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Receding Horizon Control for Distributed Receding Horizon Control Distributed Receding Horizon Controloffor for Distributed Energy Management a Hybrid Energy Management of a Hybrid Energy Management a Hybrid ⋆ Heavy-Duty Vehicle withofAuxiliaries ⋆ Heavy-Duty Vehicle with Auxiliaries Heavy-Duty Vehicle with Auxiliaries ⋆ M.C.F. Donkers ∗∗ J.T.B.A Kessels ∗∗ ∗∗ ∗∗ ∗ ∗ M.C.F. Donkers Kessels ∗ J.T.B.A ∗∗ M.C.F. Donkers J.T.B.A Kessels S. Weiland M.C.F. Donkers J.T.B.A Kessels ∗ ∗ S. Weiland ∗ S. Weiland Weiland S. ∗ Dept. of Elect. Eng., Eindhoven University of Technology, Netherlands ∗ ∗ Dept. Elect. Eng., Eindhoven University Technology, Netherlands ∗ ∗∗ Dept. of Elect. Eng., Eindhoven University of Technology, Netherlands DAFof Control Group, of Eindhoven, Netherlands Dept. ofTrucks Elect. NV, Eng.,Vehicle Eindhoven University of Technology, Netherlands ∗∗ ∗∗ Trucks NV, Vehicle Control Group, Eindhoven, Netherlands ∗∗ DAF DAF Trucks NV, Vehicle Control Group, Eindhoven, Netherlands DAF Trucks NV, Vehicle Control Group, Eindhoven, Netherlands Abstract: In this paper, a real-time and distributed solution to Complete Vehicle Energy Abstract: this a and solution to Vehicle Abstract: In this paper, a real-time and distributed solution to Complete Vehicle Energy Management is presented using receding control horizon in combination with a Abstract: In In(CVEM) this paper, paper, a real-time real-time andadistributed distributed solution to Complete Complete Vehicle Energy Energy Management (CVEM) is presented using a receding control horizon in combination Management (CVEM) is dual presented using aa receding receding control horizon in combination combination with dual decomposition. The decomposition allows the CVEM optimization problemwith to beaaa Management (CVEM) is presented using control horizon in with dual decomposition. The dual decomposition allows the CVEM optimization problem to dual decomposition. The dual decomposition allows the CVEM optimization problem to be solved by solving several optimization problems. receding horizon control problem dual decomposition. Thesmaller dual decomposition allows theThe CVEM optimization problem to be be solved by solving several smaller optimization problems. The receding horizon control problem solved by solving several smaller optimization problems. The receding horizon control problem is formulated with variable sample intervals, allowing for large prediction horizons with only a solved by solving several smaller optimization problems. The receding horizon control problem is with intervals, allowing large prediction horizons with only is formulated formulated with variable sample intervals, allowing for for large prediction horizons with only aaa limited number of variable decision sample variables and constraints. The receding horizon control problem is formulated with variable sample intervals, allowing for large prediction horizons with only limited decision and constraints. The horizon problem limited number of decision variables and heavy-duty constraints. vehicle, The receding receding horizon control problem is solvednumber for a of case study variables of a hybrid equipped withcontrol a high-voltage limited number of decision variables and constraints. The receding horizon control problem is solved for a case study of a hybrid heavy-duty vehicle, equipped with a high-voltage is solved for a case study of a hybrid heavy-duty vehicle, equipped with a high-voltage battery system and a refrigerated semi-trailer. Simulations demonstrate that close to optimal is solved for a case study of a hybrid heavy-duty vehicle, equipped with a high-voltage battery system and a refrigerated semi-trailer. Simulations demonstrate that close to battery system and a refrigerated semi-trailer. Simulations demonstrate that close to optimal performance in terms of fuel consumption is obtained. The average execution time is 11.4 ms battery system and a refrigerated semi-trailer. Simulations demonstrate that close to optimal optimal performance terms fuel is The average time performance in inthat terms ofproposed fuel consumption consumption is obtained. obtained. The real-time average execution execution time is is 11.4 11.4 ms ms demonstrating theof solution method is indeed implementable. performance in terms of fuel consumption is obtained. The average execution time is 11.4 ms demonstrating demonstrating that the proposed solution method is indeed real-time implementable. demonstrating that that the the proposed proposed solution solution method method is is indeed indeed real-time real-time implementable. implementable. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Optimal control, Energy management system, Receding horizon control, Keywords: Optimal control, Energy management system, Receding horizon control, Keywords: Energy management system, Receding DistributedOptimal control, control, Predictive control, Hybrid Electric Vehicles Keywords: Optimal control, Energy management system, Receding horizon horizon control, control, Distributed control, Predictive control, Hybrid Electric Vehicles Distributed control, Predictive control, Hybrid Electric Vehicles Distributed control, Predictive control, Hybrid Electric Vehicles 1. INTRODUCTION et al., 2014; Schepmann and Vahidi, 2011; Borhan et al., 1. INTRODUCTION et Schepmann Vahidi, 2011; Borhan al., 1. et al., al., 2014; 2014; Schepmann and and Vahidi,use 2011; Borhan et etconal., 2012). All aforementioned methods a centralized 1. INTRODUCTION INTRODUCTION et al., 2014; Schepmann and Vahidi, 2011; Borhan et al., 2012). All aforementioned methods use a centralized con2012). All aforementioned methods use a centralized control approach to solve the energy management problemconfor Hybrid technology enables vehicles to make a powersplit 2012). All aforementioned methods use a centralized approach to solve energy for Hybrid technology enables vehicles to make powersplit approach to solve the energy management problem for Hybrid enables to aa atrol hybrid vehicle By addingproblem more decibetweentechnology the primary powervehicles device, e.g., an internal com- trol trol approach to without solve the theauxiliaries. energy management management problem for Hybrid technology enables vehicles to make make a powersplit powersplit asion hybrid vehicle without auxiliaries. By adding more decibetween the primary power device, e.g., an internal coma hybrid vehicle without auxiliaries. By adding more decibetween the primary power device, e.g., an internal comvariables and states to the problem, as with CVEM, bustion engine and the secondary power device, e.g., an between the primary power device, e.g., an internal com- a hybrid vehicle without auxiliaries. By adding more decision variables states problem, as bustion engine and the secondary power device, e.g., an sioncentralized variables and and states to to the problem, as with with CVEM, CVEM, bustion engine power an the controller canthe become computationally deelectric machine, so the thatsecondary the fuel consumption cane.g., be revariables and states to the problem, as with CVEM, bustion engine and and the secondary power device, device, e.g., an sion the centralized controller can become computationally deelectric machine, so that the fuel consumption can be rethe centralized controller can become computationally deelectric machine, so fuel can be remanding and is controller not scalable the complete duced. However, powerthe flows the vehicle are not centralized canwithout becomerevising computationally deelectric machine,the so that that the fuelinconsumption consumption can belimre- the manding and is not scalable without revising the complete duced. However, the power flows in the vehicle are not limmanding and is not scalable without revising the complete duced. However, the power flows in the vehicle are not limcontroller. ited to However, the propulsion system only. Particularly fornot heavyduced. the power flows in the vehicle are lim- manding and is not scalable without revising the complete controller. ited to the propulsion system only. Particularly for heavyited the system only. heavydutyto a significant amount of power is for consumed controller. ited tovehicles, the propulsion propulsion system only. Particularly Particularly for heavy- controller. For this reason, distributed solutions for energy manduty vehicles, a significant amount of power is consumed duty vehicles, aa significant of is by auxiliary systems, such asamount a refrigerated semi-trailer, an For duty vehicles, significant amount of power power is consumed consumed reason, solutions for manFor this reason, distributed solutions for energy management to distributed appear. Many interesting solutions for For this this start reason, distributed solutions for energy energy manby auxiliary systems, such as a refrigerated semi-trailer, an by auxiliary systems, such as a refrigerated semi-trailer, an air auxiliary supply system andsuch coolant As global efficiency by systems, as asystems. refrigerated semi-trailer, an agement start to appear. Many interesting solutions agement start to appear. Many interesting solutions for distributed control can be found from other fields, see e.g., start to appear. Many interesting solutions for for air supply system and coolant systems. As global efficiency air supply system and coolant As efficiency at vehicle is not by optimizing of the agement air supply level system andguaranteed coolant systems. systems. As global globaleach efficiency distributed can be from fields, see distributedetcontrol control can for be found found from other other fields, see e.g., e.g., (Stephens al., 2015) distributed energy demand side distributed control can be found from other fields, see e.g., at vehicle level is not guaranteed by optimizing each of the at vehicle level is not guaranteed by optimizing each of the components separately, energy management needs to be at vehicle level is not guaranteed by optimizing each of the (Stephens et for demand side (Stephens et al., 2015) for distributed energy demand side management and2015) (Fardad et al., 2010)energy for optimal control (Stephens et al., al., 2015) for distributed distributed energy demand side components separately, energy management needs to be components separately, energy to done on a complete vehicle level.management We refer to needs this desired components separately, energy management needs to be be of management and (Fardad et al., 2010) for optimal control management and (Fardad et al., 2010) for optimal control vehicle formations. The application of distributed somanagement and (Fardad et al., 2010) for optimal control done on a complete vehicle level. We refer to this desired done on a complete vehicle level. We refer to this desired energyonmanagement strategylevel. as Complete done a complete vehicle We refer Vehicle to this Energy desired of vehicle formations. The application of distributed of vehicle formations. The application of distributed solution methods to the CVEM problem in the automotive of vehicle formations. The application of distributed sosoenergy management strategy as Complete Vehicle Energy energy management strategy as Vehicle Energy Management (CVEM), see (Kessels et al., 2012). energy management strategy as Complete Complete VehicleBehavior Energy lution to CVEM problem in lutionis methods methods to the the CVEM problem in the the automotive automotive field only recently starting to attract attention. In (Chen lution methods to the CVEM problem in the automotive Management (CVEM), see (Kessels et al., 2012). Behavior Management (CVEM), see (Kessels et al., 2012). Behavior of each auxiliary component is generally unique and each Management (CVEM), see (Kessels et al., 2012). Behavior field is recently starting to In field is only recently starting to attract attention. In (Chen et al., 2014), a real-time game-theoretic is only only recently startingimplementable to attract attract attention. attention. In (Chen (Chen of each auxiliary component is generally unique and each of each is unique and auxiliary adds at component least one state and decision to field of each auxiliary auxiliary component is generally generally uniquevariable and each each et al., 2014), a real-time implementable game-theoretic et al., 2014), a real-time implementable game-theoretic approach to CVEM is shown, where the drive cycle is only et al., 2014), a real-time implementable game-theoretic auxiliary adds at least one state and decision variable to auxiliary adds at least one state and decision variable to the CVEM problem. state variables include auxiliary adds at leastExamples one stateofand decision variable to approach to CVEM is shown, where the drive cycle is only approach to CVEM is shown, where the drive cycle is only predicted over a short horizon. In (Nguyen et al., 2014), approach to CVEM is shown, where the drive cycle is only the CVEM problem. Examples of state variables include the CVEM Examples state include temperature in the refrigerated semi-trailer, air pressure the CVEM problem. problem. Examples of of state variables variables include predicted over a short horizon. In (Nguyen et al., 2014), predicted over aa short short horizon. In (Nguyen (Nguyen with et al., al.,MPC 2014), a game-theoretic approach in combination is predicted over horizon. In et 2014), temperature in the refrigerated semi-trailer, air pressure temperature in refrigerated semi-trailer, in the air supply system and temperature of air thepressure fluid in aa game-theoretic temperature in the the refrigerated semi-trailer, air pressure approach in combination with MPC is game-theoretic approach in combination with MPC is used to arrive at a distributed solution, while real-time a game-theoretic approach in combination with MPC is in the air supply system and temperature of the fluid in in air thethe engine coolantsystem system.and in the air supply supply system and temperature temperature of of the the fluid fluid in in used to arrive at a distributed solution, while real-time used to arrive at a distributed solution, while real-time implementation is not considered. In (Nilsson et al., 2015), used to arrive at a distributed solution, while real-time the engine coolant system. the engine coolant system. the engine coolant system. considered. In et implementation isisnot not considered. In (Nilsson (Nilsson et al., al., 2015), 2015), the computationis distributed using the Alternating DiWell-known real-time methods for energy management are implementation implementation is not considered. In (Nilsson et al., 2015), the computation is distributed using the Alternating DiWell-known real-time methods for energy management are the computation is distributed using the Alternating DiWell-known real-time for management are rection Method ofisMultipliers ideas from equivalent consumption minimization strategies (ECMS) computation distributed(ADMM), using the while Alternating DiWell-known real-time methods methods for energy energy management are the rection Method of Multipliers (ADMM), while ideas from equivalent consumption minimization strategies (ECMS) rection Method of Multipliers (ADMM), while ideas from equivalent consumption minimization strategies (ECMS) ECMS are used to calculate the equivalent costs at a suwith various adaptive mechanisms on drive cy- rection Method of Multipliers (ADMM), while ideas from equivalent consumption minimizationbased strategies (ECMS) ECMS are used to calculate the equivalent costs at a suwith various adaptive mechanisms based on drive cyECMS are used to calculate the equivalent costs at a with various adaptive mechanisms based on drive cypervisory level. In (Romijn et al., 2014), the computation cle prediction, driving patterns recognition and state-ofsuwith various adaptive mechanisms based on drive cy- ECMS are used to calculate the equivalent costs at a supervisory level. In (Romijn et al., 2014), the computation cle prediction, driving patterns recognition and state-ofpervisory level.via In a(Romijn (Romijn et al., al., 2014), 2014), the the computation cle prediction, patterns recognition is distributed dual decomposition without using a charge feedback,driving see (Onori and Serrao, 2011)and andstate-ofthe ref- pervisory level. In et computation cle prediction, driving patterns recognition and state-ofis distributed via aaa dual decomposition using charge feedback, see (Onori and Serrao, 2011) and the refdistributed via dual decomposition without using a charge see (Onori Serrao, and refsupervisory level allows drive cycle without prediction to bea erencesfeedback, therein. To anticipation on future events is distributed viawhich dual decomposition without using a charge feedback, seeimprove (Onori and and Serrao, 2011) 2011) and the the ref- is supervisory level which allows drive cycle prediction to be erences therein. To improve anticipation on future events supervisory level which allows drive cycle prediction to be erences therein. To improve anticipation on future events readily taken into account. using road preview information, a significant amount of supervisory level which allows drive cycle prediction to be erences therein. To improve anticipation on future events readily taken into using road preview information, significant amount of taken into account. using road aa significant of work can bepreview found oninformation, methods that based onamount (stochasreadily taken into account. account. using road preview information, a are significant amount of readily This paper proposes a computationally efficient implemenwork can be found on methods that are based on (stochaswork can found methods that based tic) Model e.g., on (Di(stochasCairano This work can be bePredictive found on on Control methods(MPC), that are aresee based on (stochaspaper proposes efficient This paper proposes computationally efficient implementation of the method presented in (Romijn et implemenal., 2014). This paper proposes aaa computationally computationally efficient implementic) Model Predictive Control (MPC), see e.g., (Di Cairano tic) Model Predictive Control (MPC), see e.g., (Di Cairano tic) Model Control (MPC), e.g., Cairano ⋆ tation of the method presented in (Romijn et 2014). tation of the method presented in (Romijn et al., 2014). This workPredictive has received financial supportsee from the(Di FP7 of the The dual decomposition allows the large-scale optimizatation of the method presented in (Romijn et al., al., 2014). ⋆ ⋆ This work has received financial support from the FP7 of the The dual decomposition allows the large-scale optimizaEuropean Commission under the grant CONVENIENT (312314). This work has received financial support from the FP7 of the ⋆ The dual dual decomposition decomposition allows allows the the large-scale large-scale optimizaoptimizaThis work has received financial support from the FP7 of the The T.C.J. T.C.J. T.C.J. T.C.J.

Romijn ∗∗ ∗ Romijn Romijn Romijn ∗

European Commission under the grant CONVENIENT (312314). European European Commission Commission under under the the grant grant CONVENIENT CONVENIENT (312314). (312314).

Copyright 2015 IFAC 203 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright 2015 IFAC 203 Copyright © 2015 IFAC 203 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 203Control. 10.1016/j.ifacol.2015.10.029

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Ef

Pf

ICE

Pp Pr

Est Eth

Pst Pth

High-voltage battery Refrigerated semi-trailer

Pb Pe

the number of decision variables n. Note that in (Romijn et al., 2014), we had that τ (ℓ) = 1 for all ℓ ∈ {1, . . . , n}. The application of the dual decomposition to the CVEM problem with variable sample intervals remains the same and details are given in (Romijn et al., 2014). A summary of the notation, the component models and the problem formulation are given in this section.

Pbr Gearbox

Pw Electric machine

Pem

Pld 2.1 Notation

Fig. 1. Hybrid powertrain, including an internal combustion engine (ICE), an electric machine (EM), a highvoltage battery and a refrigerated semi-trailer. tion problem to be solved by solving several smaller optimization problems. Furthermore, the approach is scalable in terms of adding components. However, CVEM, as in (Romijn et al., 2014), is solved for a relatively long horizon with a fixed sample interval, which allows computing the optimal solution for CVEM offline. This solution method is unsuitable for real-time implementation. To resolve this issue, we propose to use the receding horizon principle and apply the same dual decomposition to this problem. Computational limitations due to real-time implementation restrict the maximum allowable number of decision variables. Therefore, varying sample intervals will be used in our problem formulation to create a large prediction horizon, while having only a limited number of decision variables and constraints. The approach is similar to move blocking, see, e.g., (Cagienard et al., 2007), but is computationally less expensive due to a reduction in constraints. The receding horizon control approach in this paper is in line with distributed model predictive control, see, e.g., (Maestre and Negenborn, 2014). Owing to the problem formulation, a new implementation is proposed. 2. TOPOLOGY AND PROBLEM FORMULATION We reconsider in this paper the case study of (Romijn et al., 2014), consisting of a heavy-duty hybrid vehicle that includes an internal combustion engine (ICE), an electric machine (EM), a high-voltage battery and a refrigerated semi-trailer. The topology is schematically shown in Fig. 1, in which Pf and Pp denote the ICE’s fuel and mechanical power, respectively, Pe and Pem the EM’s electrical and mechanical power, respectively, Pb and Pst the battery’s electrical and stored chemical power, respectively, Pld and Pth the refrigerated semitrailer’s electrical and thermal power, respectively, Pbr and Pr is the mechanical brake power and requested drive power, respectively and Est denotes the battery state of energy and Eth denotes the thermal energy in the refrigerated semi-trailer. The main objective is to minimize the predicted cumulative fuel consumption, at each time instant k ∈ N, given by n � J(k) = τ (ℓ) m ˙ f (Pp (k + ℓ − 1|k)), (1) ℓ=1

subject to dynamics and conversion efficiencies of the components in Fig. 1. In this expression, n is the number of decision variables, Pp (k + ℓ − 1|k) is the predicted engine power at time instant k + ℓ − 1 using information available at time instant k and τ (ℓ) ∈ R+ is a variable sample� interval. In this formulation, the prediction horizon n Np = ℓ=1 τ (ℓ) ∈ R+ can be large, without increasing

204

The CVEM problem can be formulated as a static optimization problem, as was done in (Romijn et al., 2014). To formulate this problem over a receding horizon, we introduce the following notation: Pi (k) = [Pi (k|k), . . . , Pi (k + n − 1|k)]T ∈ Rn , (2a) Em (k) = [Em (k + 1|k), . . . , Em (k + n|k)]T ∈ Rn , (2b) αh,j (k) = [αh,j (k|k), . . . , αh,j (k + n − 1|k)]T ∈ Rn , (2c) (2d) τ = [τ (1), . . . , τ (n)]T ∈ Rn , for k ∈ N, i ∈ {f, p, em, e, b, ld, st, th br, r}, j ∈ {0, 1, 2}, m ∈ {st, th} and h ∈ {p, em}. In this notation, ℓ|k denotes decisions of variables Pi or predictions of states Em at (discrete) time instant ℓ based on information at (discrete) time instant k. Speed-dependent efficiency coefficients αh,j (ℓ|k) = fh,j (ω(ℓ|k)), are used to model the ICE and the EM, respectively, with ω(ℓ|k) the predicted engine speed. It is assumed that the EM runs at the same speed as the ICE. The functions fh,j (ω(ℓ|k)) are not part of the optimization problem because engine speed is predicted and can therefore be of any type. Because of the variable sample interval τ (ℓ) = t(k + ℓ) − t(k +ℓ−1), we need to revise the formulation of the energy buffer state dynamics when compared to (Romijn et al., 2014). The energy in the high-voltage battery and the refrigerated semi-trailer are represented by a first-order differential equation, i.e., d ˜ m Pm (t), Em (t) = A˜m Em (t) + B (3) dt

˜ m are scalars. This for m ∈ {st, th} and where A˜m and B differential equation allows us to make a prediction of Em (k + ℓ|k) = Em (t(k + ℓ)), for a given initial condition Em (k+ℓ−1|k) = Em (t(k+ℓ−1)), by using the convolution integral ˜

Em (k + ℓ|k) = eAm τ (ℓ) Em (k + ℓ − 1|k) � τ (ℓ) ˜ ˜m Pm (t(k + ℓ) − s)ds. (4) + e Am s B 0

If we restrict the power Pm to be piecewise constant, i.e., Pm (k + ℓ − 1|k) = Pm (s) for s ∈ [t(k + ℓ − 1), t(k + ˜ ℓ)] and ℓ ∈ {1, . . . , n} and define Am (ℓ) = eAm τ (ℓ) and � τ (ℓ) A˜ s ˜m ds, then expression (4) can be Bm (ℓ) = 0 e m B grouped for all ℓ ∈ {1, . . . , n} and written as Em (k) = Φm Em (k) + Γm Pm (k), (5) with Bm (1)

Γm

0

Bm (2) Am (2)Bm (1) . . = . n n−1 � A (ℓ)B (1) �A (ℓ)B m

ℓ=2

m

m

ℓ=2

m (2)

... .. . ..

.

0

0 , (n)

0

. . . Bm

(6a)

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Φm =

T Am (1)

(Am (2)Am (1))

T

... (

n ℓ=1

Am (ℓ))

T

T

,

(6b)

n

for m ∈ {st, th} with ℓ=1 Am (ℓ) = Am (n)Am (n − 1) . . . Am (1). Note that, if τ (ℓ) in (4) is sufficiently small, the solution of (3) can be approximated with a forward Euler approximation instead of (4), since it holds that ˜ eAm τ (ℓ) ≈ (1 + τ (ℓ)A˜m ). Then, Am (ℓ) = 1 + τ (ℓ)A˜m and ˜m , for l ∈ {1, . . . , n}, which leads to the Bm (ℓ) = τ (ℓ)B result presented in (Romijn et al., 2014). Finally, in this paper the notation 1 and 0 will be used to denote a vector of appropriate size with all elements equal to 1 and 0, respectively. The notation diag(w) ∈ Rp×p will be used to denote a diagonal matrix with entries from the vector w ∈ Rp on its diagonal. The notation I will be used to denote the identity matrix of appropriate size. 2.2 Problem Formulation The objective of CVEM is to minimize the cumulative fuel consumption given by (1) subject to the constraints, dynamics and conversion efficiencies of the components in Fig. 1. Let P(k) = [Pf (k) Pp (k) Pem (k) Pe (k) Pst (k) Pb (k) Pth (k) Pld (k) Pbr (k)] ∈ Rn×9 denote the matrix with the total power distribution over the decision variables. Then, the CVEM problem can be formalized as a static optimization problem at every time instant k ∈ N, which is given by min τ T Pf (k), (7) P(k)

subject to the efficiencies of the power convertors Pf (k) = diag(αp,2 (k)) diag(Pp (k))Pp (k) (8a) + diag(αp,1 (k))Pp (k) + αp,0 (k), Pe (k) = diag(αem,2 (k))diag(Pem (k))Pem (k) + diag(αem,1 (k))Pem (k)+αem,0 (k), (8b) (8c) Pb (k) = Pst (k) − βst diag(Pst (k))Pst (k), 1 (8d) Pld (k) = ηth (Pth (k) − βth diag(Pth (k))Pth (k)), and subject to the power constraints Pi,min Pi (k) Pi,max , (8e) for i ∈ {p, em, st, th} and subject to the power balance constraints given by the topology in Fig. 1,

205

high-voltage battery and refrigerated semitrailer, Est (k) and Eth (k), respectively, are given by (5) with A˜st = 0, ˜ th = −1, for some ˜st = −1, A˜th = −h/Cth and B B given heat transfer coefficient h > 0 and thermal capacity Cth > 0. Note that the optimization problem (7) subject to (8) is a Quadratically Constrained Quadratic Program, which can be hard to solve in real-time. For further details on the origin of the optimization problem, the reader is referred to (Romijn et al., 2014). Remark 1: Due to the variable sample intervals, the predicted states (5) are only constrained via (8h) at each sample instant, i.e., at t(k + ℓ − 1) for ℓ ∈ {1, . . . , n}. To assure that the intersample behaviour satisfies the constraints, we require that the eigenvalues of A˜m in (3) are real, which holds for the high-voltage battery and for the refrigerated semi-trailer. This is a restriction of using variable sample intervals when compared to move blocking, see e.g., (Cagienard et al., 2007). In case of move blocking, the amount of decision variables is reduced by fixing the inputs to be constant over a certain number of sample intervals. The number of state constraints however, is not reduced. As a result, move blocking requires significantly more state constraints, which is computationally less efficient. Remark 2: Equation (8i) introduces a constraint that enforces the energy content in the buffer at the end of the prediction horizon to be at a given reference value. This constraint is moved with the receding horizon so in principle this constraint is never satisfied. For short horizons, this constraint is restricting and often implemented as a soft constraint, see e.g., (Schepmann and Vahidi, 2011). This will add an extra weighted term to the objective function in (7), which introduces a tuning parameter. Instead, in this paper the prediction horizon is taken long enough, so that the effect of the tracking constraint is weakened, leaving no tuning parameter in the problem formulation. 3. DISTRIBUTED SOLUTION FOR CVEM In the previous section, we have defined the CVEM problem as a receding horizon optimization problem. In this section, we will propose a distributed solution for this optimization problem using a dual decomposition, see (Boyd and Vandenberghe, 2004).

Pr (k) − Pp (k) − Pem (k) = Pbr (k) 0,

(8f)

3.1 Lagrange Dual Formulation

Pe (k) − Pb (k) − Pld (k) = 0,

(8g)

In (Romijn et al., 2014), the problem is decomposed on system level, so that after decomposition, each of the smaller optimization problems is related to one of the systems in the vehicle, i.e., the ICE, the EM, the high-voltage battery, the refrigerated semi-trailer and the vehicle itself. For each of the systems, we can define a cost function related to the ‘energy losses’ of a system, given by J(Pi1 , Pi2 ) = τ T (Pi1 − Pi2 ), (9) for i1 ∈ {f, e, st, r}, i2 ∈ {p, em, b, ld, br}, where Pi1 can be seen as the output power and Pi2 can be seen as the input power of the system. The difference represents the energy losses. Then, under the assumption that (8f), (8g) and (8i) holds, the objective function (7) can be rewritten as (10) τ T Pf (k) = J(Pf (k), Pp (k)) + J(Pe (k), Pem (k)) + J(Pst (k), Pb (k)) + J(0, Pld (k)) + J(Pr (k), Pbr (k)).

and subject to the constraints on the energy buffer state dynamics (5) given by Em,min Em (k) Em,max (8h) for m ∈ {st, th}, with all such inequalities to be understood entry-wise and finally subject to a tracking constraint, i.e., a constraint at the end of the prediction horizon, given by Em (k + n|k) = Em,ref , (8i) for m ∈ {st, th} and some given Em,ref . In (8c) and (8d), ηth > 0 is a given thermal efficiency coefficient and βth > 0, βst > 0 are given conversion efficiency coefficients. In the optimization problem (7)-(8), the requested drive power Pr (k) is assumed to be given as it is predicted with preview information available at time instant k. The energy in the 205

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Note that for the refrigerated semi-trailer, all the power flowing into the refrigerated semi-trailer is converted into heat and eventually lost to the environment such that a 0 appears in the cost function. Instrumental for the decomposition is the notion of the so-called ‘partial Lagrangian’ (Boyd and Vandenberghe, 2004). It is obtained by augmenting the objective function (7) with only the constraints that are defined in decision variables for more than one system, i.e., the power balance constraints (8f) and (8g). The local constraints, (8), except for (8f) and (8g), defined in decision variables for one system only, will be directly passed to the partial Lagrange dual function defined below. For the CVEM under consideration, the partial Lagrangian at time instant k ∈ N is given by L(P(k), λ(k), ν(k), Pr (k)) = τ T Pf (k) + λ(k)T (Pr (k) −Pp (k)−Pem (k)) + ν T (k)(Pe (k)−Pb (k)−Pld (k)), (11) in which λ(k) ∈ Rn+ and ν(k) ∈ Rn are Lagrange multipliers associated with the power balance constraints (8f) and (8g), respectively. In Fig. 1, it is shown that these constraints correspond with the two points where mechanical power and electrical power is aggregated. The interesting economic interpretation of the Lagrange multipliers is that they can be seen as prices for which systems can buy or sell power at these selected points.

Table 1. Optimization matrices i

Hi (k)

fi (k)

p diag(τ ) diag(αp,2 (k)) em diag(τ +ν(k)) diag(αem,2 (k)) st th

βst diag(τ + ν(k)) βth diag(τ + ν(k)) η

diag(αp,1 (k))τ − τ − λ(k) diag(αem,1 (k))(τ +ν(k))− τ − λ(k) −ν(k) − η1 (τ + ν(k)) th

ble in variables related to each system, which allows us to evaluate the Lagrange dual function (12) in a distributed fashion. This is explained in the next section. 3.2 Minimizing the Lagrange Dual Function The minimization over P(k), as in (12), can be done in a distributed fashion. To show this, observe that (12), subject to (8), except for (8f) and (8g), can be decomposed as follows g(λ(k), ν(k), Pr (k)) = gp (λ(k)) + gem (λ(k), ν(k)) + gst (ν(k)) + gth (ν(k)) + gr (λ(k), Pr (k)), (14) in which gp (λ(k)) := min J Pf (k),Pp (k) −λT (k)Pp (k), (15a) Pf (k),Pp (k)

subject to (8a), (8e) is the dual function related to the ICE, gem (λ(k), ν(k)) := min J(Pem (k), Pe (k)) Pe (k),Pem (k)

For the optimization problem (7)-(8) and the partial Lagrangian (11), the partial Lagrange dual function is given by g(λ(k), ν(k),Pr (k)) = min L(P(k), λ(k), ν(k), Pr (k)), (12)

−λ(k)T Pem (k)+ν T (k)Pe (k), (15b) subject to (8b) and (8e) is the dual function related to the EM, gst (ν(k)) := min J(Pst (k), Pb (k))−ν T (k)Pb (k), (15c)

subject to the local constraints given by (8), except for (8f) and (8g). The solution to the CVEM optimization problem (7)-(8) at each time instant k ∈ N is found via the following iterative procedure:

subject to (8c), (8e), (8g), (8h) is the dual function related to the high-voltage battery, J(0, Pld (k))−ν T (k)Pld (k), (15d) gth (ν(k)) := min

P(k)

Algorithm 1: • Initialize the iteration counter j := 0. • Obtain the energy buffer states Est (k|k) and Eth (k|k). • Initialize the dual variables λ0 (k) and ν 0 (k). while (Pr (k) − Pjp (k) − Pjem (k) > ǫλ & −ǫν > Pje (k) − Pjb (k)−Pjld (k) > ǫν & j jmax ) • Evaluate the partial Lagrange dual function (12), subject to (8), except for (8f) and (8g), to obtain Pj (k). • Take a dual gradient step over λ(k) and ν(k) λj+1 (k)=max{0, λj (k)+Sλ (Pr (k)−Pjp (k)−Pjem (k))}, j+1

j

(k)+Sν (Pje (k)−Pjb (k)−Pjld (k)),

ν (k) = ν (13) where Sλ and Sν are diagonal matrices with appropriately chosen step sizes. • The iteration counter is increased by one: j := j + 1. end • Set ¯j = j. Whenever the stop criterion is satisfied for some j = ¯j jmax , the first element of the calculated control sequence ¯ is applied, i.e., Pj (k|k). Then, the procedure is repeated at the next time instant k + 1. Because of the particular decomposition of the objective function (10), the partial Lagrangian (11) becomes separa206

Pst (k),Pb (k)

Pth (k),Pld (k)

subject to (8d), (8e), (8g) and (8h) is the dual function related to the refrigerated semi-trailer and gr (λ(k), Pr (k)) := min J(Pr (k), Pbr (k))+λT (k)Pr (k). Pbr (k)0

(15e) is the dual function related to the vehicle. The dual functions (15a) and (15b) can be evaluated by solving a Quadratic Program (QP), i.e., min

Pi (k)

Pi (k)T Hi (k)Pi (k) + fiT (k)Pi (k),

(16a)

subject to Pi,min Pi (k) Pi,max , (16b) for i ∈ {p, em} where the matrices Hp (k), fp (k), Hem (k) and fem (k) are given in Table 1. Note that, in order for Hem (k) to be strictly positive definite, we require that ν(k) > −τ . The dual functions (15c) and (15d) can be evaluated by solving the QP (16a) subject to (16b) and subject to Ei,min Φi Ei (k|k) + Γi Pi (k) Ei,max , (16c) Ei (k + n|k) = Ei,ref , (16d) for i ∈ {st, th}, where the matrices Hst (k), fst (k), Hth (k) and fth (k) are also given in Table 1. Prediction of the requested power Pr (k) is given by preview information and as such evaluation of the dual function (15e) reduces to deciding on the use of mechanical

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brake power Pbr (k). This is straightforward, as mechanical brake power is only necessary when the power balance constraint (8f) holds with inequality which holds when the Lagrange multiplier λ(ℓ|k) is zero for time instant ℓ as a result of the complementary slackness condition, see e.g., (Boyd and Vandenberghe, 2004). Then, the mechanical braking power at time instant ℓ must be chosen such that the inequality holds with equality. Finally, as each of the dual functions can be written as a QP, they can be evaluated explicitly or using a numerically very efficient QP solver, e.g., CVXGEN (Mattingley and Boyd, 2012). This solver can handle only a limited number of constraints which is the motivating reason to use variable sample intervals above move blocking. 4. NUMERICAL EXAMPLE In this section, simulation results are presented to illustrate the proposed distributed receding horizon control solution for CVEM of a heavy-duty hybrid vehicle with a refrigerated semi-trailer. The algorithm is simulated over two parts of a larger Pan European Drive Cycle, denoted by PEDC1 and PEDC2. In this example, the road load, i.e., the exact power request Pˆr before the gearbox with the exact engine speed ω ˆ are given for each of the drive cycles implying that the gear shift strategy is known. 4.1 Preview Information Usually, only a limited amount of information about the future road load is known that can be used for solving the receding horizon control problem. In practice, the future road load can be predicted if information about the vehicle, such as the vehicle mass and future information about vehicle velocity and road grade is known. For city driving, it can be difficult to predict even a few seconds ahead, while high-way driving at constant velocity could allow accurate prediction for a few kilometers ahead. To limit the scope of this research, the predicted road load in this paper is derived from the exact road load (Pˆr , ω ˆ ). For each interval τ (ℓ), we predict the road load by two different methods • Method M1: Pr (k + ℓ − 1) = ω(k + ℓ − 1) = • Method M2:

1 τ (ℓ) 1 τ (ℓ)

τ (ℓ)

Pˆr (t(k + ℓ) − s)ds, (17a)

0

τ (ℓ)

ω ˆ (t(k + ℓ) − s)ds, (17b)

0

Pr (k + ℓ − 1) = ω(k + ℓ − 1) =

1 N 1 N

N

Pˆr (t)dt,

(18a)

ω ˆ (t)dt,

(18b)

0

N

0

for ℓ ∈ {1, . . . , n}, where t(k + ℓ) = t(k + ℓ − 1) + τ (ℓ) and N is the length of the entire drive cycle. This should not be confused with Np , which is the length of the prediction horizon. The difference between the two methods is that Method 1 averages the road load over interval τ (ℓ), which yields a more accurate prediction than Method 2, which averages the road load over the entire drive cycle. To 207

207

Table 2. Various prediction scenarios. Sc.

Np

1a 1b 2a 2b 3

250 250 250 250 650

s s s s s

τ

for ℓ ∈ N

for ℓ = n

[ 1 1 1 1 1 1 1 1 1 241 ]T [ 1 1 1 1 1 1 1 1 1 241 ]T

M1 M1 M1 M1 M1

M1 M2 M1 M2 M2

[ 1 1 3 3 6 6 12 12 [ 1 1 3 3 6 6 12 12 [ 1 1 3 3 6 6 12 12

103 103 ]T 103 103 ]T 103 503 ]T

demonstrate the proposed distributed receding horizon control strategy, 5 scenarios are simulated with different prediction information. The number of decision variables is limited to n = 10 to ensure real-time implementation. The scenarios are defined in Table 2 in which N = {1, . . . , n−1} and a visual representation is given in Fig. 2. The variable sample intervals in each of the scenarios are arbitrarily chosen and in practice will depend on the systems available in the truck, like GPS and radar. Finally, Scenario 0 is defined as a baseline scenario. This scenario is the optimal solution over the entire drive cycle and corresponds to the results of (Romijn et al., 2014). 4.2 Simulation Results In Fig. 3, the simulation results for some of the scenarios are plotted for PEDC1. The temperature in the refrigerth ated semi-trailer is given by Tth = Tamb−E Cth , with Tamb the ambient temperature. The tracking reference for the energy in the battery is given by 50 % of the effective battery capacity. Note that not all scenarios are depicted in Fig. 3, because most of them have similar state trajectories as can be seen from Scenario 1a and 2a which have the same prediction horizon length. In case the prediction horizon becomes longer, as in Scenario 3, the state trajectories become different. In Table 3, the fuel consumption reduction ∆m ˙ f over the drive cycle is shown for all the considered scenarios, when compared to not having a hybrid powertrain with a constant energy consumption of the refrigerated semitrailer. Also, the difference between the energy in the battery between the start and the end of the drive cycle is given and is denoted by ∆Est . As with receding horizon control, it is difficult to constrain the energy at the end of the drive cycle and causes ∆Est to not be equal to zero. Nevertheless, it is shown that all scenarios perform very well and with good prediction information (see Scenario 2a, 2b and 3) close to optimal. For PEDC2, the potential fuel reduction is far less than for PEDC1, which is caused by the fact that there is less braking energy available in PEDC2. Extending the prediction horizon, as in Scenario 3, does not result in less fuel consumption for these examples. In general, the length of the prediction horizon should be chosen such that the energy in the battery can always satisfy the reference tracking constraint, while recuperating the maximum amount of braking energy. If the prediction horizon is too short, the battery will not allow maximum recuperation of brake energy in order to satisfy the reference tracking constraint. This will always have a negative effect on the fuel consumption reduction. 4.3 Computation Time Performance with respect to computation time and the amount of iterations j is important to assess the impli-

Drive cycle Scenario 1a Scenario 1b Scenario 2a Scenario 3

1380 1360 1340 1320 600

650

700

750

Time [ s ]

800

850

Fig. 2. Various prediction scenarios.

12000 10000 8000

Scenario 0 Scenario 1a Scenario 2a Scenario 3

6000 4000

5.5 5.4

0

2000

4000 6000 Time [ s ]

Fig. 3. State trajectories.

Table 3. Fuel consumption reduction results. Scenario 0 1a 1b 2a 2b 3

PEDC1 ∆Est [J] ∆m ˙ f [%] -43 1150 309 104 57 58

-6.97 -6.63 -6.39 -6.80 -6.76 -6.71

PEDC2 ∆Est [J] ∆m ˙ f [%] -19 1125 2449 97 78 54

-3.74 -3.14 -3.10 -3.43 -3.43 -3.43

cations for real-time implementation. In Fig. 4, the computation time and the number of iterations are given as function of time for Scenario 3 simulated on a standard laptop with a 2.2 GHz i7 processor. The average computation time is only 11.4 ms and the average number of iterations is 11.8. These numbers demonstrate the potential for implementing the algorithm on an embedded platform. In principle, each of the QPs can be solved on a different platform. This requires that at each time instant k, information about the Lagrange multipliers λ(k), ν(k) and information about the power distribution P(k) needs to be communicated j times between each platform. This might limit the maximum allowable number of iterations. To assess the consequence of communication constraints, Scenario 3 is simulated on PEDC1 with a maximum of 30 iterations. In this case, the potential fuel consumption reduction is -6.72 % instead of -6.71 % (see Table 3). This shows that the performance is not necessarily affected if the amount of iterations is limited. 5. CONCLUSION In this paper, a real-time and distributed solution for Complete Vehicle Energy Management is presented using a receding horizon in combination with a dual decomposition. The results show that the computational load of the proposed solution is small and that the performance is close to optimal in terms of fuel consumption. Moreover, the results are obtained without a need for parameter tuning and rely completely on component models and prediction information. Due to this strong dependency, an interesting extension to this research amounts to investigating the robustness of the algorithm against modeling and prediction errors. REFERENCES Borhan, H., Vahidi, A., Phillips, A.M., Kuang, M.L., Kolmanovsky, I.V., and Di Cairano, S. (2012). Mpc-based energy management of a power-split hybrid electric vehicle. IEEE Trans. Control Syst. Technol., 20(3), 593–603. 208

8000

10000

Computation time [ms]

0

100

14000

50

0 Nr. of iterations [−]

s

200

−200

Engine speed [ rpm ]

Battery energy E [ J ]

400

Temperature [ ° C]

Power request Pr [ kW ]

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100

Time [s]

50

0

0

2000

4000 6000 Time instant k [−]

8000

10000

Fig. 4. Comp. time and iterations.

Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press, Cambridge. Cagienard, R., Grieder, P., Kerrigan, E.C., and Morari, M. (2007). Move blocking strategies in receding horizon control. J. of Process Control, 17(6), 563–570. Chen, H., Kessels, J., Donkers, M., and Weiland, S. (2014). Game-theoretic approach for complete vehicle energy management. In Proc. IEEE Vehicle Power and Propulsion Conf., 1–6. Di Cairano, S., Bernardini, D., Bemporad, A., and Kolmanovsky, I.V. (2014). Stochastic mpc with learning for driver-predictive vehicle control and its application to hev energy management. IEEE Trans. Control Syst. Technol., 22(3), 1018–1031. Fardad, M., Lin, F., and Jovanovic, M.R. (2010). On the dual decomposition of linear quadratic optimal control problems for vehicular formations. In Proc. IEEE Conf. on Decision and Control, 6287–6292. Kessels, J., Martens, J., van den Bosch, P., and Hendrix, W. (2012). Smart vehicle powernet enabling complete vehicle energy management. In Proc. IEEE Vehicle Power and Propulsion Conf., 938–943. Maestre, J.M. and Negenborn, R.R. (2014). Distributed model predictive control made easy. Springer. Mattingley, J. and Boyd, S. (2012). Cvxgen: a code generator for embedded convex optimization. Optimization and Engineering, 13(1), 1–27. Nguyen, K.D., Bideaux, E., Pham, M.T., and Le Brusq, P. (2014). Game theoretic approach for electrified auxiliary management in high voltage network of hev/phev. In Proc. Int. Electric Vehicle Conf., 1–8. Nilsson, M., Johannesson, L., and Askerdal, M. (2015). ADMM applied to energy management of ancilliary systems in trucks. In Proc. American Control Conf. Onori, S. and Serrao, L. (2011). On adaptive-ecms strategies for hybrid electric vehicles. In Proc. of the Int. Scientific Conf. on Hybrid and Electric Vehicles, Malmaison, France. Romijn, T., Donkers, M., Kessels, J., and Weiland, S. (2014). A dual decomposition approach to complete energy management for a heavy-duty vehicle. In Proc. IEEE Conf. on Decision and Control, 3304–3309. Schepmann, S. and Vahidi, A. (2011). Heavy vehicle fuel economy improvement using ultracapacitor power assist and preview-based mpc energy management. In Proc. American Control Conf., 2707–2712. Stephens, E.R., Smith, D.B., and Mahanti, A. (2015). Game theoretic model predictive control for distributed energy demand-side management. IEEE Trans. on Smart Grid, 6(3), 1394–1402.

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