- Email: [email protected]
Robust Feedback Model Predictive Control of Sea Wave Energy Converters
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Control Proceedings of the the 20th World World Congress The International Federation of Automatic Proceedings of theJuly 20th World Congress Control The of Toulouse, France, The International International Federation of Automatic Automatic Control Toulouse, France,Federation July 9-14, 9-14, 2017 2017 Available online at www.sciencedirect.com The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
ScienceDirect
PapersOnLine 50-1 (2017) 141–146 Robust IFAC Feedback Model Predictive Control Robust Feedback Model Predictive Control Robust Control of Feedback Sea WaveModel EnergyPredictive Converters of Sea Wave Energy Converters of Sea Wave ∗∗Energy∗∗ Converters ∗ ∗∗ ∗
Siyuan Siyuan Zhan Zhan ∗ Wei Wei He He Guang Guang Li Li ∗ Wei He ∗∗ ∗∗ Guang Li ∗ ∗ Siyuan Zhan Siyuan Zhan ∗ Wei He ∗∗ Guang Li ∗ Siyuan Zhan Wei He Guang Li ∗ ∗ Queen Mary University of London, London, E1 4NS, UK ∗ Queen Mary University of London, London, E1 4NS, UK ∗ Queen Mary University of London, London, E1 4NS, UK (e-mail: [email protected], [email protected]). Mary University of London, E1 UK ∗ Queen (e-mail: [email protected], [email protected]). Queen Mary University of London, London, London, E1 4NS, 4NS, UK China ∗∗ [email protected], [email protected]). ∗∗ University(e-mail: of Science and Technology Beijing, Beijing 100083, (e-mail: [email protected], [email protected]). University of Science and Technology Beijing, Beijing 100083, China ∗∗ (e-mail: [email protected], [email protected]). ∗∗ University of Science and Technology Beijing, Beijing 100083, China (Email: [email protected]) and Technology Beijing, Beijing 100083, ∗∗ University of Science (Email: [email protected]) University of Science and Technology Beijing, Beijing 100083, China China (Email: [email protected]) (Email: [email protected]) (Email: [email protected]) Abstract: Abstract: This This paper paper presents presents a a robust robust feedback feedback model model predictive predictive control control (MPC) (MPC) strategy strategy for for Abstract: This paper presents a robust feedback model predictive control (MPC) strategy for a sea wave energy converter (WEC). The control objective is to maximise the energy output Abstract: This paper presents a robust feedback model predictive control (MPC) strategy for a sea wave This energy converter (WEC). Thefeedback control model objective is to maximise the energy output Abstract: paper presents a robust predictive control (MPC) strategy for a sea wave energy The control objective maximise the energy output while reducing the converter potential (WEC). risk of of device device damage subjectis toto sea wave excitation excitation forces as a sea energy converter (WEC). The control objective is to maximise the energy output while reducing the potential risk damage subject to sea wave forces as a sea wave wave energy converter (WEC). The control objective isto tosea maximise the paper energyforces output while reducing the potential risk of device damage subject wave excitation as persistent external disturbance. The robust feedback MPC proposed in this has the while reducing the potential risk of device damage subject to sea wave excitation forces as persistent external disturbance. The robust feedback MPC proposed in this paper has the while reducing the disturbance. potential riskThe ofthe device damage subject to sea wave excitation forces as persistent external robust MPC in this paper has the following advantages: robust feasibility and stability be (ii) persistent external disturbance. The robust feedback MPC proposed in this paper has the following prominent prominent advantages: (i) (i) the robustfeedback feasibility and proposed stability can can be guaranteed. guaranteed. (ii) persistent external disturbance. The robust feedback MPC proposed in this paper has the following prominent advantages: (i) the robust feasibility and stability can be guaranteed. (ii) the robust robustprominent MPC method method does not not rely on the the feasibility real time time and wavestability prediction. These features can can following advantages: (i) the robust can These be (ii) the MPC does rely on real wave prediction. features following prominent advantages: (i) rely the generation robust feasibility stability be guaranteed. guaranteed. (ii) the robust MPC method does on the real time wave prediction. These features can not only only reduce the unit cost ofnot energy but alsoand increase thecan reliability of the the MPC MPC the robust MPC method does not rely on the real time wave prediction. These features can not reduce the unit cost of energy generation but also increase the reliability of the robust MPC method does not rely on the real time wave prediction. These features can not only reduce the unit cost of energy generation but also increase the reliability of the MPC controller. not only controller. not only reduce reduce the the unit unit cost cost of of energy energy generation generation but but also also increase increase the the reliability reliability of of the the MPC MPC controller. controller. controller. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Wave Wave energy energy converters converters (WEC); (WEC); Robust Robust model model predictive predictive control control (MPC); (MPC); Robust Robust Keywords: Wave energy converters (WEC); Robust model predictive control (MPC); feasibility; Robust constraint satisfaction Keywords: Wave energy converters (WEC); Robust model predictive control (MPC); Robust Robust feasibility; Robust constraint satisfaction Keywords: Wave energy converters (WEC); Robust model predictive control (MPC); Robust feasibility; Robust constraint satisfaction feasibility; Robust constraint satisfaction feasibility; Robust constraint satisfaction 1. technique, 1. INTRODUCTION INTRODUCTION technique, e.g. e.g. deterministic deterministic sea sea wave wave prediction prediction (DSWP) (DSWP) 1. INTRODUCTION technique, e.g. deterministic sea wave prediction (DSWP) is combined with dynamic programming (DP). Li 1. INTRODUCTION technique, e.g. deterministic sea wave prediction (DSWP) is combined with dynamic programming (DP). Li and and 1. INTRODUCTION technique, e.g.with deterministic sea wave prediction (DSWP) is combined dynamic programming (DP). Li and Belmont (2014) further developed a MPC with convexified is combined with dynamic programming (DP). Li and Belmont (2014) further developed a MPC with convexified Ocean waves provide an enormous source of renewable is combined with dynamic programming (DP). Li and Ocean waves provide an enormous source of renewable Belmont developed aa MPC convexified objectives(2014) whichfurther significantly reduced thewith online compuBelmont (2014) further developed MPC with convexified Ocean provide an source of renewable objectives which significantly reduced the online compuenergy. waves Extensive research has been been focused focused on harnessing Ocean waves provide an enormous enormous source on of harnessing renewable Belmont (2014) further developed a MPC with convexified energy. Extensive research has objectives which significantly reduced the online compuOcean waves provide an enormous source of renewable tational burden. However, although significant improveobjectives which significantly reduced the online compuenergy. Extensive research has been focused on harnessing tational burden. However, although significant improvewave energy for decades, especially after the oil crisis in the energy. Extensive research has been beenafter focused oncrisis harnessing which significantly reduced the the onlineimprovecompuwave energy for decades, especially the oil in the objectives tational burden. energy. Extensive research has focused on harnessing ments been made increasing tational burden. However, although significant improvewave energy for decades, especially after the oil crisis in the ments have have been However, made on on although increasingsignificant the computational computational 1970s .. Two of the main advantages of ocean wave energy wave energy for decades, especially after the oil crisis in the tational burden. However, although significant improve1970s Two of the main advantages of ocean wave energy ments have been made on increasing the computational wave energy for decades, especially thepersistence oilwave crisis inand the ments speed and and the energy absorbing efficiency, most existing existing havethe been madeabsorbing on increasing increasing the computational computational 1970s .. Two of main advantages of ocean speed energy efficiency, most over wind wind energy and solar energy after are its 1970s Two of the theand main advantages of its ocean wave energy energy have been made on the over energy solar energy are persistence and ments speed and the energy absorbing efficiency, most existing 1970s . Two of the main advantages of ocean wave energy WEC MPC strategies have several drawbacks which limit speed and the energy absorbing efficiency, most existing over wind energy and solar energy are its persistence and WEC MPC strategies have several drawbacks which limit high wind energyenergy intensity (Falnes, 2007).are Despite the fact fact that that over and(Falnes, solar energy energy its persistence persistence and speed and the energy have absorbing efficiency, most existing high energy intensity 2007). Despite the WEC MPC strategies several drawbacks which limit over wind energy and solar are its and their applications WEC MPC strategies have several drawbacks which limit high energy intensity (Falnes, 2007). Despite the fact that their applications many types of sea wave energy converters (WECs) have high energy intensity (Falnes, 2007). Despite(WECs) the fact fact have that WEC MPC strategies have several drawbacks which limit manyenergy types intensity of sea wave energy converters their applications high (Falnes, 2007). Despite the that their applications many types of sea wave energy converters (WECs) have been invented and tested to extract the energy from ocean many types of sea wave energy converters (WECs) have (1) Due to their applications been invented and tested to extract the energy from ocean (1) Due to the the immatureness immatureness of of the the wave wave prediction prediction techtechmany types ofand sea tested wave energy (WECs) have been to extract the energy from ocean waves,invented the unit unit cost utilising waveconverters energy is relatively relatively high (1) Due to the immatureness of the wave prediction techbeen invented and tested to extract the energy from ocean niques, the requirement of the wave-by-wave predicwaves, the cost utilising wave energy is high (1) Due to the immatureness of the wave prediction techniques, the requirement of the wave-by-wave predicbeen invented and tested to extract the energy from ocean (1) Due to the immatureness of theand wave prediction techwaves, the unit wave energy compared other renewable energies. niques, the of wave-by-wave predicwaves, the with unit cost cost utilising wave energy is is relatively relatively high high tion the maintenance cost. compared with otherutilising renewable energies. niques, the requirement requirement of the theand wave-by-wave prediction increase increase the installation installation maintenance cost. waves, the unit cost utilising wave energy is relatively high niques, the requirement of the wave-by-wave prediccompared with other renewable energies. tion increase the installation and maintenance cost. compared with other renewable energies. The prediction inaccuracy can result in constraint tion increase the installation and maintenance cost. The prediction inaccuracy can result in constraint To increase increasewith the other efficiency of WECs, WECs, various control control methmethcompared renewable energies. tion increase the installation and maintenance cost. To the efficiency of various The prediction can constraint violations, which will the WEC The prediction inaccuracy can result indevice. constraint To the efficiency of various control methviolations, whichinaccuracy will damage damage theresult WECin device. ods have maximise the To increase the designed efficiency to of WECs, WECs, various controloutput methThe prediction inaccuracy can result in constraint odsincrease have been been designed to maximise the energy energy output violations, which will damage the WEC device. To increase the efficiency of WECs, various control meth(2) The feasibility problem of MPC for WECs has violations, which will damage the WEC device. ods have been designed to maximise the energy output (2) The feasibility problem of MPC for WECs has not not while reducing the risk of device damage. Early WEC conods have been designed to maximise the energy output violations, which will The damage the for WEC device. whilehave reducing the risk of to device damage. Early WEC con(2) The feasibility problem of MPC WECs has not ods been designed maximise the energy output been fully addressed. presence of excitation wave (2) The feasibility problem of MPC for WECs has not while reducing the risk of device damage. Early WEC conbeen fully addressed. The presence of excitation wave trollers are mostly passive controllers based on impedance while reducing the risk of device damage. Early WEC con(2) The feasibility problem of MPC for WECs has not trollers are mostly passive controllers based on impedance been addressed. The excitation while reducing the risk of is device damage. WEC conforcesfully as persistent persistent disturbances, if of not properlywave conbeen fully addressed.disturbances, The presence presenceif of excitation wave trollers are mostly controllers based on forces as not properly conmatching principal,that matching theEarly resonance fretrollers areprincipal,that mostly passive passiveis controllers based on impedance impedance been fully addressed. The presence of excitation wave matching matching the resonance freforces as persistent disturbances, if not properly controllers are mostly passive controllers based on impedance sidered in the controller design, can cause catastrophforces as persistent disturbances, if not properly conmatching principal,that is matching the resonance fresidered in the controller design, can cause catastrophquency of a WEC with the dominant frequency of the matching principal,that is matching the resonance freforces as persistent disturbances, if not properly conquency of principal,that a WEC with is thematching dominantthe frequency of frethe sidered the controller cause matching resonance ic of MPC. sidered infor theimplementation controller design, design, can cause catastrophcatastrophquency of aa WEC with the dominant frequency of the ic failure failurein for implementation of can MPC. incoming ocean waves. Although the efficiency has been quency of WEC with the dominant frequency of the sidered in the controller design, can cause catastrophincoming ocean waves. Although the efficiency has been ic failure for implementation of MPC. quency of a WEC with the dominant frequency of the ic failure for implementation of MPC. incoming ocean waves. Although the efficiency has been improved using these methods for incoming ocean waves. Although the efficiency efficiency has been been A to resolve the proposed ic failure implementation of MPC. improved by by using these Although methods especially especially for conceptual conceptual A method method to for resolve the drawbacks drawbacks proposed by by Zhan Zhan incoming ocean waves. the has improved by using these methods especially regular waves, waves, the operation safety cannot for be conceptual effectively A method to resolve the drawbacks proposed by Zhan improved by using these methods especially for conceptual et al. (2016) is to design a special linear optimal controller regular the operation safety cannot be effectively A method to resolve the drawbacks proposed by Zhan et al. (2016) is to design a special linear optimal controller improved by using these methods especially for conceptual A method tois to resolve the special drawbacks proposed by Zhan regular operation safety cannot handled in irregular sea conditions. al. regular waves, the operation safety cannot be be effectively effectively et (LOC) to the and use invariant handled waves, in real real the irregular sea wave wave conditions. et al. (2016) (2016) is to to design design special linear linear optimal controller (LOC) to increase increase the aaaefficiency efficiency and optimal use the the controller invariant regular waves, the operation safety cannot be effectively et al. (2016) is design special linear optimal controller handled in real irregular sea wave conditions. (LOC) to increase the efficiency and use the invariant handled in real irregular sea wave conditions. set theory theory to analyse analyse theefficiency potentialand constraint violations. (LOC) to increase increase thethe use the theviolations. invariant set to potential constraint More recently, recently, the optimisation-based optimisation-based controllers have have (LOC) handled in real irregular sea wave conditions. to the efficiency and use invariant More the controllers set theory to analyse the potential constraint violations. However, although the method can be effective to reduce set theory to analyse the potential constraint violations. More recently, the optimisation-based controllers have However, although the method can be effective to reduce proven their effectiveness in the WEC control problems. More recently, the optimisation-based optimisation-based controllers have set theory to analyse the potential constraint violations. provenrecently, their effectiveness in the WEC control problems. However, although the method can be effective to reduce More the controllers have the cost and to guarantee safety of the device, it However, although the method can be effective to reduce proven their effectiveness in the WEC control problems. the cost and to guarantee safety of the device, it can can Li et al. (2012); Fusco and Ringwood (2013) showed that proven their effectiveness in the WEC control problems. although thebecause method canLOC be the effective to reduce Li et al.their (2012); Fusco andinRingwood (2013) showed that However, the cost and to guarantee safety of device, it can proven effectiveness the WEC control problems. be very conservative the only works when the cost and to guarantee safety of the device, it can Li et al. (2012); Fusco and Ringwood (2013) showed that be very conservative because the LOC only works when the WEC control problem, which maximises the energy Li et al. (2012); Fusco and Ringwood (2013) showed that the cost and to guarantee safety of the device, it can theetWEC control problem, which maximises the energy be very because Li al.while (2012); Fusco and (2013) that constraints are inactive. inactive. be very conservative conservative because the the LOC LOC only only works works when when the WEC control problem, which the constraints are output satisfying the Ringwood constraints due to toshowed safety con- be the WEC control problem, which maximises maximises the energy energy very conservative because the LOC only works when output while satisfying the constraints due safety conconstraints are inactive. the WEC control problem, which maximises the energy constraints are inactive. output while satisfying the constraints due to safety considerations can be formulated as a constrained optimal output whilecan satisfying the constraints constraints due to to safety safety con- constraints One way way to toare guarantee the robust robust feasibility feasibility and and robust robust inactive. the siderations be formulated as a constrained optimal One guarantee output while satisfying the due considerations can be formulated as aaof constrained optimal control problem. The requirement optimising control One way to guarantee the robust feasibility and robust siderations can be formulated as constrained optimal constraint satisfaction of the MPC problem is to derive control problem. The requirement of optimising control One way to guarantee the robust feasibility and robust constraint satisfaction of the MPC problem is to derive siderations can handling beThe formulated as aof constrained optimal One way tosatisfaction guarantee thetherobust feasibility and robust control problem. requirement optimising control objectives while both state state and input constraints constraints constraint of MPC problem is to derive control problem. The requirement of optimising control admissible control solutions from state feedback policies objectives while handling both and input constraint satisfaction of the MPC problem is to derive admissible control solutions from state feedback policies control problem. The requirement of optimising control satisfaction of thefrom MPC problem is topolicies derive objectives handling both state and constraints lead to to the thewhile development of Model Predictive Control (M- constraint admissible solutions state feedback objectives while handlingof both statePredictive and input inputControl constraints (Goulart al., Mayne et In palead development Model (Madmissible control solutions from state2000). feedback policies (Goulart et etcontrol al., 2006; 2006; Mayne et al., al., 2000). In this this paobjectives while handling both state and input constraints admissible control solutions from state feedback policies lead to the development of Model Predictive Control (MPC) of WECs. Early researches of using the MPC for (Goulart et al., 2006; Mayne et al., 2000). In this palead to the development of Model Predictive Control (Mper, based on the availability of the current wave meaPC) of WECs. Early researches of using the MPC for (Goulart et al., 2006; Mayne et al., 2000). In this paper, based on the availability of the current wave mealead to the development of Model Predictive Control (M(Goulart et al., 2006; Mayne et al., 2000). In this paPC) WECs. Early of using MPC for WEC (Hals Cretel et based on the the wave meaPC) ofcontrol WECs.include Early researches researches of 2011; using the the MPC for per, surements, aa specific feedback MPC to WECof control include (Hals et et al., al., 2011; Cretel et al., al., per, based we on design the availability availability ofrobust the current current wave measurements, we design specificof robust feedback MPC to PC) WECs. Early researches of using the MPC for per, based on the availability of the current wave meaWEC control include (Hals et al., 2011; Cretel et al., 2011; ofBrekken, Brekken, 2011). Especially, Li et al. (2012) (2012) showed surements, we design aa specific robust feedback MPC to WEC control include (Hals et al., 2011; Cretel et al., 2011; 2011). Especially, Li et al. showed surements, we design specific robust feedback MPC to WEC control include (Hals et al., 2011; Cretel et al., 2011; Brekken, Li al. (2012) showed that energy output can prediction 2011; Brekken, 2011). Especially, Li et et if al.wave (2012) showed surements, we design a specific robust feedback MPC to that the the energy 2011). outputEspecially, can be be doubled doubled if wave prediction 2011; Brekken, 2011). Especially, Li et al. (2012) showed that the energy output can be doubled if wave prediction that the energy output can be doubled if wave prediction that the energy output can be doubled if wave prediction
Copyright © © 2017 2017 IFAC IFAC 143 Copyright 143 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 143 Copyright ©under 2017 responsibility IFAC 143Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 143 10.1016/j.ifacol.2017.08.024
Proceedings of the 20th IFAC World Congress 142 Siyuan Zhan et al. / IFAC PapersOnLine 50-1 (2017) 141–146 Toulouse, France, July 9-14, 2017
maximise the energy output while overcoming the aforementioned disadvantages. The device to be investigated in this paper reflects the working principle of a point absorber PB150 developed by OPT (2011), Inc. To present the WEC control problem, we use the models similar to those used in (Li and Belmont, 2014; Li et al., 2012). The schematic diagram of the device is shown in Figure 1. Fig. 2. Force diagram of the point absorber Notations Rn denotes the space of real n-dimensional vectors. [a1 , a2 , . . . , aN ] denotes [aT1 aT2 . . . aTN ]T . For the subsets A ⊂ Rn and B ⊂ Rn , the Minkowski set addition is defined by A + B {a + b : a ∈ A, b ∈ B}. The P -subtraction is defined by A ∼ B {a ∈ Rn : a + b ∈ A, ∀b ∈ B}. For A ⊂ Rn and matrix M of compatible dimensions, M A {M a : a ∈ A}. 2. MODEL SETUP Fig. 1. Schematic diagram of a point absorber The excitation wave force drives the float, resulting in a relative heave motion between the piston fixed to the buoy and the cylinder fixed to the sea bed. This relative motion creates fluid flow, which drives a hydraulic motor that links to a power generator. zw and zv represent the ocean surface level and the heave position of the middle point of the buoy respectively. The control input is the q-axis current in the generatorside power converter, to control the electric torque of the generator (Zhong and Weiss, 2011). The torque is proportional to the force fu acting on the piston. For ease of presentation, we directly use the force fu as the control input. By defining the direction as shown in Figure 2, the power output of the WEC at time t is expressed as P (t) = −fu z˙v and the energy absorbed during period [T1 , T2 ] is expressed as T2 P (t)dt (1) T1
Due to safety concerns, we restrict the float’s heave motion so that it can not be either completely submerged by sea water or jump out of the sea water. This constraint can be expressed as |zv − zw | ≤ Φmax (2a) where Φmax is assumed to be equal to half the height of the float without loss of generality. The WEC is also subject to control input limitation |fu | ≤ umax (2b) The controller design objective is to maximise the energy (1) subject to the state constraint (2a) and input constraint (2b). The remaining paper is organised as follows. In Section 2, the dynamic model of a WEC is represented as a state space model. The feedback MPC is formulated in Section 3. In Section 4, a numerical simulation is provided to demonstrate the efficacy of the proposed method. Finally, the paper is concluded in Section 5.
144
For ease of presentation and without loss of generality, we adopt the model in (Li and Belmont, 2014) without involving the terms for friction and viscosity forces. The dynamic model of a single WEC illustrated in Fig. 2 can be described as (3) ms z¨v = −fs − fr + fu Here ms is the float mass. The buoyancy force is fs := ks (zv − zw ) where the hydrostatic stiffness ks = ρgs, with ρ as water density, g as standard gravity, and s as the cross sectional area of the float. The radiation force is fr = m∞ z¨v + D(z˙v − z˙w ) where m∞ is the added mass and D is the damping ratio. Note that the damping ratio D is a frequency-dependent term, which can be represented by a convolution term (Falnes, 2007). Here we assume D as a constant term resulting in a second order system for demonstration purpose; the proposed methods can be applied to higher order WEC model with D as a frequency-dependent term without loss of generality. Note that for a more accurate model, an extra convolution kernel needs to be used to represent the dynamics from wave motion to wave excitation force (Yu and Falnes, 1995). By defining state variable x := [zv − zw , z˙v ], external disturbance w := z˙w , control input u = fu , the state space model becomes x˙ = Ac x + Buc u + Bwc w where 0 −1 1 0 Ac = ks D Bwc = D Buc = 1 (4) − m m m m By discretising the system with sampling time Ts = 0.02s, we have the discrete time model x(k + 1) = Ax(k) + Bu u(k) + Bw w(k) (5) 3. FEEDBACK MPC FORMULATION The system described in (5) is a linear time invariant system subject to state constraints x ∈ X, input constraints u ∈ U and bounded disturbance w ∈ W for safety
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Siyuan Zhan et al. / IFAC PapersOnLine 50-1 (2017) 141–146
143
operation purpose. These constraint sets are represented as (6a) X {x ∈ R2 , |x1 | ≤ Φmax , |x2 | ≤ x2,max }
there are two main obstacles to the implementation of the optimisation (9) from the dependence of the future disturbances (the future wave profile) and the infinite dimensions.
(6c) W {x ∈ R, |w| ≤ wmax } where x2,max ,umax ,wmax are the maximal buoy heave velocity, maximal control input and maximal magnitude of the first derivative of wave’s heave motion (as disturbance) respectively. Assumption 1. The states x1 , x2 and the disturbance w are available at the current time t.
3.1 Constraints Restriction Approach
In many papers, robust feedback MPC algorithms (Lee and Kouvaritakis, 2000; Mayne et al., 2005; Chisci et al., 2001) adopt the feedback law u(x) = Kx x + v. In this paper, based on the availability of current disturbance and state information, we extend their result by designing a combined state and disturbance feedback control policy (7) u(x, w) = Kx x + Kw w + v where Kx and Kw are the pre-designed LOC that maximise the energy output of the unconstrained system while guaranteeing systems stability, which can be derived using the method by Zhan et al. (2016). v is the optimisation variable which is used to cope with the constraints: v = 0 when constraints are inactive, and v = 0 when constraints become active. Using the same concept from Chisci ∞ et al. (2001), it can be shown that minimisation of i=0 v(i)2 amounts to maximisation of energy output.
However, designing the feedback controller directly with the nominal system (10) while ignoring all the disturbances (i.e. the nominal predictive control) can cause infeasibility and instability problems. To preserve robust feasibility and constraints satisfaction, following Chisci et al. (2001); Rossiter et al. (1998), we replace the original constraints (9d) with restricted ones that account for the disturbance.
U {x ∈ R, |u| ≤ umax }
(6b)
For completeness, we recall the result from (Zhan et al., 2016). The LOC without considering constraints for a WEC can be calculated by (8a) u = K x x + Kw w where (8b) Kx = − (R + BuT SBu )−1 (T + BuT SA) BuT SBu )−1 BuT SBw
Kw = − (R + (8c) and S is the solution of the following discrete algebraic Ricatti equation S = AT SA + Q (8d) −(AT SBu + T T )(R + BuT SBu )−1 (BuT SA + T ) and Q, T, R are cost coefficients of the LOC objective T ∞ 1 xk Q T T xk (8e) J= uk T R 2 uk k=0
The idea of the feedback MPC policy proposed here is to convert the constrained energy maximisation problem to a convex constrained deviation regulation problem ∞ v 2 (k + i) (9a) min [v(k),v(k+1),... ]
i=0
s.t. x(k + i + 1) = Ax(k + i) + Bu u(k + i) + Bw w(k + i) (9b) u(k + i) = Kx x(k + i) + Kw w(k + i) + v(k + i) (9c) u(k + i) ∈ U, x(k + i) ∈ X, ∀w(k + i) ∈ W ∀i = 0, 1, . . . (9d)
For safety and efficiency operation purpose, the state and control input constraints (9d) must be robustly satisfied and the energy output need to be optimised. However, 145
In order to remove the future disturbance term w in (9), we introduce the nominal (disturbance-free) system corresponding to (5). x ¯(k + 1) = A¯ x(k) + Bu u ¯(k) (10) where x ¯, u ¯ denote the state and control input when w = 0.
The system (9b) with feedback MPC policy (7) can be rephrased as (11) x(k + 1) = AK x(k) + Bu v(k) + BK w(k) where AK = A + Bu Kx , BK = Bw + Bu Kw . Remark 2. With a proper choice of the LOC cost coefficients Q,T ,R, the system with u = Kx x + Kw w is stable, (i.e. AK has all its eigenvalues strictly inside the unit circle). Theorem 3. (Robust constraints satisfaction). If there is a control sequence [v(k), v(k + 1), . . . ] solving the following optimisation problem (9), the sequence [v(k), v(k + 1), . . . ] also satisfies all the constraints in (9d). ∞ v 2 (k + i) (12a) min [v(k),v(k+1),... ]
i=0
s.t. x ¯(k) = x(k) (12b) ¯(k) + Bu v(k) + BK w(k) (12c) x ¯(k + 1) = AK x x ¯(k + i + 1) = AK x ¯(k + i) + Bu v(k + i) i = 1, 2, . . . (12d) ¯(k) + Kw w(k) + v(k) (12e) u ¯(k) = Kx x u ¯(k + i) = Kx x ¯(k + i) + v(k + i) i = 1, 2, . . . (12f) u ¯(k) ∈ U, x ¯(k) ∈ X (12g) u ¯(k + i) ∈ Ui , x ¯(k + i) ∈ Xi , ∀i = 1, 2, . . . (12h) where Xi = X ∼ Di , Ui = U ∼ Kx Di ∼ Kw W and i−2 Di AjK BK W (12i) j=0
Proof. The difference between the real state and the nominal state can be expressed by x(k) = x ¯(k) (13a) i−2 j AK BK w(j + 1) i ≥ 1 (13b) x(k + i) = x ¯(k + i) + j=0
u(k) = u ¯(k) u(k + i) = u ¯(k + i) + Kw w(k + i) i−2 + Kx AjK BK w(j + 1) i ≥ 1 j=0
(13c)
(13d)
Proceedings of the 20th IFAC World Congress 144 Siyuan Zhan et al. / IFAC PapersOnLine 50-1 (2017) 141–146 Toulouse, France, July 9-14, 2017
Combining (13) with (12), we have if (12g) and(12h) are satisfied, (9d) is satisfied, which completes the proof. Theorem 4. (Recursive feasibility). If the control problem (9) is feasible for initial state x(k), the feedback MPC u = Kx x + Kw w + v will always be feasible for all future state. Proof. The proof can be completed in a similar way to (Chisci et al., 2001) 3.2 Terminal Constraints To make the feedback MPC with infinite prediction horizon (12) practically implementable, it is necessary to assume v(k + i) = 0 for i > N and design a terminal constraint. Chisci et al. (2001) shows that the terminal constraint can be related to the maximal output admissible set (MOAS) Σ {x ∈ R2 , AK x ∈ Xk , Kx AiK x ∈ Uk , ∀i > 0} (14) Remark 5. Since the state and input constraints X and U are polytopes, then the maximal output admissible set Σ is a polytope and can be finitely determined (Kolmanovsky and Gilbert, 1995). Assumption 6. There exists s MOAS for (11) with the state constraint X, the input constraint U, the disturbance bound W and the feedback control policy u = Kx x + Kw w for i > N . Theorem 7. If Assumption 6 holds, the infinite horizon feedback MPC problem (9) is equivalent to the following finite horizon MPC N =1 v 2 (k + i) (15a) min [v(k),v(k+1),...,v(k+N −1)]
i=0
s.t. x ¯(k) = x(k) (15b) ¯(k) + Bu v(k) + BK w(k) (15c) x ¯(k + 1) = AK x x ¯(k + i + 1) = AK x ¯(k + i) + Bu v(k + i) i = 1, . . . , N − 1 (15d) ¯(k) + Kw w(k) + v(k) (15e) u ¯(k) = Kx x u ¯(k + i) = Kx x ¯(k + i) + v(k + i) i = 1, . . . , N − 1 (15f) u ¯(k) ∈ U, x ¯(k) ∈ X (15g) u ¯(k + i) ∈ Ui , x ¯(k + i) ∈ Xi , ∀i = 1, . . . , N − 1 (15h) x ¯(k + N ) ∈ Σ ∼ DN (15i) where Σ and DN are defined in (14) and (12i) respectively.
Proof. The proof can be completed in a similar way to (Chisci et al., 2001) Remark 8. The feedback MPC problem can be formulated to a quadratic programming (QP), which will be discussed in subsection 3.3. Remark 9. The terminal constraints guaranteed that the terminal state will lay in the MOAS and will remain in the MOAS for all i > N . 3.3 Implementation
The predicted state trajectory from nominal system (12d) is (17a) x ¯(k + i) = AiK x(k) + Mi v(k) where i−2 Mi = Ai−1 (17b) K B u AK B u . . . B u 0 . . . 0
and the predicted control sequence is u ¯(k + i) = Kx AiK x(k) + Ni v(k) (17c) where i−2 Ni = Kx Ai−1 K Bu Kx AK Bu . . . Kx Bu 1 0 . . . 0 (17d) Remark 10. Since X, U, W are polytopes, the state constraints Xi , control input constraints Ui and the terminal constraint Σ ∼ DN are all polytopes and can be expressed by (18a) Xk {x ∈ R2 , fx,k x ≤ gx,k } Uk {u ∈ R, fu,k x ≤ gu,k }
(18b)
(18c) Σ ∼ DN {x ∈ R , fx,Σ x ≤ gx,Σ } where f and g are corresponding support vectors of the polytopes. 2
The objective function can be expressed as T
J v(k) v(k) The state and input constraints can be written as Γx v(k) + Λx x(k) + Πx w(k) ≤ ηx Γu v(k) + Λu x(k) + Πu w(k) ≤ ηu fx,0 fx,0 M0 fx,1 AK fx,1 M1 Λx = Γx = .. .. . . fx,N −1 MN −1 fu,0 N0 fu,1 N1 Γu = .. . fu,N −1 NN −1 0 fx,1 BK f A B x,2 K K Πx = .. .
−1 fx,N −1 AN K fu,0 Kx fu,1 Kx AK Λx = .. .
−2 fx,N −1 AN BK K
−1 fu,N −1 Kx AN K fu,0 Kw fu,1 Kx BK Πu = fu,2 Kx AK BK .. .
(19) (20a) (20b)
(20c)
(20d)
−2 fu,N −1 Kx AN BK K (20e)
gx,0 gx,0 gx,1 gu,1 ηu = . (20f) ηx = . .. .. gx,N −1 gu,N −1 and terminal constraint N −1 fx,Σ MN v(k) + fx,Σ AN BK w(k) ≤ gx,Σ K x(k) + fx,Σ AK (20g) where Mi ,Ni are defined in (17b) and (17d), f and g are defined in (18). Then we have the QP problem min (19) s.t. (20) (21) v(k)
In this subsection, we convert the feedback MPC problem (15) to a QP problem. Let v(k) denote the control sequence v(k) = [v(k), . . . , v(k + N − 1)] (16) 146
4. NUMERICAL SIMULATION In this section, we present the simulation result using the feedback MPC strategy described in (20) and compare it
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Siyuan Zhan et al. / IFAC PapersOnLine 50-1 (2017) 141–146
145
0.0195 . We find that eigenvalues of AK are strictly inside 0.0495 the unit circle.
Figure 3 shows that with the designed maximal disturbance w = 1.0m/s, the terminal constraints shrink as the prediction horizon N increases. That is because the larger N will require more margin to account for future disturbance to guarantee the terminal state to stay strictly within the MOAS. Figure 4 shows that with predictive horizon N = 5, the larger maximal disturbances will result in smaller terminal constraints (i.e. the feedback MPC get more conservative as we designed for larger maximal disturbances).
Fig. 3. Terminal constraints for different predictive horizon with wmax = 1.0m/s with the unconstrained linear optimal control case. The model we used here has similar dynamics to the 2nd order model in (Li et al., 2012). The parameters for the WEC model are Stiffness Mass of the float Frequency independent added mass Constant coefficient term Input magnitude constraint The buoy heave position limit The buoy heave velocity limit
k = 6.39 × 105 N/m ms = 1 × 104 kg ma = 7 × 104 kg D = 2 × 105 Nm/s umax = 3 × 105 N Φmax = 1.2m x2,max = 10m/s
With sampling time Ts = 0.02s, the discrete model parameters are 0 −0.0195 0.9984 0.0195 Bw = A= Bu = 0.0024 0.0503 −0.1557 0.9497
Fig. 5. The wave amplitude and its derivative data used in simulations
Fig. 6. The control input and buoy heave motion using the unconstrained LOC
Fig. 4. Terminal constraints for different wmax with N = 5 0 0 , 0 50 T = [0 1], R = 0.05. We have Kx = [0.5629 −26.4890], 0.9984 −0.0188 Kw = −0.3509 and AK = , BK = 0.1544 0.8851
The weights of the LOC are chosen as Q =
147
The wave amplitude and its derivative data used in simulations are shown in Figure 5. Figure 6 shows the control input and heave motion trajectories using u = Kx x+Kw w. We can find that state constraint is violated around 7s when we use the unconstrained LOC. Figure 7 and Figure 8 show the control input and heave motion trajectories and the power and energy generated respectively. We can see the state and input constraints of the feedback MPC are satisfied for the whole period
Proceedings of the 20th IFAC World Congress 146 Siyuan Zhan et al. / IFAC PapersOnLine 50-1 (2017) 141–146 Toulouse, France, July 9-14, 2017
Fig. 7. The control input and buoy heave motion using the feedback MPC
Fig. 8. Energy output of feedback MPC and the energy reversal is avoided. The energy output is 1.0562 × 108 J. 5. CONCLUSION We have proposed a novel feedback MPC strategy for WECs which uses the information of both the current states and disturbance for feedback. This novel MPC policy does not require future wave predictions so that the installation and operation cost of WECs is reduced. The novel feedback MPC can guarantee system’s constraint satisfaction and feasibility under bounded but persistent disturbances. The advantages of low cost and reliability make MPC for WECs more attractive. Since the restricted constraints and the pre-designed linear feedback part can be calculated off-line, the feedback MPC policy does not add significant on-line computational burden. In the simulation, we also shows that the feedback MPC is more robust than the corresponding LOC in terms of handling constraints. The theoretical support and time simulation provide engineers much confidence when designing the noval feedback MPC for WECs. REFERENCES Brekken, T.K. (2011). On model predictive control for a point absorber wave energy converter. In PowerTech, 148
2011 IEEE Trondheim, 1–8. IEEE. Chisci, L., Rossiter, J.A., and Zappa, G. (2001). Systems with persistent disturbances: predictive control with restricted constraints. Automatica, 37(7), 1019–1028. Cretel, J.A., Lightbody, G., Thomas, G.P., and Lewis, A.W. (2011). Maximisation of energy capture by a waveenergy point absorber using model predictive control. IFAC Proceedings Volumes, 44(1), 3714–3721. Falnes, J. (2007). A review of wave-energy extraction. Marine Structures, 20(4), 185–201. Fusco, F. and Ringwood, J.V. (2013). A simple and effective real-time controller for wave energy converters. IEEE Transactions on sustainable energy, 4(1), 21–30. Goulart, P.J., Kerrigan, E.C., and Maciejowski, J.M. (2006). Optimization over state feedback policies for robust control with constraints. Automatica, 42(4), 523– 533. Hals, J., Falnes, J., and Moan, T. (2011). Constrained optimal control of a heaving buoy wave-energy converter. Journal of Offshore Mechanics and Arctic Engineering, 133(1), 011401. Kolmanovsky, I. and Gilbert, E.G. (1995). Maximal output admissible sets for discrete-time systems with disturbance inputs. In American Control Conference, Proceedings of the 1995, volume 3. IEEE. Lee, Y.I. and Kouvaritakis, B. (2000). Robust receding horizon predictive control for systems with uncertain dynamics and input saturation. Automatica, 36(10), 1497–1504. Li, G. and Belmont, M.R. (2014). Model predictive control of sea wave energy converters–part i: A convex approach for the case of a single device. Renewable Energy, 69, 453–463. Li, G., Weiss, G., Mueller, M., Townley, S., and Belmont, M.R. (2012). Wave energy converter control by wave prediction and dynamic programming. Renewable Energy, 48, 392–403. Mayne, D.Q., Rawlings, J.B., Rao, C.V., and Scokaert, P.O. (2000). Constrained model predictive control: Stability and optimality. Automatica, 36(6), 789–814. Mayne, D.Q., Seron, M.M., and Rakovi´c, S. (2005). Robust model predictive control of constrained linear systems with bounded disturbances. Automatica, 41(2), 219– 224. OPT (2011). Ocean power technologies, pb 150 power buoy, http://www.oceanpowertechnologies.com/pb150-gallery/. Technical report, PB150. Rossiter, J.A., Kouvaritakis, B., and Rice, M. (1998). A numerically robust state-space approach to stablepredictive control strategies. Automatica, 34(1), 65–73. Yu, Z. and Falnes, J. (1995). State-space modelling of a vertical cylinder in heave. Applied Ocean Research, 17(5), 265–275. Zhan, S., Huijberts, H., Na, J., and Li, G. (2016). Optimal controller design and constraints analysis for wave energy converters. In 11th International Conference on Control, Proceedings of the 2016. UKACC. Zhong, Q.C. and Weiss, G. (2011). Synchronverters: Inverters that mimic synchronous generators. IEEE Transactions on Industrial Electronics, 58(4), 1259– 1267.
ScienceDirect
PapersOnLine 50-1 (2017) 141–146 Robust IFAC Feedback Model Predictive Control Robust Feedback Model Predictive Control Robust Control of Feedback Sea WaveModel EnergyPredictive Converters of Sea Wave Energy Converters of Sea Wave ∗∗Energy∗∗ Converters ∗ ∗∗ ∗
Siyuan Siyuan Zhan Zhan ∗ Wei Wei He He Guang Guang Li Li ∗ Wei He ∗∗ ∗∗ Guang Li ∗ ∗ Siyuan Zhan Siyuan Zhan ∗ Wei He ∗∗ Guang Li ∗ Siyuan Zhan Wei He Guang Li ∗ ∗ Queen Mary University of London, London, E1 4NS, UK ∗ Queen Mary University of London, London, E1 4NS, UK ∗ Queen Mary University of London, London, E1 4NS, UK (e-mail: [email protected], [email protected]). Mary University of London, E1 UK ∗ Queen (e-mail: [email protected], [email protected]). Queen Mary University of London, London, London, E1 4NS, 4NS, UK China ∗∗ [email protected], [email protected]). ∗∗ University(e-mail: of Science and Technology Beijing, Beijing 100083, (e-mail: [email protected], [email protected]). University of Science and Technology Beijing, Beijing 100083, China ∗∗ (e-mail: [email protected], [email protected]). ∗∗ University of Science and Technology Beijing, Beijing 100083, China (Email: [email protected]) and Technology Beijing, Beijing 100083, ∗∗ University of Science (Email: [email protected]) University of Science and Technology Beijing, Beijing 100083, China China (Email: [email protected]) (Email: [email protected]) (Email: [email protected]) Abstract: Abstract: This This paper paper presents presents a a robust robust feedback feedback model model predictive predictive control control (MPC) (MPC) strategy strategy for for Abstract: This paper presents a robust feedback model predictive control (MPC) strategy for a sea wave energy converter (WEC). The control objective is to maximise the energy output Abstract: This paper presents a robust feedback model predictive control (MPC) strategy for a sea wave This energy converter (WEC). Thefeedback control model objective is to maximise the energy output Abstract: paper presents a robust predictive control (MPC) strategy for a sea wave energy The control objective maximise the energy output while reducing the converter potential (WEC). risk of of device device damage subjectis toto sea wave excitation excitation forces as a sea energy converter (WEC). The control objective is to maximise the energy output while reducing the potential risk damage subject to sea wave forces as a sea wave wave energy converter (WEC). The control objective isto tosea maximise the paper energyforces output while reducing the potential risk of device damage subject wave excitation as persistent external disturbance. The robust feedback MPC proposed in this has the while reducing the potential risk of device damage subject to sea wave excitation forces as persistent external disturbance. The robust feedback MPC proposed in this paper has the while reducing the disturbance. potential riskThe ofthe device damage subject to sea wave excitation forces as persistent external robust MPC in this paper has the following advantages: robust feasibility and stability be (ii) persistent external disturbance. The robust feedback MPC proposed in this paper has the following prominent prominent advantages: (i) (i) the robustfeedback feasibility and proposed stability can can be guaranteed. guaranteed. (ii) persistent external disturbance. The robust feedback MPC proposed in this paper has the following prominent advantages: (i) the robust feasibility and stability can be guaranteed. (ii) the robust robustprominent MPC method method does not not rely on the the feasibility real time time and wavestability prediction. These features can can following advantages: (i) the robust can These be (ii) the MPC does rely on real wave prediction. features following prominent advantages: (i) rely the generation robust feasibility stability be guaranteed. guaranteed. (ii) the robust MPC method does on the real time wave prediction. These features can not only only reduce the unit cost ofnot energy but alsoand increase thecan reliability of the the MPC MPC the robust MPC method does not rely on the real time wave prediction. These features can not reduce the unit cost of energy generation but also increase the reliability of the robust MPC method does not rely on the real time wave prediction. These features can not only reduce the unit cost of energy generation but also increase the reliability of the MPC controller. not only controller. not only reduce reduce the the unit unit cost cost of of energy energy generation generation but but also also increase increase the the reliability reliability of of the the MPC MPC controller. controller. controller. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Wave Wave energy energy converters converters (WEC); (WEC); Robust Robust model model predictive predictive control control (MPC); (MPC); Robust Robust Keywords: Wave energy converters (WEC); Robust model predictive control (MPC); feasibility; Robust constraint satisfaction Keywords: Wave energy converters (WEC); Robust model predictive control (MPC); Robust Robust feasibility; Robust constraint satisfaction Keywords: Wave energy converters (WEC); Robust model predictive control (MPC); Robust feasibility; Robust constraint satisfaction feasibility; Robust constraint satisfaction feasibility; Robust constraint satisfaction 1. technique, 1. INTRODUCTION INTRODUCTION technique, e.g. e.g. deterministic deterministic sea sea wave wave prediction prediction (DSWP) (DSWP) 1. INTRODUCTION technique, e.g. deterministic sea wave prediction (DSWP) is combined with dynamic programming (DP). Li 1. INTRODUCTION technique, e.g. deterministic sea wave prediction (DSWP) is combined with dynamic programming (DP). Li and and 1. INTRODUCTION technique, e.g.with deterministic sea wave prediction (DSWP) is combined dynamic programming (DP). Li and Belmont (2014) further developed a MPC with convexified is combined with dynamic programming (DP). Li and Belmont (2014) further developed a MPC with convexified Ocean waves provide an enormous source of renewable is combined with dynamic programming (DP). Li and Ocean waves provide an enormous source of renewable Belmont developed aa MPC convexified objectives(2014) whichfurther significantly reduced thewith online compuBelmont (2014) further developed MPC with convexified Ocean provide an source of renewable objectives which significantly reduced the online compuenergy. waves Extensive research has been been focused focused on harnessing Ocean waves provide an enormous enormous source on of harnessing renewable Belmont (2014) further developed a MPC with convexified energy. Extensive research has objectives which significantly reduced the online compuOcean waves provide an enormous source of renewable tational burden. However, although significant improveobjectives which significantly reduced the online compuenergy. Extensive research has been focused on harnessing tational burden. However, although significant improvewave energy for decades, especially after the oil crisis in the energy. Extensive research has been beenafter focused oncrisis harnessing which significantly reduced the the onlineimprovecompuwave energy for decades, especially the oil in the objectives tational burden. energy. Extensive research has focused on harnessing ments been made increasing tational burden. However, although significant improvewave energy for decades, especially after the oil crisis in the ments have have been However, made on on although increasingsignificant the computational computational 1970s .. Two of the main advantages of ocean wave energy wave energy for decades, especially after the oil crisis in the tational burden. However, although significant improve1970s Two of the main advantages of ocean wave energy ments have been made on increasing the computational wave energy for decades, especially thepersistence oilwave crisis inand the ments speed and and the energy absorbing efficiency, most existing existing havethe been madeabsorbing on increasing increasing the computational computational 1970s .. Two of main advantages of ocean speed energy efficiency, most over wind wind energy and solar energy after are its 1970s Two of the theand main advantages of its ocean wave energy energy have been made on the over energy solar energy are persistence and ments speed and the energy absorbing efficiency, most existing 1970s . Two of the main advantages of ocean wave energy WEC MPC strategies have several drawbacks which limit speed and the energy absorbing efficiency, most existing over wind energy and solar energy are its persistence and WEC MPC strategies have several drawbacks which limit high wind energyenergy intensity (Falnes, 2007).are Despite the fact fact that that over and(Falnes, solar energy energy its persistence persistence and speed and the energy have absorbing efficiency, most existing high energy intensity 2007). Despite the WEC MPC strategies several drawbacks which limit over wind energy and solar are its and their applications WEC MPC strategies have several drawbacks which limit high energy intensity (Falnes, 2007). Despite the fact that their applications many types of sea wave energy converters (WECs) have high energy intensity (Falnes, 2007). Despite(WECs) the fact fact have that WEC MPC strategies have several drawbacks which limit manyenergy types intensity of sea wave energy converters their applications high (Falnes, 2007). Despite the that their applications many types of sea wave energy converters (WECs) have been invented and tested to extract the energy from ocean many types of sea wave energy converters (WECs) have (1) Due to their applications been invented and tested to extract the energy from ocean (1) Due to the the immatureness immatureness of of the the wave wave prediction prediction techtechmany types ofand sea tested wave energy (WECs) have been to extract the energy from ocean waves,invented the unit unit cost utilising waveconverters energy is relatively relatively high (1) Due to the immatureness of the wave prediction techbeen invented and tested to extract the energy from ocean niques, the requirement of the wave-by-wave predicwaves, the cost utilising wave energy is high (1) Due to the immatureness of the wave prediction techniques, the requirement of the wave-by-wave predicbeen invented and tested to extract the energy from ocean (1) Due to the immatureness of theand wave prediction techwaves, the unit wave energy compared other renewable energies. niques, the of wave-by-wave predicwaves, the with unit cost cost utilising wave energy is is relatively relatively high high tion the maintenance cost. compared with otherutilising renewable energies. niques, the requirement requirement of the theand wave-by-wave prediction increase increase the installation installation maintenance cost. waves, the unit cost utilising wave energy is relatively high niques, the requirement of the wave-by-wave prediccompared with other renewable energies. tion increase the installation and maintenance cost. compared with other renewable energies. The prediction inaccuracy can result in constraint tion increase the installation and maintenance cost. The prediction inaccuracy can result in constraint To increase increasewith the other efficiency of WECs, WECs, various control control methmethcompared renewable energies. tion increase the installation and maintenance cost. To the efficiency of various The prediction can constraint violations, which will the WEC The prediction inaccuracy can result indevice. constraint To the efficiency of various control methviolations, whichinaccuracy will damage damage theresult WECin device. ods have maximise the To increase the designed efficiency to of WECs, WECs, various controloutput methThe prediction inaccuracy can result in constraint odsincrease have been been designed to maximise the energy energy output violations, which will damage the WEC device. To increase the efficiency of WECs, various control meth(2) The feasibility problem of MPC for WECs has violations, which will damage the WEC device. ods have been designed to maximise the energy output (2) The feasibility problem of MPC for WECs has not not while reducing the risk of device damage. Early WEC conods have been designed to maximise the energy output violations, which will The damage the for WEC device. whilehave reducing the risk of to device damage. Early WEC con(2) The feasibility problem of MPC WECs has not ods been designed maximise the energy output been fully addressed. presence of excitation wave (2) The feasibility problem of MPC for WECs has not while reducing the risk of device damage. Early WEC conbeen fully addressed. The presence of excitation wave trollers are mostly passive controllers based on impedance while reducing the risk of device damage. Early WEC con(2) The feasibility problem of MPC for WECs has not trollers are mostly passive controllers based on impedance been addressed. The excitation while reducing the risk of is device damage. WEC conforcesfully as persistent persistent disturbances, if of not properlywave conbeen fully addressed.disturbances, The presence presenceif of excitation wave trollers are mostly controllers based on forces as not properly conmatching principal,that matching theEarly resonance fretrollers areprincipal,that mostly passive passiveis controllers based on impedance impedance been fully addressed. The presence of excitation wave matching matching the resonance freforces as persistent disturbances, if not properly controllers are mostly passive controllers based on impedance sidered in the controller design, can cause catastrophforces as persistent disturbances, if not properly conmatching principal,that is matching the resonance fresidered in the controller design, can cause catastrophquency of a WEC with the dominant frequency of the matching principal,that is matching the resonance freforces as persistent disturbances, if not properly conquency of principal,that a WEC with is thematching dominantthe frequency of frethe sidered the controller cause matching resonance ic of MPC. sidered infor theimplementation controller design, design, can cause catastrophcatastrophquency of aa WEC with the dominant frequency of the ic failure failurein for implementation of can MPC. incoming ocean waves. Although the efficiency has been quency of WEC with the dominant frequency of the sidered in the controller design, can cause catastrophincoming ocean waves. Although the efficiency has been ic failure for implementation of MPC. quency of a WEC with the dominant frequency of the ic failure for implementation of MPC. incoming ocean waves. Although the efficiency has been improved using these methods for incoming ocean waves. Although the efficiency efficiency has been been A to resolve the proposed ic failure implementation of MPC. improved by by using these Although methods especially especially for conceptual conceptual A method method to for resolve the drawbacks drawbacks proposed by by Zhan Zhan incoming ocean waves. the has improved by using these methods especially regular waves, waves, the operation safety cannot for be conceptual effectively A method to resolve the drawbacks proposed by Zhan improved by using these methods especially for conceptual et al. (2016) is to design a special linear optimal controller regular the operation safety cannot be effectively A method to resolve the drawbacks proposed by Zhan et al. (2016) is to design a special linear optimal controller improved by using these methods especially for conceptual A method tois to resolve the special drawbacks proposed by Zhan regular operation safety cannot handled in irregular sea conditions. al. regular waves, the operation safety cannot be be effectively effectively et (LOC) to the and use invariant handled waves, in real real the irregular sea wave wave conditions. et al. (2016) (2016) is to to design design special linear linear optimal controller (LOC) to increase increase the aaaefficiency efficiency and optimal use the the controller invariant regular waves, the operation safety cannot be effectively et al. (2016) is design special linear optimal controller handled in real irregular sea wave conditions. (LOC) to increase the efficiency and use the invariant handled in real irregular sea wave conditions. set theory theory to analyse analyse theefficiency potentialand constraint violations. (LOC) to increase increase thethe use the theviolations. invariant set to potential constraint More recently, recently, the optimisation-based optimisation-based controllers have have (LOC) handled in real irregular sea wave conditions. to the efficiency and use invariant More the controllers set theory to analyse the potential constraint violations. However, although the method can be effective to reduce set theory to analyse the potential constraint violations. More recently, the optimisation-based controllers have However, although the method can be effective to reduce proven their effectiveness in the WEC control problems. More recently, the optimisation-based optimisation-based controllers have set theory to analyse the potential constraint violations. provenrecently, their effectiveness in the WEC control problems. However, although the method can be effective to reduce More the controllers have the cost and to guarantee safety of the device, it However, although the method can be effective to reduce proven their effectiveness in the WEC control problems. the cost and to guarantee safety of the device, it can can Li et al. (2012); Fusco and Ringwood (2013) showed that proven their effectiveness in the WEC control problems. although thebecause method canLOC be the effective to reduce Li et al.their (2012); Fusco andinRingwood (2013) showed that However, the cost and to guarantee safety of device, it can proven effectiveness the WEC control problems. be very conservative the only works when the cost and to guarantee safety of the device, it can Li et al. (2012); Fusco and Ringwood (2013) showed that be very conservative because the LOC only works when the WEC control problem, which maximises the energy Li et al. (2012); Fusco and Ringwood (2013) showed that the cost and to guarantee safety of the device, it can theetWEC control problem, which maximises the energy be very because Li al.while (2012); Fusco and (2013) that constraints are inactive. inactive. be very conservative conservative because the the LOC LOC only only works works when when the WEC control problem, which the constraints are output satisfying the Ringwood constraints due to toshowed safety con- be the WEC control problem, which maximises maximises the energy energy very conservative because the LOC only works when output while satisfying the constraints due safety conconstraints are inactive. the WEC control problem, which maximises the energy constraints are inactive. output while satisfying the constraints due to safety considerations can be formulated as a constrained optimal output whilecan satisfying the constraints constraints due to to safety safety con- constraints One way way to toare guarantee the robust robust feasibility feasibility and and robust robust inactive. the siderations be formulated as a constrained optimal One guarantee output while satisfying the due considerations can be formulated as aaof constrained optimal control problem. The requirement optimising control One way to guarantee the robust feasibility and robust siderations can be formulated as constrained optimal constraint satisfaction of the MPC problem is to derive control problem. The requirement of optimising control One way to guarantee the robust feasibility and robust constraint satisfaction of the MPC problem is to derive siderations can handling beThe formulated as aof constrained optimal One way tosatisfaction guarantee thetherobust feasibility and robust control problem. requirement optimising control objectives while both state state and input constraints constraints constraint of MPC problem is to derive control problem. The requirement of optimising control admissible control solutions from state feedback policies objectives while handling both and input constraint satisfaction of the MPC problem is to derive admissible control solutions from state feedback policies control problem. The requirement of optimising control satisfaction of thefrom MPC problem is topolicies derive objectives handling both state and constraints lead to to the thewhile development of Model Predictive Control (M- constraint admissible solutions state feedback objectives while handlingof both statePredictive and input inputControl constraints (Goulart al., Mayne et In palead development Model (Madmissible control solutions from state2000). feedback policies (Goulart et etcontrol al., 2006; 2006; Mayne et al., al., 2000). In this this paobjectives while handling both state and input constraints admissible control solutions from state feedback policies lead to the development of Model Predictive Control (MPC) of WECs. Early researches of using the MPC for (Goulart et al., 2006; Mayne et al., 2000). In this palead to the development of Model Predictive Control (Mper, based on the availability of the current wave meaPC) of WECs. Early researches of using the MPC for (Goulart et al., 2006; Mayne et al., 2000). In this paper, based on the availability of the current wave mealead to the development of Model Predictive Control (M(Goulart et al., 2006; Mayne et al., 2000). In this paPC) WECs. Early of using MPC for WEC (Hals Cretel et based on the the wave meaPC) ofcontrol WECs.include Early researches researches of 2011; using the the MPC for per, surements, aa specific feedback MPC to WECof control include (Hals et et al., al., 2011; Cretel et al., al., per, based we on design the availability availability ofrobust the current current wave measurements, we design specificof robust feedback MPC to PC) WECs. Early researches of using the MPC for per, based on the availability of the current wave meaWEC control include (Hals et al., 2011; Cretel et al., 2011; ofBrekken, Brekken, 2011). Especially, Li et al. (2012) (2012) showed surements, we design aa specific robust feedback MPC to WEC control include (Hals et al., 2011; Cretel et al., 2011; 2011). Especially, Li et al. showed surements, we design specific robust feedback MPC to WEC control include (Hals et al., 2011; Cretel et al., 2011; Brekken, Li al. (2012) showed that energy output can prediction 2011; Brekken, 2011). Especially, Li et et if al.wave (2012) showed surements, we design a specific robust feedback MPC to that the the energy 2011). outputEspecially, can be be doubled doubled if wave prediction 2011; Brekken, 2011). Especially, Li et al. (2012) showed that the energy output can be doubled if wave prediction that the energy output can be doubled if wave prediction that the energy output can be doubled if wave prediction
Copyright © © 2017 2017 IFAC IFAC 143 Copyright 143 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 143 Copyright ©under 2017 responsibility IFAC 143Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 143 10.1016/j.ifacol.2017.08.024
Proceedings of the 20th IFAC World Congress 142 Siyuan Zhan et al. / IFAC PapersOnLine 50-1 (2017) 141–146 Toulouse, France, July 9-14, 2017
maximise the energy output while overcoming the aforementioned disadvantages. The device to be investigated in this paper reflects the working principle of a point absorber PB150 developed by OPT (2011), Inc. To present the WEC control problem, we use the models similar to those used in (Li and Belmont, 2014; Li et al., 2012). The schematic diagram of the device is shown in Figure 1. Fig. 2. Force diagram of the point absorber Notations Rn denotes the space of real n-dimensional vectors. [a1 , a2 , . . . , aN ] denotes [aT1 aT2 . . . aTN ]T . For the subsets A ⊂ Rn and B ⊂ Rn , the Minkowski set addition is defined by A + B {a + b : a ∈ A, b ∈ B}. The P -subtraction is defined by A ∼ B {a ∈ Rn : a + b ∈ A, ∀b ∈ B}. For A ⊂ Rn and matrix M of compatible dimensions, M A {M a : a ∈ A}. 2. MODEL SETUP Fig. 1. Schematic diagram of a point absorber The excitation wave force drives the float, resulting in a relative heave motion between the piston fixed to the buoy and the cylinder fixed to the sea bed. This relative motion creates fluid flow, which drives a hydraulic motor that links to a power generator. zw and zv represent the ocean surface level and the heave position of the middle point of the buoy respectively. The control input is the q-axis current in the generatorside power converter, to control the electric torque of the generator (Zhong and Weiss, 2011). The torque is proportional to the force fu acting on the piston. For ease of presentation, we directly use the force fu as the control input. By defining the direction as shown in Figure 2, the power output of the WEC at time t is expressed as P (t) = −fu z˙v and the energy absorbed during period [T1 , T2 ] is expressed as T2 P (t)dt (1) T1
Due to safety concerns, we restrict the float’s heave motion so that it can not be either completely submerged by sea water or jump out of the sea water. This constraint can be expressed as |zv − zw | ≤ Φmax (2a) where Φmax is assumed to be equal to half the height of the float without loss of generality. The WEC is also subject to control input limitation |fu | ≤ umax (2b) The controller design objective is to maximise the energy (1) subject to the state constraint (2a) and input constraint (2b). The remaining paper is organised as follows. In Section 2, the dynamic model of a WEC is represented as a state space model. The feedback MPC is formulated in Section 3. In Section 4, a numerical simulation is provided to demonstrate the efficacy of the proposed method. Finally, the paper is concluded in Section 5.
144
For ease of presentation and without loss of generality, we adopt the model in (Li and Belmont, 2014) without involving the terms for friction and viscosity forces. The dynamic model of a single WEC illustrated in Fig. 2 can be described as (3) ms z¨v = −fs − fr + fu Here ms is the float mass. The buoyancy force is fs := ks (zv − zw ) where the hydrostatic stiffness ks = ρgs, with ρ as water density, g as standard gravity, and s as the cross sectional area of the float. The radiation force is fr = m∞ z¨v + D(z˙v − z˙w ) where m∞ is the added mass and D is the damping ratio. Note that the damping ratio D is a frequency-dependent term, which can be represented by a convolution term (Falnes, 2007). Here we assume D as a constant term resulting in a second order system for demonstration purpose; the proposed methods can be applied to higher order WEC model with D as a frequency-dependent term without loss of generality. Note that for a more accurate model, an extra convolution kernel needs to be used to represent the dynamics from wave motion to wave excitation force (Yu and Falnes, 1995). By defining state variable x := [zv − zw , z˙v ], external disturbance w := z˙w , control input u = fu , the state space model becomes x˙ = Ac x + Buc u + Bwc w where 0 −1 1 0 Ac = ks D Bwc = D Buc = 1 (4) − m m m m By discretising the system with sampling time Ts = 0.02s, we have the discrete time model x(k + 1) = Ax(k) + Bu u(k) + Bw w(k) (5) 3. FEEDBACK MPC FORMULATION The system described in (5) is a linear time invariant system subject to state constraints x ∈ X, input constraints u ∈ U and bounded disturbance w ∈ W for safety
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Siyuan Zhan et al. / IFAC PapersOnLine 50-1 (2017) 141–146
143
operation purpose. These constraint sets are represented as (6a) X {x ∈ R2 , |x1 | ≤ Φmax , |x2 | ≤ x2,max }
there are two main obstacles to the implementation of the optimisation (9) from the dependence of the future disturbances (the future wave profile) and the infinite dimensions.
(6c) W {x ∈ R, |w| ≤ wmax } where x2,max ,umax ,wmax are the maximal buoy heave velocity, maximal control input and maximal magnitude of the first derivative of wave’s heave motion (as disturbance) respectively. Assumption 1. The states x1 , x2 and the disturbance w are available at the current time t.
3.1 Constraints Restriction Approach
In many papers, robust feedback MPC algorithms (Lee and Kouvaritakis, 2000; Mayne et al., 2005; Chisci et al., 2001) adopt the feedback law u(x) = Kx x + v. In this paper, based on the availability of current disturbance and state information, we extend their result by designing a combined state and disturbance feedback control policy (7) u(x, w) = Kx x + Kw w + v where Kx and Kw are the pre-designed LOC that maximise the energy output of the unconstrained system while guaranteeing systems stability, which can be derived using the method by Zhan et al. (2016). v is the optimisation variable which is used to cope with the constraints: v = 0 when constraints are inactive, and v = 0 when constraints become active. Using the same concept from Chisci ∞ et al. (2001), it can be shown that minimisation of i=0 v(i)2 amounts to maximisation of energy output.
However, designing the feedback controller directly with the nominal system (10) while ignoring all the disturbances (i.e. the nominal predictive control) can cause infeasibility and instability problems. To preserve robust feasibility and constraints satisfaction, following Chisci et al. (2001); Rossiter et al. (1998), we replace the original constraints (9d) with restricted ones that account for the disturbance.
U {x ∈ R, |u| ≤ umax }
(6b)
For completeness, we recall the result from (Zhan et al., 2016). The LOC without considering constraints for a WEC can be calculated by (8a) u = K x x + Kw w where (8b) Kx = − (R + BuT SBu )−1 (T + BuT SA) BuT SBu )−1 BuT SBw
Kw = − (R + (8c) and S is the solution of the following discrete algebraic Ricatti equation S = AT SA + Q (8d) −(AT SBu + T T )(R + BuT SBu )−1 (BuT SA + T ) and Q, T, R are cost coefficients of the LOC objective T ∞ 1 xk Q T T xk (8e) J= uk T R 2 uk k=0
The idea of the feedback MPC policy proposed here is to convert the constrained energy maximisation problem to a convex constrained deviation regulation problem ∞ v 2 (k + i) (9a) min [v(k),v(k+1),... ]
i=0
s.t. x(k + i + 1) = Ax(k + i) + Bu u(k + i) + Bw w(k + i) (9b) u(k + i) = Kx x(k + i) + Kw w(k + i) + v(k + i) (9c) u(k + i) ∈ U, x(k + i) ∈ X, ∀w(k + i) ∈ W ∀i = 0, 1, . . . (9d)
For safety and efficiency operation purpose, the state and control input constraints (9d) must be robustly satisfied and the energy output need to be optimised. However, 145
In order to remove the future disturbance term w in (9), we introduce the nominal (disturbance-free) system corresponding to (5). x ¯(k + 1) = A¯ x(k) + Bu u ¯(k) (10) where x ¯, u ¯ denote the state and control input when w = 0.
The system (9b) with feedback MPC policy (7) can be rephrased as (11) x(k + 1) = AK x(k) + Bu v(k) + BK w(k) where AK = A + Bu Kx , BK = Bw + Bu Kw . Remark 2. With a proper choice of the LOC cost coefficients Q,T ,R, the system with u = Kx x + Kw w is stable, (i.e. AK has all its eigenvalues strictly inside the unit circle). Theorem 3. (Robust constraints satisfaction). If there is a control sequence [v(k), v(k + 1), . . . ] solving the following optimisation problem (9), the sequence [v(k), v(k + 1), . . . ] also satisfies all the constraints in (9d). ∞ v 2 (k + i) (12a) min [v(k),v(k+1),... ]
i=0
s.t. x ¯(k) = x(k) (12b) ¯(k) + Bu v(k) + BK w(k) (12c) x ¯(k + 1) = AK x x ¯(k + i + 1) = AK x ¯(k + i) + Bu v(k + i) i = 1, 2, . . . (12d) ¯(k) + Kw w(k) + v(k) (12e) u ¯(k) = Kx x u ¯(k + i) = Kx x ¯(k + i) + v(k + i) i = 1, 2, . . . (12f) u ¯(k) ∈ U, x ¯(k) ∈ X (12g) u ¯(k + i) ∈ Ui , x ¯(k + i) ∈ Xi , ∀i = 1, 2, . . . (12h) where Xi = X ∼ Di , Ui = U ∼ Kx Di ∼ Kw W and i−2 Di AjK BK W (12i) j=0
Proof. The difference between the real state and the nominal state can be expressed by x(k) = x ¯(k) (13a) i−2 j AK BK w(j + 1) i ≥ 1 (13b) x(k + i) = x ¯(k + i) + j=0
u(k) = u ¯(k) u(k + i) = u ¯(k + i) + Kw w(k + i) i−2 + Kx AjK BK w(j + 1) i ≥ 1 j=0
(13c)
(13d)
Proceedings of the 20th IFAC World Congress 144 Siyuan Zhan et al. / IFAC PapersOnLine 50-1 (2017) 141–146 Toulouse, France, July 9-14, 2017
Combining (13) with (12), we have if (12g) and(12h) are satisfied, (9d) is satisfied, which completes the proof. Theorem 4. (Recursive feasibility). If the control problem (9) is feasible for initial state x(k), the feedback MPC u = Kx x + Kw w + v will always be feasible for all future state. Proof. The proof can be completed in a similar way to (Chisci et al., 2001) 3.2 Terminal Constraints To make the feedback MPC with infinite prediction horizon (12) practically implementable, it is necessary to assume v(k + i) = 0 for i > N and design a terminal constraint. Chisci et al. (2001) shows that the terminal constraint can be related to the maximal output admissible set (MOAS) Σ {x ∈ R2 , AK x ∈ Xk , Kx AiK x ∈ Uk , ∀i > 0} (14) Remark 5. Since the state and input constraints X and U are polytopes, then the maximal output admissible set Σ is a polytope and can be finitely determined (Kolmanovsky and Gilbert, 1995). Assumption 6. There exists s MOAS for (11) with the state constraint X, the input constraint U, the disturbance bound W and the feedback control policy u = Kx x + Kw w for i > N . Theorem 7. If Assumption 6 holds, the infinite horizon feedback MPC problem (9) is equivalent to the following finite horizon MPC N =1 v 2 (k + i) (15a) min [v(k),v(k+1),...,v(k+N −1)]
i=0
s.t. x ¯(k) = x(k) (15b) ¯(k) + Bu v(k) + BK w(k) (15c) x ¯(k + 1) = AK x x ¯(k + i + 1) = AK x ¯(k + i) + Bu v(k + i) i = 1, . . . , N − 1 (15d) ¯(k) + Kw w(k) + v(k) (15e) u ¯(k) = Kx x u ¯(k + i) = Kx x ¯(k + i) + v(k + i) i = 1, . . . , N − 1 (15f) u ¯(k) ∈ U, x ¯(k) ∈ X (15g) u ¯(k + i) ∈ Ui , x ¯(k + i) ∈ Xi , ∀i = 1, . . . , N − 1 (15h) x ¯(k + N ) ∈ Σ ∼ DN (15i) where Σ and DN are defined in (14) and (12i) respectively.
Proof. The proof can be completed in a similar way to (Chisci et al., 2001) Remark 8. The feedback MPC problem can be formulated to a quadratic programming (QP), which will be discussed in subsection 3.3. Remark 9. The terminal constraints guaranteed that the terminal state will lay in the MOAS and will remain in the MOAS for all i > N . 3.3 Implementation
The predicted state trajectory from nominal system (12d) is (17a) x ¯(k + i) = AiK x(k) + Mi v(k) where i−2 Mi = Ai−1 (17b) K B u AK B u . . . B u 0 . . . 0
and the predicted control sequence is u ¯(k + i) = Kx AiK x(k) + Ni v(k) (17c) where i−2 Ni = Kx Ai−1 K Bu Kx AK Bu . . . Kx Bu 1 0 . . . 0 (17d) Remark 10. Since X, U, W are polytopes, the state constraints Xi , control input constraints Ui and the terminal constraint Σ ∼ DN are all polytopes and can be expressed by (18a) Xk {x ∈ R2 , fx,k x ≤ gx,k } Uk {u ∈ R, fu,k x ≤ gu,k }
(18b)
(18c) Σ ∼ DN {x ∈ R , fx,Σ x ≤ gx,Σ } where f and g are corresponding support vectors of the polytopes. 2
The objective function can be expressed as T
J v(k) v(k) The state and input constraints can be written as Γx v(k) + Λx x(k) + Πx w(k) ≤ ηx Γu v(k) + Λu x(k) + Πu w(k) ≤ ηu fx,0 fx,0 M0 fx,1 AK fx,1 M1 Λx = Γx = .. .. . . fx,N −1 MN −1 fu,0 N0 fu,1 N1 Γu = .. . fu,N −1 NN −1 0 fx,1 BK f A B x,2 K K Πx = .. .
−1 fx,N −1 AN K fu,0 Kx fu,1 Kx AK Λx = .. .
−2 fx,N −1 AN BK K
−1 fu,N −1 Kx AN K fu,0 Kw fu,1 Kx BK Πu = fu,2 Kx AK BK .. .
(19) (20a) (20b)
(20c)
(20d)
−2 fu,N −1 Kx AN BK K (20e)
gx,0 gx,0 gx,1 gu,1 ηu = . (20f) ηx = . .. .. gx,N −1 gu,N −1 and terminal constraint N −1 fx,Σ MN v(k) + fx,Σ AN BK w(k) ≤ gx,Σ K x(k) + fx,Σ AK (20g) where Mi ,Ni are defined in (17b) and (17d), f and g are defined in (18). Then we have the QP problem min (19) s.t. (20) (21) v(k)
In this subsection, we convert the feedback MPC problem (15) to a QP problem. Let v(k) denote the control sequence v(k) = [v(k), . . . , v(k + N − 1)] (16) 146
4. NUMERICAL SIMULATION In this section, we present the simulation result using the feedback MPC strategy described in (20) and compare it
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Siyuan Zhan et al. / IFAC PapersOnLine 50-1 (2017) 141–146
145
0.0195 . We find that eigenvalues of AK are strictly inside 0.0495 the unit circle.
Figure 3 shows that with the designed maximal disturbance w = 1.0m/s, the terminal constraints shrink as the prediction horizon N increases. That is because the larger N will require more margin to account for future disturbance to guarantee the terminal state to stay strictly within the MOAS. Figure 4 shows that with predictive horizon N = 5, the larger maximal disturbances will result in smaller terminal constraints (i.e. the feedback MPC get more conservative as we designed for larger maximal disturbances).
Fig. 3. Terminal constraints for different predictive horizon with wmax = 1.0m/s with the unconstrained linear optimal control case. The model we used here has similar dynamics to the 2nd order model in (Li et al., 2012). The parameters for the WEC model are Stiffness Mass of the float Frequency independent added mass Constant coefficient term Input magnitude constraint The buoy heave position limit The buoy heave velocity limit
k = 6.39 × 105 N/m ms = 1 × 104 kg ma = 7 × 104 kg D = 2 × 105 Nm/s umax = 3 × 105 N Φmax = 1.2m x2,max = 10m/s
With sampling time Ts = 0.02s, the discrete model parameters are 0 −0.0195 0.9984 0.0195 Bw = A= Bu = 0.0024 0.0503 −0.1557 0.9497
Fig. 5. The wave amplitude and its derivative data used in simulations
Fig. 6. The control input and buoy heave motion using the unconstrained LOC
Fig. 4. Terminal constraints for different wmax with N = 5 0 0 , 0 50 T = [0 1], R = 0.05. We have Kx = [0.5629 −26.4890], 0.9984 −0.0188 Kw = −0.3509 and AK = , BK = 0.1544 0.8851
The weights of the LOC are chosen as Q =
147
The wave amplitude and its derivative data used in simulations are shown in Figure 5. Figure 6 shows the control input and heave motion trajectories using u = Kx x+Kw w. We can find that state constraint is violated around 7s when we use the unconstrained LOC. Figure 7 and Figure 8 show the control input and heave motion trajectories and the power and energy generated respectively. We can see the state and input constraints of the feedback MPC are satisfied for the whole period
Proceedings of the 20th IFAC World Congress 146 Siyuan Zhan et al. / IFAC PapersOnLine 50-1 (2017) 141–146 Toulouse, France, July 9-14, 2017
Fig. 7. The control input and buoy heave motion using the feedback MPC
Fig. 8. Energy output of feedback MPC and the energy reversal is avoided. The energy output is 1.0562 × 108 J. 5. CONCLUSION We have proposed a novel feedback MPC strategy for WECs which uses the information of both the current states and disturbance for feedback. This novel MPC policy does not require future wave predictions so that the installation and operation cost of WECs is reduced. The novel feedback MPC can guarantee system’s constraint satisfaction and feasibility under bounded but persistent disturbances. The advantages of low cost and reliability make MPC for WECs more attractive. Since the restricted constraints and the pre-designed linear feedback part can be calculated off-line, the feedback MPC policy does not add significant on-line computational burden. In the simulation, we also shows that the feedback MPC is more robust than the corresponding LOC in terms of handling constraints. The theoretical support and time simulation provide engineers much confidence when designing the noval feedback MPC for WECs. REFERENCES Brekken, T.K. (2011). On model predictive control for a point absorber wave energy converter. In PowerTech, 148
2011 IEEE Trondheim, 1–8. IEEE. Chisci, L., Rossiter, J.A., and Zappa, G. (2001). Systems with persistent disturbances: predictive control with restricted constraints. Automatica, 37(7), 1019–1028. Cretel, J.A., Lightbody, G., Thomas, G.P., and Lewis, A.W. (2011). Maximisation of energy capture by a waveenergy point absorber using model predictive control. IFAC Proceedings Volumes, 44(1), 3714–3721. Falnes, J. (2007). A review of wave-energy extraction. Marine Structures, 20(4), 185–201. Fusco, F. and Ringwood, J.V. (2013). A simple and effective real-time controller for wave energy converters. IEEE Transactions on sustainable energy, 4(1), 21–30. Goulart, P.J., Kerrigan, E.C., and Maciejowski, J.M. (2006). Optimization over state feedback policies for robust control with constraints. Automatica, 42(4), 523– 533. Hals, J., Falnes, J., and Moan, T. (2011). Constrained optimal control of a heaving buoy wave-energy converter. Journal of Offshore Mechanics and Arctic Engineering, 133(1), 011401. Kolmanovsky, I. and Gilbert, E.G. (1995). Maximal output admissible sets for discrete-time systems with disturbance inputs. In American Control Conference, Proceedings of the 1995, volume 3. IEEE. Lee, Y.I. and Kouvaritakis, B. (2000). Robust receding horizon predictive control for systems with uncertain dynamics and input saturation. Automatica, 36(10), 1497–1504. Li, G. and Belmont, M.R. (2014). Model predictive control of sea wave energy converters–part i: A convex approach for the case of a single device. Renewable Energy, 69, 453–463. Li, G., Weiss, G., Mueller, M., Townley, S., and Belmont, M.R. (2012). Wave energy converter control by wave prediction and dynamic programming. Renewable Energy, 48, 392–403. Mayne, D.Q., Rawlings, J.B., Rao, C.V., and Scokaert, P.O. (2000). Constrained model predictive control: Stability and optimality. Automatica, 36(6), 789–814. Mayne, D.Q., Seron, M.M., and Rakovi´c, S. (2005). Robust model predictive control of constrained linear systems with bounded disturbances. Automatica, 41(2), 219– 224. OPT (2011). Ocean power technologies, pb 150 power buoy, http://www.oceanpowertechnologies.com/pb150-gallery/. Technical report, PB150. Rossiter, J.A., Kouvaritakis, B., and Rice, M. (1998). A numerically robust state-space approach to stablepredictive control strategies. Automatica, 34(1), 65–73. Yu, Z. and Falnes, J. (1995). State-space modelling of a vertical cylinder in heave. Applied Ocean Research, 17(5), 265–275. Zhan, S., Huijberts, H., Na, J., and Li, G. (2016). Optimal controller design and constraints analysis for wave energy converters. In 11th International Conference on Control, Proceedings of the 2016. UKACC. Zhong, Q.C. and Weiss, G. (2011). Synchronverters: Inverters that mimic synchronous generators. IEEE Transactions on Industrial Electronics, 58(4), 1259– 1267.