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Flatness-based Model Predictive Control for the Fuel Optimization of Hybrid Electric Vehicles
Preprints, 5th IFAC Conference on Nonlinear Model Predictive Preprints, 5th IFAC Conference on Nonlinear Model Predictive Preprints, 5th 5th IFAC IFAC Conference Conference on on Nonlinear Nonlinear Model Model Predictive Predictive Control Preprints, Control Control September 17-20, 2015. Seville, SpainAvailable online at www.sciencedirect.com Control September 17-20, 2015. Seville, Spain September September 17-20, 17-20, 2015. 2015. Seville, Seville, Spain Spain
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IFAC-PapersOnLine 48-23 (2015) 464–470
Flatness-based Model Predictive Control Flatness-based Model Predictive Control Flatness-based Model Predictive Control for the Fuel Optimization of Hybrid for the Fuel Optimization of Hybrid for the Fuel Optimization of Hybrid Electric Vehicles Electric Electric Vehicles Vehicles ∗ ∗∗ Martina Joˇ ssevski, ∗ D. Abel ∗∗ Martina evski, ∗∗ Martina Joˇ Joˇ evski, ∗∗ D. D. Abel Abel ∗∗ Martina Joˇ ssevski, D. Abel ∗ ∗ Institute of Automatic Control, RWTH Aachen University, 52074 ∗ Institute of Automatic Control, RWTH Aachen University, 52074 ∗ Institute ofGermany Automatic Control,[email protected]) RWTH Aachen Aachen University, University, 52074 52074 Aachen, (( e-mail: Institute of Automatic Control, RWTH Aachen, Germany [email protected]) ∗∗ Aachen, Germany (( e-mail: e-mail: [email protected]) Institute of Automatic Control, RWTH Aachen University Aachen, Germany e-mail: [email protected]) ∗∗ of Automatic Control, RWTH Aachen University ∗∗ ∗∗ Institute Institute Institute of of Automatic Automatic Control, Control, RWTH RWTH Aachen Aachen University University
Abstract: In this paper aa nonlinear model predictive controller for the optimization of the Abstract: In this nonlinear model predictive controller for the optimization of Abstract: In vehicle this paper paper aefficiency nonlinear model predictive controller forthe theconcept optimization of the the hybrid electric fuel is introduced. The idea is to use of differential Abstract: In this paper a nonlinear model predictive controller for the optimization of the hybrid electric vehicle fuel efficiency is introduced. The idea is to use the concept of differential hybrid electric electric vehicle fuel fuel efficiency isofintroduced. introduced. The idea idea is to to use the concept concept of differential differential flatness to determine the evolution system’s state across the prediction horizon without hybrid vehicle efficiency is The is use the of flatness to determine the evolution of system’s state across the prediction horizon without flatness to determine evolution of state the prediction horizon without numerical The entire nonlinear optimization problem expressed as aa function of flatness tointegration. determine the the evolution of system’s system’s state across across the is prediction horizon without numerical integration. The entire nonlinear optimization problem is expressed as function of integration. The entire nonlinear optimization problem is expressed as a function of anumerical fictious output, called the flat output and its first derivative. The optimal distribution numerical integration. The entire nonlinear optimization problem is expressed as a function of a output, called the output its first derivative. The distribution of a fictious fictious output, called the flat flat output and and its is first derivative. Thebyoptimal optimal distribution of the requested torque between propulsion devices then determined aa static optimization. a fictious output, called the flat output and its first derivative. The optimal distribution of the requested torque between propulsion devices is then determined by static optimization. the requested requested torque between propulsion devices is then then is determined by aa static static optimization. The flatness-based model predictive controller (FMPC) compared to linear time-varying the torque between propulsion devices is determined by a optimization. The flatness-based model predictive controller (FMPC) is compared to aa linear time-varying The flatness-based model predictive compared MPC (LTV-MPC) which applies the controller linearized (FMPC) model of of is the plant to to to predict thetime-varying future state state The flatness-based model applies predictive controller (FMPC) isthe compared to a linear linear time-varying MPC (LTV-MPC) which the linearized model plant predict the future MPC (LTV-MPC) which applies the linearized model of the plant to predict the future state and output trajectories. Both strategies are evaluated on standard driving cycles. To proof the MPC (LTV-MPC) which applies the linearized model of the plant to predict the future state and trajectories. Both strategies are on driving cycles. To proof the and output output trajectories. Both strategies are evaluated evaluated on standard standard drivingare cycles. To and proof the concept it is assumed that the velocity profile and driver torque demand known exact and output trajectories. Both strategies are evaluated on standard driving cycles. To proof the concept it is assumed that the velocity profile and driver torque demand are known and exact concept it is assumed that the velocity profile and driver torque demand are known and exact over a predefined prediction horizon. concept it is assumed that the velocity profile and driver torque demand are known and exact over a predefined prediction horizon. over over a a predefined predefined prediction prediction horizon. horizon. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: hybrid electric vehicles, Keywords: hybrid hybrid electric vehicles, vehicles, energy energy management, management, nonlinear nonlinear MPC, MPC, flatness, flatness, B-splines B-splines Keywords: Keywords: hybrid electric electric vehicles, energy energy management, management, nonlinear nonlinear MPC, MPC, flatness, flatness, B-splines B-splines 1. INTRODUCTION optimal control problem: Dynamic Programming (DP), 1. INTRODUCTION INTRODUCTION optimal control Dynamic Programming 1. optimal methods control problem: problem: Dynamic Programming (DP), (DP), 1. INTRODUCTION indirect and direct methods. optimal control problem: Dynamic Programming (DP), indirect indirect methods methods and and direct direct methods. methods. indirect methods and direct methods. Prompted by increasing energy and environmental chalPrompted by by increasing increasing energy energy and and environmental environmental chalchal- Dynamic Programming is widely applied in the energy Prompted is applied in the lenges, powertrain hybridization remains a major major future future Prompted by increasing energy and environmental chal- Dynamic Dynamic Programming Programming is itwidely widely applied inoptimal the energy energy lenges, powertrain hybridization remains a management of HEVs as calculates the fuel Dynamic Programming is widely applied in the energy lenges, powertrain hybridization remains aa major future management of HEVs as it calculates the optimal fuel development trend in the automotive industry. Compared lenges, powertrain hybridization remains major future management by of going HEVsrecursively as it it calculates calculates the in optimal fuel development trend in the automotive industry. Compared consumption backwards time using management of HEVs as the optimal fuel development trend in the automotive industry. Compared consumption by going recursively backwards in time using to the conventional vehicles, hybrid electric vehicles (HEV) development trend vehicles, in the automotive industry. Compared consumption by going recursively backwards in time using to the conventional hybrid electric vehicles (HEV) Bellman’s principle of optimality (Bertsekas [2000]). It consumption by going recursively backwards in time using to the conventional vehicles, hybrid electric vehicles (HEV) principle of optimality (Bertsekas [2000]). It are augmented by an alternative mover, usually an electric to the conventional vehicles, hybrid electric vehicles (HEV) Bellman’s Bellman’s principle of optimality (Bertsekas [2000]). It are augmented by an alternative mover, usually an electric guarantees the global optimality in case when the driving Bellman’s principle of optimality (Bertsekas [2000]). It are augmented by an alternative mover, usually an electric guarantees the global optimality in case when the driving motor and/or generator. The additional degree of freedom, are augmented by an alternative mover, usually an electric guarantees the global global optimality inascase case when the the method driving motor and/or generator. The additional degree of freedom, cycle is perfectly known. However a non-causal guarantees the optimality in when driving motor and/or generator. The additional degree of freedom, cycle is perfectly known. However as a non-causal method introduced by the electric branch, gives the possibility to motor and/or The additional of freedom, cycle is perfectly known. However as aa non-causal non-causal method introduced bygenerator. the electric electric branch, givesdegree the possibility possibility to cycle it is not applicable in real applications and is mainly is perfectly known. However as method introduced by the branch, gives the to is not applicable in real applications and is mainly increase the fuel economy of HEVs, e.g. by operating the introduced by the electric branch, gives the possibility to it it is not applicable in real applications and is mainly increase the fuel economy of HEVs, e.g. by operating the applied as a benchmark solution to assess the performance it is not applicable in real applications and is mainly increase the fuel economy of HEVs, e.g. by operating the applied as a benchmark solution to assess the performance internal combustion engine in the most efficient region or increase the fuel economy of HEVs, e.g. by operating the applied as a benchmark solution to assess the performance internal combustion engine in the most efficient region or or tune the parameters of other HEV control strategies applied as a benchmark solution to assess the performance internal combustion engine in the most efficient region or parameters of other control strategies by recuperating the kinetic kinetic energy whichefficient would otherwise otherwise internal combustion engine energy in the most region or or or tune tune the the and parameters of [2009], other HEV HEV control strategies by recuperating recuperating the which would (Sundstrom Guzzella Guzzella and Sciaretta or tune the parameters of other HEV control strategies by the kinetic energy which would otherwise (Sundstrom and Guzzella [2009], Guzzella and Sciaretta be wasted in heat by using the friction brakes. by recuperating the kinetic energy which would otherwise (Sundstrom and Guzzella [2009], Guzzella and Sciaretta be wasted in heat by using the friction brakes. [2013]). (Sundstrom and Guzzella [2009], Guzzella and Sciaretta be [2013]). be wasted wasted in in heat heat by by using using the the friction friction brakes. brakes. [2013]). [2013]). Essentially, there are two possibilities to improve the effiEssentially, there there are are two two possibilities possibilities to to improve improve the the effieffi- In addition to DP which finds global optimal fuel consumpEssentially, In addition to DP which finds global optimal fuel consumpciency of the hybrid component sizing and the Essentially, there arepowertrain: two possibilities to improve the effiaddition to which finds global optimal fuel ciency of the hybrid powertrain: component sizing and the In tion over an priori given driving indirect methods In addition toaa DP DP which finds globalcycle, optimal fuel consumpconsumpciency of the hybrid powertrain: component sizing and the tion over an priori given driving cycle, indirect methods application of energy management strategies. Component ciency of the hybrid powertrain: component sizing and the tion over an aa priori given driving cycle, indirect methods application of energy management strategies. Component convert the optimal control problem into a boundary value tion over an priori given driving cycle, indirect methods application of energy management strategies. Component convert the optimal control problem into a boundary value sizing is employed to optimally determine dimensions of application of energy management strategies. Component convert the the optimal control problem into aa boundary boundary value sizing is employed to optimally determine dimensions of problem which is then solved numerically by solving the convert optimal control problem into value sizing is employed to optimally determine dimensions of problem which is then solved numerically by solving the the HEV powertrain components so that fuel consumption, sizing is employed to optimally determine dimensions of problemofwhich which is then then solved numerically numerically bystart solving the the HEV powertrain components so that fuel consumption, system differential equations which satisfy and end problem is solved by solving the the HEV powertrain components so that fuel consumption, system of differential equations which satisfy start and end vehicle weight and cost are minimized. Under the asthe HEV powertrain components so that fuel consumption, system of differential equations which satisfy start and end vehicle weight and cost are minimized. Under the aspoint conditions. In (Serrao et al. [2009]), as an indirect system of differential equations which satisfy start and end vehicle weight and cost are Under the asconditions. In (Serrao et al. [2009]), as an indirect sumption that the vehicle has already vehicle weight andhybrid cost electric are minimized. minimized. Under thebeen as- point point conditions. In et [2009]), as sumption that the the hybrid electric vehicle has has already been method, Pontryagin Minimum Principle is used point conditions. In (Serrao (Serrao et al. al. [2009]),(PMP) as an an indirect indirect sumption that hybrid electric vehicle already been method, Pontryagin Minimum Principle (PMP) used optimally designed, further improvements with regards to sumption that the hybrid electric vehicle has already been method, Pontryagin Minimum Principle function (PMP) is isand used optimally designed, further improvements with regards to to minimize at each instant Hamiltonian to method, Pontryagin Minimum Principle (PMP) is used optimally designed, further improvements with regards to to minimize at each instant Hamiltonian function and the fuel economy can be achieved by energy management optimally designed, further improvements with regards to to minimize minimize atoptimal each instant instant Hamiltonian function and to to the fuel economy can be achieved by energy management find a global trajectory for the control inputs. to at each Hamiltonian function and to the fuel can achieved by management find aa global optimal trajectory for the control inputs. strategies. The objective the energy management stratthe fuel economy economy can be be of achieved by energy energy management find global optimal trajectory for the control inputs. strategies. The objective of the energy management stratIn literature, Pontryagin Minimum Principle (Kim et al. find a global optimal trajectory for the control inputs. strategies. The ofconsumption the stratIn literature, Pontryagin Minimum Principle (Kim et al. egy is to minimize the fuel by determining the strategies. The objective objective the energy energy management management stratliterature, Pontryagin Minimum Principle (Kim egy is is to to minimize minimize the fuel fuelofconsumption consumption by determining determining the In [2011]) and the Equivalent Consumption In literature, Pontryagin Minimum PrincipleMinimization (Kim et et al. al. egy the by the [2011]) and the Equivalent Consumption Minimization energy flow split between the engine and the alternative egy is to minimize the fuel consumption by determining the [2011]) and and the isEquivalent Equivalent Consumption Minimization energy flow split between the engine and the alternative Strategy, which derived from the PMP, are frequently [2011]) the Consumption Minimization energy flow split between the engine and the alternative is derived the are frequently power source while meeting powertrain constraints. energy flow split between engine and the alternative Strategy, Strategy, which which isstrategies derived from from the PMP, PMP, are frequently power source source while meetingthe powertrain constraints. presented control (Musardo et al. [2005], Serrao Strategy, which is derived from the PMP, are frequently power presented control strategies (Musardo et al. [2005], Serrao power source while while meeting meeting powertrain powertrain constraints. constraints. presented control strategies (Musardo et al. [2005], Serrao et al. [2009], Onori et al. [2010], Ambuhl and Guzzella presented control strategies (Musardo et al. [2005], Serrao From the aspect of optimal control theory, the energy et al. [2009], Onori et al. [2010], Ambuhl and Guzzella From the aspect of optimal control theory, the energy et al. [2009], Onori et al. [2010], Ambuhl and Guzzella From the aspect of optimal control theory, the energy [2009]). et al. [2009], Onori et al. [2010], Ambuhl and Guzzella management problem of hybrid electric vehicles is formuFrom the aspect of optimal control theory, the energy management problem problem of of hybrid hybrid electric vehicles vehicles is is formuformu- [2009]). [2009]). management lated as aa finite-time optimalvehicles control is problem. management problemconstrained of hybrid electric electric formu- [2009]). lated as finite-time constrained optimal control problem. Direct methods differ from the indirect approaches in the lated as aa finite-time constrained optimal control problem. Direct methods differ from the indirect approaches in the Generally there are three main approaches to address the lated as finite-time constrained optimal control problem. Direct methods differ from the indirect approaches in the Generally there are three main approaches to address the way that control (and state) trajectories of the original Direct methods differ from the indirect approaches in the Generally that control (and state) trajectories of the original Generally there there are are three three main main approaches approaches to to address address the the way way that that control control (andare state) trajectories ofthe theoptimizaoriginal optimization problem parametrized and way (and state) trajectories of the original This work was financially supported by German Scientific Founoptimization problem are parametrized and the optimiza This work was financially supported by German Scientific Founoptimization problem are to parametrized and the the optimizaoptimiza tion problem converted aa finite-dimensional nonlinear optimization problem are parametrized and This work financially dation This (Deutsche work was was Forschungsgemeinschaft). financially supported supported by by German German Scientific Scientific FounFountion problem is is converted to finite-dimensional nonlinear dation (Deutsche Forschungsgemeinschaft). tion tion problem problem is is converted converted to to aa finite-dimensional finite-dimensional nonlinear nonlinear dation (Deutsche (Deutsche Forschungsgemeinschaft). Forschungsgemeinschaft). dation Copyright 2015 IFAC 465 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 465 Copyright 2015 IFAC 465 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 465Control. 10.1016/j.ifacol.2015.11.322
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programming problem (NLP). Direct single shooting, direct multiple shooting and direct collocation reside to this group of numerical optimization methods. In (Dosthosseini et al. [2011]) the application of a direct method for HEV energy management is presented. 1.1 Main Contribution In this contribution we use the idea which is similar to that of direct approaches to design an energy management controller for HEVs. Using a specified functional form, the states, control inputs and outputs of the systems are approximated and the optimization problem is converted to a NLP. The optimization problem is stated as nonlinear model predictive control scheme which optimizes the fuel consumption and the electric storage usage over a certain portion of the predefined driving cycle. The objective function is minimized under a set of state and control constraints which result from the time-varying operational limits of the powertrain components. However in our approach the property of differential flatness (Hagenmeyer and Delaleau [2008], Devos and L´evine [2006], Utz et al. [2007], J. A. De Don´ a [2009], Suryawan et al. [2012]) is used to parametrize the system and to transform the optimization task into a NLP. Applying the flat property, the system dynamics are implicitly satisfied and the differential equations are not integrated to obtain state predictions over a prediction horizon. Elimination of the system’s differential equations leads to a static optimization problem. In addition the constrained optimal control problem for the original nonlinear system instead for the linearized system, as in LTVMPC (Borhan et al. [2012], Joˇsevski [2014b]), is solved. The nonlinear model predictive controller is designed by casting the original formulation into a static nonlinear programming problem. This approach can be seen as an alternative to direct multiple shooting method. The paper is organized as follows. The assumed hybrid powertrain is presented in section 2. In section 3, we briefly introduce preliminaries regarding flatness and flatnessbased optimization. The non-flat formulation of the model predictive controller, nonlinear system transformation, flat output parametrization as well as the resulting flatness based MPC control problem are described in section 4. Finally simulation results are presented in section 5, while section 6 concludes the paper. 2. VEHICLE MODEL In this case study a parallel hybrid electric vehicle configuration is discussed in which the internal combustion engine (ICE) and electric machine (EM) are mounted on the same shaft and separated by the clutch (Fig. 1). This configuration allows combined as well as pure electric and pure thermal driving. When the clutch is closed both devices propel the vehicle and their speed is directly proportional to the vehicles velocity. To model the vehicle, a quasi-static forward approach is used. Thereby, vehicle parameters as well as quasi-static maps for propulsion components and energy storage are obtained from the automotive simulation software ADVISOR (Wipke et al. [1999]). 466
Fig. 1. Parallel hybrid electric vehicle structure 2.1 Powertrain Model It is presumed that the desired velocities and thus the torque demands at the wheels are known in advance, i.e. they act as an exogenous disturbance input to the system. The torque Treq,wh which is requested to follow the given velocity profile v(t) when only longitudinal vehicle dynamics are considered is defined by: 1 Treq,wh = rwh ( ρCd Av 2 + me v˙ + mg(fr cosθ + sinθ))(1) 2 where v represents the vehicles velocity, θ is the road inclination, ρ is the air density, Cd is the drag coefficient, A is the vehicle’s cross section, fr is the rolling resistance coefficient, m is the vehicle mass and me is the effective vehicle mass including inertia of the rotating parts. The influence of road inclination which leads to additional acceleration or deceleration is neglected, i.e. the road grade θ is considered to be zero. In order to meet drivability constraint, the demanded torque at the wheels Treq,wh has to be equal to the actual total torque at wheel. Thereby, in the considered parallel hybrid architecture, the powertrain wheel torque at the wheels Twh is provided by the following sum: Twh = ηgb Rgb (Tice + Tem )
(2)
where Tice and Tem are the torque of the internal combustion engine (ICE) and the electric machine (EM) respectively. While in previous work (Joˇsevski [2014b]) the author considered the torque of friction brakes to be control input of supervisory strategy and thus the usage of friction brakes was optimized as well, here the braking torque is exempt from the controller design. The friction brakes are applied only when regenerative braking is not possible because the charging limit in the electric branch of the hybrid powertrain has been reached. Therefore the pre-transmission torque, i.e. torque at the input stage of the transmission Tgb = Twh /(ηgb Rgb ) is divided between the motor and the engine. The gear ratio of the current gear, including differential gear, is denoted by Rgb while ηgb describes the transmission efficiency which is assumed to be constant. In our control approach we place attention on the optimization of the continuous control variables, i.e. torque request and consider that gear switching events are known. Furthermore it is assumed that the torque responses are instantaneous and thus the dynamic behavior of the ICE and EM are not part of the simplified model.
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2.2 Internal Combustion Engine
2.4 Battery
The instantaneous fuel consumption m ˙ f in g/s of the engine is evaluated from a quasi-static fuel map which nonlinearly depends on the engine torque Tice and the angular velocity ωice : m˙ f (Tice , ωice ) = fm˙ f (Tice , ωice ) (3) An analytical model of the engine fuel consumption which is used in the controller design, expresses the fuel consumption rate as a third order polynomial: 3 3 j i m˙ f (Tice , ωice ) ≈ aij · ωice · Tice (4)
In the context of energy management control strategies, the complete dynamics of the HEV system are reduced to the dynamics of the energy buffer, i.e. in this study a battery. The battery pack is modeled by a zero-th order equivalent circuit consisting of cell modules connected in series. The HEV system dynamics are given by the battery SoC time variation according to: dx ˙ = −Ibatt /Qbatt = SoC (8) dt where Ibatt represents the battery terminal current and Qbatt the nominal cell capacity. Following the definition for the battery power Pem and Kirchhoff’s second law for the equivalent circuit model: 2 (9) Pem = Uoc (x)Ibatt − Rint (x)Ibatt the battery current is expressed as a function of Pem so that the SoC dynamics can be written as: Uoc (x) − Uoc (x) − 4Rint (x)Pem x˙ = − (10) 2Rint (x)Qbatt where Uoc (x) is the open circuit voltage of the battery and Rint (x) is the equivalent internal resistance (both are functions of state of charge x). For controller design, we use a simple integrator model which is well known to be appropriate for this purpose (Cairano et al. [2013]), i.e. Pem (11) x˙ = − Uoc Qbatt where the Uoc is assumed to be constant. Both propulsion movers are subject to physical operation limits, such as minimum and maximum speed and torque. The torque limits change with respect to the corresponding angular velocities. Additionally local constraints are imposed on the battery SoC. To protect the battery from an excessive wear its SoC should be maintained within a specified range. In the following we assume that the usable range of the battery is restricted to 50 - 70%, i.e. x ∈ [0.5, 0.7].
i=0 j=0
Thereby, the rotational speed of the engine is defined by: v Rgb (5) ωice = ωwh Rgb = rwh where wwh stands for wheel angular speed of the vehicle. In the parallel configuration, when the clutch is closed and both motor and engine are mechanically linked to the wheels, the motor speed ωem is equal to the engine speed ωice . 2.3 Electric Machine Comparably to the fuel mass flow m ˙ f , the EM power is defined as a nonlinear function of the motor torque Tem and speed ωem : Pem (Tem , ωem ) = fem (Tem , ωem ). (6) However instead of using an polynomial approximation as in (Joˇsevski [2014b]), in this work the electric power is analytically modeled using a cosine approximation of the following form: Pem (Tem , ωem ) ≈ (ωem + e)(a + b cos(Tem /c + d)) (7) − π ≤ Tem /c + d ≤ 0 where the coefficients a, b, c, d, e are defined as shown in Fig. 2. For the considered electric machine, the feasible values of Tem are in a range for which the cosine argument remains in the interval [−π, 0]. A cosine approximation is used in order to uniquely express Tem as a function of the flat output (see section 3) which cannot be done with the polynomial approximation for the applied quasi-static map.
x105
Electric power [W]
2.5
1 0.5 0 -0.5 -1 600 0
200 motor speed [rad/s]
0
-300
-200
100
200
3.1 Definition of Flatness Differential flatness, originally introduced by (Fliess et al. [1995]), is a property of some nonlinear systems which allows that all system variables can be parametrized in terms of a finite number of certain, potentially non physical, variables and their time derivatives. Definition 1.1 A nonlinear dynamic system Σ : x˙ = f (x, u) (12) y = g(x, u) (13) where f and g are sufficiently smooth, is said to be differentially flat if there exist a set of auxiliary variables yf = h(x), yf ∈ Rm denoted as flat outputs such that all states x ∈ Rn and inputs u ∈ Rm of the original nonlinear system can be expressed in terms of these variables and a finite number r of their time derivatives: (r−1) ) (14) x = ψx (yf , y˙ f , . . . yf
2 1.5
400
3. FLAT DYNAMIC SYSTEMS
300
-100 motor torque [Nm]
Fig. 2. Cosine approximation of the electric power map. Coefficient values: a = 261.8, b = 410, c = 305, d = −2.263, e = 21.91. 467
(r)
u = ψu (yf , y˙ f , . . . yf ) (15) The relative degree r, 0 ≤ r ≤ n, where n is the order of the system, denotes how many times the flat output yf has
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to be differentiated until the input u appears explicitly. For SISO systems, the relative degree is defined according to (Nijmeijer and van der Schaft [1995]) and (Isidori [1995]). Definition 1.2 A nonlinear SISO system (12)-(13) has the relative degree r at point x0 if the following condition is fulfilled: ∂ i ∂ r Lf h = 0, i ∈ 0, . . . r − 1, L h = 0 (16) ∂u ∂u f for all admissible u and all x in the neighborhood of x0 . The operator Lf represents the Lie derivative along the vector field f . The components of yf are differentially independent if there is no differential equation of the form: (m)
ϕ(yf , y˙ f , . . . yf
)=0
(17)
which they satisfy. This restriction is equivalent to the condition that the number of the flat outputs is equal to the number of control inputs, i.e dim(yf ) = dim(u) = m. The parametrization which results from the flat property enables to express a complete state information analytically. Thereby flatness does not imply that the system will be converted to a linear system. Furthermore, in the optimization context flatness property can be used to boil down the original nonlinear control problem to a static optimization problem in which the system states are not obtained by the integration but are rather parametrized using flat outputs (Mahadevan et al. [2001], Oldenburg and Marquardt [2002]). 3.2 Flatness-based Optimization If the nonlinear system in the following dynamic optimization problem tf J(x0 (t), u(t)) = Φ(x, u)|t=tf + L(x, u)dt min x(t),u(t)
subject to
0
x˙ = f (x, u), t ∈ [t0 , tf ], c(x, u) ≥ 0 x(t0 ) = x0
(18) where J is the cost function, Φ the terminal cost, c the nonlinear constraints, and x0 the initial state, is flat, the optimization can be written as an optimization problem problem in flat coordinates tf (r) (r) Φ (yf , y˙ f . . . yf )|t=tf + L (yf , y˙ f . . . yf )dt min yf (t)
0
(19)
subject to
(r)
c(yf , y˙ f . . . yf ) ≥ 0, t ∈ [t0 , tf ]
(20)
with algebraic instead of differential equation constraints. Thereby the dynamic model constraints are entirely eliminated from the optimization problem. The m auxiliary variables and their r time derivatives can now be interpreted as optimization variables of the reformulated optimization problem (Oldenburg and Marquardt [2002]). In order to determine the optimal trajectory for the control input, the reformulated continuous time optimization task is approximated through parametrization of flat outputs. The flat outputs are parametrized in the following manner: 468
yf,i (t) =
Mi
Θik (t)Zik ,
467
i = 1, . . . m
(21)
k=0
where Θik (t) denote the basis functions, Zik , the corresponding coefficients while Mi + 1 is the number of basis functions chosen to represent each flat output. Flat output parametrization will render the optimization problem to the following nonlinear programming problem:
Φ (Z, t)|t=tf +
min Z
Nopt −1
L (Z, tj )
(22)
j=0
subject to c(Z, tj ) ≥ 0, t = t0 + (j − 1)Ts where Nopt is the number of time steps, Z the optimization vector containing the coefficients of the basis functions and Ts the controller sample time. 4. FLATNESS-BASED MPC CONTROLLER The aim of the energy management strategy designed in this section is to minimize the fuel consumption of the hybrid powertrain and to optimize the battery usage while keeping the system within its operation limits. Before we formulate the nonlinear MPC based on the flatness property, we briefly provide an overview of the HEV model and general MPC controller structure. Thereby only the system model is provided in continuous form, while for the MPC formulation discrete representation is given, where k will denote discrete time. 4.1 Non-Flat HEV Optimization Problem Regarding the simplified vehicle model introduced in section 2, the nonlinear prediction model of the hybrid powertrain in its state-space representation can be written as: x˙ = f (x, u, w) (23) y = g(x, u) (24) where x = SoC denotes the state, u = Tice the control T ˙ f SoC] the controlled outputs and w = input, y = [m Treq,wh the current driver torque request. Here only Tice is used as control input as Tem can directly be derived when Tice is known. The general MPC-based HEV optimization problem, i.e. without using a flat representation, is expressed as the minimization of the following cost function: J(∆u, s ) =
Hp −1
j=0
+
H u −1 j=0
e(k+j|k) 2Q + e(k+Hp |k) 2QHp
(25)
∆u(k+j|k) 2R + αs · s
subject to system dynamics (23)-(24), input constraints: (26) Tice,(k+j|k) ≤ Tice,max (ωice,(k+j|k) ) (27) Tice,(k+j|k) ≥ Tice,min (ωice,(k+j|k) ) (28) Tem,(k+j|k) ≤ Tem,max (ωem,(k+j|k) ) (29) Tem,(k+j|k) ≥ Tem,min (ωem,(k+j|k) ) with j = 0, ..., Hu − 1 and state constraints: (30) SoCmin − s ≤ SoC(k+j|k) ≤ SoCmax + s (31) s ≥ 0 where j = 1, ..., Hp . The quadratic cost (25) penalizes the deviation of controlled variables from their reference values
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e(k+j|k) = y(k+j|k) − yref , the change of control input as well as the usage of the slack variable s . The reference trajectories of the controlled outputs yref are assumed to be constant over the prediction horizon. Constraints (26)(29) are imposed to enforce that Tice and Tem remain in the admissible operating range. The ICE and EM maximum and minimum torque change in dependence of their respective angular velocities as we already noted in section 2. To guarantee the feasibility of the optimization problem constraints on SoC (30)-(31) have been implemented as soft constraints using slack variables s . The weighting matrices of the MPC are defined as: Q = diag(qm˙ f , qSoC ), QHp = diag(qm˙ f ,Hp , qSoC,Hp ) and R = r∆Tice . Additionally the second term of the performance index denotes terminal cost which is imposed at the end of the prediction horizon. This term forces the SoC towards its reference at the end of the prediction. As a result of the optimization procedure the optimal control sequence ∆u∗ = [∆u∗(k|k) , ..., ∆u∗(k+Hu −1|k) ]T is calculated but in accordance to the receding horizon paradigm only the first element is used to obtain the control input which is actually applied to the system. 4.2 Flat Output and HEV System Parametrization As already mentioned we would like to design a nonlinear model predictive controller for HEVs. Instead to linearize the system (23)-(24) at every operating point, and use the linearized model to calculate predictions of systems variables we will keep the nonlinear structure of the system and first introduce a transformation to flat coordinates which we will use later to design a nonlinear model predictive controller. The nonlinear HEV system (23)-(24) is flat with the battery state of charge x = SoC being the flat output yf : x = yf (32) Pem (33) x˙ = y˙ f = − Uoc Qbatt As the order of the system is equal to the highest considered derivative of the flat output (r = n = 1), the system does not exhibit uncontrollable internal dynamics (Graichen [2006]). To express the EM and ICE torque a cosine approximation (7) of the EM power Pem and SoC dynamics (11) are used and the following relations are obtained: Tice = Tgb − Tem (34) Tem = −c arccos(φ) − cd (35) a y˙ f Uoc Qbatt − , −1≤φ≤1 (36) φ=− (ωem + e)b b Similarly, the controlled outputs, i.e. the fuel consumption rate and the state of charge, can be expressed in terms of yf and y˙ f as: 3 3 i m˙ f = aij · ωice · (Tgb − Tem )j (37) i=0 j=0
=
3 3 i=0 j=0
i aij · ωice · (Tgb − ψu (y˙ f ))j
SoC = yf When all system variables are parametrized by the flat output the MPC performance index (25), input and state 469
constraints (26)-(30) can be defined only in dependence of the SoC, i.e the flat output. 4.3 Flat Output Parametrization using B-splines In order to formulate the nonlinear flatness-based optimal control problem, the flat output SoC and its first deriva˙ are parametrized using B-spline basis functions tive SoC Bj,d (t) of order d, i.e. yf (t) = yf (t0 ) +
M j=0
Bj,d (t) · Zj ,
(38)
where M + 1 denotes the number of control points Zj which correspond to the basis function coefficients. The B-spline curve yf (t) has the property to be C (d−2) con˙ tinuous. As we have to parameterize SoC and SoC, we have chosen d = 3. The basis functions Bj,d (t) are defined with respect to a so called knot vector Υ = (τ0 , τ1 , . . . , τN ) with a monotonically increasing knot sequence τ0 ≤ τ1 ≤ . . . ≤ τN . Let t0 , t1 , . . . , tHp denote the equally spaced discrete time steps of the prediction horizon. Then, the corresponding open uniform knot vector is defined as Υ = (t0 , t0 , t0 , t1 , ...tHp −1 , tHp , tHp , tHp ) where the first and last knot have a multiplicity of d to define an open spline curve. With N + 1 being the number of knots and d the spline degree, the number M +1 of control points can be derived using the relation M = N + 2 · (d − 1). In this contribution, we use the following recursion scheme to obtain the basis functions 1 if τj ≤ t ≤ τj+1 , j = 0, . . . , N (39) Bj,1 (t) = 0 otherwise t − τj τj+d − t Bj,d−1 (t) + Bj+1,d−1 (t) Bj,d (t) = τj+d−1 − τj τj+d − τj+1 (40) Using the spline approximation the SoC is parametrized as a linear combination of the basis functions and its propagation over the prediction horizon can be provided in matrix form as follows: SoC(k|k)
B0,d (t0 ) . . . BM,d (t0 )
Z0
SoC(k+1|k) B0,d (t1 ) . . . BM,d (t1 ) Z1 = . + SoC(t0 ) .. .. .. .. . . . SoC(k+Hp |k)
B0,d (tHp ) . . . BM,d (tHp )
ZM
(41)
Now the optimization problem (25)-(31) can be converted to the discrete optimization problem where the control points are optimization variables. The cost function as well as the imposed control and state constraints become a function of those parameters. In addition, equality constraint referring to system dynamics (23)-(24) vanishes from the controller design,i.e. through the flat parametrization the system dynamic equation becomes implicitly satisfied. The nonlinear optimization problem is solved numerically be evaluating the controlled outputs and control inputs at discrete time steps. The number of optimization variables in the flatness-based MPC algorithm is equal to M + 1 + nslack , where M + 1 is the number of control points while nslack denotes a number of slack variables introduced to ensure the feasibility of the control problem. The slack variables are introduced for SoC, Tel and the constraints on cosine argument.
2015 IFAC NMPC September 17-20, 2015. Seville, Spain
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469
Driving Profile Reference
PMPC
Driving Profile
30
FMPC
Velocity [m/s]
Velocity [m/s]
40 30 20 10
20 10 Reference
0
0
PMPC
FMPC
50
Torques PMPC [Nm]
Torques PMPC [Nm]
Control Inputs
0
-50 EM
-100
0 -50 EM
PMPC
FMPC
300 200 100 0 PMPC
upper SoC limit
FMPC
SoC [-]
0.60 0.55 lower SoC limit
0.50 0
200
600 Time t [s]
800
1000
1200
EM
Control Outputs
400
PMPC
FMPC
PMPC
FMPC
300 200 100 0 upper SoC limit
0.65 0.60 0.55 lower SoC limit
0
Fig. 3. Simulation results: NEDC Driving Cycle
ICE
-50
0.50 400
EM
0
0.70
0.65
ICE
50
-100
Fuel consumption [g]
Fuel consumption [g]
ICE
Control Outputs
400
0.70
SoC [-]
Torques FMPC [Nm]
Torques FMPC [Nm]
50
-100
0
-50
ICE
-100
50
100
200
300
400 500 Time t [s]
600
700
Fig. 4. Simulation results: FTP Highway Driving Cycle
5. SIMULATION RESULTS The flatness based energy management controller, designed in section 4, is used in simulation runs in combination with standard driving cycles and the obtained results are presented in Fig 3. and 4. The results are compared to those generated when LTV-MPC controller is applied. In the following we will refer to this controller as prescient MPC (PMPC). To proof the concept, both flatness-based MPC and prescient MPC have specified future information about the driving cycle over the prediction horizon. The values for controller weights are set to: qm˙ f = 1, qSoC = 400, qm˙ f ,Hp = 1, qSoC,Hp = 2000, r∆Tice = 0.01 and αs = 105 . Additionally, prediction and control horizon are set to Hp = Hu = 10. The controller sampling time is Ts = 1 s. The controller sampling time is selected so that the system behavior over a longer prediction horizon can be considered. However this leads to limitations regarding drivability, i.e the driver torque demand can only be delivered each second. To overcome this problem we proposed a hierarchical control scheme in (Joˇsevski [2014a]) in which the dynamics of the propulsion machines have additionally been considered to enable a continuous instead of stepwise torque supply. The reference SoC is constant over the driving cycle SoCref = 0.6, while its limits are defined as: SoCmin = 0.5 and SoCmax = 0.7. Constraints on the state of charge (30)-(31) and cosine constraints (36) are 470
softened so that the HEV energy management optimization problem stays always feasible. For prescient MPC a quadratic programming solver qpOASES (Ferreau [2012]) is used. Flatness MPC problem is solved using sequential quadratic programming (SQP) with a maximum of 100 iterations. Thereby in each iteration qpOASES is applied to solve the local QP problem. In addition to plots in Fig. 3 and Fig. 4 related to the simulations on standard driving cycles, Table 1 provides an overview on the HEV fuel economy, for both NMPC and FMPC controllers. Thereby the values related to the absolute fuel consumption over the corresponding driving cycle, battery charge difference between the end and the beginning of the driving cycle as well as equivalent fuel consumption are shown. The equivalent fuel consumption is computed by converting ∆SoC into fuel and adding it to the fuel consumption. The first value in each column refers to the nominal PMPC controller, while the second value is related to the FMPC.For the provided parameter setting, application of the FMPC controller results in the improvement of fuel economy on both examined driving cycles. Thereby the equivalent fuel consumption when flatness-based MPC is used is 5.67% lower for the NEDC and 5.08% lower for the FTP Highway compared to the corresponding values for the PMPC controller. As shown in Table 2, the average simulation time of the control structure with the flatnessbased MPC on Intel i5 2.6 GHz notebook was 21.1 ms
2015 IFAC NMPC 470 September 17-20, 2015. Seville, Spain
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Table 1. PMPC vs. FMPC control results Drive Cycle
mf (g)
∆ SoC gain/loss
Equiv.cons. (g)
NEDC FTP HW
332.3/346 330.5/338.8
+3.33/+7.33 +0.7/+3.83
307.41/289.97 325.83/309.28
Table 2. Complexity and Computational Issues
Number of opt. variables Number of constraints Max. execution time NEDC Mean. execution time NEDC Max. execution time FTP HW Mean. execution time FTP HW
MPC
FMPC
11 61
15 93
15.4 ms 4.1 ms 42.7 ms 4 ms
556.8 ms 21.1 ms 368.3 ms 13.6 ms
for simulations on NEDC and 13.6 ms for FTP Highway simulations. 6. CONCLUSION We proposed a flatness MPC to address an energy management problem of hybrid electric vehicles. The controller represents an alternative approach to direct multiple shooting to design a nonlinear model predictive controller for fuel economy optimization. On studied driving cycles FMPC resulted in improved fuel economy compared to the LTV-MPC controller. The HEV considered in this work applies the SoC as the only state while the engine torque is used as control input. Future work will include the definition of the flatness based MPC for MIMO HEV systems, by applying the velocity - apart from SoC - as additional state and by extending the input vector by the braking torque. REFERENCES D. Ambuhl and L. Guzzella. Predictive reference signal generator for hybrid electric vehicles. IEEE Transactions on Vehicular Technology, 58(9):4730–4740, 2009. Dimitri P. Bertsekas. Dynamic Programming and Optimal Control. Athena Scientific, 2nd edition, 2000. H. Borhan, A. Vahidi, A.M. Phillips, M.L. Kuang, I.V. Kolmanovsky, and S. Di Cairano. MPC-based energy management of a power-split hybrid electric vehicle. IEEE Transactions on Control Systems Technology, 20 (3):593–603, 2012. S.D. Cairano, D. Bernardini, A. Bemporad, and I.V. Kolmanovsky. Stochastic MPC with learning for driverpredictive vehicle control and its application to hev energy management. IEEE Transactions on Control Systems Technology,, PP(99):1–1, 2013. T. Devos and J. L´evine. A flatness-based nonlinear predictive approach for crane control. In IFAC Workshop on NMPC for Fast Systems, 2006. R. Dosthosseini, A.Z. Kouzani, and F. Sheikholeslam. Direct method for optimal power management in hybrid electric vehicles. International Journal of Automotive Technology, 12(6):943–950, 2011. H. J. Ferreau. qpOASES User’s Manual (Version 3.0beta). Optimization in Engineering Center (OPTEC) and Department of Electrical Engineering, KU Leuven, 2012. M. Fliess, J. L´evine, and P. Rouchon. Flatness and defect of nonlinear systems: Introductory theory and examples. International Journal of Control, 61:1327–1361, 1995. 471
K. Graichen. Feedforward control design for finite-time transition problems of nonlinear systems with input and output constraints. PhD Dissertation, 2006. L. Guzzella and A. Sciaretta. Vehicle Propulsion Systems. Springer, 2013. V. Hagenmeyer and E. Delaleau. Continuous-time nonlinear flatness-based predictive control: an exact feedforward linearisation setting with an induction drive example. International Journal of Control, 2008. A. Isidori. Nonlinear Control Systems. Springer-Verlag, New York, third edition, 1995. M. M. Seron J. L´evine J. A. De Don´a, F. Suryawan. A Flatness-Based Iterative Method for Reference Trajectory Generation in Constrained NMPC. Springer Verlag, 2009. M. Joˇsevski. Multi-time Scale Model Predictive Control Framework for Energy Management of Hybrid Electric Vehicles. In IEEE Conference on Decision and Control, 2014a. M. Joˇsevski. Energy Management of Parallel Hybrid Electric Vehicles Based on Stochastic Model Predictive Control. In IFAC World Congress, 2014b. N. Kim, S. Cha, and H. Peng. Optimal control of hybrid electric vehicles based on pontryagin’s minimum principle. IEEE Transactions on Control Systems Technology, 19(5):1279–1287, 2011. R. Mahadevan, S.K. Agrawal, and F.J. Doyle. Differential flatness based nonlinear predictive control of fed-batch bioreactors. Control Engineering Practice, 9(8):889 – 899, 2001. Advanced Control of Chemical Processes. C. Musardo, G. Rizzoni, and B. Staccia. A-ECMS: An adaptive algorithm for hybrid electric vehicle energy management. In 44th IEEE Conference on Decision and Control and European Control Conference., 2005. H. Nijmeijer and A. van der Schaft. Nonlinear Dynamical Control Systems. Springer-Verlag New York, Inc., 1995. J. Oldenburg and W. Marquardt. Flatness and higher order differential model representations in dynamic optimization. Computers & Chemical Engineering, 2002. S. Onori, L. Serrao, and G. Rizzoni. Adaptive equivalent consumption minimization strategy for hybrid electric vehicles. In Proceedings of the ASME Dynamic Systems and Control Conference, 2010. L. Serrao, S. Onori, and G. Rizzoni. ECMS as a realization of pontryagin’s minimum principle for HEV control. In American Control Conference, 2009. O. Sundstrom and L. Guzzella. A generic dynamic programming matlab function. In IEEE Control Applications, (CCA) and Intelligent Control, (ISIC), 2009. F. Suryawan, J. De Don´a, and M. Seron. Splines and polynomial tools for flatness-based constrained motion planning. International Journal of Systems Science, 2012. T. Utz, V. Hagenmeyer, and B. Mahn. Comparative evaluation of nonlinear model predictive and flatnessbased two-degree-of-freedom control design in view of industrial application. Journal of Process Control, 2007. K.B. Wipke, M.R. Cuddy, and S.D. Burch. ADVISOR 2.1: a user-friendly advanced powertrain simulation using a combined backward/forward approach. IEEE Transactions on Vehicular Technology, 48(6):1751–1761, 1999.
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Flatness-based Model Predictive Control Flatness-based Model Predictive Control Flatness-based Model Predictive Control for the Fuel Optimization of Hybrid for the Fuel Optimization of Hybrid for the Fuel Optimization of Hybrid Electric Vehicles Electric Electric Vehicles Vehicles ∗ ∗∗ Martina Joˇ ssevski, ∗ D. Abel ∗∗ Martina evski, ∗∗ Martina Joˇ Joˇ evski, ∗∗ D. D. Abel Abel ∗∗ Martina Joˇ ssevski, D. Abel ∗ ∗ Institute of Automatic Control, RWTH Aachen University, 52074 ∗ Institute of Automatic Control, RWTH Aachen University, 52074 ∗ Institute ofGermany Automatic Control,[email protected]) RWTH Aachen Aachen University, University, 52074 52074 Aachen, (( e-mail: Institute of Automatic Control, RWTH Aachen, Germany [email protected]) ∗∗ Aachen, Germany (( e-mail: e-mail: [email protected]) Institute of Automatic Control, RWTH Aachen University Aachen, Germany e-mail: [email protected]) ∗∗ of Automatic Control, RWTH Aachen University ∗∗ ∗∗ Institute Institute Institute of of Automatic Automatic Control, Control, RWTH RWTH Aachen Aachen University University
Abstract: In this paper aa nonlinear model predictive controller for the optimization of the Abstract: In this nonlinear model predictive controller for the optimization of Abstract: In vehicle this paper paper aefficiency nonlinear model predictive controller forthe theconcept optimization of the the hybrid electric fuel is introduced. The idea is to use of differential Abstract: In this paper a nonlinear model predictive controller for the optimization of the hybrid electric vehicle fuel efficiency is introduced. The idea is to use the concept of differential hybrid electric electric vehicle fuel fuel efficiency isofintroduced. introduced. The idea idea is to to use the concept concept of differential differential flatness to determine the evolution system’s state across the prediction horizon without hybrid vehicle efficiency is The is use the of flatness to determine the evolution of system’s state across the prediction horizon without flatness to determine evolution of state the prediction horizon without numerical The entire nonlinear optimization problem expressed as aa function of flatness tointegration. determine the the evolution of system’s system’s state across across the is prediction horizon without numerical integration. The entire nonlinear optimization problem is expressed as function of integration. The entire nonlinear optimization problem is expressed as a function of anumerical fictious output, called the flat output and its first derivative. The optimal distribution numerical integration. The entire nonlinear optimization problem is expressed as a function of a output, called the output its first derivative. The distribution of a fictious fictious output, called the flat flat output and and its is first derivative. Thebyoptimal optimal distribution of the requested torque between propulsion devices then determined aa static optimization. a fictious output, called the flat output and its first derivative. The optimal distribution of the requested torque between propulsion devices is then determined by static optimization. the requested requested torque between propulsion devices is then then is determined by aa static static optimization. The flatness-based model predictive controller (FMPC) compared to linear time-varying the torque between propulsion devices is determined by a optimization. The flatness-based model predictive controller (FMPC) is compared to aa linear time-varying The flatness-based model predictive compared MPC (LTV-MPC) which applies the controller linearized (FMPC) model of of is the plant to to to predict thetime-varying future state state The flatness-based model applies predictive controller (FMPC) isthe compared to a linear linear time-varying MPC (LTV-MPC) which the linearized model plant predict the future MPC (LTV-MPC) which applies the linearized model of the plant to predict the future state and output trajectories. Both strategies are evaluated on standard driving cycles. To proof the MPC (LTV-MPC) which applies the linearized model of the plant to predict the future state and trajectories. Both strategies are on driving cycles. To proof the and output output trajectories. Both strategies are evaluated evaluated on standard standard drivingare cycles. To and proof the concept it is assumed that the velocity profile and driver torque demand known exact and output trajectories. Both strategies are evaluated on standard driving cycles. To proof the concept it is assumed that the velocity profile and driver torque demand are known and exact concept it is assumed that the velocity profile and driver torque demand are known and exact over a predefined prediction horizon. concept it is assumed that the velocity profile and driver torque demand are known and exact over a predefined prediction horizon. over over a a predefined predefined prediction prediction horizon. horizon. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: hybrid electric vehicles, Keywords: hybrid hybrid electric vehicles, vehicles, energy energy management, management, nonlinear nonlinear MPC, MPC, flatness, flatness, B-splines B-splines Keywords: Keywords: hybrid electric electric vehicles, energy energy management, management, nonlinear nonlinear MPC, MPC, flatness, flatness, B-splines B-splines 1. INTRODUCTION optimal control problem: Dynamic Programming (DP), 1. INTRODUCTION INTRODUCTION optimal control Dynamic Programming 1. optimal methods control problem: problem: Dynamic Programming (DP), (DP), 1. INTRODUCTION indirect and direct methods. optimal control problem: Dynamic Programming (DP), indirect indirect methods methods and and direct direct methods. methods. indirect methods and direct methods. Prompted by increasing energy and environmental chalPrompted by by increasing increasing energy energy and and environmental environmental chalchal- Dynamic Programming is widely applied in the energy Prompted is applied in the lenges, powertrain hybridization remains a major major future future Prompted by increasing energy and environmental chal- Dynamic Dynamic Programming Programming is itwidely widely applied inoptimal the energy energy lenges, powertrain hybridization remains a management of HEVs as calculates the fuel Dynamic Programming is widely applied in the energy lenges, powertrain hybridization remains aa major future management of HEVs as it calculates the optimal fuel development trend in the automotive industry. Compared lenges, powertrain hybridization remains major future management by of going HEVsrecursively as it it calculates calculates the in optimal fuel development trend in the automotive industry. Compared consumption backwards time using management of HEVs as the optimal fuel development trend in the automotive industry. Compared consumption by going recursively backwards in time using to the conventional vehicles, hybrid electric vehicles (HEV) development trend vehicles, in the automotive industry. Compared consumption by going recursively backwards in time using to the conventional hybrid electric vehicles (HEV) Bellman’s principle of optimality (Bertsekas [2000]). It consumption by going recursively backwards in time using to the conventional vehicles, hybrid electric vehicles (HEV) principle of optimality (Bertsekas [2000]). It are augmented by an alternative mover, usually an electric to the conventional vehicles, hybrid electric vehicles (HEV) Bellman’s Bellman’s principle of optimality (Bertsekas [2000]). It are augmented by an alternative mover, usually an electric guarantees the global optimality in case when the driving Bellman’s principle of optimality (Bertsekas [2000]). It are augmented by an alternative mover, usually an electric guarantees the global optimality in case when the driving motor and/or generator. The additional degree of freedom, are augmented by an alternative mover, usually an electric guarantees the global global optimality inascase case when the the method driving motor and/or generator. The additional degree of freedom, cycle is perfectly known. However a non-causal guarantees the optimality in when driving motor and/or generator. The additional degree of freedom, cycle is perfectly known. However as a non-causal method introduced by the electric branch, gives the possibility to motor and/or The additional of freedom, cycle is perfectly known. However as aa non-causal non-causal method introduced bygenerator. the electric electric branch, givesdegree the possibility possibility to cycle it is not applicable in real applications and is mainly is perfectly known. However as method introduced by the branch, gives the to is not applicable in real applications and is mainly increase the fuel economy of HEVs, e.g. by operating the introduced by the electric branch, gives the possibility to it it is not applicable in real applications and is mainly increase the fuel economy of HEVs, e.g. by operating the applied as a benchmark solution to assess the performance it is not applicable in real applications and is mainly increase the fuel economy of HEVs, e.g. by operating the applied as a benchmark solution to assess the performance internal combustion engine in the most efficient region or increase the fuel economy of HEVs, e.g. by operating the applied as a benchmark solution to assess the performance internal combustion engine in the most efficient region or or tune the parameters of other HEV control strategies applied as a benchmark solution to assess the performance internal combustion engine in the most efficient region or parameters of other control strategies by recuperating the kinetic kinetic energy whichefficient would otherwise otherwise internal combustion engine energy in the most region or or or tune tune the the and parameters of [2009], other HEV HEV control strategies by recuperating recuperating the which would (Sundstrom Guzzella Guzzella and Sciaretta or tune the parameters of other HEV control strategies by the kinetic energy which would otherwise (Sundstrom and Guzzella [2009], Guzzella and Sciaretta be wasted in heat by using the friction brakes. by recuperating the kinetic energy which would otherwise (Sundstrom and Guzzella [2009], Guzzella and Sciaretta be wasted in heat by using the friction brakes. [2013]). (Sundstrom and Guzzella [2009], Guzzella and Sciaretta be [2013]). be wasted wasted in in heat heat by by using using the the friction friction brakes. brakes. [2013]). [2013]). Essentially, there are two possibilities to improve the effiEssentially, there there are are two two possibilities possibilities to to improve improve the the effieffi- In addition to DP which finds global optimal fuel consumpEssentially, In addition to DP which finds global optimal fuel consumpciency of the hybrid component sizing and the Essentially, there arepowertrain: two possibilities to improve the effiaddition to which finds global optimal fuel ciency of the hybrid powertrain: component sizing and the In tion over an priori given driving indirect methods In addition toaa DP DP which finds globalcycle, optimal fuel consumpconsumpciency of the hybrid powertrain: component sizing and the tion over an priori given driving cycle, indirect methods application of energy management strategies. Component ciency of the hybrid powertrain: component sizing and the tion over an aa priori given driving cycle, indirect methods application of energy management strategies. Component convert the optimal control problem into a boundary value tion over an priori given driving cycle, indirect methods application of energy management strategies. Component convert the optimal control problem into a boundary value sizing is employed to optimally determine dimensions of application of energy management strategies. Component convert the the optimal control problem into aa boundary boundary value sizing is employed to optimally determine dimensions of problem which is then solved numerically by solving the convert optimal control problem into value sizing is employed to optimally determine dimensions of problem which is then solved numerically by solving the the HEV powertrain components so that fuel consumption, sizing is employed to optimally determine dimensions of problemofwhich which is then then solved numerically numerically bystart solving the the HEV powertrain components so that fuel consumption, system differential equations which satisfy and end problem is solved by solving the the HEV powertrain components so that fuel consumption, system of differential equations which satisfy start and end vehicle weight and cost are minimized. Under the asthe HEV powertrain components so that fuel consumption, system of differential equations which satisfy start and end vehicle weight and cost are minimized. Under the aspoint conditions. In (Serrao et al. [2009]), as an indirect system of differential equations which satisfy start and end vehicle weight and cost are Under the asconditions. In (Serrao et al. [2009]), as an indirect sumption that the vehicle has already vehicle weight andhybrid cost electric are minimized. minimized. Under thebeen as- point point conditions. In et [2009]), as sumption that the the hybrid electric vehicle has has already been method, Pontryagin Minimum Principle is used point conditions. In (Serrao (Serrao et al. al. [2009]),(PMP) as an an indirect indirect sumption that hybrid electric vehicle already been method, Pontryagin Minimum Principle (PMP) used optimally designed, further improvements with regards to sumption that the hybrid electric vehicle has already been method, Pontryagin Minimum Principle function (PMP) is isand used optimally designed, further improvements with regards to to minimize at each instant Hamiltonian to method, Pontryagin Minimum Principle (PMP) is used optimally designed, further improvements with regards to to minimize at each instant Hamiltonian function and the fuel economy can be achieved by energy management optimally designed, further improvements with regards to to minimize minimize atoptimal each instant instant Hamiltonian function and to to the fuel economy can be achieved by energy management find a global trajectory for the control inputs. to at each Hamiltonian function and to the fuel can achieved by management find aa global optimal trajectory for the control inputs. strategies. The objective the energy management stratthe fuel economy economy can be be of achieved by energy energy management find global optimal trajectory for the control inputs. strategies. The objective of the energy management stratIn literature, Pontryagin Minimum Principle (Kim et al. find a global optimal trajectory for the control inputs. strategies. The ofconsumption the stratIn literature, Pontryagin Minimum Principle (Kim et al. egy is to minimize the fuel by determining the strategies. The objective objective the energy energy management management stratliterature, Pontryagin Minimum Principle (Kim egy is is to to minimize minimize the fuel fuelofconsumption consumption by determining determining the In [2011]) and the Equivalent Consumption In literature, Pontryagin Minimum PrincipleMinimization (Kim et et al. al. egy the by the [2011]) and the Equivalent Consumption Minimization energy flow split between the engine and the alternative egy is to minimize the fuel consumption by determining the [2011]) and and the isEquivalent Equivalent Consumption Minimization energy flow split between the engine and the alternative Strategy, which derived from the PMP, are frequently [2011]) the Consumption Minimization energy flow split between the engine and the alternative is derived the are frequently power source while meeting powertrain constraints. energy flow split between engine and the alternative Strategy, Strategy, which which isstrategies derived from from the PMP, PMP, are frequently power source source while meetingthe powertrain constraints. presented control (Musardo et al. [2005], Serrao Strategy, which is derived from the PMP, are frequently power presented control strategies (Musardo et al. [2005], Serrao power source while while meeting meeting powertrain powertrain constraints. constraints. presented control strategies (Musardo et al. [2005], Serrao et al. [2009], Onori et al. [2010], Ambuhl and Guzzella presented control strategies (Musardo et al. [2005], Serrao From the aspect of optimal control theory, the energy et al. [2009], Onori et al. [2010], Ambuhl and Guzzella From the aspect of optimal control theory, the energy et al. [2009], Onori et al. [2010], Ambuhl and Guzzella From the aspect of optimal control theory, the energy [2009]). et al. [2009], Onori et al. [2010], Ambuhl and Guzzella management problem of hybrid electric vehicles is formuFrom the aspect of optimal control theory, the energy management problem problem of of hybrid hybrid electric vehicles vehicles is is formuformu- [2009]). [2009]). management lated as aa finite-time optimalvehicles control is problem. management problemconstrained of hybrid electric electric formu- [2009]). lated as finite-time constrained optimal control problem. Direct methods differ from the indirect approaches in the lated as aa finite-time constrained optimal control problem. Direct methods differ from the indirect approaches in the Generally there are three main approaches to address the lated as finite-time constrained optimal control problem. Direct methods differ from the indirect approaches in the Generally there are three main approaches to address the way that control (and state) trajectories of the original Direct methods differ from the indirect approaches in the Generally that control (and state) trajectories of the original Generally there there are are three three main main approaches approaches to to address address the the way way that that control control (andare state) trajectories ofthe theoptimizaoriginal optimization problem parametrized and way (and state) trajectories of the original This work was financially supported by German Scientific Founoptimization problem are parametrized and the optimiza This work was financially supported by German Scientific Founoptimization problem are to parametrized and the the optimizaoptimiza tion problem converted aa finite-dimensional nonlinear optimization problem are parametrized and This work financially dation This (Deutsche work was was Forschungsgemeinschaft). financially supported supported by by German German Scientific Scientific FounFountion problem is is converted to finite-dimensional nonlinear dation (Deutsche Forschungsgemeinschaft). tion tion problem problem is is converted converted to to aa finite-dimensional finite-dimensional nonlinear nonlinear dation (Deutsche (Deutsche Forschungsgemeinschaft). Forschungsgemeinschaft). dation Copyright 2015 IFAC 465 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 465 Copyright 2015 IFAC 465 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 465Control. 10.1016/j.ifacol.2015.11.322
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465
programming problem (NLP). Direct single shooting, direct multiple shooting and direct collocation reside to this group of numerical optimization methods. In (Dosthosseini et al. [2011]) the application of a direct method for HEV energy management is presented. 1.1 Main Contribution In this contribution we use the idea which is similar to that of direct approaches to design an energy management controller for HEVs. Using a specified functional form, the states, control inputs and outputs of the systems are approximated and the optimization problem is converted to a NLP. The optimization problem is stated as nonlinear model predictive control scheme which optimizes the fuel consumption and the electric storage usage over a certain portion of the predefined driving cycle. The objective function is minimized under a set of state and control constraints which result from the time-varying operational limits of the powertrain components. However in our approach the property of differential flatness (Hagenmeyer and Delaleau [2008], Devos and L´evine [2006], Utz et al. [2007], J. A. De Don´ a [2009], Suryawan et al. [2012]) is used to parametrize the system and to transform the optimization task into a NLP. Applying the flat property, the system dynamics are implicitly satisfied and the differential equations are not integrated to obtain state predictions over a prediction horizon. Elimination of the system’s differential equations leads to a static optimization problem. In addition the constrained optimal control problem for the original nonlinear system instead for the linearized system, as in LTVMPC (Borhan et al. [2012], Joˇsevski [2014b]), is solved. The nonlinear model predictive controller is designed by casting the original formulation into a static nonlinear programming problem. This approach can be seen as an alternative to direct multiple shooting method. The paper is organized as follows. The assumed hybrid powertrain is presented in section 2. In section 3, we briefly introduce preliminaries regarding flatness and flatnessbased optimization. The non-flat formulation of the model predictive controller, nonlinear system transformation, flat output parametrization as well as the resulting flatness based MPC control problem are described in section 4. Finally simulation results are presented in section 5, while section 6 concludes the paper. 2. VEHICLE MODEL In this case study a parallel hybrid electric vehicle configuration is discussed in which the internal combustion engine (ICE) and electric machine (EM) are mounted on the same shaft and separated by the clutch (Fig. 1). This configuration allows combined as well as pure electric and pure thermal driving. When the clutch is closed both devices propel the vehicle and their speed is directly proportional to the vehicles velocity. To model the vehicle, a quasi-static forward approach is used. Thereby, vehicle parameters as well as quasi-static maps for propulsion components and energy storage are obtained from the automotive simulation software ADVISOR (Wipke et al. [1999]). 466
Fig. 1. Parallel hybrid electric vehicle structure 2.1 Powertrain Model It is presumed that the desired velocities and thus the torque demands at the wheels are known in advance, i.e. they act as an exogenous disturbance input to the system. The torque Treq,wh which is requested to follow the given velocity profile v(t) when only longitudinal vehicle dynamics are considered is defined by: 1 Treq,wh = rwh ( ρCd Av 2 + me v˙ + mg(fr cosθ + sinθ))(1) 2 where v represents the vehicles velocity, θ is the road inclination, ρ is the air density, Cd is the drag coefficient, A is the vehicle’s cross section, fr is the rolling resistance coefficient, m is the vehicle mass and me is the effective vehicle mass including inertia of the rotating parts. The influence of road inclination which leads to additional acceleration or deceleration is neglected, i.e. the road grade θ is considered to be zero. In order to meet drivability constraint, the demanded torque at the wheels Treq,wh has to be equal to the actual total torque at wheel. Thereby, in the considered parallel hybrid architecture, the powertrain wheel torque at the wheels Twh is provided by the following sum: Twh = ηgb Rgb (Tice + Tem )
(2)
where Tice and Tem are the torque of the internal combustion engine (ICE) and the electric machine (EM) respectively. While in previous work (Joˇsevski [2014b]) the author considered the torque of friction brakes to be control input of supervisory strategy and thus the usage of friction brakes was optimized as well, here the braking torque is exempt from the controller design. The friction brakes are applied only when regenerative braking is not possible because the charging limit in the electric branch of the hybrid powertrain has been reached. Therefore the pre-transmission torque, i.e. torque at the input stage of the transmission Tgb = Twh /(ηgb Rgb ) is divided between the motor and the engine. The gear ratio of the current gear, including differential gear, is denoted by Rgb while ηgb describes the transmission efficiency which is assumed to be constant. In our control approach we place attention on the optimization of the continuous control variables, i.e. torque request and consider that gear switching events are known. Furthermore it is assumed that the torque responses are instantaneous and thus the dynamic behavior of the ICE and EM are not part of the simplified model.
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2.2 Internal Combustion Engine
2.4 Battery
The instantaneous fuel consumption m ˙ f in g/s of the engine is evaluated from a quasi-static fuel map which nonlinearly depends on the engine torque Tice and the angular velocity ωice : m˙ f (Tice , ωice ) = fm˙ f (Tice , ωice ) (3) An analytical model of the engine fuel consumption which is used in the controller design, expresses the fuel consumption rate as a third order polynomial: 3 3 j i m˙ f (Tice , ωice ) ≈ aij · ωice · Tice (4)
In the context of energy management control strategies, the complete dynamics of the HEV system are reduced to the dynamics of the energy buffer, i.e. in this study a battery. The battery pack is modeled by a zero-th order equivalent circuit consisting of cell modules connected in series. The HEV system dynamics are given by the battery SoC time variation according to: dx ˙ = −Ibatt /Qbatt = SoC (8) dt where Ibatt represents the battery terminal current and Qbatt the nominal cell capacity. Following the definition for the battery power Pem and Kirchhoff’s second law for the equivalent circuit model: 2 (9) Pem = Uoc (x)Ibatt − Rint (x)Ibatt the battery current is expressed as a function of Pem so that the SoC dynamics can be written as: Uoc (x) − Uoc (x) − 4Rint (x)Pem x˙ = − (10) 2Rint (x)Qbatt where Uoc (x) is the open circuit voltage of the battery and Rint (x) is the equivalent internal resistance (both are functions of state of charge x). For controller design, we use a simple integrator model which is well known to be appropriate for this purpose (Cairano et al. [2013]), i.e. Pem (11) x˙ = − Uoc Qbatt where the Uoc is assumed to be constant. Both propulsion movers are subject to physical operation limits, such as minimum and maximum speed and torque. The torque limits change with respect to the corresponding angular velocities. Additionally local constraints are imposed on the battery SoC. To protect the battery from an excessive wear its SoC should be maintained within a specified range. In the following we assume that the usable range of the battery is restricted to 50 - 70%, i.e. x ∈ [0.5, 0.7].
i=0 j=0
Thereby, the rotational speed of the engine is defined by: v Rgb (5) ωice = ωwh Rgb = rwh where wwh stands for wheel angular speed of the vehicle. In the parallel configuration, when the clutch is closed and both motor and engine are mechanically linked to the wheels, the motor speed ωem is equal to the engine speed ωice . 2.3 Electric Machine Comparably to the fuel mass flow m ˙ f , the EM power is defined as a nonlinear function of the motor torque Tem and speed ωem : Pem (Tem , ωem ) = fem (Tem , ωem ). (6) However instead of using an polynomial approximation as in (Joˇsevski [2014b]), in this work the electric power is analytically modeled using a cosine approximation of the following form: Pem (Tem , ωem ) ≈ (ωem + e)(a + b cos(Tem /c + d)) (7) − π ≤ Tem /c + d ≤ 0 where the coefficients a, b, c, d, e are defined as shown in Fig. 2. For the considered electric machine, the feasible values of Tem are in a range for which the cosine argument remains in the interval [−π, 0]. A cosine approximation is used in order to uniquely express Tem as a function of the flat output (see section 3) which cannot be done with the polynomial approximation for the applied quasi-static map.
x105
Electric power [W]
2.5
1 0.5 0 -0.5 -1 600 0
200 motor speed [rad/s]
0
-300
-200
100
200
3.1 Definition of Flatness Differential flatness, originally introduced by (Fliess et al. [1995]), is a property of some nonlinear systems which allows that all system variables can be parametrized in terms of a finite number of certain, potentially non physical, variables and their time derivatives. Definition 1.1 A nonlinear dynamic system Σ : x˙ = f (x, u) (12) y = g(x, u) (13) where f and g are sufficiently smooth, is said to be differentially flat if there exist a set of auxiliary variables yf = h(x), yf ∈ Rm denoted as flat outputs such that all states x ∈ Rn and inputs u ∈ Rm of the original nonlinear system can be expressed in terms of these variables and a finite number r of their time derivatives: (r−1) ) (14) x = ψx (yf , y˙ f , . . . yf
2 1.5
400
3. FLAT DYNAMIC SYSTEMS
300
-100 motor torque [Nm]
Fig. 2. Cosine approximation of the electric power map. Coefficient values: a = 261.8, b = 410, c = 305, d = −2.263, e = 21.91. 467
(r)
u = ψu (yf , y˙ f , . . . yf ) (15) The relative degree r, 0 ≤ r ≤ n, where n is the order of the system, denotes how many times the flat output yf has
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to be differentiated until the input u appears explicitly. For SISO systems, the relative degree is defined according to (Nijmeijer and van der Schaft [1995]) and (Isidori [1995]). Definition 1.2 A nonlinear SISO system (12)-(13) has the relative degree r at point x0 if the following condition is fulfilled: ∂ i ∂ r Lf h = 0, i ∈ 0, . . . r − 1, L h = 0 (16) ∂u ∂u f for all admissible u and all x in the neighborhood of x0 . The operator Lf represents the Lie derivative along the vector field f . The components of yf are differentially independent if there is no differential equation of the form: (m)
ϕ(yf , y˙ f , . . . yf
)=0
(17)
which they satisfy. This restriction is equivalent to the condition that the number of the flat outputs is equal to the number of control inputs, i.e dim(yf ) = dim(u) = m. The parametrization which results from the flat property enables to express a complete state information analytically. Thereby flatness does not imply that the system will be converted to a linear system. Furthermore, in the optimization context flatness property can be used to boil down the original nonlinear control problem to a static optimization problem in which the system states are not obtained by the integration but are rather parametrized using flat outputs (Mahadevan et al. [2001], Oldenburg and Marquardt [2002]). 3.2 Flatness-based Optimization If the nonlinear system in the following dynamic optimization problem tf J(x0 (t), u(t)) = Φ(x, u)|t=tf + L(x, u)dt min x(t),u(t)
subject to
0
x˙ = f (x, u), t ∈ [t0 , tf ], c(x, u) ≥ 0 x(t0 ) = x0
(18) where J is the cost function, Φ the terminal cost, c the nonlinear constraints, and x0 the initial state, is flat, the optimization can be written as an optimization problem problem in flat coordinates tf (r) (r) Φ (yf , y˙ f . . . yf )|t=tf + L (yf , y˙ f . . . yf )dt min yf (t)
0
(19)
subject to
(r)
c(yf , y˙ f . . . yf ) ≥ 0, t ∈ [t0 , tf ]
(20)
with algebraic instead of differential equation constraints. Thereby the dynamic model constraints are entirely eliminated from the optimization problem. The m auxiliary variables and their r time derivatives can now be interpreted as optimization variables of the reformulated optimization problem (Oldenburg and Marquardt [2002]). In order to determine the optimal trajectory for the control input, the reformulated continuous time optimization task is approximated through parametrization of flat outputs. The flat outputs are parametrized in the following manner: 468
yf,i (t) =
Mi
Θik (t)Zik ,
467
i = 1, . . . m
(21)
k=0
where Θik (t) denote the basis functions, Zik , the corresponding coefficients while Mi + 1 is the number of basis functions chosen to represent each flat output. Flat output parametrization will render the optimization problem to the following nonlinear programming problem:
Φ (Z, t)|t=tf +
min Z
Nopt −1
L (Z, tj )
(22)
j=0
subject to c(Z, tj ) ≥ 0, t = t0 + (j − 1)Ts where Nopt is the number of time steps, Z the optimization vector containing the coefficients of the basis functions and Ts the controller sample time. 4. FLATNESS-BASED MPC CONTROLLER The aim of the energy management strategy designed in this section is to minimize the fuel consumption of the hybrid powertrain and to optimize the battery usage while keeping the system within its operation limits. Before we formulate the nonlinear MPC based on the flatness property, we briefly provide an overview of the HEV model and general MPC controller structure. Thereby only the system model is provided in continuous form, while for the MPC formulation discrete representation is given, where k will denote discrete time. 4.1 Non-Flat HEV Optimization Problem Regarding the simplified vehicle model introduced in section 2, the nonlinear prediction model of the hybrid powertrain in its state-space representation can be written as: x˙ = f (x, u, w) (23) y = g(x, u) (24) where x = SoC denotes the state, u = Tice the control T ˙ f SoC] the controlled outputs and w = input, y = [m Treq,wh the current driver torque request. Here only Tice is used as control input as Tem can directly be derived when Tice is known. The general MPC-based HEV optimization problem, i.e. without using a flat representation, is expressed as the minimization of the following cost function: J(∆u, s ) =
Hp −1
j=0
+
H u −1 j=0
e(k+j|k) 2Q + e(k+Hp |k) 2QHp
(25)
∆u(k+j|k) 2R + αs · s
subject to system dynamics (23)-(24), input constraints: (26) Tice,(k+j|k) ≤ Tice,max (ωice,(k+j|k) ) (27) Tice,(k+j|k) ≥ Tice,min (ωice,(k+j|k) ) (28) Tem,(k+j|k) ≤ Tem,max (ωem,(k+j|k) ) (29) Tem,(k+j|k) ≥ Tem,min (ωem,(k+j|k) ) with j = 0, ..., Hu − 1 and state constraints: (30) SoCmin − s ≤ SoC(k+j|k) ≤ SoCmax + s (31) s ≥ 0 where j = 1, ..., Hp . The quadratic cost (25) penalizes the deviation of controlled variables from their reference values
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e(k+j|k) = y(k+j|k) − yref , the change of control input as well as the usage of the slack variable s . The reference trajectories of the controlled outputs yref are assumed to be constant over the prediction horizon. Constraints (26)(29) are imposed to enforce that Tice and Tem remain in the admissible operating range. The ICE and EM maximum and minimum torque change in dependence of their respective angular velocities as we already noted in section 2. To guarantee the feasibility of the optimization problem constraints on SoC (30)-(31) have been implemented as soft constraints using slack variables s . The weighting matrices of the MPC are defined as: Q = diag(qm˙ f , qSoC ), QHp = diag(qm˙ f ,Hp , qSoC,Hp ) and R = r∆Tice . Additionally the second term of the performance index denotes terminal cost which is imposed at the end of the prediction horizon. This term forces the SoC towards its reference at the end of the prediction. As a result of the optimization procedure the optimal control sequence ∆u∗ = [∆u∗(k|k) , ..., ∆u∗(k+Hu −1|k) ]T is calculated but in accordance to the receding horizon paradigm only the first element is used to obtain the control input which is actually applied to the system. 4.2 Flat Output and HEV System Parametrization As already mentioned we would like to design a nonlinear model predictive controller for HEVs. Instead to linearize the system (23)-(24) at every operating point, and use the linearized model to calculate predictions of systems variables we will keep the nonlinear structure of the system and first introduce a transformation to flat coordinates which we will use later to design a nonlinear model predictive controller. The nonlinear HEV system (23)-(24) is flat with the battery state of charge x = SoC being the flat output yf : x = yf (32) Pem (33) x˙ = y˙ f = − Uoc Qbatt As the order of the system is equal to the highest considered derivative of the flat output (r = n = 1), the system does not exhibit uncontrollable internal dynamics (Graichen [2006]). To express the EM and ICE torque a cosine approximation (7) of the EM power Pem and SoC dynamics (11) are used and the following relations are obtained: Tice = Tgb − Tem (34) Tem = −c arccos(φ) − cd (35) a y˙ f Uoc Qbatt − , −1≤φ≤1 (36) φ=− (ωem + e)b b Similarly, the controlled outputs, i.e. the fuel consumption rate and the state of charge, can be expressed in terms of yf and y˙ f as: 3 3 i m˙ f = aij · ωice · (Tgb − Tem )j (37) i=0 j=0
=
3 3 i=0 j=0
i aij · ωice · (Tgb − ψu (y˙ f ))j
SoC = yf When all system variables are parametrized by the flat output the MPC performance index (25), input and state 469
constraints (26)-(30) can be defined only in dependence of the SoC, i.e the flat output. 4.3 Flat Output Parametrization using B-splines In order to formulate the nonlinear flatness-based optimal control problem, the flat output SoC and its first deriva˙ are parametrized using B-spline basis functions tive SoC Bj,d (t) of order d, i.e. yf (t) = yf (t0 ) +
M j=0
Bj,d (t) · Zj ,
(38)
where M + 1 denotes the number of control points Zj which correspond to the basis function coefficients. The B-spline curve yf (t) has the property to be C (d−2) con˙ tinuous. As we have to parameterize SoC and SoC, we have chosen d = 3. The basis functions Bj,d (t) are defined with respect to a so called knot vector Υ = (τ0 , τ1 , . . . , τN ) with a monotonically increasing knot sequence τ0 ≤ τ1 ≤ . . . ≤ τN . Let t0 , t1 , . . . , tHp denote the equally spaced discrete time steps of the prediction horizon. Then, the corresponding open uniform knot vector is defined as Υ = (t0 , t0 , t0 , t1 , ...tHp −1 , tHp , tHp , tHp ) where the first and last knot have a multiplicity of d to define an open spline curve. With N + 1 being the number of knots and d the spline degree, the number M +1 of control points can be derived using the relation M = N + 2 · (d − 1). In this contribution, we use the following recursion scheme to obtain the basis functions 1 if τj ≤ t ≤ τj+1 , j = 0, . . . , N (39) Bj,1 (t) = 0 otherwise t − τj τj+d − t Bj,d−1 (t) + Bj+1,d−1 (t) Bj,d (t) = τj+d−1 − τj τj+d − τj+1 (40) Using the spline approximation the SoC is parametrized as a linear combination of the basis functions and its propagation over the prediction horizon can be provided in matrix form as follows: SoC(k|k)
B0,d (t0 ) . . . BM,d (t0 )
Z0
SoC(k+1|k) B0,d (t1 ) . . . BM,d (t1 ) Z1 = . + SoC(t0 ) .. .. .. .. . . . SoC(k+Hp |k)
B0,d (tHp ) . . . BM,d (tHp )
ZM
(41)
Now the optimization problem (25)-(31) can be converted to the discrete optimization problem where the control points are optimization variables. The cost function as well as the imposed control and state constraints become a function of those parameters. In addition, equality constraint referring to system dynamics (23)-(24) vanishes from the controller design,i.e. through the flat parametrization the system dynamic equation becomes implicitly satisfied. The nonlinear optimization problem is solved numerically be evaluating the controlled outputs and control inputs at discrete time steps. The number of optimization variables in the flatness-based MPC algorithm is equal to M + 1 + nslack , where M + 1 is the number of control points while nslack denotes a number of slack variables introduced to ensure the feasibility of the control problem. The slack variables are introduced for SoC, Tel and the constraints on cosine argument.
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469
Driving Profile Reference
PMPC
Driving Profile
30
FMPC
Velocity [m/s]
Velocity [m/s]
40 30 20 10
20 10 Reference
0
0
PMPC
FMPC
50
Torques PMPC [Nm]
Torques PMPC [Nm]
Control Inputs
0
-50 EM
-100
0 -50 EM
PMPC
FMPC
300 200 100 0 PMPC
upper SoC limit
FMPC
SoC [-]
0.60 0.55 lower SoC limit
0.50 0
200
600 Time t [s]
800
1000
1200
EM
Control Outputs
400
PMPC
FMPC
PMPC
FMPC
300 200 100 0 upper SoC limit
0.65 0.60 0.55 lower SoC limit
0
Fig. 3. Simulation results: NEDC Driving Cycle
ICE
-50
0.50 400
EM
0
0.70
0.65
ICE
50
-100
Fuel consumption [g]
Fuel consumption [g]
ICE
Control Outputs
400
0.70
SoC [-]
Torques FMPC [Nm]
Torques FMPC [Nm]
50
-100
0
-50
ICE
-100
50
100
200
300
400 500 Time t [s]
600
700
Fig. 4. Simulation results: FTP Highway Driving Cycle
5. SIMULATION RESULTS The flatness based energy management controller, designed in section 4, is used in simulation runs in combination with standard driving cycles and the obtained results are presented in Fig 3. and 4. The results are compared to those generated when LTV-MPC controller is applied. In the following we will refer to this controller as prescient MPC (PMPC). To proof the concept, both flatness-based MPC and prescient MPC have specified future information about the driving cycle over the prediction horizon. The values for controller weights are set to: qm˙ f = 1, qSoC = 400, qm˙ f ,Hp = 1, qSoC,Hp = 2000, r∆Tice = 0.01 and αs = 105 . Additionally, prediction and control horizon are set to Hp = Hu = 10. The controller sampling time is Ts = 1 s. The controller sampling time is selected so that the system behavior over a longer prediction horizon can be considered. However this leads to limitations regarding drivability, i.e the driver torque demand can only be delivered each second. To overcome this problem we proposed a hierarchical control scheme in (Joˇsevski [2014a]) in which the dynamics of the propulsion machines have additionally been considered to enable a continuous instead of stepwise torque supply. The reference SoC is constant over the driving cycle SoCref = 0.6, while its limits are defined as: SoCmin = 0.5 and SoCmax = 0.7. Constraints on the state of charge (30)-(31) and cosine constraints (36) are 470
softened so that the HEV energy management optimization problem stays always feasible. For prescient MPC a quadratic programming solver qpOASES (Ferreau [2012]) is used. Flatness MPC problem is solved using sequential quadratic programming (SQP) with a maximum of 100 iterations. Thereby in each iteration qpOASES is applied to solve the local QP problem. In addition to plots in Fig. 3 and Fig. 4 related to the simulations on standard driving cycles, Table 1 provides an overview on the HEV fuel economy, for both NMPC and FMPC controllers. Thereby the values related to the absolute fuel consumption over the corresponding driving cycle, battery charge difference between the end and the beginning of the driving cycle as well as equivalent fuel consumption are shown. The equivalent fuel consumption is computed by converting ∆SoC into fuel and adding it to the fuel consumption. The first value in each column refers to the nominal PMPC controller, while the second value is related to the FMPC.For the provided parameter setting, application of the FMPC controller results in the improvement of fuel economy on both examined driving cycles. Thereby the equivalent fuel consumption when flatness-based MPC is used is 5.67% lower for the NEDC and 5.08% lower for the FTP Highway compared to the corresponding values for the PMPC controller. As shown in Table 2, the average simulation time of the control structure with the flatnessbased MPC on Intel i5 2.6 GHz notebook was 21.1 ms
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Table 1. PMPC vs. FMPC control results Drive Cycle
mf (g)
∆ SoC gain/loss
Equiv.cons. (g)
NEDC FTP HW
332.3/346 330.5/338.8
+3.33/+7.33 +0.7/+3.83
307.41/289.97 325.83/309.28
Table 2. Complexity and Computational Issues
Number of opt. variables Number of constraints Max. execution time NEDC Mean. execution time NEDC Max. execution time FTP HW Mean. execution time FTP HW
MPC
FMPC
11 61
15 93
15.4 ms 4.1 ms 42.7 ms 4 ms
556.8 ms 21.1 ms 368.3 ms 13.6 ms
for simulations on NEDC and 13.6 ms for FTP Highway simulations. 6. CONCLUSION We proposed a flatness MPC to address an energy management problem of hybrid electric vehicles. The controller represents an alternative approach to direct multiple shooting to design a nonlinear model predictive controller for fuel economy optimization. On studied driving cycles FMPC resulted in improved fuel economy compared to the LTV-MPC controller. The HEV considered in this work applies the SoC as the only state while the engine torque is used as control input. Future work will include the definition of the flatness based MPC for MIMO HEV systems, by applying the velocity - apart from SoC - as additional state and by extending the input vector by the braking torque. REFERENCES D. Ambuhl and L. Guzzella. Predictive reference signal generator for hybrid electric vehicles. IEEE Transactions on Vehicular Technology, 58(9):4730–4740, 2009. Dimitri P. Bertsekas. Dynamic Programming and Optimal Control. Athena Scientific, 2nd edition, 2000. H. Borhan, A. Vahidi, A.M. Phillips, M.L. Kuang, I.V. Kolmanovsky, and S. Di Cairano. MPC-based energy management of a power-split hybrid electric vehicle. IEEE Transactions on Control Systems Technology, 20 (3):593–603, 2012. S.D. Cairano, D. Bernardini, A. Bemporad, and I.V. Kolmanovsky. Stochastic MPC with learning for driverpredictive vehicle control and its application to hev energy management. IEEE Transactions on Control Systems Technology,, PP(99):1–1, 2013. T. Devos and J. L´evine. A flatness-based nonlinear predictive approach for crane control. In IFAC Workshop on NMPC for Fast Systems, 2006. R. Dosthosseini, A.Z. Kouzani, and F. Sheikholeslam. Direct method for optimal power management in hybrid electric vehicles. International Journal of Automotive Technology, 12(6):943–950, 2011. H. J. Ferreau. qpOASES User’s Manual (Version 3.0beta). Optimization in Engineering Center (OPTEC) and Department of Electrical Engineering, KU Leuven, 2012. M. Fliess, J. L´evine, and P. Rouchon. Flatness and defect of nonlinear systems: Introductory theory and examples. International Journal of Control, 61:1327–1361, 1995. 471
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