Model Predictive Control of a Nonlinear System with Known Scheduling Variable

Transcription

1 Proeedings of the 17th Nordi Proess Control Worshop Tehnial University of Denmar, Kgs Lyngby, Denmar Model Preditive Control of a Nonlinear System with Known Sheduling Variable Mahmood Mirzaei Niels Kjølstad Poulsen Hans Henri Niemann Department of Informatis and Mathematial Modeling, Tehnial University of Denmar, Denmar, mmir@immdtud, np@immdtud) Department of Eletrial Engineering, Tehnial University of Denmar, Denmar, hhn@eletrodtud) Abstrat: Model preditive ontrol MPC) of a lass of nonlinear systems is onsidered in this paper We will use Linear Parameter Varying LPV) model of the nonlinear system By taing the advantage of having future values of the sheduling variable, we will simplify state predition Consequently the ontrol problem of the nonlinear system is simplified into a quadrati programming Wind turbine is hosen as the ase study and we hoose wind speed as the sheduling variable Wind speed is measurable ahead of the turbine, therefore the sheduling variable is nown for the entire predition horizon Keywords: Model preditive ontrol, linear parameter varying, nonlinear systems, wind turbines, LIDAR measurements 1 INTRODUCTION Model preditive ontrol MPC) has been an ative area of researh and has been suessfully applied on different appliations in the last deades Qin and Badgwell 1996)) The reason for its suess is its straightforward ability to handle onstraints Moreover it an employ feedforward measurements in its formulation and an easily be extended to MIMO systems However the main drawba of MPC was its on-line omputational omplexity whih ept its appliation to systems with relatively slow dynamis for a while Fortunately with the rapid progress of fast omputations, better optimization algorithms, off-line omputations using multi-parametri programming Baoti 25)) and dediated algorithms and hardware, its appliations have been extended to even very fast dynamial systems suh as DC-DC onverters Geyer 25)) Basially MPC uses a model of the plant to predit its future behavior in order to ompute appropriate ontrol signals to ontrol outputs/states of the plant To do so, at eah sample time MPC uses the urrent measurement of outputs/states and solves an optimization problem The result of the optimization problem is a sequene of ontrol inputs of whih only the first element is applied to the plant and the proedure is repeated at the next sample time with new measurements Maiejowsi 22)) This approah is alled reeding horizon ontrol Therefore basi elements of MPC are: a model of the plant to predit its future, a ost funtion whih reflets ontrol objetives, onstraints on inputs and states/outputs, an optimization algorithm and the reeding horizon priniple Depending on the type of the model, the ontrol problem is alled linear MPC, hybrid MPC, nonlinear MPC et Nonlinear MPC This wor is supported by the CASED Projet funded by grant DSF of the Danish Counil for Strategi Researh is normally omputationally very expensive and generally there is no guarantee that the solution of the optimization problem is a global optimum In this wor we extend the idea of linear MPC using linear parameter varying LPV) systems to formulate a tratable preditive ontrol of nonlinear systems To do so, we use future values of a disturbane to the system that ats as a sheduling variable in the model However there are some assumptions that restrit our solution to a speifi lass of problems The sheduling variable is assumed to be nown for the entire predition horizon And the operating point of the system mainly depends on the sheduling variable 2 PROPOSED METHOD Generally the nonlinear dynamis of a plant ould be modeled as the following differene equation: x +1 = fx, u, d ) 1) With x, u and d as states, inputs and disturbanes respetively Using the nonlinear model, the nonlinear MPC problem ould be formulated as: N 1 min lx N ) + lx +i, u +i ) 2) u i= Subjet to x +1 = fx, u, d ) 3) u +i U 4) ˆx +i X 5) Where l denotes some arbitrary norm and U and X show the set of aeptable inputs and states As it was mentioned beause of the nonlinear model, this problem is omputationally too expensive One way to avoid this problem is to linearize around an equilibrium point of the system and use linearized model instead of the nonlinear model However for some plants assumption of linear 163

2 Proeedings of the 17th Nordi Proess Control Worshop Tehnial University of Denmar, Kgs Lyngby, Denmar model does not hold for long predition horizons as the plant operating point hanges, for example based on some disturbanes that at as a sheduling variable An example ould be a wind turbine for whih wind speed ats as a sheduling variable and hanges the operating point of the system 21 Linear MPC formulation The problem of linear MPC ould be formulated as: N 1 min x N Qf + x +i Q + u +i R 6) u,u 1,,u N 1 i= Subjet to x +1 = Ax + Bu 7) u +i U 8) ˆx +i X 9) Assuming that we use norms 1, 2 and the optimization problem beomes onvex providing that the sets U and X are onvex Convexity of the optimization problem maes it tratable and guarantees that the solution is the global optimum The problem above is based on a single linear model of the plant around one operating point However below we formulate our problem using linear parameter varying systems LPV) in whih the sheduling variable is nown for the entire predition horizon 22 Linear Parameter Varying systems Linear Parameter Varying LPV) systems are a lass of linear systems whose parameters hange based on a sheduling variable Study of LPV systems was motivated by their use in gain-sheduling ontrol of nonlinear systems Aparian et al 1995)) LPV systems are able to handle hanges in the dynamis of the system by parameter varying matries Definition LPV systems) let Z denote disrete time We define the following LPV systems: x +1 = Aγ )x + Bγ )u 1) n γ n γ Aγ ) = A j γ,j Bγ ) = B j γ,j 11) j=1 j=1 Whih Aγ ) and Bγ ) are funtions of the sheduling variable γ The variables x R nx, u R nu, and γ R nγ are the state, the ontrol input and the sheduling variable respetively 23 Problem formulation Using the above definition, the linear parameter varying LPV) model of the nonlinear system with disturbanes is of the following form: x +1 = Aγ ) x + Bγ )ũ + B d γ ) d 12) This model is formulated based on deviations from the operating point However we need the model to be formulated in absolute values of inputs, states and disturbanes Beause in our problem the steady state point hanges as a funtion of the sheduling variable and we need to introdue a variable to apture its bahavior In order to rewrite the state spae model in the absolute form we use: x = x x 13) ũ = u u 14) d = d d 15) x, u and d are values of states, inputs and disturbanes at the operating point Therefore the LPV model beomes: x +1 = Aγ )x x ) + Bγ )u u ) + B d γ )d d ) + x +1 Whih ould be written as: with 16) x +1 = Aγ )x + Bγ )u + B d γ )d + λ 17) λ = x +1 Aγ )x Bγ )u B d γ )d 18) Now having the LPV model of the system we proeed to ompute state preditions In linear MPC predited states at step n is: x +n = A n x + A i Bu +) i for i= n = 1, 2,, N 19) However in our method the predited state is also a funtion of sheduling variable Γ n = γ +1, γ +2, γ +n ) T for n = 1, 2,, N 1 and we assume that the sheduling variable is nown for the entire predition Therefore the predited state ould be written as: x +1 γ ) = Aγ )x + Bγ )u + B d γ )d + λ 2) And for n Z, n 1: x +n+1 Γ n ) = Aγ +i )x i=n 1 + j= i=n j 1 + j= i=n j + j= i=n j Aγ +i ) Aγ +i ) Aγ +i ) ) ) ) Bγ +j )u +j B d γ +j )d +j λ +) j 21) +Bγ +n )u +n + B d γ +n )d +n + λ +n Using the above formulas we write down the staed predited states whih beomes: with X = ΦΓ)x + H u Γ)U + H d Γ)D + Φ λ Γ)Λ 22) X = x +1 x +2 x +N ) T 23) U = u u +1 u +N 1 ) T 24) D = d d +1 d +N 1 ) T 25) Γ = γ γ +1 γ +N 1 ) T 26) Λ = λ λ +1 λ +N 1 ) T 27) In order to summarize formulas for matries Φ, Φ λ, H u and H d, we define a new funtion as: n ψm, n) = Aγ +i ) 28) i=m Therefore the matries beome: 164

3 Proeedings of the 17th Nordi Proess Control Worshop Tehnial University of Denmar, Kgs Lyngby, Denmar ψ1, 1) ψ2, 1) ΦΓ) = ψ3, 1) ψn, 1) I ψ1, 1) I Φ λ Γ) = ψ2, 1) ψ2, 2) I ψn 1, 1) ψn 1, 2) ψn 1, 3) I Bγ ) ψ1, 1)Bγ ) Bγ +1 ) H uγ) = ψ2, 1)Bγ ) ψ2, 2)Bγ +1 ) ψn 1, 1)Bγ ) ψn 1, 2)Bγ +1 ) Bγ N 1 ) B d γ ) ψ1, 1)B d γ ) B d γ +1 ) H d Γ) = ψ2, 1)B d γ ) ψ2, 2)B d γ +1 ) ψn 1, 1)B d γ ) ψn 1, 2)B d γ +1 ) B d γ N 1 ) After omputing the state preditions as funtions of ontrol inputs 22), we an write down the optimization problem similar to a linear MPC problem as a quadrati program: min U Subjet to: X T QX + U T RU U U X X 3 CASE STUDY 29) The ase study here is a wind turbine Wind turbine ontrol is a hallenging problem as the dynamis of the system hanges based on wind speed whih has a stohasti nature The method that we propose here is to use wind speed as a sheduling variable With the advanes in LIDAR tehnology Harris et al 26)) it is possible to measure wind speed ahead of the turbine and this enables us to have the sheduling variable of the plant for the entire predition horizon 31 Modeling Nonlinear model For modeling purposes, the whole wind turbine an be divided into 4 subsystems: Aerodynamis subsystem, mehanial subsystem, eletrial subsystem and atuator subsystem The aerodynami subsystem onverts wind fores into mehanial torque and thrust on the rotor The mehanial subsystem onsists of drivetrain, tower and blades Drivetrain transfers rotor torque to eletrial generator Tower holds the naelle and withstands the thrust fore And blades transform wind fores into toque and thrust The generator subsystem onverts mehanial energy to eletrial energy and finally the blade-pith and generator-torque atuator subsystems are part of the ontrol system To model the whole wind turbine, models of these subsystems are obtained and at the end they are onneted together A wind model is obtained and augmented with the wind turbine model to be used for wind speed estimation Figure 1 shows the basi subsystems and their interations The dominant dynamis of the wind turbine ome from its flexible struture Several degrees of freedom ould be onsidered to model the flexible struture, but for ontrol design mostly just a few important degrees of freedom are onsidered In figure 2 basi degrees of freedom whih are normally being onsidered in the design model are shown However in this wor we only onsider two degrees of freedom, namely the rotational degree of freedom DOF) and drivetrain torsion Nonlinearity of the wind turbines mostly omes from its aerodynamis Blade element momentum BEM) theory Hansen 28)) is used to numerially alulate aerodynami torque and thrust on the wind turbine This theory explains how torque and thrust are related to wind speed, blade pith angle and rotational speed of the rotor In steady state, ie disregarding dynami inflow, the following formulas an be used to alulate aerodynami torque and thrust Q r = 1 1 ρπr 2 v 2 ω ec 3 p θ, ω, v e ) 3) r Q t = 1 2 ρπr2 v 2 ec t θ, ω, v e ) 31) In whih Q r and Q t are aerodynami torque and thrust, ρ is the air density, ω r is the rotor rotational speed, v e is the effetive wind speed, C p is the power oeffiient and C t is the thrust fore oeffiient The absolute angular position of the rotor and generator are of no interest to us, therefore we use ψ = θ r θ g instead whih is the drivetrain torsion Having aerodynami torque and modeling drivetrain with a simple mass-spring-damper, the whole system equation with 2 degrees of freedom beomes: J r ω r = Q r ω r ω g N g ) ψ 32) N g J g ) ω g = ω r ω g N g ) + ψ N g Q g 33) ψ = ω r ω g N g 34) P e = Q g ω g 35) In whih J r and J g are rotor and generator moments of inertia, ψ is the drivetrain torsion, and are the drivetrain damping and stiffness fators respetively lumped in the low speed side of the shaft and P e is the generated eletrial power For numerial values of these parameters and other parameters given in this paper, we refer to Jonman et al 29)) θ in Pith Servo Q in Gen Servo θ Q Wind v fw v e Aerodynamis Q r Drivetrain ω g ω r Q g Generator Fig 1 Wind turbine subsystems v t F T P out Tower 165

4 Proeedings of the 17th Nordi Proess Control Worshop Tehnial University of Denmar, Kgs Lyngby, Denmar LPV model Colleting all the disussed models, matries of the state spae model beome: aγ)! 1 Aγ) = C= 1 2 N J Qg Ng Jg Ng Jg g g ) b1 γ)! 1 Bγ) = D= Jg ωg 43) T Fig 2 Basi degrees of freedom Linearized model As it was mentioned in the previous setion, wind turbines are nonlinear systems A basi approah to design ontrollers for nonlinear systems is to linearize them around some operating points For a wind turbine, the operating points on the quasi-steady Cp and Ct urves are nonlinear funtions of rotational speed ωr, blade pith θ and wind speed v To get a linear model of the system we need to linearize around these operating points Rotational speed and blade pith are measurable with enough auray, however this is not the ase for the effet of wind on the rotor Wind speed hanges along the blades and with azimuth angle angular position) of the rotor This is beause of wind shear and tower shadow and stohasti spatial distribution of the wind field Therefore a single wind speed does not exist to be used and measured for finding the operating point We bypass this problem by defining a fititious variable alled effetive wind speed ve ) whih shows the effet of wind in the rotor dis on the wind turbine In our two DOFs model only the aerodynami torque Qr ) and eletri power Pe ) are nonlinear Taylor expansion is used to linearize them Qr Qr Qr Qr ω, θ, ve ) = ω + θ + ve 36) ω θ ve a Pe = b1 Pe Pe ωg + Qg ωg Qg {z} Qg T In whih x = ωr ωg ψ), u = θ Qg ) and y = T ωr ωg Pe ) are states, inputs and outputs respetively In the matrix B, parameter b1 is unertain Therefore the unertain linear state spae model beomes: x = Aγ)x + Bγ)u y = Cx + Du 32 Control objetives The most basi ontrol objetive of a wind turbine is to maximize aptured power during the life time of the wind turbine This means trying to maximize aptured power when wind speed is below its rated value This is also alled maximum power point traing MPPT) However when wind speed is above rated, ontrol objetive beomes regulation of the outputs around their rated values while trying to minimize dynami loads on the struture These objetives should be ahieved against flutuations in wind speed whih ats as a disturbane to the system In this wor we have onsidered operation of the wind turbine in above rated full load region) Therefore we try to regulate rotational speed and generated power around their rated values and remove the effet of wind speed flutuations 33 Offset free ontrol b2 37) ωg For the sae of simpliity in notations we use Qr, Pe, θ, ω and ve instead of Qr, Pe, θ, ω and ve around the operating points from now on Using the linearized aerodynami torque, the 2 DOFs linearized model beomes: a ω r = ωr + ωg ψ + b1 θ + b2 ve 38) Qg ω g = ωr 2 ωg + ψ 39) Ng Jg Ng Jg Ng Jg Jg ωg ψ = ωr 4) Ng Pe = Qg ωg + ωg Qg 41) A more detailed desription of the model and linearization is given in Mirzaei et al 211)) 166 Persistent disturbanes and modeling error an ause an offset between measured outputs and desired outputs To avoid this problem we have employed an offset free referene traing approah see Muse and Badgwell 22) and Pannohia and Rawlings 23)) Our RMPC solves the regulation problem around the operating point However we regulate around the operating point x and u ) whih results in offset from desired outputs To avoid this problem in our ontrol algorithm we shift origin in our regulation problem to x and u instead In order to find new origins, we have augmented linear model of the plant with a disturbane model that adds fititious disturbanes to the system The fititious disturbanes ompensate the differene between measured outputs and desired outputs State spae model of the augmented system is: x +1 = A x + B u 44) y = C x + Du 45) in whih the augmented state and matries are:

5 Proeedings of the 17th Nordi Proess Control Worshop Tehnial University of Denmar, Kgs Lyngby, Denmar Table 1 Performane omparison between gain sheduling approah and linear MPC Parameters Proposed approah Linear MPC SD of ω r RPM) SD of P e Watts) Mean value of P e Watts) SD of pith degrees) SD of shaft moment NM) ˆx ) +1 A Bd x = ˆd +1 Ã = A d 46) ˆp +1 A p B = B ) T C = C Cp ) 47) ˆx, ˆd and ˆp are system states, input/state and output disturbanes respetively A, B, C, D) are matries of the linearized model, B d and C p show effet of disturbanes on states and outputs respetively A d and A p show dynamis of input/state and output disturbanes For more information and how to hoose these matries we refer to Muse and Badgwell 22)) and Pannohia and Rawlings 23)) Sine the disturbanes are not measurable, an extended Kalman filter is designed to estimate them The estimated disturbanes are used to remove any offset between desired outputs and measured outputs Based on this model and estimated disturbanes, x and u whih are offset free steady state input and states an be alulated: ) ) ) A I B x C D u = Bd ˆd 48) C p ˆp After alulating these values, we simply replae x and u in 18) with x and u whih results in: λ = x +1 Aγ )x Bγ )u B d γ )d 49) As it ould be seen from the table and figures, the proposed approah gives better regulation on rotational speed and generated power smaller standard deviations) while maintaining a smaller shaft moment and pith ativity timeseonds) Fig 3 Blade-pith referene degrees, red-dashed line approah) SIMULATIONS In this setion simulation results for the obtained ontroller are presented The ontroller is implemented in MATLAB and is tested on a full omplexity FAST Jonman and 25)) model of the referene wind turbine Jonman et al 29)) Simulations are done with realisti turbulent wind speed, with Kaimal model ie 25)) as the turbulene model and TurbSim Jonman 29)) is used to generate wind profile In order to stay in the full load region, a realization of turbulent wind speed is used from ategory C of the turbulene ategories of the IEC ie 25)) with 18m/s as the mean wind speed 41 Stohasti simulations In this setion simulation results for a stohasti wind speed is presented Control inputs whih are pith referene θ in and generator reation torque referene Q in along with system outputs whih are rotor rotational speed ω r and eletrial power P e are plotted in figures 3-6 reddashed lines are results of linear MPC and solid blue lines show the results of the proposed approah) Simulation results show good regulations of generated power and rotational speed Table 1 shows a omparison of the results between the proposed approah and MPC approah based on linearization at eah sample point Henrisen 27)) 39 time seonds) Fig 4 Generator-torque referene NM, red-dashed line approah) REFERENCES 25) IEC wind turbines-part 1: Design requirements Aparian, P, Gahinet, P, and Beer, G 1995) Selfsheduled h ontrol of linear parameter-varying systems: a design example Automatia, 319), Baoti, M 25) Optimal Control of Pieewise Affine Systems a Multi-parametri Approah PhD thesis Geyer, T 25) Low Complexity Model Preditive Control in Power Eletronis and Power Systems PhD thesis Hansen, MOL 28) Aerodynamis of Wind Turbines Earthsan Harris, M, Hand, M, and Wright, A 26) LIDAR for turbine ontrol Tehnial report, National Renewable Energy Laboratory, Golden, CO 167

6 Proeedings of the 17th Nordi Proess Control Worshop Tehnial University of Denmar, Kgs Lyngby, Denmar 13 Pannohia, G and Rawlings, JB 23) Disturbane models for offset-free model-preditive ontrol AIChE Journal, 492), Qin, SJ and Badgwell, TA 1996) An overview of industrial model preditive ontrol tehnology timeseonds) Fig 5 Rotor rotational speed ω r, rpm, red-dashed line approah) timeseonds) Fig 6 Eletrial power mega watts, red-dashed line approah) Henrisen, LC 27) Model Preditive Control of a Wind Turbine Master s thesis, Tehnial University of Denmar, Informatis and Mathematial Modelling, Lyngby, Denmar Jonman, B 29) Turbsim user s guide: Version 15 Tehnial report, National Renewable Energy Laboratory, 1617 Cole Boulevard, Golden, Colorado Jonman, J, Butterfield, S, Musial, W, and Sott, G 29) Definition of a 5MW referene wind turbine for offshore system development Tehnial report, National Renewable Energy Laboratory, Golden, CO Jonman, JM and, MLB 25) Fast users guide Tehnial Report NREL/EL , National Renewable Energy Laboratory, Golden, CO Maiejowsi, J 22) Preditive ontrol with onstraints Pearson Eduation Lim, Essex Mirzaei, M, Niemann, HH, and Poulsen, NK 211) A µ-synthesis approah to robust ontrol of a wind turbine In the 5th IEEE Conferene on Deision and Control and European Control Conferene Orlando, FL, USA Muse, KR and Badgwell, TA 22) Disturbane modeling for offset-free linear model preditive ontrol Journal of Proess Control, 125),