch, and yaw, respectively. Some other notations and symbols for WEC modeling are given in Table 1.

2.1. Surge-Pitch-Heave Model of WEC Modeling in State-Space Form

Each pod in Figure 2 is modeling as a wave point absorber device. The Cummins equation for the coupled surge and pitch dynamics for a point absorber pod (assuming a local reference frame) is given by,

(1)$\left(m+A_{11}(\infty )\right){\dot{v}}_1+A_{15}(\infty ){\dot{v}}_5=-F_{r,11}\left(t\right)-F_{r,15}\left(t\right)-F_{v,1}\left(t\right)+F_{e,1}\left(t\right),$

(2)$\left(m+A_{55}(\infty )\right){\dot{v}}_5+A_{51}(\infty ){\dot{v}}_1=-F_{r,55}\left(t\right)-F_{r,51}\left(t\right)-F_{v,5}\left(t\right)-F_{hs,5}\left(t\right)-F_{p,5}\left(t\right)+F_{e,5}\left(t\right).$

The Cummins equation for the heave dynamics of a point absorber pod is given by,

(3)$\left(m+A_{33}(\infty )\right){\dot{v}}_3\left(t\right)=-F_{r,33}\left(t\right)-F_{hs,3}\left(t\right)-F_{v,3}\left(t\right)-F_{p,3}\left(t\right)+F_{e,3}\left(t\right),$

The hydrostatic, viscous damping, and radiation force terms in (1) through (3) are given by,

(4)$F_{r,ij}\left(t\right)={{\displaystyle \int }}_{-\infty }^t{K}_{ij}\left(t-\tau \right)v_{j}d\tau ,$

(5)$F_{hs,i}\left(t\right)=C_i{x}_i,$

(6)$F_{v,i}\left(t\right)=C_{d,i}{v}_i\left|v_i\right|.$

A transfer function expression can approximate the convolution integral term in (4),

(7)$F_{r,pq}\left(t\right)={{\displaystyle \int }}_{-\infty }^t{K}_{pq}\left(t-\tau \right)v_{q}d\tau \iff F_{r,pq}\left(j\omega \right)=Z_{pq}\left(j\omega \right)V_q\left(j\omega \right),$

Using the device data from WAMIT [34], we can approximate the intrinsic impedance $Z_{pq}\left(j\omega \right)$ in (7) by a second order transfer function using system identification techniques,

(8)$Z_{pq}\left(j\omega \right)=\left[j\omega \left(A_{pq}\left(j\omega \right)-A_{pq}(\infty )\right)+B_{qp}\left(j\omega \right)\right]\approx {\frac{{\alpha }_{pq,1}s+{\alpha }_{pq,0}}{s^2+{\beta }_{pq,1}s+{\beta }_{pq,0}}|}_{s=j\omega },$

Using (8) in (7) enables us to express the radiation force as a second-order transfer function,

(9)$F_{r,pq}\left(s\right)\approx \frac{{\alpha }_{pq,1}s+{\alpha }_{pq,0}}{s^2+{\beta }_{pq,1}s+{\beta }_{pq,0}}{V}_q\left(s\right),$

The transfer function expression in (9) can be converted to the state-space expressions in the observer canonical forms for each of the radiation force terms,

(10)$\left[\begin{array}{c}{\dot{\xi }}_k\left(t\right)\\ {\dot{\xi }}_{k+1}\left(t\right)\end{array}\right]=\left[\begin{array}{cc}0& 1\\ a_k& a_{k+1}\end{array}\right]\left[\begin{array}{c}{\xi }_k\left(t\right)\\ {\xi }_{k+1}\left(t\right)\end{array}\right]+\left[\begin{array}{c}{b}_k\\ b_{k+1}\end{array}\right]v_q\left(t\right),$

(11)$y_{pq}\left(t\right)=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{\xi }_k\left(t\right)\\ {\xi }_{k+1}\left(t\right)\end{array}\right]\approx F_{r,pq}\left(t\right).$

By the comparison of (9)–(11), we have, ${\alpha }_{pq,1}=b_k,{\beta }_{pq,1}=-a_{k+1},{\beta }_{pq,0}=-a_k,$ and ${\alpha }_{pq,0}=b_{k+1}-b_k{a}_{k+1}$. Making a change of variables in (1),

(12)$M_{ii}=\left(m+A_{ii}(\infty )\right),$

(13)$F_{1,net}=-F_{r,11}\left(t\right)-F_{r,15}\left(t\right)-F_{v,1}\left(t\right)+F_{e,1}\left(t\right),$

(14)$F_{5,net}=-F_{r,55}\left(t\right)-F_{r,51}\left(t\right)-C_5{x}_5-F_{v,5}\left(t\right)-F_{p,5}\left(t\right)+F_{e,5}\left(t\right).$

Using (12)–(14) in (1), we get the pitch-surge coupled model of a pod as,

(15)$\left[\begin{array}{c}{\dot{v}}_1\\ {\dot{v}}_5\end{array}\right]={\left[\begin{array}{cc}{M}_{11}& A_{15}(\infty )\\ A_{51}(\infty )& M_{55}\end{array}\right]}^{-1}\left[\begin{array}{c}{F}_{1,net}\\ F_{5,net}\end{array}\right]$

The viscous drag force term $v_i\left|v_i\right|$ in (6) is a hard nonlinearity that may lead to convergence issues for the optimization solvers. One solution is to approximate this term with a soft nonlinearity by replacing it with a smooth higher-order polynomial. A third-order polynomial approximation for $v_i\left|v_i\right|$ is used in the surge and heave direction, where the range of interest of velocity is $v_i\in \left(-1.5,1.5\right) m/sec$, and a fifth-order polynomial approximation is used for pitch direction, where the range of interest of velocity is $v_i\in \left(-0.5,0.5\right) rad/sec$. With, $p_{i,j}$ being the $j^{th}$ polynomial coefficient for $i^{th}$ degree polynomial curve fit

(16)$F_{v,i}=C_{d,i}{v}_i\left|v_i\right|\approx C_{d,i}\left(p_{3,3}{v}_i^3+p_{3,1}{v}_i\right),i=1,3,$

(17)$F_{v,5}=C_{d,5}{v}_5\left|v_5\right|\approx C_{d,5}\left(p_{5,5}{v}_5^5+p_{5,3}{v}_5^3+p_{5,1}{v}_5\right).$

The curve fits (16) and (17) are shown in Figure 3a,b, respectively. Using (15) and (3), we get a Surge-Heave-Pitch model of a pod as,

(18)$\dot{X}=AX+B_p{F}_p+B_v{F}_v+B_e{F}_e,$

where,

(19)$F_p={\left[\begin{array}{cc}{F}_{p,5}& F_{p,3}\end{array}\right]}^T,$

(20)$F_v={\left[\begin{array}{ccc}{F}_{v,1}& F_{v,5}& F_{v,3}\end{array}\right]}^T,$

(21)$F_e={\left[\begin{array}{ccc}{F}_{e,1}& F_{e,5}& F_{e,3}\end{array}\right]}^T,$

(22)$X={\left[\begin{array}{ccccccccccccccc}{v}_1& v_5& x_5& {\xi }_3& {\xi }_4& {\xi }_5& {\xi }_6& {\xi }_7& {\xi }_8& {\xi }_9& {\xi }_{10}& v_3& x_3& {\xi }_1& {\xi }_2\end{array}\right]}^T.$

and the radiation force terms are approximated by following state variables using (10),

(23)$F_{r,11}={\xi }_3,F_{r,15}={\xi }_5,F_{r,51}={\xi }_7,F_{r,55}={\xi }_9,F_{r,33}={\xi }_1.$

and,

(24)$\mathbf{A}=\left[\begin{array}{cc}\begin{array}{ccccccccccc}0& 0& -m_{15}{C}_5& -m_{11}& 0& -m_{11}& 0& -m_{15}& 0& -m_{15}& 0\\ 0& 0& -m_{55}{C}_5& -m_{51}& 0& -m_{51}& 0& -m_{55}& 0& -m_{55}& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ b_3& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ b_4& 0& 0& a_3& a_4& 0& 0& 0& 0& 0& 0\\ 0& b_5& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& b_6& 0& 0& 0& a_5& a_6& 0& 0& 0& 0\\ 0& b_7& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& b_8& 0& 0& 0& 0& 0& a_7& a_8& 0& 0\\ b_9& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ b_{10}& 0& 0& 0& 0& 0& 0& 0& 0& a_9& a_{10}\end{array}& {\mathbf{0}}_{11\times 4}\\ {\mathbf{0}}_{4\times 11}& \begin{array}{cccc}0& \frac{-C_3}{M_{33}}& \frac{-1}{M_{33}}& 0\\ 1& 0& 0& 0\\ b_1& 0& 0& 1\\ b_2& 0& a_1& a_2\end{array}\end{array}\right],$

(25)${\mathbf{B}}_{\mathbf{p}}=\left[\begin{array}{cc}-m_{15}& 0\\ -m_{55}& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& \frac{-1}{M_{33}}\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\right],{\mathbf{B}}_{\mathbf{v}}=\left[\begin{array}{ccc}-m_{11}& -m_{15}& 0\\ -m_{51}& -m_{55}& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& \frac{-1}{M_{33}}\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],{\mathbf{B}}_{\mathbf{e}}=\left[\begin{array}{ccc}{m}_{11}& m_{15}& 0\\ m_{51}& m_{55}& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& \frac{1}{M_{33}}\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],$

Putting (16) and (17) in (18) gives us a 2-DoF (heave and pitch) WEC nonlinear plant model, where the surge is coupled with the pitch and heave is a decoupled DoF. We can use this plant model as a prediction model in NMPC.

2.2. Nonquadratic WEC-PTO Model

The electrical power output from the PTO mechanism of the WEC is the difference between the mechanical power input from the waves and the losses in the PTO system. For a given PTO generator, the electrical PTO power cost functional to be maximized, including the electrical losses, is given by,

(26)$\underset{F_{p,i}}{max}{P}_{E,i}={\eta }_{Conv}\left(P_{Mechanical,i}-P_{Loss,i}\right)={\eta }_{Conv}\left(F_{p,i}{v}_i-K_{Cu}{\left[I_{p,i}\left(F_{p,i}\right)\right]}^2{R}_{\mathsf{\Omega }}\right),$

The case study scenario is taken from McCleer Power’s Linear PTO generator [3] with the PTO force–current characteristics given by Figure 4. We can approximate the experimental data in Figure 4 with a mathematical relation, such as a piecewise linear function or a nonlinear function. We have used polynomial approximation which is a smooth function. This relation is described by a third-order curve fit between the PTO current and the PTO force,

(27)$I_{p,i}\left(F_{p,i}\right)=a_{3,i}{F}_{p,i}^3+a_{2,i}{F}_{p,i}^2+a_{1,i}{F}_{p,i}+a_{0,i},$

Putting (27) in (16), we get,

(28)$\begin{array}{c}{P}_{E,i}=c_{0,i}{F}_{p,i}{v}_i-(\begin{array}{c}{c}_{1,i}{F}_{p,i}^6+c_{2,i}{F}_{p,i}^5+c_{3,i}{F}_{p,i}^4+c_{4,i}{F}_{p,i}^3+c_{5,i}{F}_{p,i}^2+c_{6,i}{F}_{p,i}+c_{7,i})\end{array}\end{array},$

The PTO cost functional surface in (28) is plotted in the PTO velocity–force plane, as shown in Figure 5. The surface plot of the mechanical PTO power,$P_{Mechanical,i}=F_{p,i}{v}_i$ is non-convex, as shown in Figure 5. However, the electrical PTO power surface, $P_{E,i}$ in (26) has a quadratic power loss term, and it gives convexity to the electrical power surface along the PTO force axis in Figure 5.

3. Implementation of NMPC for 2-DoF Heave-Pitch WEC

The optimal control problem of a WEC involves manipulating the PTO force/torque to maximize the power capture while respecting some system constraints. Various optimal control approaches have been developed, and a comprehensive review can be found in [34]. MPC is a model-based online optimal control solution, and a given NMPC problem optimizes a manipulated variable $\mathrm{u}\left(t\right)$to maximize some cost functional $\mathrm{P}(·)$ while respecting the system constraints. A special class of NMPC problems has been formulated in [30], in which the cost functional takes on a nonlinear piecewise polynomial form. Considering the case of finite horizon optimization control, we can mathematically describe the NMPC problem of such a class as,

(29)$\underset{u\left(t\right)}{maximize}P\left[t,\dot{X}\left(t\right),X\left(t\right),U\left(t\right),p\left(t\right)\right]$

(30)$\mathrm{Where}:P(·)=\{\begin{array}{cc}{P}_1(·)+{\rho }_{N,1}(·),& q_k\left(t\right)<R_1\\ P_2(·)+{\rho }_{N,2}(·),& R_1\le q_k\left(t\right)\le R_2\\ \begin{array}{c}⋮\\ P_j(·)+{\rho }_{N,j}(·),\end{array}& \begin{array}{c}⋮\\ R_{j-1}\le q_k\left(t\right)\le R_j\end{array}\end{array},$

subject to,

(31)$\mathrm{Dynamic}\mathrm{Constraints}:\mathbf{0}=g\left(t,\dot{X}\left(t\right),X\left(t\right),U\left(t\right),d\left(t\right),p\left(t\right),N\right),$

(32)$\mathrm{Boundary}\mathrm{Constraint}\mathrm{Function}:\mathbf{0}=r\left(N,X\left(0\right),U\left(0\right),X\left(N\right),U\left(N\right),p\right),$

(33)$\mathrm{Path}\mathrm{Constraints}\mathrm{Function}:0\ge s\left(t,X\left(t\right),U\left(t\right),p\left(t\right)\right).$

The description of various variables and constants in (298) through (33) is given in Table 2. The wave excitation force $F_e$ acting on the hull is considered an unmeasured system disturbance, and based on the available measurements, the controller internally estimates $F_e$.

For the 2-DoF (heave-pitch) WEC problem, the objective function to be maximized in (28) will be the sum of electrical PTO power output in the heave and pitch DoFs for each pod,

(34)$\begin{array}{c}{P}_E=P_{E,3}+P_{E,5}\end{array},$

Using the technique developed in [30], we can put (34) into the pseudo-quadratic form by defining a suitable $h_i$ vector for heave and pitch as,

(35)$h_i={\left[\begin{array}{ccccc}{F}_{p,i}^3& F_{p,i}^2& F_{p,i}& v_i& 1\end{array}\right]}^T, i=3,5$

with,

(36)$h=\left[\begin{array}{c}{h}_{\mathbf{3}}\\ h_{\mathbf{5}}\end{array}\right],$

we can reformulate (34) as,

(37)$P_E=\frac{1}{2}{h}^T\left(2\left[\begin{array}{cc}{W}_{\mathbf{3}}& \mathbf{0}\\ \mathbf{0}& W_{\mathbf{5}}\end{array}\right]\right)h=\frac{1}{2}{h}^T\left(2W\right)h,$

By using (28) in (34), the weighting matrix $W$ can be obtained by polynomial decomposition of (34) by the vector $h$ in (36) as the basis vector,

(38)$W_i=\frac{1}{2}\left[\begin{array}{ccccc}-2c_{1,i}& -c_{2,i}& 0& 0& 0\\ -c_{2,i}& -2c_{3,i}& -c_{4,i}& 0& 0\\ 0& -c_{4,i}& -2c_{5,i}& c_{0,i}& -c_{6,i}\\ 0& 0& c_{0,i}& 0& 0\\ 0& 0& -c_{6,i}& 0& -2c_{7,i}\end{array}\right], i=3,5$

The controller is implemented using an ACADO toolkit [21] following the approach developed in [3].

4. Results

The schematic diagram of the test setup is shown in Figure 6. The corresponding hardware setup is shown in Figure 7. NMPC is designed in the host machine, which generated code and deployed the controller to the Speedgoat performance real-time target machine [31], model-109100 with Intel Core i3 3.3 GHz, two cores, and 2048 MB DDR3 RAM. The Speedgoat machine is interfaced with a real-time WEC emulator machine through an Ethernet universal data port (UDP) channel. The three WEC pods in Figure 1 are assumed identical, and the same controller is implemented for each pod as shown in Figure 8, while the cross-coupling between pods is ignored for this work. The physical velocity and force constraints of the PTO mechanisms imposed as $\left|v_3\right|\le 2 m/sec$, $\left|v_5\right|\le 0.5 rad/sec$ and $\left|F_{p,i}\right|\le 400 kN$. The emulated WEC-Sim model of CENTIPOD device is shown in Figure 9.

Since the WEC pods are assumed identical with no cross-coupling, results are presented only for a single pod. The sea state of interest for WEC-Sim is given in Table 3. This particular sea state’s selection is based on the future testing site of interest for the WEC device, although the hardware testing and a more elaborated study involving other sea states are planned for the future. A step time of 0.1 sec is used for MPC formulation, close to one-tenth of the peak wave period. The performance of NMPC is compared against the linear MPC, and the analysis is performed for the linear and nonlinear hydrodynamics sea conditions.

The average electrical power output results for the heave and pitch PTOs for 2-DoF pod-1 are shown in Figure 10a and Figure 10b, respectively, for linear MPC and NMPC subjected to linear hydrodynamic conditions. Here, we consider the exponentially weighted moving average (EWMA) with the forgetting factor set to unity. The instantaneous electrical power output results corresponding to Figure 10 are shown in Figure 11. The PTO force and wave excitation force profiles for 2-DoF Pod-1 with linear and nonlinear MPC under linear hydrodynamic conditions are shown in Figure 12. The PTO velocity and displacement plots for 2-DoF Pod-1 with linear and nonlinear MPC under linear hydrodynamic conditions are shown in Figure 13.

The average and instantaneous electrical power output results under nonlinear hydrodynamics for 2-DoF Pod-1 with linear and nonlinear MPC are shown in Figure 14 and Figure 15, respectively. The comparison of average electrical PTO power output with NMPC for 1-DoF and 2-DoF Pod-1 is shown in Figure 16 under nonlinear hydrodynamic conditions.

5. Discussion

The average electrical power output results in Figure 10 and Figure 14 are summarized in Table 4. NMPC shows a better performance in terms of increased power output when compared to linear MPC. This increase in the output power becomes more prominent under nonlinear hydrodynamic conditions, which are not accounted for by the linear MPC. An overall 5% increase in power by NMPC compared to linear MPC is obtained under linear hydrodynamic conditions. NMPC obtains an overall 5% increase in total power output by pod-1 than linear MPC under linear hydrodynamic conditions and 10.6% under nonlinear hydrodynamic conditions. The corresponding task execution time (TET) stats for the real-time implementations of linear MPC and NMPC in a Speedgoat real-time machine are given in Table 5. Given the controller step time of 0.1 sec, the increase in TET for NMPC compared to linear MPC is not very significant.

The average electrical power output results per PTO for 1-DoF and 2-DoF Pod1 with NMPC from Figure 16 are summarized in Table 6. In moving from 1-DoF WEC to 2-DoF WEC, a 35% increase in output power is obtained compared to heave only, and 129% increase compared to pitch only.

The locus of electrical PTO power for linear MPC and NMPC under nonlinear hydrodynamic conditions in WEC-Sim, along with the electrical power cost functional surface from Figure 5, are shown in Figure 17.

The locus of electrical PTO power in Figure 17 traverses a trajectory on the cost manifolds and satisfies the cost objective. The cost index formulation in (15) includes a convexifying quadratic term of PTO current, making the resultant electrical PTO surface convex in Figure 5, and with a smooth PTO current profile, the close loop system tends to maintain a stable operation. If the QP problem formulated at a given sample interval is infeasible, the controller will not find a solution. This issue can be handled by monitoring the status of the QP solver during each sampling interval and selecting a suboptimal solution when the QP solver fails. An average of 35% processor load was observed per sampling interval during testing.

6. Conclusions

This article presents a real-time implementation of NMPC for a nonlinear 2-DoF WEC based on Dehlsen Associates’ CENTIPOD multi-pod WEC device, with non-ideal PTOs in the heave and pitch axes. The three pods of the WEC device are assumed identical, and a nonlinear state-space model of a single pod is developed. An NMPC controller is implemented for a 2-DoF WEC device with the cost functional based on a PTO model case study with a highly nonlinear PTO current–force characteristic. The results of the linear MPC are compared with NMPC for the sea states of interest (irregular waves with Pierson Moskowitz spectrum) under linear and nonlinear hydrodynamic conditions in WEC-Sim. The proposed methodology successfully maintained an overall feasible operation of the real-time NMPC problem in simulation as indicated by the status port of the NMPC QP-solver.

An average of 35% processor load was observed per sampling interval during testing. An overall 5% increase in total power output by a single pod is obtained by NMPC compared to linear MPC under linear hydrodynamic conditions and 10.6% under nonlinear hydrodynamic conditions. Moreover, a 35% increase in net output power is obtained by the 2-DoF WEC device compared to the 1-DoF heave only, and a 129% increase compared to the 1-DoF pitch only. While the result reflects only a single sea state, the improvement is likely to be reflected similarly in annual energy production (AEP). The AEP would have a substantive impact on the levelized cost of energy (LCOE). The present work did not consider the cross-coupling between the three pods of the CENTIPOD device. The cross-coupling would be investigated in future work with anticipation of a further increase in the captured power for the sea conditions where the cross-coupling effect is no longer negligible.

Author Contributions

Conceptualization, A.M.; methodology, A.S.H., T.K.A.B. and A.M.; software, A.M. and A.S.H.; validation, A.S.H., T.K.A.B. and A.M.; formal analysis, A.S.H.; investigation, A.S.H., T.K.A.B. and A.M.; resources, T.K.A.B. and A.M.; data curation, A.S.H. and A.M.; writing—A.S.H.; writing—review and editing, T.K.A.B. and A.M.; visualization, A.S.H.; supervision, T.K.A.B.; project administration, A.M. and T.K.A.B.; funding acquisition, A.M. and T.K.A.B. All authors have read and agreed to the published version of the manuscript.

Funding

This work was funded under the TEAMER program by U.S. Department of Energy’s Energy Efficiency and Renewable Energy (EERE) office within the Water Power Technologies Office (WPTO).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Acknowledgments

The authors would like to thank Dehlsen Associates, LLC and McCleer Power for their technical support and for providing the experimental data.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. Image of the Dehlsen Associates, LLC, 1:35-scale CENTIPOD WEC.

Figure 2. Degrees of freedom for dynamic modeling of CENTIPOD WEC: (a) baseline configuration; (b) model with mooring lines.

Figure 3. Polynomial approximations of the quadratic drag term vi|vi|: (a) 3rd order curve fit for heave and surge axes; (b) 5th order curve fit for pitch axis.

Figure 4. Polynomial curve fitting to the PTO force-current experimental data for a PTO generator.

Figure 5. Mechanical and electrical PTO power surface plot in PTO velocity-force plane.

Figure 6. Schematic diagram of the test setup.

Figure 7. Hardware test setup.

Figure 8. NMPC controller for 2-DoF 3-pod CENTIPOD WEC.

Figure 9. WEC-Sim model of Dehlsen’s 2-DoF CENTIPOD device with heave and pitch PTOs for each pod.

Figure 10. Average electrical PTO power output for 2-DoF Pod-1 with linear and nonlinear MPC under linear hydrodynamic conditions in WEC-Sim and |Fpto|≤400 kN: (a) Pod-1 Heave PTO; (b) Pod-1 Pitch PTO.

Figure 11. Instantaneous electrical PTO power output for 2-DoF Pod-1 with linear and nonlinear MPC under linear hydrodynamic conditions in WEC-Sim and |Fpto|≤400 kN: (a) Pod-1 Heave PTO; (b) Pod-1 Pitch PTO.

Figure 12. The PTO force and wave excitation force profiles for 2-DoF Pod-1 with linear and nonlinear MPC under linear hydrodynamic conditions in WEC-Sim and |Fpto|≤400 kN: (a) Pod-1 Heave PTO; (b) Pod-1 Pitch PTO.

Figure 13. The PTO velocity and displacement plots for 2-DoF Pod-1 with linear and nonlinear MPC under linear hydrodynamic conditions in WEC-Sim and |Fpto|≤400 kN: (a) Pod-1 Heave PTO; (b) Pod-1 Pitch PTO.

Figure 14. Average electrical PTO power output for 2-DoF Pod-1 with linear and nonlinear MPC under Nonlinear hydrodynamic conditions in WEC-Sim and |Fpto|≤400 kN: (a) Pod-1 Heave PTO; (b) Pod-1 Pitch PTO.

Figure 15. Instantaneous electrical PTO power output for 2-DoF Pod-1 with linear and nonlinear MPC under nonlinear hydrodynamic conditions in WEC-Sim and |Fpto|≤400 kN: (a) Pod-1 Heave PTO; (b) Pod-1 Pitch PTO.

Figure 16. Average electrical PTO power output for 1-DoF and 2-DoF Pod-1 with nonlinear MPC under nonlinear hydrodynamic conditions in WEC-Sim and |Fpto|≤400 kN: (a) Pod-1 Heave PTO; (b) Pod-1 Pitch PTO.

Figure 17. The locus of electrical PTO power on the electrical power cost functional surface for linear MPC and NMPC under nonlinear hydrodynamic conditions in WEC-Sim and |Fpto|≤400 kN: (a) Pod-1 Heave PTO; (b) Pod-1 Pitch PTO.

Table 1

Notations and symbols for WEC modeling.

Variable	Description
$v_i$	Velocity (Linear or Angular) in $i^{th}$ DoF
$x_i$	Displacement (Linear or Angular) in $i^{th}$ DoF
${\xi }_i$	Intermediate State variables for radiation force State-Space approximation
$F_{r,pq}$	Radiation force in $p^{th}$ DoF due to velocity in $q^{th}$ DoF
$F_{hs,i}$	Hydrostatic force in $i^{th}$ DoF
$F_{v,i}$	Viscous drag force in $i^{th}$ DoF
$F_{e,i}$	Wave excitation force in $i^{th}$ DoF
$F_{p,i}$	PTO force in$i^{th}$ DoF
m	Mass of the float
$A_{pq}(\infty )$	Added mass at the infinite frequency in $p^{th}$DoF due to acceleration in $q^{th}$DoF
$C_i$	The hydrostatic restoring coefficient in $i^{th}$DoF
$C_{vd,i}$	Viscous drag coefficient in $i^{th}$DoF
$A_{qp}$	Frequency-dependent added mass in $p^{th}$DoF due to acceleration in $q^{th}$DoF
$B_{qp}$	Frequency-dependent damping in $p^{th}$DoF due to velocity in $q^{th}$DoF
$K_{pq}$	Radiation force impulse response without infinite frequency added mass
$Z_{qp}$	WEC Intrinsic impedance response in $p^{th}$DoF due to velocity in $q^{th}$DoF
$a_i$	Polynomial coefficients
$c_{i,j}$	Polynomial coefficients for cost functional
$I_{p,i}$	$i^{th}$PTO current
${\eta }_{Conv}$	PTO converter efficiency
$K_{Cu}$	PTO generator copper loss constant
$R_{\mathsf{\Omega }}$	PTO generator winding resistance

Table 2

Symbols and notations for NMPC formulation.

Variable	Description
$N$	Prediction horizon
$X$	State vector
${\rho }_{N,i}$	Finite horizon terminal cost penalty or Mayer terms
$P_i(·)$	Some Nonlinear functions or Lagrange terms
$p$	A column vector of time-varying parameters
$U$	PTO Force manipulated variable vector, $F_p$($N$)
$d$	Excitation force disturbance vector, $F_e$($N$)
$q_k\left(t\right)$	Cost functional scheduling variable
$R_i$	Some real numbers, such that $R_{k+1}>R_k$

Table 3

Sea states for WEC-Sim simulation.

WEC-Sim Simulation Parameter	Value
Significant Wave Height [m]	2.5
Peak Period [s]	8
Wave Spectrum Type	Pierson Moskowitz (PM)
Wave Class	Irregular

Table 4

Average electrical power output per PTO for 2-DoF Pod1 with linear MPC and NMPC.

			Average Electrical Power [kW]
ControlAlgorithm	Linear Hydrodynamic Conditions			Nonlinear Hydrodynamic Conditions
	Heave	Pitch	Total	Heave	Pitch	Total
Linear MPC	57	35	92	70	52	122
Nonlinear MPC	60	37	97	79	56	135

Table 5

Real-time timings stats for Linear MPC vs. Nonlinear MPC.

	Task Execution Time (TET) [sec]
Control Algorithm	1-DoF Heave	1-DoF Pitch	2-DoF Heave and Pitch
Linear MPC	$2.12\times {10}^{-4}$	$2.67\times {10}^{-4}$	$5.21\times {10}^{-4}$
Nonlinear MPC	$3.05\times {10}^{-4}$	$3.21\times {10}^{-4}$	$6.14\times {10}^{-4}$

Table 6

Average electrical power output per PTO for 1-DoF and 2-DoF Pod1 with NMPC.

	Average Electrical Power [kW]
	1-DoF WEC		2-DoF WEC
Axis	Heave	Pitch	Heave and Pitch
Heave	98	0	78
Pitch	0	58	55
Net Power	98	58	133