Hybrid Vehicle Control and Optimization with a New Mathematical Method

5th IFAC Conference on 5th IFAC Conference on 5th IFACand Conference on Engine Powertrain Simulation and Modeling 5th Conference on 5th IFAC IFACand Conference onControl, Engine Powertrain Control, Simulation and online Modeling Available at www.sciencedirect.com Engine and Powertrain Control,20-22, Simulation Changchun, China, September 2018 and Engine and Powertrain Simulation and Modeling Modeling 5th IFAC Conference onControl, Engine and Powertrain Control, Simulation Changchun, China, September 20-22, 2018 Changchun, China, China, September September 20-22, 20-22, 2018 and Modeling Changchun, 2018 Engine and Powertrain Control,20-22, Simulation Changchun, China, September 2018 and Modeling Changchun, China, September 20-22, 2018

ScienceDirect

IFAC PapersOnLine 51-31 (2018) 201–206

Hybrid Vehicle Control and Optimization Hybrid Vehicle Control and Optimization Hybrid Vehicle Control and Optimization Hybrid Vehicle Control and Optimization Hybrid Vehicle Control and Optimization with a New Mathematical Method Hybrid Vehicle Control and Optimization with a New Mathematical Method with a New Mathematical Method with a New Mathematical with a New Mathematical Method Method with a New Method ∗Mathematical ∗ Nariaki Tateiwa ∗ Nozomi Hata ∗ Akira Tanaka ∗∗

∗ ∗ Nariaki Tateiwa Hata Akira ∗ Nozomi ∗ Nariaki Tateiwa Nozomi Hata∗∗∗∗∗ Takashi Akira Tanaka Tanaka ∗ ∗ ∗ Nozomi ∗ Nariaki Tateiwa Hata Akira Tanaka Takashi Nakayama Yoshida Wakamatsu ∗ ∗ Nariaki Tateiwa Nozomi Hata Akira Tanaka ∗ Akihiro ∗∗ Takashi ∗ Takashi Nakayama Akihiro Yoshida Wakamatsu ∗ ∗ ∗ ∗ ∗ Takashi Nakayama Akihiro Yoshida Takashi Wakamatsu ∗ Katsuki ∗∗∗ ∗ Takashi Nakayama Akihiro Yoshida Takashi Wakamatsu Nariaki Tateiwa Nozomi Hata Akira Tanaka Fujisawa ∗∗ Takashi Nakayama ∗ Katsuki Akihiro Yoshida Takashi Wakamatsu ∗∗ Fujisawa ∗∗ ∗ ∗ Katsuki Fujisawa ∗∗ Fujisawa Takashi Nakayama Katsuki Akihiro Yoshida Takashi Wakamatsu Katsuki Fujisawa ∗∗ ∗ Katsuki Fujisawa of Mathematics, Kyushu University, Fukuoka, Japan ∗ ∗ Graduate Graduate School School of Mathematics, Kyushu University, Fukuoka, Japan ∗ School of Mathematics, Kyushu University, Fukuoka, Japan ∗ Graduate Graduate School of Mathematics, Kyushu University, Fukuoka, Japan ({n-tateiwa, n.hata, akirat.tanaka, takashi.n, akihiro.yoshida.916, Graduate School of Mathematics, Kyushu University, Fukuoka, Japan ({n-tateiwa, n.hata, akirat.tanaka, takashi.n, akihiro.yoshida.916, ∗ ({n-tateiwa, n.hata, akirat.tanaka, takashi.n, akihiro.yoshida.916, ({n-tateiwa, n.hata, akirat.tanaka, takashi.n, akihiro.yoshida.916, Graduate School of Mathematics, Kyushu University, Fukuoka, Japan tkc.w}@kyudai.jp) ({n-tateiwa, n.hata, akirat.tanaka, takashi.n, akihiro.yoshida.916, tkc.w}@kyudai.jp) tkc.w}@kyudai.jp) ∗∗ ({n-tateiwa, n.hata, akirat.tanaka, tkc.w}@kyudai.jp) takashi.n, akihiro.yoshida.916, Institute of Mathematics for Industry, Kyushu University, Fukuoka ∗∗ tkc.w}@kyudai.jp) ∗∗ Institute of Mathematics for Industry, Kyushu University, Fukuoka ∗∗ Institute of Mathematics for Industry, Kyushu University, Fukuoka ∗∗ Institute of for Industry, Kyushu University, Fukuoka tkc.w}@kyudai.jp) & AIST-Tokyo Tech Real Big-Data Computation Institute of Mathematics Mathematics forWorld Industry, Kyushu University,Open Fukuoka AIST-Tokyo Tech Real World Big-Data Computation Open ∗∗ & & AIST-Tokyo Tech Real World Big-Data Computation Open & AIST-Tokyo Tech Real World Big-Data Computation Open Institute of Mathematics for Industry, Kyushu University, Fukuoka Innovation Laboratory, Tokyo, Japan ([email protected]). & AIST-Tokyo Tech Real World Big-Data Computation Open Innovation Laboratory, Tokyo, Japan ([email protected]). Innovation Laboratory, Tokyo, JapanBig-Data ([email protected]). Innovation Laboratory, Japan & AIST-Tokyo TechTokyo, Real World Computation Open Innovation Laboratory, Tokyo, Japan ([email protected]). ([email protected]). Innovation Laboratory, Tokyo, Japan ([email protected]). Abstract: For For hybrid hybrid electric electric vehicle vehicle (HEV) (HEV) systems, systems, studies studies using using model-based model-based simulators simulators Abstract: Abstract: For hybrid electric vehicle (HEV) systems, studies using model-based simulators Abstract: For hybrid electric vehicle (HEV) systems, studies using model-based simulators have been actively conducted. The vehicle powertrain simulator makes it easier to evaluate the Abstract: For hybrid electric vehicle (HEV) systems, studies using model-based simulators have been been actively actively conducted. conducted. The The vehicle vehicle powertrain powertrain simulator simulator makes makes it it easier easier to to evaluate evaluate the have the have been actively conducted. The vehicle powertrain simulator makes it easier to evaluate the Abstract: For hybrid electric vehicle (HEV) systems, studies using model-based simulators powertrain system. In this paper, we utilize a Toyota Hybrid System (THS) simulator to obtain have been actively conducted. The vehicle powertrain simulator makes it easier to evaluate the powertrain system. In this paper, we utilize aa Toyota Hybrid System (THS) simulator to obtain powertrain system. In this paper, we utilize Toyota Hybrid System (THS) simulator to obtain powertrain system. In this paper, we utilize a Toyota Hybrid System (THS) simulator to obtain have been actively conducted. The vehicle powertrain simulator makes it easier to evaluate the a long-term control that optimizes the fuel efficiency when the vehicle speed over a certain powertrain system. In this paper, we utilize a Toyota Hybrid System (THS) simulator to obtain a long-term control that optimizes the fuel efficiency when the vehicle speed over a certain a long-term control that optimizes the fuel efficiency when the vehicle speed over aatoshortest certain a long-term control that optimizes the fuel efficiency when the vehicle speed over certain powertrain system. In this paper, we utilize a Toyota Hybrid System (THS) simulator obtain period is given. Our proposed method obtains optimal long-term control by solving the a long-term control that optimizes the fuel efficiency when thecontrol vehicleby speed over ashortest certain period is given. Our proposed method obtains optimal long-term solving the period is given. Our proposed method obtains optimal long-term solving the shortest period is Our proposed method obtains optimal long-term control by solving the a long-term control thatof the fuel efficiency when thecontrol vehicleaaby over certain path problem with charge (SOC) constraints constructing graph expressing the period is given. given. Ourstate proposed method obtains optimalafter long-term control byspeed solving theashortest shortest path problem with state ofoptimizes charge (SOC) constraints after constructing graph expressing the path problem with state of charge (SOC) constraints after constructing a graph expressing the path problem with state of charge (SOC) constraints after constructing aabygraph expressing the period is given. Our proposed method obtains optimal long-term control solving the shortest transition of the fuel and battery consumption. We also propose aa search method for vehicle path problem with state of charge (SOC) constraints after constructing graph expressing the transition of the fuel and battery consumption. We also propose search method for vehicle transition of the fuel and battery consumption. We also propose aa of search method for vehicle transition of the fuel and battery consumption. We also propose search method for vehicle path problem with state of charge (SOC) constraints after constructing a graph expressing the control using bicubic spline interpolation without the preparation a controller. We finally transition of the fuel and battery consumption. We also propose a search method for vehicle control using bicubic spline interpolation without the preparation of a controller. We finally control using bicubic spline interpolation without the preparation of aa controller. We finally control using bicubic spline interpolation without the preparation of controller. We finally transition of the fuel and battery consumption. We also propose a search method for vehicle remove almost all edges from a graph by 97.2% at most through the utilization of 0-1 integer control almost using bicubic spline interpolation without the preparation of utilization a controller. We integer finally remove all edges from a by 97.2% at most through the of 0-1 remove almost all edges from aa graph graph by 97.2% at most through the utilization of 0-1 integer remove almost all from by at most through the utilization of integer control using bicubic spline interpolation without preparation ofoptimal a controller. We finally linear which enables a 3.88x speedup in obtaining vehicle control. removeprogramming, almost all edges edges from a graph graph by 97.2% 97.2% at the most throughthe the utilization of 0-1 0-1 integer linear programming, which enables a 3.88x speedup in obtaining the optimal vehicle control. linear programming, which enables a 3.88x speedup in obtaining the optimal vehicle control. linear programming, which enables a 3.88x speedup in obtaining the optimal vehicle control. remove almost all edges from a graph by 97.2% at most through the utilization of 0-1 integer linear whichFederation enables aof 3.88x speedup in obtaining optimal © 2018,programming, IFAC (International Automatic Control) Hosting bythe Elsevier Ltd. vehicle All rightscontrol. reserved. linear programming, which enables a 3.88x speedup insimulation, obtaining the optimal vehicle control. Keywords: Hybrid vehicles, Graph theory, Computer Optimal control, Function Keywords: Hybrid vehicles, Graph theory, Computer simulation, Optimal control, Function Keywords: Hybrid vehicles, Graph theory, Computer simulation, Optimal control, Function Keywords: vehicles, theory, approximation, Optimization problem Keywords: Hybrid Hybrid vehicles, Graph Graph theory, Computer Computer simulation, simulation, Optimal Optimal control, control, Function Function approximation, Optimization problem approximation, Optimization problem approximation, Optimization problem Keywords: Hybrid vehicles, Graph theory, Computer simulation, Optimal control, Function approximation, Optimization problem approximation, Optimization problem 1. INTRODUCTION §3.1, 3.2 Discretize the driving cycle, SOC, 1. INTRODUCTION §3.1, 3.2 1. INTRODUCTION INTRODUCTION Discretize the driving cycle, SOC, §3.1, 3.2 and the 1. Discretize the driving cycle, SOC, state of vehicle to reduce the problem §3.1, Discretize the cycle, SOC, 1. INTRODUCTION §3.1, 3.2 3.2 and the state to problem Discretize the driving driving cycle,the SOC, and the the state of of vehicle vehicle to reduce reduce the problem 1. INTRODUCTION and state of vehicle to reduce the problem §3.1, 3.2 and the Discretize the driving cycle,the SOC, In the transportation field, several studies have focused on state of vehicle to reduce problem In the transportation field, several studies have focused on In the transportation field, several studies have focused on and the state of vehicle to reduce the problem §4 §5 In the transportation field, several studies have focused on new automotive green technologies having less dependence §4 In the transportation field, several studies have focused on §5 new automotive green technologies having less dependence §4 §5 Reduce new automotive green technologies having less dependence §4 §5 Optimize the Reduce new automotive green technologies having less dependence In the transportation field, several studies have focused on on fossil fuels. Hybrid electric vehicles (HEVs) have both §4 §5 Reduce Optimize the new automotive green technologies having less dependence search space and on fossil fuels. Hybrid electric vehicles (HEVs) have both Reduce Optimize the vehicle control on fossil fuels. Hybrid electric vehicles (HEVs) have both search space and and Optimize the Reduce §4 §5 on fossil fuels. Hybrid electric vehicles (HEVs) have both new automotive green technologies having less dependence search space vehicle control electric motors and a conventional engine, and can modOptimize the computation search space and on fossil fuels. Hybrid electric vehicles (HEVs) have both vehicle control electric motors and a conventional engine, and can modby solving NLP Reduce vehicle control computation search space and electric motors and a conventional engine, and can modOptimize the computation by solving NLP vehicle control electric motors and a conventional engine, and can modon fossil fuels. Hybrid electric vehicles (HEVs) have both time erate global warming. The electric motor provides addicomputation by solving NLP electric motors and a conventional engine, and can modsearch space and time by solving NLP computation erate global warming. The electric motor provides addivehicle control time erate global warming. The electric motor provides addiby solving NLP time erate warming. The electric motor provides addielectric motors a conventional and can modtional power to and assist the engine in engine, accelerating, passing, computation time erate global global warming. Theengine electric motor provides addiby solving NLP tional power to the in accelerating, passing, tional powerhills, to assist assist the engine inconserves accelerating, passing, time tional power to assist the engine in accelerating, passing, erate global warming. The electric motor provides addior climbing which not only fuel but also tional power to assist the engine in accelerating, passing, §3.3 or climbing hills, which not only conserves fuel but also or climbing hills, which not only conserves fuel but also §3.3 or which not only conserves fuel but also §3.3 tional power to assist theand engine accelerating, passing, produces lesshills, SOx, NOx, CO emissions. To §3.3 or climbing climbing hills, which not only22 in conserves fuelaccelerate but also Construct TSTG produces less SOx, NOx, CO emissions. To accelerate §3.3 produces less SOx,development, NOx, and and CO To accelerate Construct TSTG 2 emissions. emissions. To accelerate produces less SOx, NOx, and CO or climbing hills, which not only conserves fuel but also Construct TSTG the pace of HEV we propose a method for 2 which represents §3.3 Construct TSTG produces less SOx, NOx, and CO emissions. To accelerate 2 the pace of HEV development, we propose a method for which represents Construct TSTG the pace of HEV development, we propose a method for which represents the pace of HEV development, we propose a method for produces less SOx, NOx, and CO emissions. To accelerate the transition of evaluating the powertrain quickly and accurately in terms which represents 2 the pace of HEV development, we propose a method for Construct TSTG the transition of which represents evaluating the powertrain quickly and accurately in terms the transition of evaluating the powertrain quickly and accurately in terms vehicle state the transition of evaluating the powertrain quickly and accurately in terms the pace of HEV development, we propose a method for of the fuel consumption. which represents vehicle state the transition evaluating the powertrain quickly and accurately in terms vehicle state stateof of the fuel consumption. vehicle of the fuel consumption. the transition of of the fuel consumption. evaluating the powertrain quickly and accurately in terms vehicle state of the fuel consumption. For the experiment evaluation, we utilized a Toyota Hybrid state §3.4vehicle Solve SOC-constrained shortest path For the experiment evaluation, we utilized a Toyota Hybrid of the fuel consumption. §3.4 For the experiment experiment evaluation, we(2008)) utilized simulator a Toyota Toyota Hybrid Solve shortest path §3.4 For the evaluation, we utilized a Solve SOC-constrained SOC-constrained shortest path System (THS) (Liu and Peng devel§3.4 and obtain the vehicle control For the experiment evaluation, we(2008)) utilized simulator a Toyota Hybrid Hybrid Solve SOC-constrained shortest path System (THS) (Liu and Peng devel§3.4 problem problem and obtain the vehicle control Solve SOC-constrained shortest path System (THS) (Liu and Peng (2008)) simulator develproblem and and obtain obtain the the vehicle vehicle control System (THS) (Liu and Peng (2008)) simulator develFor the experiment evaluation, we utilized a Toyota Hybrid oped by the Toyota Motor Corporation. We obtain the problem control §3.4 problem System (THS) (Liu and Peng (2008)) simulator develSolve SOC-constrained shortest path and obtain the vehicle control oped by the Toyota Motor Corporation. We obtain the oped by the Toyota Motor Corporation. We obtain the oped by the Toyota Motor Corporation. We obtain the System (THS) (Liu and Peng (2008)) simulator developtimal vehicle control to minimize the cumulative fuel problem and obtain the vehicle control oped by the Toyota Motor Corporation. We obtain the optimal vehicle control to minimize the cumulative fuel Fig. 1. Flow chart of our proposed method optimal control to minimize the cumulative optimal vehicle control to the cumulative fuel Fig. 1. Flow chart of our proposed method oped by vehicle the under Toyota Motor Corporation. We obtain fuel consumption two constraints. One to restrict the Fig. 1. Flow chart of our proposed method optimal vehicle control to minimize minimize the is cumulative fuel consumption under two constraints. One is to restrict the Fig. 1. Flow chart of our proposed method consumption under two constraints. One is to restrict the Fig. 1. Flow chart of our proposed method consumption under two constraints. One is to restrict the optimal vehicle control to minimize the cumulative fuel difference in the initial and final states of charge (SOCs), consumption under two constraints. One is to restrict the difference in the initial and final states of charge (SOCs), Fig. 1. Flow chart of our proposed method difference in the initial and final states of charge (SOCs), difference in the initial and final states of charge (SOCs), consumption under two constraints. One is to restrict the and the other is to keep the SOC greater than or equal difference in theis initial and final states of charge (SOCs), and the other to keep the SOC greater than or equal utilized dynamic programing (DP) by preparing several and the other is to keep the SOC greater than or equal and other to the SOC greater than equal dynamic (DP) by several difference in theis initial and states of engine charge (SOCs), to aa the constant value for the sake of the starting utilized dynamic programing programing (DP) by preparing preparing several and the other is to keep keep thefinal SOC greater than or or equal utilized to constant value for the sake of the engine starting utilized dynamic programing (DP) by preparing several vehicle controllers. However, the finiteness of the driving to a constant value for the sake of the engine starting utilized dynamic programing (DP) by preparing several to aa the constant value for the the sake ofgreater the engine starting vehicle controllers. However, the finiteness of the driving and other is to keep the SOC than or equal system, the air conditioner, and other electric equipment. vehicle controllers. However, the finiteness of the driving to constant value for sake of the engine starting system, the air conditioner, and other electric equipment. vehicle controllers. However, the finiteness of the driving utilized dynamic programing (DP) by preparing several controllers limits the search space, and therefore, there system, the air airexperiments conditioner, and other electric equipment. vehicle controllers. However, the finiteness of the driving system, the conditioner, and other electric equipment. controllers limits the search space, and therefore, there to a constant value for the sake of the engine starting We conducted on two driving cycles, the New controllers limits the search space, and therefore, there system, the airexperiments conditioner,on and other electric equipment. We conducted two driving cycles, the New controllers limits the search space, and therefore, there vehicle controllers. However, the finiteness of the driving is a risk of overlooking a better solution. To avoid this, We conducted experiments on two driving cycles, the New controllers limits the search space, and therefore, there We conducted experiments on two driving cycles, the New is a risk of overlooking a better solution. To avoid this, system, the air conditioner, and other electric equipment. European Driving Cycle (NEDC) and Worldwide harmois a risk of overlooking a better solution. To avoid this, We conducted experiments on two driving cycles, the New European Driving Cycle (NEDC) and Worldwide harmois a risk of overlooking a better solution. To avoid this, controllers limits the search space, and therefore, there we utilize bicubic spline interpolation (BSI) and nonlinear European Driving Cycle (NEDC) and Worldwide harmois a risk of overlooking a better solution. To avoid this, European Driving Cycle (NEDC) and Worldwide harmowe utilize bicubic spline interpolation (BSI) and nonlinear We conducted experiments on two driving cycles, the New nized Light vehicle Test Cycle Class 3b (WLTC). Note that we utilize bicubic spline interpolation (BSI) and nonlinear European Driving Cycle (NEDC) and Worldwide harmonized Light vehicle Test Cycle Class 3b (WLTC). Note that we utilize bicubic spline interpolation (BSI) and nonlinear is a risk of overlooking a better solution. To avoid this, programming (NLP) to obtain the optimal vehicle control nized Light vehicle Test Cycle Class 3b (WLTC). Note that we utilize bicubic spline interpolation (BSI) and nonlinear nized Light vehicle Test Cycle Class 3b (WLTC). Note that (NLP) to obtain the optimal vehicle control European CycleCycle (NEDC) and Worldwide harmothe proposed techniques can be applied to other driving programming (NLP) to obtain the optimal vehicle control nized LightDriving vehicle Test Class 3b (WLTC). Note that programming the proposed techniques can be applied to other driving programming (NLP) to obtain the optimal vehicle control we utilize bicubic spline interpolation (BSI) and nonlinear without a controller. We also utilize 0-1 integer linear the proposed techniques can be applied to other driving programming (NLP) toWe obtain the optimal vehicle control the proposed techniques can be to driving aa controller. also utilize 0-1 integer linear nized Light vehicle Test Cycle 3b (WLTC). Note that without cycles. without also utilize 0-1 integer linear the proposed techniques can Class be applied applied to other other driving cycles. without aa controller. controller. We also utilize 0-1 integer linear (NLP) toWe obtain the optimal vehicle control programming (0-1ILP), which can close the gap between cycles. without controller. We also utilize 0-1 integer linear cycles. programming (0-1ILP), which can close the gap between the proposed techniques can be applied to other driving programming (0-1ILP), which can close the gap between cycles. programming (0-1ILP), which can close the gap between without a controller. We also utilize 0-1 integer linear Other previous studies (Borrellia et al. (2005) and Lin et the initial and final SOCs, whereas the primitive DP programming (0-1ILP), which can close the gap between Other studies (Borrellia et al. (2005) and Lin et the initial and final SOCs, whereas the primitive DP cycles. previous Other previous studies (Borrellia et al. (2005) and Lin et the initial and final SOCs, whereas the primitive DP Other previous studies (Borrellia et al. (2005) and Lin et the initial and final SOCs, the DP programming (0-1ILP), whichwhereas canwe close theprimitive gap between al. (2003)) have discretized the driving cycle and SOC, and cannot do so easily. Moreover, propose a pre-solve Other previous studies (Borrellia et al. (2005) and Lin et the initial and final SOCs, whereas the primitive DP al. (2003)) have discretized the driving cycle and SOC, and cannot do so easily. Moreover, we propose a pre-solve al. (2003)) have discretized the driving cycle and SOC, and cannot do so easily. Moreover, we propose a pre-solve al. (2003)) have discretized the driving cycle and SOC, and cannot do so easily. Moreover, we propose a pre-solve Other previous studies (Borrellia et al. (2005) and Lin et the initial and final SOCs, whereas the primitive DP al. (2003)) have discretized the driving cycle and SOC, and cannot do so easily. Moreover, we propose a pre-solve al. (2003)) have discretized the driving cycle and SOC, and cannot do so easily. Moreover, we propose a pre-solve 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2018 IFAC 224 2

1.5 2 2 1 1.5 2 1.5 0.5 1 2 1.5

1 0 0.5 1.5 1 0.5 -0.5 0 2 1 0.5 2500 0 -0.5 1.5 0.5 2000 25000 -0.5 25001 0 2000 -0.5 2500 2000 0.5 -0.5 2000 2500 0

2000

-0.5 2500

120

100

1500

120

80

1000

60

100

120

80

100

120

1500

1000

60

80

100

1500

1000

60

80

1500

1000

60

1500

1000

60

120

2000

100

1500

80

1000

Copyright 2018 224 Peer review© under responsibility of International Federation of Automatic Control. Copyright © 2018 IFAC IFAC 224 Copyright © 224 Copyright © 2018 2018 IFAC IFAC 224 10.1016/j.ifacol.2018.10.037 Copyright © 2018 IFAC 224

60

120

100

80

IFAC E-CoSM 2018 202 Changchun, China, September 20-22, 2018Nariaki Tateiwa et al. / IFAC PapersOnLine 51-31 (2018) 201–206

technique to reduce the search space and computation time in 0-1ILP. Our proposed method is summarized in Fig. 1. The remainder of this paper is organized as follows. We briefly describe the THS simulator in Section 2, and propose a method to obtain the optimal vehicle control in Section 3. There are numerous candidates of the optimal control over a cycle, and therefore, we divide the cycle into partitions and obtain the optimal controls for each partition in Section 4. We then describe a remarkable method for reducing the computation time in Section 5, and summarize the results of the numerical experiments in Section 6. 2. SIMULATOR OUTLINE In this section, we briefly outline the THS simulator. Let state be K-dimensional information of the vehicle used in the simulator, such as the engine torque, motor torque, and vehicle speed. We input state and the target torques into the simulator, and obtain state after t seconds as the output (see Fig. 2). Let state space H ⊂ RK be the set of states. Target torques Simulator

State

Target torques

State

Simulator

Target torques Simulator

t seconds Fig. 2. Brief sketch of THS simulator 3. GLOBAL OPTIMAL CONTROL SEARCH USING CONSTRAINED SHORTEST PATH PROBLEM The goal of our research is to obtain the optimal vehicle control over a cycle that satisfies SOC-constraints. There exist numerous candidates of the optimal control over a cycle; therefore, we divide the cycle into partitions and obtain the optimal control for each partition. Based on these local controls, we construct a time-series state transition graph (TSTG), a directed multigraph whose edges represent the transition of the vehicle state in one partition. Next, we obtain the transition over the cycle by solving the SOC-constrained shortest path problem (Gabriel Y. Handler (1980)) in the TSTG. Finally, we trace the shortest path to obtain the optimal vehicle control. 3.1 Discretization of driving cycle and SOC Let T be the total cycle time. We divide the period of a cycle into N partitions, 0 = t0 < t1 < t2 < · · · < tN = T , and we define segment Ln := [tn−1 , tn )∩tZ as a discretized time interval where t indicates the period of the simulator. We also divide the range of SOC into M partitions, 0 = u0 < u1 < u2 < · · · < uM = 100, and we define SOC class Sm := [um−1 , um ) per SOC interval.

space H by the engine state and the amount of SOC; we denote state class Hk1 and Hk0 as the set of states with HV and EV driving modes whose SOC belongs to Sk , w respectively. Note that {Hm ; w ∈ {0, 1}, m ∈ {1, . . . , M }} is a partition of H. Similarly, we denote H 1 and H 0 as the set of states with HV and EV driving modes, respectively. Algorithm 1: Construction of TSTG γ > 0: Parameter related to ΔSOC ρ(v): Set of edges whose head is node v for n = 0 . . . N − 1 do for Hs ∈ Layer n with ρ(Hs ) �= ∅ do Choose an edge e in ρ(Hs ) Let I be final state of edge e We estimate ΔS = {s; the vehicle can travel Ln from I such that ΔSOC is s} sm = min ΔS sM = max ΔS S = {γs; sm < γs < sM , s ∈ Z} ∪ {sm , sM } for s ∈ S do for l ∈ {0, 1} do if P (n, I, s, s + γ, l) is feasible then Let (se , fe , J) be (ΔSOC, ΔFuel, final state) when controlling the vehicle in Ln from I according to the solution of P (n, I, s, s + γ, l). Let Ht in Layer n + 1 be the state class to which J belongs. Generate edge e from Hs to Ht with (se , fe , J). end end end end end 3.3 Construction of Time-series State Transition Graph Let ΔSOC and ΔFuel be the change in the amount of SOC and the fuel consumption in one segment, respectively. Let initial and final state be the state at the beginning and end of a segment. We construct the TSTG as follows. We first set Layer n corresponding to the boundary between Ln and Ln+1 in the TSTG. For each Layer n, we prepare nodes that represent final states of Ln . Next, we generate edges between two adjacent Layers, which represent the differences between states at the beginning and the end of the segment. 





 



3.2 Definition of state classes Let HV and EV driving modes be the driving with and without the engine running, respectively. We divide state 225

  

 





    







 



 

Fig. 3. Edge of segment Ln in TSTG

IFAC E-CoSM 2018 Changchun, China, September 20-22, 2018 Nariaki Tateiwa et al. / IFAC PapersOnLine 51-31 (2018) 201–206

w The final state is J ∈ Hm  when the vehicle travels Ln w from state I ∈ Hm such that ΔSOC is se and ΔFuel is fe (Fig. 3). Each edge can be generated when the set of parameters (se , fe , I, J) is given through solving the problem P (n, I, a, b, w) detailed in Section 4. 

We demonstrate how to generate edges in the TSTG. Let I0 be the state at the time t0 , and H0 be the state class to which I0 belongs. In addition, we prepare a dummy edge whose head is H0 for convenience, and then edges in segments L1 , . . . , LN are generated sequentially as shown in Algorithm 1 above.

203

• fe : ΔFuel of edge e ∈ E • se : ΔSOC of edge e ∈ E • α: Transformation coefficient from the SOC energy to the fuel energy If the amount of SOC at the end of the cycle is larger than that at the beginning, we should underestimate the cumulative fuel consumption, and vice versa. Therefore, we minimize the objective function (a), which represents the SOC-conversed cumulative fuel consumption. We additionally set constraint (b), called a flow conservation, to obtain a path from this 0-1ILP.

3.4 Formulation as constrained shortest path problem We need to obtain the optimal vehicle control that optimizes the fuel efficiency under two SOC-constraints: (1) maintaining the amount of SOC at the beginning of the cycle close to that at the end of the cycle, and (2) maintaining SOC larger than a static value when traveling. The optimal vehicle control can be obtained after solving the following shortest path problem with respect to the fuel consumption in TSTG, which satisfies two conditions: (1� ) the summation of ΔSOC is nearly equal to zero, and (2� ) the summation of ΔSOC is larger than the constant value at every node on the path. We additionally set a dummy node vend and dummy edges from the nodes in Layer N to vend in order to formulate this problem with a single source and single sink. This can be practically formulated as 0-1ILP on the TSTG (G = (V, E)), as shown below: minimize

�

e∈E

subject to

fe xe − α

�

e∈δ(v)

xe −

�

se xe

(a)

e∈E

�

e∈ρ(v)

xe = b(v) (∀v ∈ V ) (b)

� � � �� � � se xe � ≤ ε � � �

(1� )

e∈E

n � �

j=1 e∈Ej

Variables • xe =

se xe ≥ l − S0 (n = 1, . . . , N ) (2� )

� 1 if edge e ∈ E is included in the path 0 otherwise

Fig. 4. TSTG for NEDC where the range of Sm is 1 Fig. 4 shows an example of the TSTG and its optimal solution for NEDC. In our implementation, the driving cycle is divided into accelerating, decelerating, or constantspeed segments. The orange and blue nodes belong to HV and EV driving modes, respectively, and the red path shows the shortest path. After solving this 0-1ILP, we trace the shortest path to obtain the optimal vehicle control. We call this control tracing solution. In the tracing process, we sequentially calculate the target torques in the segment L1 , . . . , LN , which duplicates ΔSOC on the shortest path. However, this operation can incur a gap in the fuel consumption between the shortest path and the tracing solution, which leads to the exact optimal control being overlooked. Therefore, we obtain k tracing solutions corresponding to the top-k shortest paths in TSTG. The k-th shortest path can be obtained by solving 0-1ILP using the additional constraints below: � xe ≤ N − 1 (k � = 1, . . . , k − 1) e∈Uk

• Uk ⊂ E: the set of edges in the k � -th shortest path

Definitions

Fig. 5 shows that the tracing solution of the 15th shortest path has the least SOC-conversed fuel consumption of them.

• N : Number ⎧ of segments if node v ∈ V belongs to Layer 0 ⎨1 • b(v) = −1 if v = vend ⎩ 0 otherwise

En ⊂ E: Set of edges in segment Ln δ(v) ⊂ E: Set of edges whose tail is node v ∈ V ρ(v) ⊂ E: Set of edges whose head is node v ∈ V ε : Acceptable error regarding the cumulative amount of change SOC • S0 : Amount of SOC at the beginning of the cycle • l: Lower bound of battery remaining at the end of each segment

• • • •

226

4. LOCAL OPTIMAL OPERATION SEARCH USING BICUBIC SPLINE INTERPOLATION In the construction of the TSTG, it is necessary to solve the problem P : • P (n, I, a, b, w) is the problem of finding a sequence of target torques to minimize ΔFuel such that the vehicle travels Ln from state I, a ≤ ΔSOC ≤ b, and in which the final state belongs to H w .

IFAC E-CoSM 2018 204 Nariaki Tateiwa et al. / IFAC PapersOnLine 51-31 (2018) 201–206 Changchun, China, September 20-22, 2018

finally generate functions f˜ and s˜ by utilizing the de-Boor algorithm (Carl de Boor (1962)) for BSI.

1st shortest path 15th shortest path

518.85

518.75

518.70

ΔSOC

tracing solution [g]

518.80

518.65

518.46

518.47

518.48

518.49

518.50

518.51

518.52

1

1

0.5

0.5 0

-0.5 2500

518.53

path solution [g]

-0.5 2500 120

2000

Fig. 5. SOC-conversed fuel consumption of 100 shortest paths and that of its tracing solutions When solving P , it is reasonable to assume the following conditions in THS: (a) The engine torque and engine shaft speed strictly determine the fuel consumption and motor torque in the HV driving part. (b) The motor torque is uniquely determined in the EV driving part. (c) The motor torque roughly determines the amount of change in SOC. These assumptions imply that we can estimate ΔFuel and ΔSOC through the determination of the engine torque and engine shaft speed. Therefore, we propose the approximation algorithm to solve P as follows. • Step 1: Divide a segment into HV and EV driving parts (as shown in Fig 7). • Step 2: Optimize the engine torque and engine shaft speed in the HV driving part. • Step 3: Obtain the motor torque that follows the target velocity. Furthermore, we impose the following two constraints (A) and (B) to solve P : (A) There exist at most one HV and EV driving part in a single segment. (B) The engine torque and engine shaft speed are fixed during each HV driving part. Step 1 determines the driving part that satisfies constraint (A), and we solve the following optimization problem P¯ under constraint (B) in Step 2. f (x, y) minimize subject to a ≤ s(x, y) + Sev ≤ b (P¯ ) (x, y) ∈ R2+

• f (x, y) ∗1 : ΔFuel of HV driving part while traveling with engine torque x and engine shaft speed y. • s(x, y) ∗1 : ΔSOC of HV driving part while traveling with engine torque x and engine shaft speed y. • Sev ∗2 : ΔSOC in EV driving part. • a, b: Constant values automatically determined in Algorithm 1. We basically do not specify f and s, and therefore we estimate these functions using BSI. We first set a rectangle domain D ⊂ R2+ as the search space of the engine torque x and shaft speed y such that D is sufficiently large to cover the optimal solution of the problem P¯ . Next, let Ω be the set of lattice points on D. For any lattice point (x, y) ∈ Ω, we calculate the ΔFuel and ΔSOC of the HV driving part from the initial state Ihv while traveling with fixed engine torque x and fixed shaft speed y. We ∗2 If

2 1.5

0

518.60

∗1 If

2 1.5

there exists only an EV driving part, then f = s = 0. there exists only an HV driving part, then Sev = 0.

227

Th sh e en 1500 aft gi sp ne ee d

100

1000

60

The

120

2000

ue

orq ne t

80

100 1500

i

eng

80 1000

60

Fig. 6. Approximate function of Δ SOC based on BSI Fig. 6 shows an example of BSI. The left side of the figure shows the values of ΔSOCs at the lattice points on D. The X-, Y-, and Z-axes represent the engine torque, engine shaft speed, and ΔSOC, respectively. We set D as [60, 120]×[1,000, 2,500] and divide it into six by six to set the lattice points. The right side of the figure shows the surface of the function s˜(x, y) by utilizing the BSI. Because f˜ and s˜ are approximate nonlinear functions of f and s, we can convert P¯ into the following NLP. f˜(x, y) minimize subject to (P˜ ) a ≤ s˜(x, y) + Sev ≤ b (x, y) ∈ D ˜ We solve the problem P by utilizing sequential quadratic programming (Jorge Noceda and Stephen J. Wright (2006)) in Step 2, and obtain the target motor torque through the binary search in Step 3. We finally describe how to divide the segment into HV and EV driving parts in Step 1. Given the problem P (n, I, a, b, w), the initial and final driving modes are automatically determined according to the initial state I and final driving mode w. Therefore, all driving patterns can be classified into four groups ((a)–(d)), as shown in Fig. 7. Note that pattern (e) does not satisfy constraint (A), and thus we exclude such patterns.  
   


  




   


  


  




   


  


 


   

 
   

 


  

     

Fig. 7. Patterns of dividing a segment into HV and EV driving parts If the initial and final driving modes are different, we switch the driving mode only once, the timing of which is determined through the following operations. We first divide Ln into β equal parts and let Λ be the set of dividing points. For each λ ∈ Λ, we calculate Steps 2 and 3 for the driving pattern that switches the driving mode at the time λ. Based on these results, we adopt the best switch timing to minimize the fuel consumption in Λ.

IFAC E-CoSM 2018 Nariaki Tateiwa et al. / IFAC PapersOnLine 51-31 (2018) 201–206 Changchun, China, September 20-22, 2018

205

5. REDUCTION OF SEARCH SPACE We propose a technique for reducing the computation time for constructing a TSTG. The idea is to quickly construct an approximated TSTG and then utilize this graph to construct a reduced TSTG, which is defined as sieved TSTG below. Numerical experiments imply that the edges in each segment have a similar fuel consumption when they have the same ΔSOC and initial driving mode. Therefore, we obtain a graph that approximates the original TSTG through the following operations. In each segment, the edges from the EV state nodes are calculated as follows: Step 1: Choose one representative node Hr0 from H 0 . Step 2: Generate edges from Hr0 . (see the left side of Fig. 8.) Step 3: Copy these edges to other EV state nodes. (see the right side of Fig. 8.)

1 Hi+1

� se

, fe

(

�

,J

)

(se , fe , J )

…

0 Hi−1

�

, fe

(

Hi0 � se

1 Hi−1

…

� se

Hi1

0 Hi+1

0 Hi−1

(

…

Hi0

Copy Edge

Seg


�

,J

)

…

, fe

( 0 Hi+1

�

� se

(se , fe , J ) �

�

,J

)

(se , fe , J ) �

, fe

�

,J

)

(se , fe , J )

� � , fe se

�

,J

)

(

Fig. 9. Imitation Graph (upper) and Sieved TSTG (lower) for WLTC 6. NUMERICAL RESULTS

1 Hi+1

Our proposed algorithm can obtain the optimal vehicle control without the use of a controller, and can be applied not only to WLTC but also to other ones. Moreover, we achieved a reduction in the computation time while maintaining the accuracy of the fuel efficiency, as shown in the experimental results below.

Hi1 1 Hi−1

…

…

…

…

The edges from the HV state are managed in the same manner.

Seg


Fig. 8. Copy edges in Imitation Graph ˜ be the approximated graph above. Let Imitation Graph G ˜ we remove edges that are probably not By utilizing G, included in the SOC-constrained shortest path in the original TSTG, and recalculate only the weights of the remaining edges to construct the TSTG effectively. We denote the TSTG obtained by this method as a sieved TSTG. We first solve the SOC-constrained shortest path ˜ in addition, let v ∗ be the objective value problem in G; ˜ we and set the parameter a ≥ 1. For each edge e˜ of G, obtain the SOC-constrained shortest path through e˜, and let v˜ be its length. If v˜ is greater than or equal to av ∗ , e˜ is probably not contained in the SOC-constrained shortest path in the original TSTG, and thus we remove e˜ from ˜ If we cannot obtain that path, any path through e˜ is G. probably not satisfied based on the SOC-constraints, and thus we also remove e˜. Parameter “a” is applied to avoid removing edges contained within in the SOC-constrained shortest path in the TSTG. We can efficiently determine whether a path exists through each edge by computing ˜ Moreover, the constrained the connected components of G. ˜ can be computed in shortest path through each edge in G parallel. The upper half of Fig. 9 shows an example of Imitation Graph of WLTC, and the lower half shows the Sieved TSTG with parameter a = 1.001. 228

For all numerical experiments, we used a PC server (2.8 GHz Intel Core i9-7960X CPU with 16 cores, 128 GB of memory, and Windows 10 OS) and MATLAB R2017a with Simulink, Parallel Computing, and Optimization Toolboxes. We set the parameters ε = 0.01, α = 16.73, l = 10, β = 16, D = [60, 120]×[1,000, 2,500] and a = 1.001. We divided the driving cycle into accelerating, decelerating, or constant-speed segments. For each m ∈ {1, . . . , 50}, we set the range of Sm = 2. We evaluated both the fuel efficiency and the computation time of tracing solutions for the original and Sieved TSTGs with several initial SOCs and the parameter γ, as summarized in Table 1 and 2. Our reduction method operates effectively in both the graph construction and computation of the shortest paths. For instance, the computation time in (c) is 3.88x faster than that of (a), whereas the fuel efficiencies of both are quite close. In addition, as the parameter γ decreases, we can control ΔSOC more strictly for each segment, which leads to a more desirable fuel efficiency (see the difference between (c) and (d)). Fig. 10 shows the two results of (c) and (e). Because of the constraint in keeping the SOC greater than or equal to l = 10, we can see that solution (e) charged the battery more than solution (c) during the early stage. Instead, both the engine torque and shaft speed of (e) are less than those of (c) in the final part of the cycle, as shown in the upper-right part of Fig. 10. There exists a problem in that our formulated 0-1ILP guarantees that the SOC will be maintained at only ≥ 10% at the end of each segment. Actually, there exists a part in

IFAC E-CoSM 2018 206 Nariaki Tateiwa et al. / IFAC PapersOnLine 51-31 (2018) 201–206 Changchun, China, September 20-22, 2018

Table 1. The optimal tracing solution for WLTC

initial SOC [%] final SOC [%] total fuel consumption [g] total SOC conversed fuel consumption [g]

(a) TSTG initial SOC:16.3 γ:0.2

(b) TSTG initial SOC:16.3 γ:0.4

(c) Sieved TSTG initial SOC:16.3 γ:0.2

(d) Sieved TSTG initial SOC:16.3 γ:0.4

(e) Sieved TSTG initial SOC:12.0 γ:0.2

16.30 16.29 518.21

16.30 16.30 518.88

16.30 16.30 518.34

16.30 16.29 519.01

12.00 12.01 520.42

518.34

518.85

518.36

519.13

520.29

Table 2. Computation time for WLTC (a)

(b)

(c)

(d)

(e)

Construction of Imitate Graph [s] Removing Edges from Imitate Graph [s]

-

-

3,062 15,163

2,172 5,250

3,038 24,745

The number of remained edges

-

-

Construction of (Sieved) TSTG [s] Obtaining 100 Shortest Paths [s] Tracing 100 Paths [s] Total Time [s]

28,079 50,642 837 79,558(22.10h)

18,433 53,126 827 72,386(20.11h)

883 / 31,970 (2.84%) 1,222 178 883 20,508(5.70h)

1,449 / 20,792 (6.97%) 2,186 488 872 10,968(3.05h)

5,382 / 30,595 (17.6%) 6,681 14,318 857 49,639(13.79h)

Fig. 10. the optimal tracing solutions for WLTC: The Torque means engine torque and, the Shaft Speed means engine shaft speed. The gray line shows the target velocity of WLTC. which the SOC is < 10% in solution (e), which is one of our future works. ACKNOWLEDGEMENTS Our research was mainly supported by Toyota Motor Corporation, the Japan Science and Technology Agency (JST), Core Research of Evolutionary Science and Technology (CREST), and JSPS KAKENHI, Grant No. JP 16H01707. REFERENCES Jinming Liu and Huei Peng. (2008) Modeling and Control of a Power-Split Hybrid Vehicle. 2008. IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, Volume 16, No. 6. Francesco Borrellia, Mato Baoti´c, Alberto Bemporadb, Manfred Morari (2005) Dynamic programming for con229

strained optimal control of discrete-time linear hybrid systems Automatica, Volume 41, Issue 10, 1709–1721 Chan-Chiao Lin, Huei Peng, Jessy W. Grizzle. (2003) Power management strategy for a parallel hybrid electric truck IEEE Transactions on Control Systems Technology, Volume 11, Issue 6, 839–849. Gabriel Y. Handler. (1980) A dual algorithm for the constrained shortest path problem. NETWORKS, Volume 10, Issue 4, 293–309. Carl de Boor. (1962) Bicubic spline interpolation. 1962. STUDIES IN APPLIED MATHEMATICS, Volume 41, 212–218. Jorge Noceda and Stephen J. Wright. (2006) Numerical Optimization. Sequential Quadratic Programming, 529– 562, Springer, New York, second edition.