Convex optimization for design and control problems in electromobility
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1 Convex optimization for design and control problems in electromobility - Recent developments through case studies - Nikolce Murgovski Department of Signals and Systems, Chalmers University of Technology Gothenburg, Sweden February 215
2 Outline Powertrain sizing and energy management of hybrid vehicles. Case study 1: Sizing of a fuel cell hybrid vehicle. Optimization methods. CONES: Matlab code for convex optimization in electromobility studies. Case study 2: Battery longevity considerations. Case study 3: Plug-in hybrid electric vehicle (PHEV) in a series configuration. HEV in a parallel configuration. Planetary gear HEV (used in Toyota Prius). N. Scania 215 2/16
3 Powertrain sizing and energy management of hybrid vehicles Hybrid vehicles include one or more energy buffers (battery, supercapacitor, flywheel) to reduce losses. The objective of the energy management controller is to optimally arbitrate power between energy sources, when driving along a driving cycle. Vehicle velocity [km/h] Road altitude [m] Distance [km] Optimal powertrain sizing refers to sizing of energy and power units that decrease vehicle price and allow optimal vehicle operation. Driving cycle: Bus line 17 in Gothenburg. Optimization framework for simultaneous component sizing and energy management of a hybrid city bus. N. Scania 215 3/16
4 75 95 Case study 1: Sizing of a fuel cell hybrid vehicle (FCHV) Torque [Nm] Speed [rpm] Torque bounds Efficiency [%] Quasi-static model of the EM. Electrical power [kw] Original model (speed [rpm]) Approximation (speed [rpm]) Torque [Nm] Approximated EM model. 3 2 Fuel cell hybrid powertrain. EM = electric machine, FCS = fuel cell system, buffer = battery or supercapacitor. Objective: find optimal sizes of buffer and FCS, find optimal power split control, Efficiency [%] Electrical power [kw] Quasi-static model of the FCS. which minimize hydrogen consumption and investment cost. Fuel power [kw] Original model Approximation Electrical power [kw] Approximated FCS model. N. Scania 215 4/16
5 Case study 1: Sizing of a fuel cell hybrid vehicle (FCHV) Optimal results for a FCHV city bus using supercapacitor as an energy buffer: Cost [EUR/km] Parameter Value FCS size 69.3 kw Buffer size.7 kwh Total cost.28 e/km Computational time <1 s Optimal results. Parameter Value Hydrogen price 4.44 e/kg FCS price e/kwh Supercapacitor price 1 e/kwh Yearly travel distance 7 km Bus service period 2 years Yearly interest rate 5 % Prices and bus specifications..3 4 Cost [EUR/km] 1 3 Optimum FCS power [kw] Buffer energy [kwh] Buffer energy [kwh] Optimal cost for different sizes of fuel cell system and electric buffer. The shaded region illustrates infeasible component sizes. FCS power [kw] N. Scania 215 5/16
6 Case study 1: Sizing of a fuel cell hybrid vehicle (FCHV) Power [kw] Buffer state of charge [%] Further details in FCS power Buffer power Time [min] FCS and buffer power trajectories Time [min] Buffer s state of charge trajectory FCS power [kw] FCS s operating points. N. Scania 215 6/16 FCS efficiency [%] Optimal operating points Distribution [%] 7 4 Optimal operating 2 points Efficiency [%] Pack power at terminals [kw] Buffer s operating points. [1] N. Murgovski, X. Hu, L. Johannesson, B. Egardt. Combined design and control optimization of hybrid vehicles. Handbook of Clean Energy Systems. In press. State of charge [%]
7 Optimization methods Heuristic algorithms. Dynamic programming. Convex optimization. N. Scania 215 7/16
8 Optimization methods Heuristic algorithms. Dynamic programming. Convex optimization. Do not guarantee optimality. N. Scania 215 7/16
9 Optimization methods Heuristic algorithms. Dynamic programming. Convex optimization. Do not guarantee optimality. Based on optimal performance. Nonlinear, non-convex, mixed-integer problems. Computation time is exponential to the number of states. Design parameters cannot be included in the optimization. N. Scania 215 7/16
10 Optimization methods Heuristic algorithms. Dynamic programming. Convex optimization. Based on optimal performance. Short computation time. Do not guarantee optimality. Based on optimal performance. Nonlinear, non-convex, mixed-integer problems. Computation time is exponential to the number of states. Computation time is polynomial to the number of states. Design parameters can be included in the optimization. Approximations to convexify the problem. Cannot handle integer decisions. Design parameters cannot be included in the optimization. N. Scania 215 7/16 x 2 +y 2 y x
11 Optimization methods Heuristic algorithms. Dynamic programming. Convex optimization. Synergy between convex optimization and dynamic programming. Based on optimal performance. Short computation time. Do not guarantee optimality. Based on optimal performance. Nonlinear, non-convex, mixed-integer problems. Computation time is exponential to the number of states. Computation time is polynomial to the number of states. Design parameters can be included in the optimization. Approximations to convexify the problem. Cannot handle integer decisions. Design parameters cannot be included in the optimization. N. Scania 215 7/16 x 2 +y 2 y x
12 CONES: Matlab code for convex optimization in electromobility studies CONES: Convex programming framework in electromobility studies. Optimization examples with realistic vehicle design and control problems. Available online Coded in Matlab. Uses CVX, a Matlab-based modeling system for convex optimization. Examples are continuously added for powertrain design and energy management of electrified vehicles. N. Scania 215 8/16
13 Case study 2: Battery longevity considerations Consider A123 battery cell. Open circuit voltage is approximated as affine in state of charge. Degradation with respect to cell current 2 (C-rate) [1]. State of charge [%] Number of cycles until end of life Internal cell power [W] Number of cycles until end of life vs. cell power. Open circuit voltage [V] State of health derivative [1/s] Original model Affine approximation Operational region x 1 6 Battery cell open circuit voltage..8 Original model Piecewise affine approximation Internal cell power [W] Derivative of battery cell state of health. [1] Wang J, Liu P, Hicks-Garner J, Sherman E, Soukiazian S, Verbrugge M, Tataria H, Musser J, Finamore P. Cycle-life model for graphite-lifepo4 cells. J. Power Sources 211;196: N. Scania 215 9/16
14 99 Case study 2: Battery longevity considerations Optimal results for a FCHV city bus using A123 battery as an energy buffer. Parameter Value FCS size 44.1 kw Buffer size (usable) 4.4 kwh Total cost.24 e/km Computational time <1 s Optimal SOC trajectory [%] Optimal results without battery SOH model. Parameter Value FCS size 47.2 kw Buffer size (usable) 11 kwh Total cost.29 e/km Computational time 1 s Optimal results with battery SOH model and no replacements. SOC limits Without SOH model With SOH model Time [min] Optimal SOC trajectory for a battery with and without SOH model. State of charge (SOC) [%] Parameter Value Hydrogen price 4.44 e/kg FCS price e/kwh Battery price 9 e/kwh Yearly travel distance 7 km Bus service period 5 years Yearly interest rate 5 % Prices and bus specifications Battery Efficiency [%] Power limits Operating points, without SOH model Operating points, with SOH model Power at cell terminals [kw] Optimal operating points for a battery with and without SOH model N. Scania 215 1/16
15 Case study 2: Battery longevity considerations Replacing the battery incurs additional costs. (Although, in certain cases it might be optimal to replace the battery several times [3].) The supercapacitor is a better alternative for this FCHV..45 Cost [EUR/km] Number of battery replacements Cost vs. number of battery pack replacements. Total cost Cost for hydrogen Cost for battery Cost for FCS Details on convex modeling and more results in [1] L. Johannesson, N. Murgovski, S. Ebbessen, B. Egardt, E. Gelso, J. Hellgren. Including a battery state of health model in the HEV component sizing and optimal control problem. IFAC Symposium on Advances in Automotive Control, Tokyo, Japan, 213. [2] X. Hu, L. Johannesson, N. Murgovski, B. Egardt. Longevity-conscious dimensioning and power management of the hybrid energy storage system in a fuel cell hybrid electric bus. Journal of Applied Energy, vol. 137, pp , 215. [3] N. Murgovski, L. Johannesson, B. Egardt. Optimal battery dimensioning and control of a CVT PHEV powertrain. IEEE Transactions on Vehicular Technology, vol. 63, no. 5, pp , 214. N. Scania /16
16 9 Case study 3: Plug-in hybrid electric vehicle (PHEV) in a series configuration Dual buffer consisting of Saft VL 45E battery and Maxwell BCAP2 P27 supercapacitor. Can charge at 7 bus stops for 1 s, or 1 min before starting the route. Efficiency [%] Torque [knm] Torque limits Auxiliary load Fuel tank Ultracapacitor Buffer ICE Battery EGU Electric grid GEN EM Plug-in HEV powertrain in a series configuration. EGU = Engine generator unit, GEN = Generator Generator power [kw] Engine generator unit (EGU) Efficiency [%] Speed [rpm] Electric machine (EM). Velocity [km/h] Altitude [m] Fast charge docking stations Distance [km] Driving cycle with charging opportunities. N. Scania /16
17 Case study 3: Plug-in hybrid electric vehicle (PHEV) in a series configuration 2 design parameters: battery and supercapacitor size. 2 states: battery and supercapacitor SOC. Magnitude of charging power is an optimization variable. Engine is turned on when demanded power exceeds a certain threshold. Optimal results: Parameter Value Diesel price 1.6 e/l Battery price 5 e/kwh Supercapacitor price 1 e/kwh Yearly travel distance 8 km Bus service period 5 years Yearly interest rate 5 % Maximum charging power 2 kw Prices and bus specifications. Charging scenario 7 chargers 1 charger Supercapacitor energy [kwh].8.4 Usable battery energy [kwh] Total cost [e/km] Diesel fuel consumption [l/km].16 Charging power [kw] Optimal results for the two charging scenarios. N. Scania /16
18 Case study 3: Plug-in hybrid electric vehicle (PHEV) in a series configuration Supercapacitor SOC [%] Infrastructure with 7 chargers Infrastructure with 1 charger Power limits 7 chargers 1 charger Battery SOC [%] s charging intervals min charging interval 2 SOC limits Time [min] Cell power [kw] Optimal buffer operation for the two charging scenarios. The shaded region in the right plots depicts efficiency greater than 9 %. Further details in [1] N. Murgovski, L. Johannesson, A. Grauers, J. Sjöberg. Dimensioning and control of a thermally constrained double buffer plug-in HEV powertrain. 51st IEEE Conference on Decision and Control, Maui, Hawaii, 212. [2] B. Egardt, N. Murgovski, M. Pourabdollah, L. Johannesson. Electromobility studies based on convex optimization: design and control issues regarding vehicle electrification. IEEE Control Systems Magazine, vol. 34, no. 2, pp , 214. N. Scania /16
19 (P)HEV with a parallel powertrain configuration Convex optimization can also be applied to parallel HEVs. Heuristics are used for gear selection. When using continuously variable transmission (CVT), the optimization can also find the optimal gear ratio trajectory. Further details in HEV with a parallel powertrain configuration. ICE = Internal combustion engine. HEV with a continuously variable transmission (CVT). [1] M. Pourabdollah, N. Murgovski, A. Grauers, B. Egardt. Optimal sizing of a parallel PHEV powertrain. IEEE Transactions on Vehicular Technology, vol. 62, no. 6, pp , 213. [2] N. Murgovski, L. Johannesson, B. Egardt. Optimal battery dimensioning and control of a CVT PHEV powertrain. IEEE Transactions on Vehicular Technology, vol. 63, no. 5, pp , 214. N. Scania /16
20 HEV with a planetary gear Convex optimization can also be applied to HEVs with a planetary gear unit. Heuristics are used for engine on/off. Series-parallel HEV powertrain with a planetary gear as a power-split device. Toyota Prius - power split device Further details in [1] N. Murgovski, X. Hu, B. Egardt. Computationally efficient energy management of a planetary gear hybrid electric vehicle. IFAC World Congress, Cape Town, South Africa, 214. N. Scania /16
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