Control of the Wave Energy Converter ISWEC in Simulation Ricardo António Vaz Mendes Laranjeira ricardo.m.laranjeira@tecnico.ulisboa.pt Instituto Superior Técnico - Universidade de Lisboa, Lisboa, Portugal November 2016 Abstract: The control of a Wave Energy Converter is a key factor for its power extraction capabilities. This work is focused on the control of the ISWEC (Inertial Sea Wave Energy Converter), a wave energy converter that uses its gyroscopic properties to extract sea wave energy. In this work, the model of the device is presented as well as the control technique currently used. Three control strategies are developed with the objective of improving the amount of energy extracted by the device. Fractional Control, Internal Model Control and Feedback Linearization control are implemented resorting to MATLAB and Simulink. Irregular waves are considered and the performance of these control strategies is evaluated and compared to the currently in use Proportional Derivative controller. The simulations performed in this work show that the use of these controllers leads to similar power extraction. However, the use of Internal Model Control or Feedback Linearization control is advantageous as these controllers have less parameters to tune once deployed in the real environment. eywords: Wave Energy Converter; Fractional Control; Internal Model Control, Feedback Linearization Control 1. Introduction 1.1. ISWEC Wave Energy Converter: The ISWEC (Inertial Sea Wave Energy Converter) can be classified as an oscillating body wave energy converter. It is a floating system moored to the seafloor converting mechanical wave energy to electrical energy using a gyroscopic system. A full-scale prototype has been successfully launched in 2015 and it has been deployed on the shore of Pantelleria island in Italy. As a result of the interaction with the waves, the float rotates with a pitching motion. Due to the conservation of angular momentum on the flywheel, the combination of the flywheel speed φ and the pitch speed generates a torque along the axis that can be used to generate electrical power using a Power Take-Off (PTO) system [1], [2]. Figure 1 shows a schematic representation of the system: The main advantage of the ISWEC over most of the other wave energy converters is that externally it is composed only of a floating body without moving parts exposed to sea water or spray, thus achieving a better reliability and lower maintenance costs. Additionally in the presence of wave conditions too dangerous for safe operation the device can be switched off by stopping the flywheel, in which case the device floats as a dead body (like a wave measurement buoy). 1.2. ISWEC Model The ISWEC is modeled by considering two phenomena: The float dynamics and the gyroscope dynamics. These are reduced to the following set of equations [2]: = + h ( )++φcosε (1) = ε φδcosε (2) Figure 1: ISWEC schematic representation [3] In equation (1), describing the float dynamics, is the wave excitation torque, is the equivalent moment of inertia of the float that takes into account its moment of inertia and the added mass due to water dragged as it moves, h the radiation force response function, the hydrostatic stiffness and is the flywheel moment of inertia around the rotation axis. In equation (2), " is the PTO control torque and is the moment of inertia around PTO axis.
The convolution integral in equation (1) describing the radiation torque is approximated by a state space model following [4]. #$ = h ( ) & ='&+( #$ =)& These equations modeling the ISWEC system are implemented in Simulink as represented in Figure 2: Gyroscope Dynamics Force Exchange Hull Dynamics >=?; states, irregular waves are generated using the Bretschneider Spectrum: *(+)= 5 0 + / 16+ 12 3 4 5 617 8 9 /07 9 (3) Where + is the angular frequency, + / is the modal (most likely) frequency and 2 3 is the significant wave height (the mean wave height of the highest third of waves). From the spectrum, a finite number of sinusoidal waves are created, each with its own amplitude and frequency characterized by the spectrum. Each individual harmonic is assigned with a random phase. Following the linear superposition principle the total excitation force acting on the system is computed as the sum of the excitation force of each harmonic wave. Each excitation force is computed referring to the response amplitude operator of the ISWEC system. Figure 4 shows an example of a Bretchneider spectrum and the corresponding wave elevation profile generated for 2 3 =0.5m and T=5.5 s Radiation Force Figure 2: Simulink ISWEC model This model has been verified experimentally and is assumed to be an acceptable model of the system [2]. As such this model is considered as the plant model of the ISWEC system and is used as a comparison for the design of the proposed models. 1.3. Wave Data In this work, real on-site wave data is used to evaluate the system as well as computer generated irregular waves. The measured wave data was obtained from the location of the ISWEC prototype. Figure 3 shows the wave data used: Wave elevation (m) Figure 4: Bretchneider Spectrum and wave surface elevation for 2 3 =0.5 and T=5.5 Eight waves are created with the following significant heights and periods chosen to be representative of different sea states: Table 1: Significant Height and period of generated waves Wave ID 1 2 3 4 5 6 7 8 2 3 (m) 0.5 0.5 1.5 1.5 1.5 2.5 2.5 3.25 (;) 5.5 6.5 5.5 6.5 7.5 6.5 7.5 8.0 1.4. Proportional Derivative Controller Figure 3: Wave data To ensure the system experiences different sea The controller currently in use by the ISWEC system is a PD controller. The idea behind this controller is to control the PTO to behave as a spring-damper system with stiffness < and damping =. This means applying the following
control law: = < = (4) This can be modeled in Simulink using a feedback loop as shown in Figure 5: " Controller = Figure 5: Nonlinear model with PTO control law The design of the PD controller is made by searching for the pair of parameters (=,<) that optimizes the average power absorption for a given wave over a simulation of 600 seconds considering the waves described in section 1.3. This optimization is done using the fminsearch MATLAB function. fminsearch uses the Nelder- Mead algorithm to find the minimum of a multivariable function. It is a simplex-based direct search method as it uses only function values, without any derivative information. The algorithm is described in detail in [5]. To evaluate the power extracted by the ISWEC, no losses are taken into consideration, and as such the negative mechanical power on the PTO is considered: A= " Note that positive values of mechanical power represent the system injecting energy to the sea so the negative power is used. Using this optimization the =,< parameters found for each wave are shown in Table 2: Table 2: Optimized PD controller parameters Wave ID k c Power (10 0 Nm) (10 0 Nm s) (kw) 1 11.17 3.83 5.99 2 10.49 6.42 5.74 3 6.61 7.14 47.80 4 6.91 7.51 47.90 5 6.49 5.95 52.93 6 5.97 10.75 115.25 7 5.82 9.84 121.66 8 5.30 11.24 150.21 After optimizing the control law through numerical simulation, the PD parameters are stored for each sea state condition. These control < ISWEC Model parameters are then changed in the real environment according to the sea state forecast. 1.5. Linearized State Space Model The nonlinear model described in equations (1) and (2) is linearized around =0 resorting to the 1º order Taylor series expansion of its non-linear term: >cos >cos >sin ( ) >cos() > (5) Applying the linearization (5) to equations (1) and (2) leads to the linearized ISWEC model: = + #$ ++φ (6) = ε φδ (7) These equations are represented by the following State-Space model: where & ='&+(G H=)&+IG L T S S &= S MNO,G=,H= P S MNO 4 S MNO Q S JMNO 0 R L = < > 0 0 0 0 0 T S 1 0 0 0 0 0 0 0 > 0 0 = P = 4 = Q = S 0S S '= S 0 0 1 0 0 0 0 0 S 0 0 1 0 U PP U P4 U PQ U P0 S 0 0 0 0 1 0 0 0 S 0 0 0 0 0 1 0 0 S J 0 0 0 0 0 0 1 0 R L 0 0 T 1 S S S (= 0S,)=V0 1 0 0 0 0 0 0W, I=0 0S 0S 0S J 0R The output of the state-space model, considering the measured wave as input, is presented in Figure
6: Figure 6: PTO angle output Figure 6 shows that the State-Space model is a good approximation of the non-linear model for small oscillations. For greater oscillations the model starts to deviate from the non-linear model. 1.6. Proposed Models The two proposed models in this work are: 1: Output combination model and 2: Variable State matrix model. The output combination model is obtained by linearization of the nonlinear model around different working points and combining their output in the following way: H(&)= Y Z[ Z (&) Y Z, for ^=1,2,3 b (8) Where [ Z represents the ^-th linear model linearized around Z, and Y Z represents the weight of the model ^. [ Z =c = + #$ ++>cos Z = ε >cos Z (9) The weights Y Z are obtained using trapezoidal membership functions for ^=1 and ^=b and triangular membership function for every other value of ^: 1, P 4 Y P =d PTO angle (º) 4 P, P 4 0, 4 (10) 0,< Z6P j Z6P h, Y Z = Z Z6P Z Z6P i ZlP, h ZlP Z ZlP Z g 0, ZlP (11) 0, m6p m6p Y m =d, m m6p m m6p (12) 1, m The Variable State Matrix model is obtained as follows: & =' n ()&+(G (13) Where the variable state matrix A is given by: ' ()= Y Z' Z Y Z, for ^=1,2,3 b (14) With ' Z being the state matrix of the system linearized around Z, [ Z. The following linearization points were considered: Table 3: Set of linear models Model Points of linearization, (a) [0,90] (b) [0,45,90] (c) [0,15,30,45,60,75,90] (d) [0,10,20,30,40,50] (e) [0,5,10,15,20,25,35,45] Simulations are made considering the measured wave as input and considering performance parameters of Table 4: Table 4: Performance parameters MSE: Mean VAF: Variance squared error accounted for s p (H Z Hq Z ) 4 r ZtP MD: Maximum deviation u1 v4 (H Hq) v 4 w max H (H) Z Hq Z Where H Z is the output of the non-linear model and Hq is the output of the evaluated model. Table 5 shows the results for PTO angle and pitch angle: Table 5: Model evaluation Model PTO angle [W Performance Parameter MSE VAF MD State space 0.8947 0.9954 6.4381 Output combination Variable State Matrix (a) 0,8962 0,9954 6,4408 (b) 1,1727 0,9940 6,9360 (c) 0,4031 0,9979 5,2474 (d) 0,4077 0,9979 4,2679 (e) 0,3922 0,9980 3,9460 (a) 2,7490 0,9859 6,3653 (b) 0,4244 0,9978 1,8409 (c) 0,0049 1,0000 0,2017 (d) 0,0009 1,0000 0,0896 (e) 0,0002 1,0000 0,0649
Table 5: Model evaluation (cont.) Model Pitch Angle [W Performance Parameter MSE VAF MD State space 0.0713 0.9979 1.5591 Output combination Variable State Matrix (a) 0,0713 0,9979 1.5581 (b) 0,3067 0,9910 3.1662 (c) 0,0904 0,9973 2.6688 (d) 0,0927 0,9973 2.6593 (e) 0,0874 0,9974 2.2834 (a) 0,0405 0,9988 0,8944 (b) 0,0215 0,9994 0,9044 (c) 0,0438 0,9987 1.3089 (d) 0,0493 0,9986 1.3518 (e) 0,0530 0,9984 1.3790 controller. This means implementing the following control law: 3 = = } } < (16) Where the term $~ " $ ~ represents the fractional derivative of order of. Figure 7 shows the Simulink implementation. Fractional Controller < It can be seen that both models show an improvement modeling the PTO angle but only the variable state matrix model shows improvement modeling the pitch angle. " = } } 2. Controller design ISWEC Model In this section the controllers developed in this work will be presented. These are: Fractional PD controller (FPD) Internal Model Controller (IMC) Feedback Linearization Controller (FL) The design of the IMC and FL controllers differs from the FPD controller as the first controllers are designed to follow a reference and the FPD controller is optimized to absorb maximum power. In this section, the power extracting capabilities provided by the controllers is not taken into consideration. The IMC and FL will be designed first to follow a reference With this in mind, a reference for these controllers has to be defined. According to [6], wave energy is captured most efficiently when the WEC speed is in resonance with the wave excitation force, and so it is natural to consider such a reference: M()= 180 2 1061 () (15) The reference gain is chosen to ensure that it stays within reasonable amplitude values with no consideration for power extracted. 2.1. Fractional Controller The idea behind the Fractional PD controller is to change the order of the damping of the PD Figure 7: Fractional PD Controller in Simulink The fractional derivative is implemented using a CRONE (Commande Robuste d Ordre Non-Entier) 7 th order approximation. With poles and zeros within frequency range of [0.001, 1000] rad/s. More information on fractional control and the CRONE approximation can be found in [7]. 2.2. Internal Model Control The internal model control methodology ([8], [9]) is applied to the ISWEC with the following control structure: M IMC " controller Figure 8: IMC implementation ISWEC Linear model The linear model is defined as the transfer function representation of the State Space defined by (6) and (7) with zeros, poles and gain listed in Table 6:
Table 6: Zeros, Poles and Gain of model transfer function Zeros Poles Gain 0-0.051±1.193i -0.084±1.456i -0.537±1.162i 2.222 10-0.508±1.148i -5-1.671±1.627i -1.667±1.616i Since the linear model has no zeros or poles in the right-half s plane, following the IMC design procedure, an IMC controller can be defined as =N(;) 6P (17) The filter N is chosen, to make sure that is, in addition to stable and causal, proper. The filter N is chosen by simulation, evaluating the controller performance for different configurations. To test the controller performance the reference (15) is considered. The results obtained with each filter are presented in Table 7: Table 7: Controller performance for different filter parameters Filter MSE VAF MD 1 ( ;+1) 4 2 ;+1 ( ;+1) 4 10 8.47 10 Q -0.0049 278.0826 1 5.33 10 0-0.0116 996.8283 0.1 107.6043 0.5912 32.2889 0.01 1.0941 0.9887 3.2884 0.001 0.0114 0.9999 0.3432 0.0001 0.0003 1.0000 0.0666 10 4.08 10 Q -0.0221 286.9043 1 1.12 10 0 0.0303 983.1729 0.1 0.5087 0.9942 2.3203 0.01 0.0001 1.0000 0.0636 0.001 0.0001 1.0000 0.0446 0.0001 0.0001 1.0000 0.0435 From the results of Table 7 the filter N(;) is chosen: N(;)= 2 0.01;+1 (0.01;+1) 4 (18) 2.3. Feedback Linearization Control Feedback Linearization Control, in its simplest form, uses feedback to cancel the nonlinearities of a nonlinear system so that the closed-loop dynamics (;) become linear [8], [10]. Considering the equations (1) and (2) that model the system, to cancel the non-linear term, the following control action is considered: = > =?;+ O (19) Leading to the following expression: =( O >=?;+>=?;) 1 (20) =O To obtain the PTO speed: = O Leading to the following transfer function realization: (;)= L() L(O) = (;) Ž(;) =1 (21) ; A proportional Feedback controller is designed with the configuration represented in Figure 10: (;) Ž(;) (;) Figure 10: Feedback Controller The closed-loop transfer function becomes: (;) (;) = 1 ; +1 ISWEC Linearized Model (22) Equation (22) represents a first order system with time constant 1/. A gain of 100 is chosen and Figure 11 shows the system response when considering the reference (15). PTO speed (º/s) (;) O " (&) ISWEC (&) (;) Figure 9: Feedback linearization & Figure 11: System Response (overlapping reference and Response curves) This ideal behavior is present because perfect non-linearity canceling is considered in equation (20). In practice this is not possible as perfect
knowledge of the model is not available. To verify the viability of the controller in face of model uncertainties a sensibility analysis on the model parameters is performed. A 10% parameter variation is considered and the model output is compared to the model with no parameter variation. The results are shown in Table 8: Table 8: Sensitivity analysis on the model parameters Parameter Variation (%) MSE VAF MD -10 1.0x10-4 1 0.0355 10 1.3x10-4 1 0.0391 > -10 0.011 0.999 0.2971 10 0.011 0.999 0.2948-10 2.4x10-31 1 7.1x10-15 10 2.2x10-31 1 7.1x10-15 All Parameters -10 2.2x10-31 1 7.1x10-15 10 2.3x10-31 1 7.1x10-15 -10 0.0423 0.9995 0.5751 10 0.0407 0.9995 0.5692 It can be concluded that the controller is robust to model uncertainty. Table 8 shows that even the worst case (considering a 10% variation in all the model parameters) produces a maximum deviation of around 0.6º/s. This amounts to a small variation in the system output as it is two orders of magnitude smaller than the output. Table 9: Extracted Power for fractional controller Wave ID k ( 10 0 ) c ( 10 0 ) a (-) Power (kw) 1 24.60 12.43 1.81 6.47 2 20.55 10.63 1.62 5.84 3 19.65 12.55 1.63 49.34 4 14.99 9.56 1.50 48.62 5 8.79 6.24 1.21 53.06 6 9.53 11.22 1.18 115.38 7 6.21 9.85 1.02 121.66 8 3.20 11.43 0.89 150.38 3.2. Reference Definition According to [6], wave energy is captured most efficiently when the WEC is in resonance with the wave. A reference is considered where the PTO speed is in phase with the wave force defined as: M = > # (24) Another reference is proposed by looking at the behavior of the system when controlled by the optimized PD controller. Figure 12 shows that the system tends to present a PTO speed in phase with the force acting on the gyroscope defined as: =>cos (25) 3. Power Extraction In this section the power extraction capabilities of the controllers will be evaluated. For the FPD controller the same optimization is performed as for the PD controller. For the IMC and FL controllers, a reference is proposed; and their power extraction capabilities is evaluated and compared to the original PD controller. 3.1. Fractional PD controller The control law that defines the FPD controller is: 3 = = } } < (23) As stated, the same optimization using fminsearch is used to find the parameters (=,<,) that produce the greatest average power extraction over simulations of 600 seconds. The results obtained are shown in Table 9: [-] [-] Figure 12: PTO speed and Having said this, the reference considered is defined as: M = (26) The gain is chosen through simulation to optimize power extraction, using the fminsearch optimization function. 3.2.1 Internal Model Control Table 10 shows the gains and power obtained for the set of irregular waves following the defined references.
Table 10: Extracted Power for IMC controller Power (kw) Wave ID M M M M 1 0.76 0.4 2.31 4.92 2 0.77 0.41 2.34 5.05 3 0.84 0.53 20.54 43.64 4 0.96 0.53 21.42 44.82 5 0.73 0.69 18.25 47.96 6 0.72 1.01 50.68 116.48 7 0.54 1.04 40.41 121.99 8 0.43 0.99 36.3 156.09 3.2.2 Feedback Linearization Control The same analysis is performed for the Feedback Linearization Controller. Results are shown in Table 11: Table 11: Extracted Power for FL controller Power Wave (kw) ID M M M M 1 0.76 0.4 2.31 4.92 2 0.76 0.41 2.33 5.05 3 0.84 0.52 20.54 43.61 4 0.96 0.53 21.38 44.78 5 0.73 0.67 18.2 47.87 6 0.72 0.67 50.62 116.16 7 0.54 0.74 40.34 120.14 8 0.43 0.78 36.22 155.92 Table 10 and Table 11 show similar results for both controllers. However a significant increase in performance can be seen choosing M over M. It can be concluded that the choice of reference has a much greater impact than the choice of controller. 3.3. Power Extraction Comparison The power extracted by each controller is presented in Table 12: Table 12: Mean absorbed power Wave ID PD FPD IMC FL 1 5.99 6.47 4.92 4.92 2 5.74 5.84 5.05 5.05 3 47.8 49.34 43.64 43.61 4 47.9 48.62 44.82 44.78 5 52.93 53.06 47.96 47.87 6 115.25 115.38 116.48 116.16 7 121.66 121.66 121.99 120.14 8 150.21 150.38 156.09 155.92 These results are presented in graphical form, with the power extracted by each controller being scaled by the power absorbed by the PD controller. 1.2 1 0.8 0.6 0.4 0.2 0 Figure 13- Dimensionless mean absorbed power Looking at Table 12 and Figure 13 it can be concluded that the FPD controller is the controller that provides the best power extraction to the ISWEC, followed by the PD controller. The reference based controllers (IMC and FL) show a small decrease in extracted power. This can be explained because the design of the PD and FPD controller is fundamentally different than the design of the IMC and FL controllers. The first set of controllers are tuned resorting to extensive simulation to find the appropriate set of parameters that achieve the maximum power while the second set of controllers are tuned to follow a desired PTO speed reference. 4. Conclusion Power Extracion 1 2 3 4 5 6 7 8 PD FPD IMC FL From the simulations performed, some conclusions can be drawn about the designed controllers. First it can be concluded that all three designed controllers provide the ISWEC system with similar power extraction capabilities. Second it can be concluded that IMC and FL controllers developed are able to control the ISWEC to system to follow a desired reference. Aside from good reference tracking they present the system with good disturbance rejection. These two controllers show similar power extraction, with the correct reference definition showing a much greater impact in extracted power over the controller choice. The models developed in section 1.6 through linearization over various working points led to a small increase in modeling performance over the nonlinear zone of the system. This leads to a small
increase of power extracted for the waves with greater significant height. The FPD shows an improvement over the current PD controller as expected, as the PD controller can be seen as a particular case of the FPD with one fixed parameter. The improvement observed is not significant enough to justify the increase in controller complexity while keeping the same shortcomings of the PD controller in the need of extensive simulation to obtain the right parameters. The main advantage of the IMC and FL controllers over the PD controller is in the reduction of tuning parameters, as after the design of any controller through simulation, there must be some tuning of the control parameters in the real environment. This tuning procedure can be very difficult and time consuming as every change in parameters as to be evaluated, and during this time the sea state is changing. To conclude, the final choice of control strategy should be made from experimental results, as the performance of the controllers is expected to be worse in the real environment. Because the PD and FPD controllers are tuned by the use of simulation (without any physical meaning), it is expected that these controllers deteriorate more than the reference based controllers. For the reference based controllers it is expected that the IMC would outperform the FL controller as the nonlinearities can never be so accurately canceled in the real environment as in simulation. pp. 112 147, 1998. [6] J. Falnes, Ocean waves and Oscillating systems. Cambridge University Press, 2004. [7] D. Valério and J. Sà da Costa, Introduction to single-input, single-output fractional control, IET Control Theory Appl., vol. 5, no. June 2010, p. 1033, 2011. [8] W. S. Levine, The Control Handbook, Control Handb., p. 1566, 1996. [9] D. Rivera, Internal model control: a comprehensive view, Arizona State Univ., 1999. [10] J. Slotine and W. Li, Applied Nonlinear Control. Prentice Hall, 1991. Bibliography [1] G. Bracco, E. Giorcelli, and G. Mattiazzo, ISWEC: A gyroscopic mechanism for wave power exploitation, Mech. Mach. Theory, vol. 46, no. 10, pp. 1411 1424, 2011. [2] G. Bracco, ISWEC : a Gyroscopic Wave Energy Converter, Politecnico di Torino, 2010. [3] M. Raffero, M. Martini, B. Passione, G. Mattiazzo, E. Giorcelli, and G. Bracco, Stochastic Control of Inertial Sea Wave Energy Converter, vol. 2015, 2014. [4] T. Pérez and T. I. Fossen, Time-vs. frequency-domain Identification of parametric radiation force models for marine structures at zero speed, Model. Identif. Control, vol. 29, no. 1, pp. 1 19, 2008. [5] J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, Convergence Properties of the Nelder--Mead Simplex Method in Low Dimensions, SIAM J. Optim., vol. 9, no. 1,