OPTIMAL DESIGN OF HYBRID ELECTRIC VEHICLE FOR FUEL ECONOMY

Transcription

1 OPTIMAL DESIGN OF HYBRID ELECTRIC VEHICLE FOR FUEL ECONOMY By Archit Rastogi Mingxuan Zhang Pulkit Agrawal Vasu Goel ME Winter 2012 Final Report ABSTRACT With the increase in fuel prices and growing concern regarding the availability of oil reserves, we need to find means to reduce fuel consumption in vehicles. Reducing the BSFC (Brake Specific Fuel Consumption) and heat losses between the engine and radiator and other thermal management systems will results in considerable saving of energy. Also reduction in fuel consumption directly correlates to reduction in greenhouse gas emissions. Therefore we seek to investigate and optimize vehicle system propulsion sources and their corresponding thermal management devices in order to maximize the overall fuel economy. Optimization of engine design will have a major positive impact on the overall objective because any improvement in engine design will directly result in better fuel economy and performance. Ideal Atkinson cycle engine has been considered for optimization study subject to packaging and other constraints. The engine variables present some very interesting tradeoffs which are further discussed in detail in the report. 1 P a g e

2 Contents SYSTEM INTRODUCTION... 6 I) ENGINE SUBSYSTEM Pulkit Agrawal ) INTRODUCTION:... 7 ASSUMPTIONS ) NOMENCLATURE ) MATHEMATICAL MODELS... 9 ATKINSON ENGINE... 9 OBJECTIVE FUNCTION CONSTRAINTS SUMMARY MODEL: FUNCTIONAL DEPENDENCY TABLE ) MODEL ANALYSIS MONOTONICITY ANALYSIS DESIGN OF EXPERIMENTS ) NUMERICAL RESULTS MATLAB RESULTS SUMMARIZED RESULT: OPTIMUS RESULTS PARAMETRIC STUDY N- ENGINE SPEED (RPM) ANALYSIS WITH NEW UPPER BOUND ON PARAMETER T 3 (PEAK TEMPERATURE). 21 PARAMETER T 3 WITH RELAXED UPPER BOUND ON F (0.5 FROM 0.2) CONCLUSION II) RADIATOR SUBSYSTEM Vasu Goel INTRODUCTION ) PROBLEM STATEMENT ASSUMPTIONS & METHODOLOGY FOLLOWED ) NOMENCLATURE P a g e

3 3) MATHEMATICAL MODEL A) INEQUALITY CONSTRAINTS B) BOUNDS C) DESIGN VARIABLES D) PARAMETERS E) FEASIBLE POINT F) SUMMARY OF OPTIMIZATION MODEL G) OPTIMIZATION MODEL IN NEGATIVE NULLITY FORM ) MODEL ANALYSIS A) MONOTONICITY TABLE B) WELL BOUNDEDNESS: C) SCALING: D) FIRST ORDER AND SECOND ORDER DOE (see APPENDIX-II for plots): E) ACTIVITY TABLE F) FUNCTIONAL DEPENDENCY TABLE ) NUMERICAL RESULTS A) STARTING POINTS B) DIFFERENT ALGORITHMS: C) OPTIMAL SOLUTION: D) MATLAB RESULTS AND PLOTS E) RESULT ANALYSIS F) PARAMETRIC STUDY (See APPENDIX-II for plots of Parametric Study) ) CONCLUSIONS III) BATTERY SUBSYSTEM Archit Rastogi ) INTRODUCTION ) NOMENCLATURE ) MATHEMATICAL MODEL OBJECTIVE FUNCTION: CONSTRAINTS P a g e

4 DESIGN VARIABLES PARAMETERS MODEL SUMMARY ) MODEL ANALYSIS FUNCTIONAL DEPENDENCY TABLE MONOTONICITY ANALYSIS DESIGN OF EXPERIMENTS ) NUMERICAL RESULTS: SCALING FINAL RESULT : CONCLUSIONS PARAMETRIC STUDIES EFFECT OF MASS EFFECT OF MAXIMUM BATTERY CAPACITY EFFECT OF MAXIMUM DISCHARGING RATE EFFECT OF MINIMUM CHARGING RATE EFFECT OF MINIMUM CHARGING CURRENT EFFECT OF MAXIMUM DISCHARGING CURRENT EFFECT OF CHANGING THE BOUNDS: ACKNOWLEDGEMENTS IV) BATTERY THERMAL MANAGEMENT Mingxuan Zhang ) PROBLEM STATEMENT ) NOMENCLATURE ) MATHEMATICAL MODEL OBJECTIVE FUNCTION CONSTRAINTS DESIGN VARIABLES AND PARAMETERS ) OPTIMIZATION MODEL ANALYSIS MONOTONICITY ANALYSIS: P a g e

5 5) NUMERICAL RESULTS FIRST ORDER RELATION PLOTS: SECOND ORDER: ) SYSTEM-LEVEL TRADEOFFS V) SYSTEM LEVEL OPTIMIZATION ) PROBLEM STATEMENT ) NOMENCLATURE ) MATHEMATICAL MODEL OBJECTIVE FUNCTION: VARIABLES CONSTRAINTS SUMMARY OF THE OPTIMIZATION MODEL: TRADE-OFFS ) NUMERICAL RESULTS US06 DRIVE CYCLE US06 DRIVE CYCLE UDDS (URBAN) DRIVE CYCLE HWFET (HIGHWAY) DRIVE CYCLE RESULT ANALYSIS ) REFERENCES VI) APPENDIX I: A) MATLAB CODE Engine Subsystem (PULKIT GUPTA) B) MATLAB CODE - Radiator Subsystem (VASU GOEL) C) MATLAB CODE Battery Subsystem (ARCHIT RASTOGI) D) MATLAB CODE Battery Thermal Management Subsystem (MINGXUAN ZHANG) VII) APPENDIX II A) Radiator Subsystem: First Order Design of Experiments B) Radiator Subsystem: Second Order Design of Experiments C) Radiator Subsystem: Parametric Study Curves P a g e

6 SYSTEM INTRODUCTION A hybrid electric vehicle system consists of two power sources. One is a conventional internal combustion engine and the second one is battery. For such a system the thermal management system is also quite different from the conventional vehicles. Apart from radiator that is required to dissipate the waste heat from the engine, the battery pack also requires an altogether separate cooling system. This is termed as the battery thermal management system in this study. The system comprises engine, radiator, battery and battery thermal management system. The overall system problem aims to minimize the fuel consumption of a hybrid vehicle and maximum the power output. The power consumed by auxiliary devices like radiator pump, radiator fan reduces the fuel economy of the vehicle as engine has to supply more power and hence requires more fuel. Moreover the heat transfer loss in engine and battery system is pretty significant and has to be minimized in order to increase the fuel economy and performance of the vehicle. For this study, the subsystem level optimization goal for engine subsystem is to minimize the brake specific fuel consumption (bsfc). Subsystem level optimization goal for radiator is to maximize the heat dissipation capacity of the radiator. Subsystem level optimization goal for battery is to maximize the power available from the battery. Similarly, subsystem level optimization goal for battery thermal management system is to minimize the power consumed in the auxiliary fan that is used to cool the battery. 1. Engine Subsystem Pulkit Gupta 2. Radiator Subsystem Vasu Goel 3. Battery Subsystem Archit Rastogi 4. Battery Thermal Management System Mingxuan Zhang 6 P a g e

7 I) ENGINE SUBSYSTEM Pulkit Agrawal 1) INTRODUCTION: This subsystem focuses on minimizing the overall fuel consumption of an Atkinson engine based on ideal thermodynamic Otto cycle optimization. In Atkinson cycle the intake valve closing is delay so as to decrease the compression work. But it also results in backflow of fuel air mixture to the intake manifold and thus the energy content inside the cylinder decreases which lead to lower peak pressure and temperature and hence lower expansion work. This tradeoff in the intake valve closing makes it and interesting problem to study for optimization. Variables Bore, Stroke, Compression Ratio, k (point of IVC), f (residual mass fraction) Some of the expected tradeoffs are: 1) Stroke On increasing the stroke the power output will increase and thus BSFC will decrease. But increasing the stroke also increases the Mean piston speed, which is directly proportional to FMEP and thus friction losses increases which results in decrease in power output. Also at low speed heat losses increases substantially while at high speed friction losses increases. Thus an optimum speed needs to be obtained for maximum efficiency. 2) Bore The bore to stroke ratio has to been in a certain range. If the ratio is one then the design is called as square cylinder design and if it is greater than 1 it is over square design. Thus the tradeoffs involved in bore size design will be very similar to the one discussed above for stoke length. 3) On closing the intake valve late, compression work decreases but on the other hand net heat content inside cylinder decreases because some of the mass backflows in the intake manifold. Thus optimum IVC point has to be determined for maximum power output. 4) On increasing the trapped residual mass fraction the intake temperature increases which leads to higher peak temperature and thus more work output. But also the residual mass occupies space inside the cylinder and reduces the amount of fresh charge inside the cylinder. ASSUMPTIONS 1) The engine model considered is based on ideal gas cycle for simplicity and thus the results obtained may be different from actual engine data. 2) The stress on the mechanical parts due to temperature and pressure has been neglected for the sake of simplicity. 3) The value of gamma has been assumed to be constant, 1.4, for a mixture of ideal air and fuel (iso-octane in this case). In reality gamma is a function of temperature. 7 P a g e

8 2) NOMENCLATURE Variables Symbol Description Units B Bore Diameter m CR Compression Ratio ---- f Residual Mass Fraction ---- k Optimal Intake valve closing ---- S Stroke m Other Intermediate Variables Symbol Description Units A Area of combustion chamber m 2 FMEP Friction Mean Effective Pressure kpa h Convective Heat Transfer coefficient kj/k-m 2 m Total mixture mass in cylinder kg mf mass of fuel in cylinder kg P b Brake Power kw U p Mean Piston Speed m/s Vc Clearance Volume m3 Vd Sweep Volume m3 Vo Volume at Intake Valve closing m3 Wg Gross Work kj Wnet Net Work output kj Wpump Pumping work kj Parameters Symbol Description Value Units AFR Air to Fuel Ratio nc No. of cylinders 4 unit nr No. of revolutions per cycle 2 rev/cycle N Revolutions per minute (RPM) 3000 rev/min Pex Exhaust pressure 150 kpa Pin Intake Pressure 100 kpa Qlhv Lower Heating Value of fuel kj/kg Qloss Heat loss to the walls 1.5 kj/kg R Universal gas constant kj/kg-k Tcool Coolant Temperature 353 Kelvin 8 P a g e

9 Tex Exhaust Temperature 450 Kelvin Tin Intake Temperature 300 Kelvin ϒ Ratio of Specific Heat Table 1: Engine Subsystem: Nomenclature 3) MATHEMATICAL MODELS ATKINSON ENGINE Fig. 1: Atkinson Cycle Here the performance of an air standard Atkinson cycle with heat-transfer loss, friction is analyzed. According to (P-V) diagram in figure 1 and 2, process (1-2) is an adiabatic (isentropic) compression then heat is added in process (2-3) at a constant volume. Process (3-4) is an adiabatic (isentropic) expansion and the last process (4-1) is heat rejection which takes place at constant pressure. Point 1 shows the Intake Valve Closing and it can be written empirically in terms of Vc and Vd+ Vc. Vo = k (Vc+Vd) + (1-k)Vc, where k = 0 to 1 The sweep volume or displacement volume of engine cylinder depends on Bore and Stroke. The clearance volume depends on the displacement volume and compression ratio. Larger the bore and stoke, more will the displacement volume. Mean piston speed is given by: V d = 2 B S 4 U p, V c = 2SN Work from state 0-1: 0W1 P ( ) in V0 V1 V d CR 1 9 P a g e

10 The temperature at state 1 is calculated based on residual mass fraction and Temperature of Intake and Exhaust manifold. The residual trapped mass increases the temperature of the intake air. Pin 1 T1 (1 f ) Ti fte[(1 )(1 ) P ex The total mass of mixture trapped inside the cylinder can be calculated at state 1 using ideal gas law. PV mrt Process 1-2 is reversible adiabatic process and thus the temperature and pressure at state 2 can be found using ideal gas law for isentropic process. ( V ) T2 T1 V 1 1 1, c ( V ) P2 P1 V Compression work done from state 1-2 can be calculated using the following: c W mc ( ) v T T Process 2-3 is heat addition at constant volume. The total heat released by the working fluid is given by the mass flow rate of fuel times the Lower Heating value of fuel. But some of the heat released by the fuel is lost by the cylinder to the radiator through the walls. Temperature at state 3 can be calculated from the difference between the heat released by the fuel and heat loss to the walls. The coefficient of convective heat transfer has been calculated using Woschni s equation. Q Q mc T T, Qadd m f QLHV ( ) add loss v 3 2 Q A h( T T ), loss p gas walls hb U pb m a( ), k h 3.26B P T U There will be pumping losses or pumping work because the engine is throttled and the exhaust pressure is higher than the intake pressure. W V ( P P ) pump d in ex Wnet Wgross Wpump IMEP is defined as the net indicated work divided by displacement volume. 10 P a g e

11 Friction losses inside the cylinder cannot be modeled empirically as it is specific for each engine and depends on a lot of factors like the number of piston rings, lubrication, stroke, operating RPM etc. FMEP can be approximated very close to the actual value using the equation found online. FMEP IMEP( U p ) The actual FMEP model is defined by the modified Chen Flynn (1965) correlation seen below. Where A cf, B cf, C cf, Q cf are the Chen Flynn coefficients inputted by the user, P max is the peak cylinder pressure, S fact is the speed factor, RPM is the engine speed and stroke is the cylinder stroke. [ ( ) ( ) ( ) ] On increasing the stroke and RPM, mean piston speed increases and so does the mean piston speed which reduces the net output at the shaft and thus brake power decreases. BMEP IMEP FMEP Brake Power is given by: P b Vd BMEP nc N n 60 r OBJECTIVE FUNCTION The objective function is to minimize the Brake Specific Fuel Consumption of Engine subject to the geometric constraints of engine design. mf min BSFC P CONSTRAINTS 1) More the size of bore, more will be the power produced and but the weight of the engine will also increase which will result in lower fuel economy. There the size of Bore diameter must have an upper and lower bound. B 2) The Bore diameter will also be limited by the length of the engine as a packaging constraint. The length of engine should not be more than 400mm. The minimum distance between two cylinders is taken as 20% of the bore diameter. b 11 P a g e

12 1.2 nc B 3) Ratio of Bore to Stroke has an upper bound and lower bound. If the bore to stroke ratio is 1 then engine is said to be having a square cylinder. For B/S > 1, cylinder is called oversquare. 0.7 B/S 4) Higher the compression ratio, more will the peak pressure inside the cylinder and thus more work and power will be produced. But with higher peak pressure, knocking inside the cylinder increases. Therefore Compression Ratio must have a upper and lower bound. 7 CR 5) The knock limited compression ratio is given by Heywood using the following equation: CR 6) The value of k, which gives the point of optimal intake Valve Closing, must be between zero and one. k 7) The residual mass fraction, f, trapped inside the cylinder after exhaust valve closing cannot be more than 20%. f 8) The mean piston speed has an upper and lower bound as given in the book, Fundamentals of Internal Combustion Engines by John Heywood. U p SUMMARY MODEL: min BSFC f(b, S, CR, k, f) subject to: g1: B g2: 1.2(nc/2)*B g3: 85 - B g4: B 1.4S g5: 0.7S B g6: 2SN 15 g7: 8 2SN g8: CR 14 g9: 7 CR g10 CR B g11: k -1 g12: -k g13: f g14: - f 12 P a g e

13 FUNCTIONAL DEPENDENCY TABLE B S CR k f Pin Pex Tin Tex N nc nr Qloss Qlhv R ϒ Obj. Func. g1 X X X X X X X X X X X X X X X X X g2 X X g3 X g4 X X g5 X X g6 X X g7 X X g8 g9 X X g10 X X g11 g12 g13 g14 X X X X Table 2: Engine Subsystem: Functional Dependency Table 13 P a g e

14 4) MODEL ANALYSIS MONOTONICITY ANALYSIS B S CR k f Obj. Func. - U U U + g1 + g2 + g3 - g4 + - g5 - + g6 + g7 - g8 + g9 - g g11 + g12 - g13 + g14 - Table 3: Engine Subsystem: Monotonicity Analysis The monotonicity table shows that all the variables are bounded both above and below by atleast one constraint and hence the optimization problem is well bounded. ACTIVE CONSTRAINT From the monotonicity table, we can see that the objective function is increasing w.r.t the variable, f. g14 is the only constraint that bounds f from below. Therefore constraint g14 is active. CONSTRAINT REDUNDANCY Objective functionis decreasing w.r.t B (Bore diameter).therefore one out of g1, g2, g4 and g10 constraints must be active. Taking g2 to be active, and n c = 4 cylinders, the value of B come out to be 166. Constraint g1 imposes a stricter constraint and thus constraint g2 is redundant. The monotonicity w.r.t the other three variables cannot be determined because the function is giving different trends of graph in DOE for different nominal values. 14 P a g e

15 SCALING OF VARIABLES To ensure that the model runs smoothly and reaches optima, all the variables were scaled to be between 0-1. All variables except CR were already between 0-1. Therefore CR was also scaled to be in-between 0.07 to OPTIMUS MODEL OPTIMUS model was created with intent to verify the results. Fig. 2: Engine Subsystem: Optimization model in Optimus The above OPTIMUS model shows the four input variables and the objective function in output along with 14 constraints. MATLAB code is executed by the central Action 1 block in the figure shown. DESIGN OF EXPERIMENTS Design of experiments was performed using Matlab to see how the function behaves with respect to the variables. Both 1 st order and 2 nd order DOE were performed and the results obtained are shown below. 1 ST ORDER DOE Nominal Values- x_nom = [ ] 15 P a g e

16 225 Objective v/s B 230 Objective v/s S B Objective v/s CR CR Objective v/s f S Objective v/s k k f Fig. 3: Engine Subsystem: First order DOE 2 ND ORDER DOE Earlier in the progress report, 2 nd order DOE was shown using MATLAB plots. OPTIMUS provides better plots with more flexibility to study the 2 nd and higher order interactions between the variables. A slider bar easily allows us to observe how the plot of objective function with 2 variables changes on changing some other 3 rd or 4 th variable. Therefore DOE was again carried out using OPTIMUS. The plots are shown below. 16 P a g e

17 Fig. 4: Engine Subsystem:Second order DOE 5) NUMERICAL RESULTS MATLAB RESULTS The optimization was done using medium-scale: SQP (Sequential Quadratic programming), Quasi Newton, line search algorithm and also Interior-point on Matlab using fmincon function. Results are shown below for different starting point using both of these algorithm. 17 P a g e

18 Initial Guess Algorithm Optimized Value of variables Obj. Fun. X0= [B S CR k f] B S CR k f BSFC No. of iteration [ ] SQP Interiorpoint [ ] SQP Interiorpoint [ ] SQP Interiorpoint [ ] SQP Interiorpoint [ ] SQP Interiorpoint Table 4: Engine Subsystem: Starting Point Results RESULT ANALYSIS Five different starting points or initial guess gave similar results. Also two different algorithms were used and both resulted in exactly the same solution. The solution converged and all constraints were satisfied. Local minimum was found that satisfied all the constraints. Also the value of eigenvectors of Hessian for all the above optimization cases were greater than or equal to zero. Thus, this is an optimum point and the Hessian is positive semi definite. Two constraints are active. Variable B is at the lower bound and hence constraint g1 is active. Constraint g14 is also active as expected from monotonicity analysis. Also the value of Lagrange multipliers (lambda) from fmincon solver confirms that constraint g1 and g14 is active. Lambda for g1 is and for g14 is Large positive value of lambda at these two lower bounds suggests that it is active. First order optimality point value from fmincon solver: firstorderopt: e-004, is very small and positive and suggests that KKT conditions are satisfied. Thus it is an optimum solution. SUMMARIZED RESULT: Bore = 85 cm Stroke = 61cm CR = 8.7 k = 0.21 f = 0 BSFC = 197 g/kw-hr 18 P a g e

19 OPTIMUS RESULTS Exp.Num Bore Stroke CR k f BSFC Results 94 mm 80mm g/kW-h Table 5: Engine Subsystem: Results from Optimus The result from OPTIMUS shows that the solutions converged to a BSFC value of 201 after 56 experiments. The variable f is 0 as expected from the monotonicity analysis. The results from MATLAB & OPTIMUS are both very close and the variables are also mostly same. Fig. 5: Engine Subsystem: First order DOE 19 P a g e

20 PARAMETRIC STUDY N- ENGINE SPEED (RPM) Parameter Objective func. Optimized Value of variables N (RPM) BSFC (g/kw-h) B S CR k f Table 6: Engine Subsystem: Parametric Study for Engine Speed (rpm) On changing the parameter N(engine speed, rpm), optimum changes. BSFC increases with parameter N while CR first increases till 4500 rpm and then again decreases at higher rpm. BSFC increases because friction losses (FMEP) and heat losses increases which leads to more fuel consumption. 250 Objective v/s N 0.12 B v/s N S v/s N CR v/s N k(ivc) v/s N f(residual mass fraction) v/s N Fig. 6: Engine Subsystem: Parametric study with respect to engine speed (rpm) 20 P a g e

21 ANALYSIS WITH NEW UPPER BOUND ON PARAMETER T 3 (PEAK TEMPERATURE) T 3 (Peak Temp.) Objective func. Optimized Value of variables Upper Bound BSFC (g/kw-h) B S CR k f Table 7: Engine Subsystem: Parametric study for upper bound on peak temperature Earlier there was no bound on the peak temperature inside the cylinder. This parametric study is to see the effects of introducing a bound on the peak temperature on the objective function and variables. Results show that with higher upper bound on peak temperature we get better BSFC values. The peak temperature inside the cylinder is limited by the knocking which is measured by the ringing intensity. This new bound on the model is actually a nonlinear constraint.t 3 f(b, S, CR, k, f). It was not included in the original model to keep things simplified and was mentioned at starting in the assumptions. We can also see the for the value of upper bound from 2500 to 3100 K, the variable f achieves optimum at its upper bound. 400 Objective v/s T3 2 B v/s T B v/s T CR v/s T k(ivc) v/s T f(residual mass fraction) v/s T Fig. 7: Engine Subsystem: Parametric study with respect to peak temperature T 3 21 P a g e

22 PARAMETER T 3 WITH RELAXED UPPER BOUND ON F (0.5 FROM 0.2) T 3 (Peak Temp.) Objective func. Optimized Value of variables Upper Bound BSFC (g/kw-h) B S CR k f Table 8: Engine Subsystem: Parametric study for peak temperature T 3 with relaxed upper bound on f On relaxing the upper bound of variable f, we can see that the optimized value of BSFC decreases for the same limit on peak temperature as in previous case. The trends of the plot are the same as in previous case, but the magnitudes have changed. Residual mass fraction decreases at higher peak temperature which is justified because if f increases then quantity of fresh charge decreases which results in lower net energy input in the system and consequently lower peak temperature and vice versa. 300 Objective v/s T3 2 B v/s T B v/s T CR v/s T k(ivc) v/s T f(residual mass fraction) v/s T Fig.8: Engine Subsystem: Parametric study for peak temperature T 3 with relaxed upper bound on f 22 P a g e

23 CONCLUSION The optimized BSFC value of 197 g/kw-h is very practical and physically intuitive. The parametric tradeoffs were as expected and are summarized again below. BSFC increases because friction losses increase at higher rpm. CR decreases till 3500 rpm and then again increases. At higher engine speed IVC is further delayed. Residual mass fraction is always at its lower bound because of a active constraint. Higher peak temperature = more Work output = more Power = lower BSFC. Bore & Stroke does not change with T 3. Compression Ratio increases till 3100K and then again decreases. Intake valve closing is later with higher peak temperature. Variable f is at its upper bound till 3100 K and g14 constraint is no longer active. BSFC decreases from 300 to 280 g/kw-hr on relaxing the upper bound on f. Compression Ratio is constant till 3000K and increases for higher peak temperature on relaxing the upper bound on f. Variable reaches a maximum of 0.32 on relaxing the upper bound on f. 23 P a g e

24 II) RADIATOR SUBSYSTEM Vasu Goel INTRODUCTION A car radiator is a type of heat exchanger. It is designed to transfer heat from the hot coolant that flows through it to the air blown over it by the radiator fan. Most modern cars use aluminium radiators. These radiators are made by brazing thin aluminium fins to flattened aluminium tubes. The coolant flows from the inlet to the outlet through many tubes mounted in a parallel arrangement as shown in Fig. 9. The fins conduct the heat from the tubes and transfer it to the air flowing over the radiator tubes. As the coolant flows through the tubes of the radiator, heat is transferred through the fins and tube walls to the air by conduction and convection. The heat is rejected from the coolant to the air through three major thermal resistances: o Convection from the coolant to the inner surface of the tube. o Conduction through the tube wall. o Convection from the outer surface of the tube and fins to the air flowing outside. Fig. 9: Radiator Subsystem: A CAD model of radiator and a schematic of radiator tube and fins 1) PROBLEM STATEMENT About 33% of the power generated by the engine through combustion is lost as heat. Insufficient heat dissipation can result in the overheating of the engine, which leads to the breakdown of lubricating oil, metal weakening of engine parts, and significant wear between engine parts. To minimize the stress on the engine as a result of heat generation, automotive radiators must be designed so that they are compact and still maintain high levels of heat dissipation rate. Radiator should be capable of removing the heat generated by heat sources but its size is limited by the packaging space inside the vehicle. The present work aims at optimizing the design of car radiator so as to maximize the heat dissipation rate making the best use of available space and resources. 24 P a g e

25 Fig. 10: Arrangement of a Radiator in a Vehicle ASSUMPTIONS & METHODOLOGY FOLLOWED a) The temperature of the ambient air has been assumed to be constant at 25 ºC. However in real conditions it can vary between -30 ºC to 50 ºC. b) The volumetric flow rate of air and convective heat transfer coefficient of air are taken as constants for the sake of simplicity of the model. c) ε-ntu model [4] for cross flow heat exchangers is used to model the radiator. d) Infinite length fin approximation [4] has been used to model the fins. Constraints have been incorporated accordingly such that these models hold true. e) The thermal and mechanical stresses acting on various mechanical parts associated with radiator have not been taken into account. This work only aims to maximize the heat dissipation rate of radiator. 2) NOMENCLATURE Symbol Description Unit A Overall heat transfer area m 2 A a Surface area of radiator that comes in contact with air m 2 A c Surface area of radiator that comes in contact with coolant m 2 A c_fin Cross Section area of each fin m 2 A s_fin Surface are of each fin m 2 c a Specific heat capacity of air J/kg.K C a a c a Heat capacity of air J/K.s = W/K c c Specific heat capacity of coolant J/kg.K C c c c c Heat capacity for coolant J/K.s = W/K 25 P a g e

26 C max Larger of the two heat capacities amongst C a and C c J/K.s = W/K C min Smaller of the two heat capacities amongst C a and C c J/K.s = W/K C r Heat capacity ratio D h Hydraulic diameter m F gap Gap between two consecutive fins m h a Convective heat transfer coefficient of air W/m 2 K h c Convective heat transfer coefficient of coolant W/m 2 K H fin Fin height m H tube Tube height m ITD Initial temperature difference K or ºC k a Thermal conductivity of air W/m.K k c Thermal conductivity of coolant W/m.K k fin Thermal conductivity of material of fin W/m.K L rad Length of radiator m a Mass flow rate of air Kg/s c Mass flow rate of coolant Kg/s N fins Number of fins per tube NTU Number of transfer units N tube Number of coolant tubes Nu Nusselt number P fin Perimeter of fin base m Pr Prandtl number P tube Perimeter of tube m P w Wetted perimeter m Q Heat dissipation rate from radiator J/s or W Re Reynolds number T a Ambient air temperature K T c Temperature of coolant coming out of engine core K T fin Fin thickness m U Overall heat transfer coefficient W/m 2 K v a Velocity of air m/s v c Velocity of coolant m/s VOLa Volumetric flow rate of air m 3 /s VOLc Volumetric flow rate of coolant m 3 /s W fin Fin width m 26 P a g e

27 W rad Radiator width m W tube Tube width m ε Overall Effectiveness ε f Fin Effectiveness η o Overall surface efficiency associated with one tube and fin assembly μ a Coefficient of dynamic viscosity of air Pa.s μ c Coefficient of dynamic viscosity of coolant Pa.s ρ a Density of air kg/m 3 ρ c Density of coolant kg/m 3 Table 9: Radiator Subsystem: Nomenclature Subscript a is for Air Subscript c is for Coolant represents a dimensionless quantity 3) MATHEMATICAL MODEL Effectiveness NTU (ε- NTU) Method will be used to build a Mathematical Model of heat transfer in a radiator. ε- NTU Model is defined by following equations. a) Initial Temperature Difference (ITD) ITD = Coolant Temperature - Air Temperature b) Overall Heat Transfer Coefficient: The overall thermal resistance present in the system: c) Number of Transfer Units (NTU) C min = Cc or Ca whichever is smaller. d) ε- NTU Relation for a Cross-flow heat exchanger with unmixed fluids: A mathematical expression of heat exchange effectiveness vs. the number of transfer units: ( )( ) where, e) Heat Transfer Equation: The rate of conductive heat transfer Q 27 P a g e

28 Apart from the above equations the other intermediate equations used to build the complete model are as given below. a) Reynolds Number: A dimensionless modulus that represents fluid flow conditions: ( ) ρ v D μ b) Nusselt Number: It gives the ratio of convective heat transfer to conductive heat transfer over a boundary. Larger Nusselt number corresponds to greater convective heat transfer which is a characteristic of turbulent flow. ( ) c) Prandtl Number: A dimensionless modulus that relates fluid viscosity to the thermal conductivity, a low number indicates high convection. ( ) d) Relation between Nusselt Number and Reynold s Number: For turbulent flow (Reynold s number > 2300), Dittus Boelter equation can be used to establish a relation between Nusselt number and Reynolds number. This equation can be used to calculate the coefficient of heat transfer for fluids in turbulent flow: e) Hydraulic Diameter (D h ): Parameter used to equate any flow geometry to that of a round pipe. D h A) INEQUALITY CONSTRAINTS a) Area of the finned surface must be larger than unfinned area of tube [4]. Number of Fins Surface area of one fin Tube Perimeter (Radiator Length Number of fins Fin Thickness) g1: P tube (L rad N fin T fin ) - N fin A s_fin 0 where, P tube = 2 (H tube + W tube ) A s_fin = P fin H fin P fin = 2 (W fin + T fin ) b) An infinite fin approximation has been used in this study. For an infinite fin approximation to be valid, the fin should be able to dissipate more than 99 percent of the maximum possible heat that can be dissipated [1]. 28 P a g e

29 where, P fin = 2 (W fin + T fin ) A c_fin = W fin T fin g2: 2.65 m H fin 0 c) For use of fins to be justified the fin effectiveness should be greater than 2 [4]. g3: 2 ε f 0 d) NTU should be greater than 0.25 for effective heat dissipation [4]. g4: 0.25 NTU 0 e) Gap between two fins should be atleast 1.5 times the thickness of fin to allow for effective heat transfer [5]. Gap between two fins (Thickness of Fin) 1.5 g5: 1.5 T fin F gap 0 f) Total width of fins and gaps between two fins should be less than length of radiator. Number of Fins Fin Thickness + Number of Gaps Gap between two fins Length of Radiator g6: N fins T fin + (N fins 1) F gap - L rad 0 g) Total height of fins and tubes should be less than radiator width. Number of Coolant Tubes Tube Height + (Number of Coolant Tubes 1) Fin Height Width of radiator g7: N tube H tube + (N tube 1) H fin W rad 0 B) BOUNDS To avoid numerical problems like division by zero in the code, we require strict inequalities for design variables to be greater than zero. So an arbitrary small positive number ϵ is used in the following inequalities in place of zero. Most four-cylinder automobiles, depending on their size, have radiator cores that vary from19'' X 11.5'' X 0.7'' to 27'' X 17'' X 0.9'' [4] h) Length of Radiator: For length I have taken the lower limit of the above mentioned range as my upper bound so as to find maximum possible heat dissipation capacity of radiator with minimum length. Length of Radiator 0.50 meter g8: L rad P a g e

30 i) Width of Radiator: As per the above mentioned range width has been taken between 11.5 and meter Width of Radiator 0.45 meter g9: 0.30 W rad 0 g10: W rad j) Fin Height Height of fin ϵ g11: H fin +ϵ 0 k) Height of Coolant Tube Height of tube 0 g12: H tube +ϵ 0 l) Number of Fins per Tube Number of fins 0 g13: N fins +ϵ 0 m) Number of Coolant Tubes Number of tubes 0 g14: N tubes +ϵ 0 n) Thickness of Fins: The fins cannot be realistically manufactured thinner that seventhousandths of an inch or 0.178mm [2] Thickness of fins g15: T fin o) Gap between Fins Gap between Fins m g16: F gap C) DESIGN VARIABLES 1) Radiator length (L rad ) 2) Radiator width (W rad ) 3) Number of coolant tubes (N tube ) 4) Tube height (H tube ) 5) Number of fins per tube (N fins ) 6) Thickness of one fin (T fin ) 7) Fin height (H fin ) 8) Gap between fins (F gap ) 30 P a g e

31 D) PARAMETERS 1) Ambient Temperature (T a ) = 25 ºC 2) Temperature of Coolant (T c ) = 120 ºC 3) Thermal conductivity of material of fin (K fin )= 180 W/m K 4) Thermal conductivity of coolant (k c )= W/m K 5) Volumetric flow rate of air (VOL air )= 1.1 m 3 /s 6) Volumetric flow rate of coolant (VOL c )= m 3 /s 7) Convective heat transfer coefficient of air (h a ) = 580 W/m 2 K 8) Density of coolant. (ρ c )= kg/m 3 9) Dynamic viscosity of coolant (μ c )= Pa.s E) FEASIBLE POINT A feasible set of solution which satisfies all the constraints is given below. x=[.450;.350; ; 0.001; 0.02; 150; 15; 0.002]; Design Variable Value Radiator length (L rad ) 45cm Radiator width (W rad ) 35 cm Thickness of One Fin (T fin ) 200 µm Tube height (H tube ) 1 mm Fin Height(H fin ) 20 mm Number of Fins per Tube (N tube ) 150 Number of Tubes (N fins ) 15 Gap between fins (F gap ) 2 mm Table 10: Radiator Subsystem: A feasible point for the problem Values of all the inequality constraints at the above mentioned feasible point are: Constraint Value g g g g g g g Table 11: Radiator Subsystem: Constraints value at the feasible point Since all the constraints are negative, this represents a feasible solution. Value of the objective function at this point: f = Q = 69 KW. So Heat Dissipation Rate Q = 69 KW. 31 P a g e

32 F) SUMMARY OF OPTIMIZATION MODEL rad, W rad, N tube, H tube, N fins, T fin, H fin,f gap ) DESIGN VARIABLES x(1): Radiator length (L rad ) x(2): Radiator width (W rad ) x(3): Thickness of one fin (T fin ) x(4): Tube height (H tube ) x(5): Fin height (H fin ) x(6): Number of fins per tube (N fins ) x(7): Number of coolant tubes (N tube ) x(8): Gap between fins (F gap ) G) OPTIMIZATION MODEL IN NEGATIVE NULLITY FORM min Q f ( L rad, W rad, N tube, H tube, N fins, T fin, H fin,f gap ) subject to g1: P tube (L rad N fin T fin ) - N fin A s_fin 0 g2: 2.65 m H fin 0 g3: 2 ε f 0 g4: 0.25 NTU 0 g5: 1.5 T fin F gap 0 g6: N fins T fin + (N fins 1) F gap - L rad 0 g7: N tube H tube + (N tube 1) H fin W rad 0 g8: L rad g9: W rad g10: 0.30 W rad 0 g11: H fin +ϵ 0 g12: H tube +ϵ 0 g13: N fins +ϵ 0; N fins ϵ I g14: N tube +ϵ 0; N tube ϵ I g15: T fin g16: F gap ϵ a very small arbitrary positive number taken to avoid mathematical problems due to possible division by zero during iterations. 32 P a g e

33 4) MODEL ANALYSIS A) MONOTONICITY TABLE C O N S T R A I N T S B O U N D S L rad W rad T fin H tube H fin N fins N tube F gap f U + + g1 + + g2 + g3 + g4 U + g5 + g g g8 + g9 + g10 g11 g12 g13 g14 g15 g16 Table 12: Radiator Subsystem: Monotonicity Analysis + signifies increasing nature of the constraint with respect to the variable signifies decreasing nature of the constraint with respect to the variable U signifies presence of regional monotonicity in the constraint with respect to the variable Red colored signs show the active constraints B) WELL BOUNDEDNESS: a) Since the objective function value is decreasing with respect to L rad, and g1 and g8 are decreasing with respect to L rad, objective function is well bounded with respect to L rad. b) Since the objective function value is decreasing with respect to W rad, and g9 increasing with respect to W rad, objective function is well bounded with respect to W rad c) The objective function is regionally monotonic with respect to T fin,, initially the objective function decreases with increasing fin thickness upto a particular value and after that the objective function increases. But we have constraints both increasing (g2, g3 and g5) as well 33 P a g e

34 as decreasing (g1 and g15) with respect to T fin, which means that objective function is well bounded with respect to T fin. d) Since the objective function value is increasing with respect to H tube, and g12 is decreasing with respect to H tube, objective function is well bounded with respect to H tube. e) Since the objective function value is decreasing with respect to H fin and g7 is increasing with respect to H fin, objective function is well bounded with respect to H fin. f) Since the objective function value is decreasing with respect to N fins and g6 is increasing with respect to N fins, objective function is well bounded with respect to N fins. g) Since the objective function value is decreasing with respect to N tube and g7 is increasing with respect to N tube, objective function is well bounded with respect to N tube. h) Since the objective function value is increasing with respect to F gap and g5 is decreasing with respect to F gap, objective function is well bounded with respect to F gap. C) SCALING: Scaling has been done using the inbuilt scaling feature of FMINCON toolbox. The scaling command is TypicalX which is set in defined in the customized options setting of FMINCON using optimset. "TypicalX" specifies the expected scaling or order of the optimization solution. Functions such as FMINCON in the MATLAB Optimization Toolbox use this parameter in determining the optimization step size. D) FIRST ORDER AND SECOND ORDER DOE (see APPENDIX-II for plots): First order DOE was performed for the variables with respect to the objective function and constraints. In this only one variable was varied and others were kept fixed. The plots show that objective function is increasing with respect to variables like length of radiator, fin height, number of fins per tube, number of tubes. Also the objective function is decreasing with respect to variables like fin thickness, gap between fins, tube height whereas with respect to fin thickness the objective function shows regional monotonicity. From first order DOE, the problem seems to be well bound to give a solution. Second Order DOE: To further check the well boundedness and nature of design problem second order DOE were also performed by varying two variables at a time keeping others fixed. From second order DOE analysis also the system seems to be well bound and an optimal solution is expected. NOTE: See Appendix-II for first order and second order DOE curves. 34 P a g e

35 B O U N D S C O N S T R A I N T S Optimal Design of Hybrid Electric Vehicle for Fuel Economy E) ACTIVITY TABLE L rad W rad T fin H tube H fin N fins N tube F gap f = g1 * * g2 g3 * g4 g5 * * g6 * A g7 A A g8 * g9 A g10 g11 g12 A g13 g14 g15 * g16 * Table 13: Radiator Subsystem: Activity analysis A Active Constraint a. Constraint g9 is active for W rad b. Constraint g12 is active for H tube c. Constraint g7 is active for H fin d. Constraint g6 is active for N fin e. Constraint g7 is active for N tube f. For L rad either of the two constraints g1 or g8 can be active. From analysis and code results it was found that g8 is active for L rad g. For F gap either of the two constraints g5 or g16 can be active. From analysis and code results it was found that g16 is active for L rad h. For T fin either of the constraints g1, g3, g5, g6 or g15 could be active. However from model analysis and code results it was found that g15 is active for L rad. 35 P a g e

36 F) FUNCTIONAL DEPENDENCY TABLE D E S I G N V A R I A B L E S P A R A M E T E R S L rad W rad T fin H tube H fin N fins N tube F gap K fin K c T air T c VOL a VOL c f g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 g16 Table 14: Radiator Subsystem: Functional dependency table 36 P a g e

37 5) NUMERICAL RESULTS A) STARTING POINTS Five different starting points were chosen: 1) 0.45; 0.35; ; ; ; 50; 41; ) 0.01; 0.31; ; ; 0.42; 0.01; 30; ) 0.21; 0.41; ; ; 0.42; 100; 320; ) 0.21; 0.41; ; ; 0.002; 590; 320; ) 0.27; 0.38; ; ; 0.300; 990; 320; 0.02 The following values were selected for termination criteria 1) Tolerance in X:10^(-8) 2) Tolerance in Constraints: 10^(-8) 3) Tolerance in Function Value: 10^(-8), B) DIFFERENT ALGORITHMS: 1) SQP ALGORITHM seems to be most effective for this optimization problem. It converged in minimum 12 iterations which also seems to be correct as the number of design variables are 8. Start Point No. of Iterations X* Function Value 0.45; 0.35; ; ; ; 0.45; ; 0.001; ; 50; 41; ; ; 31.1; ; 0.31; ; ; ; 0.45; ; 0.001; ; 0.01; 30; ; ; 31.1; ; 0.41; ; ; ; 0.45; ; 0.001; ; 100; 320; ; ; 31.1; ; 0.41; ; ; ; 0.45; ; 0.001; ; 590; 320; ; ; 31.1; ; 0.38; ; ; 0.300; 990; 320; ; 0.45; ; 0.001; ; ; 31.17; Table 15: Radiator Subsystem: Starting point analysis for SQP algorithm 37 P a g e

38 2) ACTIVE SET ALGORITHM failed to give any solution for the second and third start point whereas as seen above SQP algorithm converges from all the start points. However from last start point active-set algorithm converges in 7 iterations whereas SQP algorithm converges in 22 iterations with that start point. Start Point No. of X* Function Iterations Value 0.45; 0.35; ; ; ; 0.45; ; 0.001; ; 50; 41; ; ; 31.1; ; 0.31; ; ; Failed to 0.42; 0.01; 30; Converge 0.21; 0.41; ; ; Failed to 0.42; 100; 320; Converge 0.21; 0.41; ; ; ; 0.45; ; 0.001; ; 590; 320; ; ; 31.1; ; 0.38; ; ; ; 0.45; ; 0.001; ; 990; 320; ; ; 31.17; Table 16: Radiator Subsystem: Starting point analysis for Active-Set algorithm 3) INTERIOR POINT ALGORITHM: Though interior point algorithm converged for every starting point, but it takes a significant number of more iterations than the SQP algorithm. Start Point No. of X* Function Iterations Value 0.45; 0.35; ; ; ; 0.45; ; 0.001; ; 50; 41; ; ; 31.1; ; 0.31; ; ; ; 0.45; ; 0.001; ; 0.01; 30; ; ; 31.1; ; 0.41; ; ; ; 0.45; ; 0.001; ; 100; 320; ; ; 31.1; ; 0.41; ; ; ; 0.45; ; 0.001; ; 590; 320; ; ; 31.1; ; 0.38; ; ; ; 0.45; ; 0.001; ; 990; 320; ; ; 31.17; Table 17: Radiator Subsystem: Starting point analysis for Interior Point algorithm 38 P a g e

39 However from all the three algorithms and from all the starting points same optimal solution is obtained. All the starting points converge to the same set of design variables and function value indicating the presence of Global Convergence. C) OPTIMAL SOLUTION: X = FVAL = The optimized result can be summarized as : Design Variable Value Radiator length (L rad ) 50 cm Radiator width (W rad ) 45 cm Thickness of One Fin (T fin ) 200 µm Tube height (H tube ) 1 mm Fin Height (H fin ) 14 mm Number of Fins per Tube (N tube ) 425 Number of Tubes (N fins ) 31 Gap between fins (F gap ) 1 mm Maximum Heat Dissipation Rate: 107 KW Table 18: Radiator Subsystem: The optimized final result 39 P a g e

40 D) MATLAB RESULTS AND PLOTS Starting_Point = Norm of First-order Iter F-count f(x) Feasibility Steplength step optimality e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-013 Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the selected value of the function tolerance, and constraints are satisfied to within the selected value of the constraint tolerance. 40 P a g e

41 <stopping criteria details> OUTPUT = iterations: 13 funccount: 126 algorithm: 'sequential quadratic programming' message: [1x785 char] constrviolation: e-017 stepsize: 1 firstorderopt: e-013 ans = FVAL = EigenVal_Hessian = 1.0e+008 * EigenVal_Hessian(1) ans = e P a g e

42 Step size First-order optimality Current point Function value Optimal Design of Hybrid Electric Vehicle for Fuel Economy >> EigenVal_Hessian(2) ans = e-008 NonLinear_Constraints = Current Point Current Function Value: Number of variables: 8 Step Size: e Iteration First-order 8 x 105 Optimality: e Iteration Iteration Fig. 11:Radiator Subsystem: Plot of variation of objective function value with each iteration 42 P a g e

43 E) RESULT ANALYSIS As can be seen from the highlighted text of results, a) Result displayed is : Local minima is found that satisfies the constraints b) Number of iterations taken = 13 c) Six eigen values of the Hessian Matrix are positive and two eigen values (the first and second eigen values) are zero. So the Hessain matrix is semi - positive definite and this signifies the presence of possible valley. This is seen to be true while doing the parametric studies. d) The values of all the nonlinear constraints are either zero or negative which means that the optimal solution obtained is feasible. e) Length of radiator and width of radiator converged to the upper bound and gap between two fins converged to the lower bound. This was expected because our objective function is to maximize the heat dissipation capacity which will increase as the length and width of radiator increase. Similarly lesser the gap between two fins, more number of fins can fit along the length of radiator thereby increasing the heat dissipation rate of radiator. F) PARAMETRIC STUDY (See APPENDIX-II for plots of Parametric Study) 1) THERMAL CONDUCTIVITY OF FIN Material &Thermal Conductivity (W/mK) Number of Iterations X* Heat Dissipation (KW) Steel: ; 0.45; ; 0.001; ; ; 93.19; Carbon Steel: ; 0.45; ; 0.001; ; ; 57.55; 0.001; Brass: ; 0.45; ; 0.001; ; ; 32.97; 0.001; Aluminium: ; 0.45; ; 0.001; ; ; 31.1; Copper: ; 0.45; ; 0.001; ; ; ; Table 19: Radiator Subsystem: Parametric study for thermal conductivity of material of fin The maximum heat dissipation rate for different materials is almost the same between 106 KW to 109 KW. However, for each material a different set of optimal design variables is obtained. This shows that 43 P a g e

44 there exists a sort of a valley in the design problem where the minimum function value is same for different sets of design variables. 2) CONVECTIVE HEAT TRANSFER COEFFICIENT OF AIR h air Number of (W/m 2 K) Iterations Heat X* Dissipation (KW) No Feasible Solution Found No Feasible Solution Found Table 20: Radiator Subsystem: Parametric study for convective heat transfer coefficient of air From parametric study we can see that even though the maximum heat dissipation rate remains almost same for various cases shown above but there are differences in the value of number of coolant tubes and the tube height. For h air <200 W/m 2 K, no feasible solution was observed and on increasing h air heat dissipation rate is seen to increase which is expected. 3) AMBIENT AIR TEMPERATURE AND COOLANT TEMPERATURE Parametric studies were also performed by varying temperature of ambient air and temperature of coolant coming into the radiator tubes. As expected the heat dissipation rate of radiator decreases as the temperature of ambient air increases and heat dissipation rate of radiator increases as the temperature of coolant coming from engine core into the radiator tubes increases. NOTE: The plots of parametric studies for radiator subsystem are at the end of APPENDIX II 44 P a g e

45 6) CONCLUSIONS a. The heat dissipation rate of radiator increases with length and width. b. Heat dissipation rate decreases with increase in fin gap. c. Heat dissipation rate increases with increase in number of fins per tube. d. Heat dissipation rate increases with increase in number of tubes. e. Heat dissipation rate decreases with increase in tube height. f. Heat dissipation rate also depends on ambient temperature and decreases with increase in ambient temperature. g. Heat dissipation rate increases with increase in the temperature of coolant coming into the radiator from the engine core. h. Heat dissipation rate also increases with increase in the volumetric flow rate of coolant and air. 45 P a g e

46 III) BATTERY SUBSYSTEM Archit Rastogi Optimal Design of Hybrid Electric Vehicle for Fuel Economy 1) INTRODUCTION Hybrid electric vehicles are those which combines the conventional power source i.e. engine with the alternate electric power source i.e. battery. The presence of electric powertrain ensures high efficiency and low emissions. The battery is used as a power storage device and motor/generator are used in conjunction for the conversion of electric energy to mechanical energy and vice-versa. In a hybrid vehicle, power requirements are met by both engine and battery. When the power command is low; the battery alone drives the vehicle. When the power demand is medium, the battery makes up for the vehicle power fluctuations such that the engine always works in efficient regions thereby lowering the fuel consumption. For high power demands engine and battery both works simultaneously. The battery requirements for the hybrid electric vehicles are manifold. Typical life of the HEV vehicle battery is about 8 years and it mainly depends on the temperature and state of the charge of the battery. The battery state of charge should be maintained within the limits in order to increase the life of the battery. Also the rate of charging and discharging should be higher so that minimum time should be spent in the charging and discharging. The battery should be able to absorb large currents and also meet the power demands particularly in the inefficient engine regions. The subsystem aims at increasing the efficiency of the vehicle by efficient utilization of battery by maximizing the power generated by the battery, maintaining the state of charge of battery at the optimum level and its size (battery size is also important as increasing the weight decreases the efficiency) as specified. Power generated can be met by using a big battery. It would also help in maintaining the state of charge within the levels as the current inflow and outflow would not be large. But increasing the size of the battery adversely affects the fuel economy. Also the open circuit voltage increases with the state of charge, so we can increase the power by increasing state of charge but after certain limit internal resistance also increase which decreases the power. Also state of charge cannot be too high or too low. SYSTEM The overall system comprises of engine, radiator, battery and battery thermal management system. The system problem aims to minimize the fuel consumption of a hybrid vehicle and maximum the power output. The power consumed by auxiliary devices like radiator, pump and fan reduces the fuel economy 46 P a g e

47 of the vehicle as engine has to supply more power and hence require more fuel. Moreover the heat transfer losses in engine and battery system are pretty significant and have to be minimized in order to increase the fuel economy and performance of the vehicle. 2) NOMENCLATURE Design Variables Symbol Description Unit n s n p soc Number of series modules Number of cells in parallel in a module State of charge P bat Power generated by battery W Pr eg Power generated by regenerative braking Table 21: Battery Subsystem: Design variables W Parameters and intermediate variables Symbol Description Unit Values soc Minimum state of charge 0.3 min soc Maximum state of charge 0.9 max m Mass of each cell g 50 mod M Maximum mass g max V Open circuit Voltage Volt oc R Battery internal resistance ohm V Nominal Voltage of each cell Volt 9 cell 47 P a g e

48 C Maximum discharging rate of max a cell Optimal Design of Hybrid Electric Vehicle for Fuel Economy Q Maximum battery capacity A-h 5 1 I Maximum discharging current A 5.5 dis max I Minimum Charging current A -2 minch C Minimum charging rate of a min cell rc Constant w Weighting factor Table 22: Battery Subsystem: Parameter and Intermediate Variables 3) MATHEMATICAL MODEL OBJECTIVE FUNCTION: The objective is to increase the power generated by the battery. We want maximum power to be supplied by the battery which in other words decreases the fuel consumption and emissions. Total power requirement is given by the drive cycle and then power requirements are met by engine and the battery. min f P P P req bat reg As Preq is taken as constant, the model for optimization can be written as min f P bat P r eg Here the assumption is made that the charging is done through Preg only. So Preg is negative for the case. Primarily we want to maximize the net power output. The battery model is modeled as a static circuit with an internal resistance R, as shown in figure 12. The open circuit voltage V oc and R is both state-dependent parameters. They are functions of the battery s state 48 P a g e

49 of charge (SOC) and temperature. The battery temperature is assumed to be constant and the temperature effect is ignored. Also the charging and discharging resistances are assumed to be same. Power generated by a single cell can be modeled as: Fig. 12: Battery Subsystem: Battery model P V I I R 2 ( oc ) On solving this quadratic equation we get 2 Voc Voc 4R P I 2R 2 Voc Voc 4R P V IR 2 Also rate of charge is given by: soc I / c bat max The open circuit voltage and battery resistance is a function of state of charge and temperature. Here for simplicity constant temperature is assumed. V R oc bat f ( soc) f ( soc) These are calculated using the experimental data and applying curve fit to it. The curve fitting is done through least square projection of order three. Also it is assumed that charging and discharging resistances are same. The curves given are for 9 volt battery cell. The battery consists of cells in parallel in a module and a number of such modules in series. 49 P a g e

50 w Open Circuit Voltage [V] Internal Resistance Optimal Design of Hybrid Electric Vehicle for Fuel Economy 8.5 Exp. Data Polynomial Fit Exp. Data Polynomial Fit SOC SOC Fig.13:Battery Subsystem: Open circuit Voltage Vs SOC Fig.14: Battery Subsystem: Internal Resistance Vs SOC Results of Curve fitting V n soc soc soc oc 2 3 s 2( ) n R soc soc soc n s ( ) p There are n s series module and n p cells in parallel in each module. Also the weighting factor has the following relation with state of charge soc(%) Fig.15:Battery Subsystem: Weighting Factor It remains unity till 80% state of charge and then linearly decreases to 0 at 90% and remains 0 afterwards 50 P a g e

51 Assumptions: 1) The power requirement is constant. The power requirements are specified by drive cycle. They are met by engine and the battery. Here only battery is taken into account and Power requirement is held constant. 2) The open circuit voltage and internal resistance are function of state of charge only. In fact they are functions of state of charge and the temperature. But the effect of temperature is ignored. 3) Charging and discharging resistances are taken as same. They are different but the values does not differ much so they are taken as same. 4) The battery is assumed to be charged with regenerative braking only. The battery is generally charged with engine also and from external sources also in case of plug -in hybrid vehicles. 5) Regenerative charging power weight is assumed to be function of state of charge only. CONSTRAINTS g1 to g10 are the bounds g11: n n m M 0 s p mod max Total mass of the battery pack should be less than the maximum allowable mass. 2 Voc Voc 4 R( Pbat ) g12: nsvcell 0 2 Voltage of the battery at any point should be less than the maximum voltage. Voltage will be maximum when battery is charging so the bound is for Preg only. g13: P n V n C Q 0 bat s cell p max Power from the battery should be less than maximum attainable power. 2 Voc Voc 4 R( Pbat ) g14 : npidis max 0 2R 2 Voc Voc 4 R(P reg ) g15: np I min ch ( ) 0 2R The charging and discharging currents should be within the maximum discharging and minimum charging limits. 51 P a g e

52 2 Voc Voc 4 R( Pbat ) g16 : Cmax 0 2RQn s 2 Voc Voc 4 R(P reg ) g17 : Cmin ( ) 0 2RQn s The charging and discharging rates should be within the limits. h1 and h2 came directly from the model. DESIGN VARIABLES 1) Ns : Number of modules in series. This is directly related to the size of the battery, the mass of the battery and voltage across the battery. 2) Np : Number of cells in parallel in a module. This is related to mass of the battery, internal resistance and current flowing through the battery. 3) Soc : This is the an important design variable as its value cannot exceed the limits. Overcharging or undercharging the battery can reduce its life. So the state of charge should be maintained within the proper limits. 4) Pbat : This is the power produced by the battery. We need to maximize this. 5) Preg : Power absorbed in regenerative braking PARAMETERS soc min, soc max, m mod, M max, V cell, max C, Q, I dis max, I, minch C min, rc 52 P a g e

53 MODEL SUMMARY min f P P P g1: 25 n 0 g2 : n 50 0 s g3: 5 n 0 g4 : n 25 0 p min r eg p req bat reg s g5: soc soc 0 g6 : soc soc 0 mod max g7 : P 0 bat g8: P bat g9 : P 0 r eg g10 : P ( 3000) 0 g11: n n m M 0 s p max 2 Voc Voc 4 R( Pbat ) g12 : nsv 2 g13: P n V n C Q 0 bat s cell p reg max cell 0 2 Voc Voc 4 R( Pbat ) g14 : npidis max 0 2R 2 Voc Voc 4 R(P reg ) g15: np I min ch ( ) 0 2R 2 Voc Voc 4 R( Pbat ) g16 : Cmax 0 2RQn 2 Voc Voc 4 R(P reg ) g17 : Cmin ( ) 0 2RQn g18: P wrc 0 s s h V n soc soc soc oc s 2( ) n h R soc soc soc n s ( ) p 53 P a g e

54 4) MODEL ANALYSIS FUNCTIONAL DEPENDENCY TABLE a) With Variables n s n p soc P bat Pr eg f * * g1 * g2 * g3 * g4 * g5 * g6 * g7 * g8 * g9 * g10 * g11 * * g12 * * g13 * * * g14 * * g15 * * g16 * * g17 * * g18 * h1 * * h2 * * * Table 23: Battery Subsystem: Functional dependency table for variables 54 P a g e

55 (b) With Parameters soc_ min soc_ max Vcel l Cma x Q m Mm ax Cmi n Imin _ch Idis_ max rc Voc R w f g1 g2 g3 g4 g5 * g6 * g7 g8 g9 g10 g11 * * g12 * * * g13 * * * g14 * * * g15 * * * g16 * * * * g17 * * * * g18 * * h1 * h2 * Table 24: Battery Subsystem: Functional dependency table for parameters 55 P a g e

56 MONOTONICITY ANALYSIS n s n p soc P bat Pr eg f (-) (-) g1 (-) g2 (+) g3 (-) g4 (+) g5 (-) g6 (+) g7 (-) g8 (+) g9 (-) g10 (+) g11 (+) (+) g12 (-) U g13 (-) (-) (+) g14 (-) U g15 (+) U g16 (-) U g17 (+) U g18 (+) Table 25: Battery Subsystem: Monotonicity Table From the monotonicity analysis we can infer that the problem is well constrained. Not much can be told about the activity of the constraints. By Monotonicity Principle 1 g8 or g13 are active with respect to Pbat. The g12,g14 and g16 may also be active if they are monotonic. g10 or g18 are active with respect to Preg. Also g15 or g17 could also be active if they are monotonic Also g5 and g6 are semi-active by monotonicity principle 2 Nothing could be said about other constraints. 56 P a g e

57 DESIGN OF EXPERIMENTS First Order (Varying only one variable at a time). The design of variable study is performed to see how the objective function and different constraints vary with respect to the design variables ( one variable at a time keeping others fixed at their nominal values) with the specified parameters to look out for the discontinuities. They also tell about the hidden monotonicity due to the presence of equality constraints Objective Function: x 104 Objective v/s Ns x 104 Objective v/s Np Ns Np x 104 Objective v/s so -3 x 104 Objective v/s Pb so -3 x 104 Objective v/s Pr Pb x Pr x 10 4 Fig.16:Battery Subsystem: Objective function Vs design Variables Monotonic with respect to Pbat and Pr. Constraints : g1 to g10 are bounds and will produce straight lines only. So main consideration is for nonlinear constraints g11 to g18 57 P a g e

58 -1 x g 1(x) v/s Ns x g 1(x) v/s Np Ns x g 1(x) v/s so Np x g 1(x) v/s Pb so x g 1(x) v/s Pr Pr Pb x 10 4 Fig.17:Battery Subsystem: g11 Vs Design Variables As seen from the Fig 18, g12 is monotonically increasing with respect to Pb. So g12 can also be active from our results of the monotonicity table. Also the function is non monotonic with respect to soc g 1 2(x) v/s Ns -269 g 1 2(x) v/s Np Ns g 1 2(x) v/s so so -268 g 1 2(x) v/s Pr Np g 1 2(x) v/s Pb Pb x Pr Fig.18:Battery Subsystem: g12 Vs Design Variables 58 P a g e

59 2 x g 3(x) v/s Ns Ns x g 3(x) v/s so so x g 3(x) v/s Pr x g 3(x) v/s Np Np 4 x g 3(x) v/s Pb Pb x Pr Fig.19:Battery Subsystem: g13 Vs Design Variables 100 g 1 4(x) v/s Ns 200 g 1 4(x) v/s Np Ns g 1 4(x) v/s so so 76 g 1 4(x) v/s Pr Np g 1 4(x) v/s Pb Pb x Pr Fig.20:Battery Subsystem: g14 Vs Design Variables By the same argument as for g12, g14 could also be active with respect to Pb 59 P a g e

60 40 g 1 5(x) v/s Ns 50 g 1 5(x) v/s Np Ns 25 g 1 5(x) v/s so Np 24 g 1 5(x) v/s Pb so 50 g 1 5(x) v/s Pr Pb x Pr Fig.21:Battery Subsystem: g15 Vs Design Variables From Monotonicity table, it was found that g15 could be active with respect to Preg if it was monotonically increasing. But from the Fig.21 we can see that it is monotonically decreasing, so it is not active. 1 g 1 6(x) v/s Ns g 1 6(x) v/s Np Ns g 1 6(x) v/s so so 2 g 1 6(x) v/s Pr Np g 1 6(x) v/s Pb Pb x Pr Fig.22:Battery Subsystem: g16 Vs Design Variables As shown in Fig.22 g16 is monotonically increasing with respect to Pbat, so it can also be active 60 P a g e

61 -0.5 g 1 7(x) v/s Ns g 1 7(x) v/s Np Ns -0.7 g 1 7(x) v/s so Np 2 0 g 1 7(x) v/s Pb so -0.6 g 1 7(x) v/s Pr Pb x Pr Fig.23:Battery Subsystem: g17 Vs Design Variables From Fig. 23, g17 is not monotonically increasing with respect to Preg. So cannot be active 2001 g 1 8(x) v/s Ns 2001 g 1 8(x) v/s Np Ns so g 1 8(x) v/s so g 1 8(x) v/s Pr Np g 1 8(x) v/s Pb Pb x Pr Fig.24:Battery Subsystem: g18 Vs Design Variables Result of monotonicity analysis and design of experiments of first order The monotonicity analysis and first order DOE results shows that Any of the g8, g12, g13, g14 or g16 could be active with respect to Pbat. g10 or g18 could be active with respect to Preg g5 and g6 are semi-active by MP2 61 P a g e

62 5) NUMERICAL RESULTS: SCALING As the values of the design variables are not of same order. For instance, soc varies from 0 to 1 and Pbat varies from to 50000, scaling was necessary. For scaling following algorithm is used. Also the value of Pre is negative so different scaling is used. X(scaled) = (X- LB)/(UB-LB) if X>0 X(scaled) = (2X-LB-UB)/(UB-LB) if X<0 The optimization is carried out for different starting points using the fmincon function in Matlab. The algorithm used is interior point algorithm and active set. The interior point algorithm has various options. Following has been chosen to get optimum results. Options for fmincon LargeScale On TolFun 1e-6 FinDiffType AlwaysHonorConstraints Central Bounds Table 26: Battery Subsystem: FMINCON settings The Tolfun value signifies Termination tolerance on the function value, a positive scalar. The default value of 1e-6 is taken. The AlwaysHonorConstraints value has been set to bounds which signifies that at every iteration bounds has been satisfied. Large scale is taken as on FinDiffType is set to central as the central difference are more accurate. The following results have been obtained which found to be same for different starting points: algorithm : interior-point x_opt : (34, 23, 0.8, , ) exitflag : 1 iterations : P a g e

63 First-order Norm of Iter F-count f(x) Feasibility optimality step e e e e-007 Local minimum is found which satisfies the constraints. Optimization Metric Options relative first-order optimality = 5.94e-007 TolFun = 1e-006 (selected) relative max(constraint violation) = 0.00e+000 TolCon = 1e-006 (default) The values of first order optimality and norm of the step as governed by KKT conditions shows that they have become small enough to make the algorithm converge The eigen values of hessian at this point are 1.0e+006 * Which are all positives or zero which shows that hessian is positive semidefinite and hence the point is a minimum. To check the activity of the constraints, langrangian multipliers are evaluated Constraints Non linear Activity g Active g12 0 g13 0 g Active g15 0 g g17 0 g18 0 Table 27: Battery Subsystem: Constraint activity The table 27 shows that g11 and g14 are active which confirms from the monotonicity analysis. 63 P a g e

64 Upper Activity bound Ns 0 Np 0 soc Active Pbat 0 Preg 9 Active Table 28: Battery Subsystem: Upper Bounds Activity This shows that upper bounds of soc and Preg are active. Different starting points give the same minima as shown in the Table 29. The minimizers are also same with small changes in Ns and Np. Design Starting Starting starting optimal upper bound Lower Variables point 1 point 2 point3 point bound Ns Np Soc Pbat Preg Objective (W) Table 29: Battery Subsystem: Different Starting Points Results Besides this active set algorithm is also tried. It is found that active set converges to same minima and minimizers but in 6 iterations only. So active set is a better algorithm for this problem as computation cost is low. Algorithm Iterations Interior Point 18 Active Set 6 FINAL RESULT : Maximum Power output from Battery : 32 KW at x = (34, 23, 0.8, , ) 64 P a g e

65 CONCLUSIONS Optimal Design of Hybrid Electric Vehicle for Fuel Economy The value of the objective function from the model agrees with the practical values. The power of approximately 30 KW is required by the vehicle going at around m/s. The number of modules comes out to be approximately 37 which is approximately equal to the Toyota Prius. The soc reaches its maximum value as the objective function is to increase power which causes the soc to reach upper bound for maximum power. The results also match with the monotonicity analysis. PARAMETRIC STUDIES The optimum and optimizer so obtained subjected to all the constraints corresponds to the value of parameters used in the formulation of the model. It is possible that if the values of these parameters are changed, optimum value of objective functions and design variable changes. So in order to see the sensitivity of optimum value to changing parameter values a parametric study is carried out. EFFECT OF MASS The effect of changing the mass of cell or effect of changing the maximum mass remains same. So only effect of changing the cell mass is studied. As the constraint on maximum mass is found to be active, the results are expected to show change in optimum value The following results are obtained: Mass of Cell Ns Np Soc Pbat Preg F Table 30: Battery Subsystem: Parametric study for mass of cell 65 P a g e

66 Maximum power output of the battery (W)) Optimal Design of Hybrid Electric Vehicle for Fuel Economy 4.2 x mass of each cell (g) Fig.25:Battery Subsystem: Effect of mass of Cell As the mass of the cell increases, the number of cells decreases for a constant maximum mass. So Ns decreases which in turn reduces the Pbat. So the Pbat decreases as the mass increases. Preg as expected from the monotonicity analysis remains constant at the upper bound. EFFECT OF MAXIMUM BATTERY CAPACITY Q Ns Np Soc Pbat Preg F Table 31: Battery Subsystem: Parametric study for maximum battery capacity Q is the maximum battery capacity. The results are shown for three values of Q. For the given problem, if Q goes below 4.8 keeping other design variables at their nominal value, the optimization is not possible. As seen from the table the objective function value remains constant as Q increases. However 66 P a g e

67 the Ns, Np and soc changes. for Q = 4.8, the soc does not hit the upper bound. So for this value of Q the upper bound of soc is not active. The value of Ns and Np changes significantly with Q EFFECT OF MAXIMUM DISCHARGING RATE Cmax Ns Np Soc Pbat Preg F Table 32: Battery Subsystem: Parametric study for maximum discharging rate As seen from the table after Cmax = 1, the value of the function remains constant. For Cmax = 0.9 the lower bound of soc becomes active. For other values, upper bound remains active. This shows that the value of Cmax should be chosen above 0.9 EFFECT OF MINIMUM CHARGING RATE Cmin Ns Np Soc Pbat Preg F Table 33: Battery Subsystem: Parametric study for minimum discharging rate Pbat,Preg, soc remains on the upper bound and hence f remains same. The value of Ns and Np however changes with Cmin EFFECT OF MINIMUM CHARGING CURRENT Imin_ch Ns P a g e

68 Np soc Pbat Preg F Table 34: Battery Subsystem: Parametric study for minimum charging rate The function value remains almost same. At -3 the value of Ns changes significantly. The soc and Pbat remains on their upper bounds. EFFECT OF MAXIMUM DISCHARGING CURRENT Idis_max Ns Np soc Pbat Preg F Table 35: Battery Subsystem: Parametric study for maximum discharging current 68 P a g e

69 Maximum power output of the battery (W)) Optimal Design of Hybrid Electric Vehicle for Fuel Economy 3.3 x Maximum Discharging Current (A) Fig.26:Battery Subsystem: Maximum Battery Power Vs Maximum Discharging Current The table shows that function optimum value depends on the value of Idis_max. It increases with the maximum discharging current. Thus the value of Idis_max should be carefully chosen. Also at Idis_max = 6 soc hits the lower bound which shows it cannot be decreased beyond that. EFFECT OF CHANGING THE BOUNDS: The value of Preg remains at the upper bound for all above cases. Thus the bounds on the Preg are changed to see the effect on the objective function and the constraints. Preg Ns Np soc Pbat Preg F Table 36: Battery Subsystem: Effect of changing the bound on Preg 69 P a g e

70 Even after changing the bounds the value of Preg remains at the upper bound. There is a little change in the value of Pbat. The value of objective function is minimum at Preg = However the value of variables changes. Conclusions from Parametric Studies The objective function optimum value is sensitive to the mass, maximum discharging current and change of bounds on the Preg. Preg almost every time reaches the upper bound. So the values of mass, maximum discharging current and Preg upper bound should be chosen wisely. Other parameters don t change optimum value of the objective function significantly but they change the value of design variables. Design Rules : 1- To increase the power output of the battery, the mass of each cell can be decreased if we want to keep the mass of the battery fixed. This actually increases the number of cells and provide large power. But the extent to which mass can be reduced without changing the characteristics of the individual cell is not much. So essentially if we want same characteristics we cannot reduce the mass of the individual cell beyond a limit. 2- The Battery power can also be increased by increasing the maximum discharging current. But that also after some value will make the system unstable ACKNOWLEDGEMENTS The battery figure and some of the data are taken from the course material of ME -566 at University of Michigan. 70 P a g e

71 IV) BATTERY THERMAL MANAGEMENT Mingxuan Zhang 1) PROBLEM STATEMENT Vehicle, as the whole system, requires balance of performance, emissions and fuel economy. While only when each subsystem work well can we have an excellent whole system. The purpose of this optimization of the battery s thermal management subsystemis to maintain a temperature such that the battery can work under its highest efficiency without a life span reducing byminimized energy consumption. One of the most successful hybrid vehicleproducts in the market is Toyota Prius. The battery thermal management part of the latest Prius is shown in Fig.27 below. Battery pack is surrounded by the thermal management system Fig.27:Battery Thermal Management System The following Fig.28 is the theoretical model of Fig.27. First, the temperature of air from outside adjusted by the air conditioner of the vehicle cabin. Then the air may be heated / cooled by the cabin. Finally, it enters the battery heat exchange part through a fan and comforts the battery pack. Fig.28:Battery Thermal Management: Theoretical model 71 P a g e

72 Another mature battery thermal management method applied by BMW on Active Hybrid 5 is called direct refrigerant-based cooling. Fig.29 below shows its theoretical concept. Liquid refrigerant is used in the evaporator and corresponding circle loop. It makes the battery thermal management system more compact with a higher cooling efficiency at the interface. Also, cooling/heating noise can be negligible compared with fans. While there is an advantage, there is always a disadvantage. Fluid refrigerant circulate with much larger power consumption. As the heat transfer direction is only from evaporator to condenser, another heating circle has to be added when battery running under heated. Fig.29:Battery Thermal Management System: Refrigerant-based cooling The battery thermal management subsystem used for this project as same as Prius, as shown below mainly includes: Entry vent, Fan, Inlet Pipe, Gaps between module, and Outlet Pipe. Other elements are not relative to heat translation. Fig.30:Battery Thermal Management System: Schematic 72 P a g e

73 There are several trade-off may need to be optimized, which may improve the system and help achieve our goal. Inside the thermal management subsystem, the trade-off is: a. The pipe size and system shape is defined by aero dynamic requirement to have a better heat transfer efficiency. However, this strange shape and its unacceptable volume potentially make the back cabin useless. b. A lower target temperature of battery pack may increase its operating time but may also waste my electricity on cooling. 2) NOMENCLATURE Name Description Unit A Width of the gap m A c Cross area the gap m 2 A tube Cross area of inlet tube m 2 B Height of the gap m Cp air Air specific heat character J/(kg K) Cp battery Battery specific heat character J/(kg K) Dh Specific diameter m E fan Total Fan energy consumption J I Battery working electric current (A) A k f Air conduct coefficient J/K m 2 L Gap length m M Number of gap m sys Thermal management system mass kg Nu Nusselt number P g Air pressure at the gap Pa P inlet Air pressure at the inlet of the fan Pa 73 P a g e

74 Power Fan Power W Pr Prandtl number - Q g Heat generate at battery per second W Q int Heat transferred per second W R Internal resistance of battery Ω R_air Air constant J/(kg K) Re Reynolds number - T b Average Battery temperature K t batt Battery wall thickness m T g Target battery temperature K Time Available battery operating time S T in Air temperature at the inlet of the fan K T m Average Tm K T out Air temperature at the outlet of gap K V air-gap air speed in the gap m/s V air-gap Air speed at the heat exchange gap (m/s) - V air-tube Air speed in the tube m/s η fan Fan Efficiency - Μ Air viscosity N s/m 2 ρ ag Gap air density kg/m 3 ρ ai Inlet air density kg/m 3 ρ air Air density kg/m 3 Table 37: Battery Thermal Management: Nomenclature 3) MATHEMATICAL MODEL In this project, A tube and other system geometry variables is related to the system volume and weight. They are also highly relative to the power requirement of the fan under same heat transfer rate.t target is the target temperature to maintain for the battery package. Setting a higher target temperature will require less cooling, which is the most commonly operating condition compared with heating. However, this will 74 P a g e

75 reduce the operating time of the battery, if the driving cycle is too busy. A trade-off decision has to be made for this variable. At this subsystem step, it is not considered as Q g is constant without driving cycle. Following are equations used to link variables, parameters and constraints.the heat exchange is calculateonly for charge condition, which is heat generation. Q g =I 2 R-IV(1-η coulomb ) (charge I<0) T batt is the temperature the battery thermal management system trying to maintain. Q int =h tot A batt (T batt -T ave-gap ) The total power consumption can be calculated by the following equations: In this mathematical model, some iteration may require according to the equations set structure. Since that would increase the project complexity to an unnecessary level, they were simplified in the process. OBJECTIVE FUNCTION The objective of this subsystem is to minimize the energy consumption and size of this battery thermal management system. All decisions of parameters at this step are based on subsystem only. They will be justified when the entire model is optimized together. 75 P a g e

76 Objective function: Minimize E fan =f (A tube, T in, m, a, b, l) The total energy consumption of the fan will be directly decided by the efficiency of the fan, cross area of the tube and the geometry of the heat exchanger. Minimize E fan =f (A tube, m, a, b, l) (Appendix II) ( ) ( ) ( ) ( ) ( ) CONSTRAINTS To optimize the battery thermal management system, following constraints are set from both basic physical laws and engineering specifications. DESIGN VARIABLES AND PARAMETERS The variables used for this project are followings: A tube, m, a, b, l. 76 P a g e

77 The parameters set for this project are given as below: η fan, ρ air-inlet, Cp batt, Cp air, ρ batt, k air, R air, Q g, T batt, P inlet, μ air, T in Parameter Table: R K air Cp batt 700 Cp air 100 P inlet 10 5 Eff 0.2 μ air ρ batt 1811 Q g 500 T batt 313 T in 298 Q int Q g Table 38: Battery Thermal Management: Table for parameters 4) OPTIMIZATION MODEL ANALYSIS MONOTONICITY ANALYSIS: In this battery thermal management subsystem, energy consumption, size and weight are optimized through the processes above. The monotonicity property of a and b is not very obvious from the expression, so some analysis has been done to prove their positive monotonicity (See Appendix I). All of the monotonicity relations are list in the following chart: E + + A tube M a b l g1 g2 g3 g4 g5 g6 + g7 + g8 g9 g10 + g Table 39: Battery Thermal Management: Monotonicity Analysis 77 P a g e

78 From above chart, we can see: g8, g9 are definitely active. One of constraints g6 and g11, one of g7 and g11 and one of g10 and g11 will be active. Possible combination are some of g6, g7, g10 active, g11 active or g6, g7, g10 active, g11 inactive. In Appendix-I (2) and Appendix-II (b), gradient projection method was applied to test if g11 is active. Since m has to be integer, gradient projection method is applied for each m in the range. The result is in Appendix-III (a), from which we can see it violate g5 and g6. So the active set here is g6 and g11. With g6 and g11, I got the new result table, which is also the final optimal solution as in the solution. Following are active constrains, which defined At, a, b, m and l. E + + A tube m a b l g6 + g7 + g8 g9 g Table 40: Battery Thermal Management: Activity table 5) NUMERICAL RESULTS With above parameters and analysis strategy, the final optimal solution for the system is At=0.02, m=10, a=0.01, b=0.008, l= Under this condition, E=118.43, m sys = (Gradient Projection Method has been apply to prove g6 is active, code is in Appendix-II-2) Table: This table shows the result from gradient projection method, objective function Power is really small at this time, but l and At results shows it violated g6 and g5. m L At g Power 1.00 (1.7074) (1.7074) P a g e

79 3.00 (1.7074) (1.7074) (1.7075) (1.7075) (1.7075) (1.7075) (1.7076) (1.7076) Table 41: Battery Thermal Management: Gradient projection method analysis As talked in analysis part, m has to be integer, so method are applied separately to each m to make sure the optimal result is reasonable. At M l g Power P a g e

80 (0.00) (0.00) Table 42: Battery Thermal Management: Table for parameters FIRST ORDER RELATION PLOTS: This figure is m-e, which shows how m=10 is picked according to constraint g7, as m also has to be integer. Fig.31: Battery Thermal Management System: Energy Consumed by fan with respect to m 80 P a g e

81 With active constraint g11, the reason why m is monotonic is because m to l relation is in its decrease region as shown in the following. Fig.32: Battery Thermal Management System: Energy Consumed by fan with respect to m SECOND ORDER: This plot is about the object function, which shows the relation between a,b and Power (other variables are at minima). It shows though a,b appear as a higher order complex term, they are still monotonic. This prove calculation in Appendix-I. Fig.33: Battery Thermal Management System: Energy Consumed by fan with respect to m 81 P a g e

82 This plot shows when g11 is active, how m and At affect object function. This proved model analysis. Fig.34: Battery Thermal Management System: Energy Consumed by fan with respect to m 6) SYSTEM-LEVEL TRADEOFFS a. Between the thermal management and other subsystem, the trade-off is: b. The engineering capacitance of battery may not reach it s maximize to avoid a special super increase in heat generation. c. The Fig.33 below shows how heat generation changes with respect to battery charge and discharge cycle. Note that the initial charging section shows the endothermic nature of the charge chemical reaction and the discharge section shows the exothermic. Fig.35: Battery Thermal Management System: Heat generation with respect to battery charge 82 P a g e

83 d. However, the heat generation increases rapidly near the end of discharge, which creates a terrible engineering problem. The cost of thermal management subsystem may increase a lot due to meeting the final special heat transfer requirement. To be more efficiency, battery designers may want to avoid that using period. e. The operating time of the battery pack may need to be adjustedaccording to itstemperature. f. Lower inlet temperature at the inlet vent will reduce the requirements of cooling air flow rate for the same heat transfer rate, but air conditioner system will has more losses. g. Mass of the battery thermal management system can also be modified with system level requirements. 83 P a g e

84 V) SYSTEM LEVEL OPTIMIZATION 1) PROBLEM STATEMENT Optimal Design of Hybrid Electric Vehicle for Fuel Economy The overall system comprises of engine, radiator, battery and battery thermal management system. The overall system problem aims to maximize the fuel economy or mpg (miles per gallon of the vehicle) over the US06 and US Urban Drive cycle (UDDS). The vehicle parameters are of Toyota Prius has been obtained from ADVISOR software developed by NREL (National Renewable Energy Laboratory) in partnership with DOE (Department of Energy). The power sources in a hybrid electric vehicle is Engine and Battery which has to meet the power demand from the drive cycle and power consumed by auxiliary devices like radiator, pump and fan. Increasing the power of Engine and battery increases the overall mass of the vehicle which adversely affects the fuel economy. Thus there is a tradeoff between the maximum power obtained in terms of fuel economy. All the variables from individual subsystems are taken as variables in the system optimization problem. Some additional equality constraints has been added to link the different subsystems. Fig. 36: System level interactions for various subsystems 84 P a g e

85 2) NOMENCLATURE Symbol Description Units dist_cycle Distance Travelled over the whole drive cycle m m Total mass kg M_batt Mass of Battery kg M_eng Mass of Engine kg M_rad Mass of Radiator kg M_veh Mass of Vehicle kg P_batt Battery Power kw Pe_max Maximum Engine Power kw P_rad Power consumed by radiator pump KW Q_rad Heat dissipated by radiator KW total_fuel Fuel consumption g/s V Speed of Vehicle mph Parameters Symbol Description Value Units A Frontal Area of Vehicle m 2 Bat_cap Battery Capacity 2 kw Cd Coefficient of Drag Cr Coefficient of Rolling Resistance g Gravitational constant 9.81 m/s 2 M_cargo Cargo Mass 136 kg M_fan Mass of Fan 13 kg M_veh Base mass of vehicle 1400 kg P_ch Charging Power 10 kw P_ev Power threshold for electric mode 20 kw SOC_t State of Charge threshold P a g e

86 3) MATHEMATICAL MODEL OBJECTIVE FUNCTION: Maximize the fuel economy or miles per gallon of the vehicle over the drive cycle. dist _ cycle( miles) mpg total _ fuel( gallons) Total Mass m Mveh Mrad Mfan Mbatt Meng M c arg o Mass_rad=Total_vol*Rho_metal M batt N * N *M s p cell 1000 Mass of fan has been assumed to be constant at 13 kg based on optimization results from battery thermal management subsystem. The mass of engine has been calculated using a linear curve fit w.r.t Power of the engine using the data collected from ADVISOR. M_eng = 2.6 (P_eng) + 26 The total resistance force to the vehicle comprises of Rolling Resistance, Aerodynamic Drag and Grade Resistance. Grade resistance has been assumed to be zero here for US06 and Urban drive cycle, UDDS. Total resistance force = F = F RR + F AD F F Cr * m* g AD RR * Cd * A* V 2 Total power required is the sum of power for acceleration, power required to overcome the resistances and power for auxiliary cooling system devices like pump and fan. P m acc F P P * Res pump fan To maximize the fuel economy we have used the concept of load leveling. It means that we want to use our battery as much as possible subject to its SOC being above a threshold value. When SOC is greater than threshold value and power demand is less than P_ev, the vehicle is in Electric Mode and all the power requirement is being met by the battery. When SOC <SOC_t, P P P eng Battery SOC at any time (t+1) depends on the battery SOC at time, t, and is given by the relation below. 86 P a g e ch 2

87 P SOC( t 1) SOC( t) batt Bat _ cap If P batt is positive, SOC decreases and battery discharges. Similarly whenp batt is negative, SOC increases and hence engine provides the power to charge the battery. VARIABLES All the variables from individual subsystem have been considered for the system level optimization. This is known as All in One (AIO) approach. In total there are 23 variables from four different subsystems. One advantage of considering all the variables in system optimization is that we can easily study the behavior of variables at both system level and compare how they differ from their subsystem optimized values. CONSTRAINTS All the constraints from different subsystems act as constraint for the system problem also. There are in total 30 constraints from the four subsystems. Additional constraints were added linking the subsystems which are given below. 1) The rate of heat dissipation of the radiator must be greater than the rate of heat loss from the engine. Thus at system level this becomes an additional inequality constraint for the radiator. NQloss n 120 c Q rad 2) Similarly the rate heat released from the battery will give the fan energy consumption for cooling the battery. 3) Power of pump, fan and battery must be greater than zero. 0 P, P, P pump fan battery 4) Maximum power of engine must be greater than maximum power of battery. P, P, P P pump fan battery eng 5) Mass of the radiator must be greater than zero. 0 M rad In total there are 38 constraints for the system optimization problem. 87 P a g e

88 SUMMARY OF THE OPTIMIZATION MODEL: Optimal Design of Hybrid Electric Vehicle for Fuel Economy Max ( ) ( ) In negative nullity form: Min ( ) subject to: g31: g32: -P pump 0 g33: -P fan 0 g34: -Pbattery 0 g35: P fan - P eng 0 g36: P pump - P eng 0 g37: P batt - P eng 0 g38: -M rad 0 TRADE-OFFS a. To increase the fuel economy, size of the battery can be increased but this in turn will increase the overall mass of the system and beyond an optimum battery size the fuel economy might actually suffer adversely due to further increase in battery size. b. The power consumption in the auxiliary devices like radiator pump, radiator fan and battery management system fan cannot be reduced below an optimum value because that may lead to reduced system performance or its failure. c. A major part of battery charging comes from the engine, so the size of the engine cannot be reduced beyond a limit and a relation should exist between the battery capacity and engine size for optimum fuel economy. 88 P a g e

89 4) NUMERICAL RESULTS US06 DRIVE CYCLE The optimization was done using medium-scale: SQP (Sequential Quadratic programming), Quasi Newton, line search algorithm and also Interior-Point on MATLAB using fmincon. Results are shown below for different starting point using both of these algorithm. Subsystems Variables Subsystem Level optimized values Optimized value on Drive Cycle US06 UDDS HWFET X X X Radiator X X X X X X X Battery thermal X management X X X X Battery X X X X X Engine X X X BSFC OBJECTIVE FUNCTION Mpg mpg miles per gallon The figures below show the drive cycle and power demand at each time step of the drive cycle. The figure on the right shows the engine power demand, fueling rate, battery power demand and State of Charge of battery during the whole drive cycle. 89 P a g e

90 Power (kw) SOC Battery power (kw) Speed (mph) Engine power (kw) Engine fueling rate (g/sec) Power (kw) SOC Battery power (kw) Speed (mph) Engine power (kw) Engine fueling rate (g/sec) Optimal Design of Hybrid Electric Vehicle for Fuel Economy US06 DRIVE CYCLE 40 US06 cycle 60 Results for the US06 cycle Time (Sec) US06 cycle Time (sec) Time (sec) Time (Sec) Time (sec) Time (sec) Fig. 37: Battery power, engine power, SOC, engine fueling rate and speed of vehicle for US06 Drive UDDS (URBAN) DRIVE CYCLE Cycle 30 US06 cycle 60 Results for the US06 cycle Time (Sec) US06 cycle Time (sec) Time (sec) Time (Sec) Time (sec) Time (sec) Fig. 38: Battery power, engine power, SOC, engine fueling rate and speed of vehicle for UDDS (Urban) Drive Cycle 90 P a g e

91 Power (kw) SOC Battery power (kw) Speed (mph) Engine power (kw) Engine fueling rate (g/sec) Optimal Design of Hybrid Electric Vehicle for Fuel Economy HWFET (HIGHWAY) DRIVE CYCLE US06 cycle Results for the US06 cycle Time (Sec) US06 cycle Time (sec) Time (sec) Time (Sec) Time (sec) Time (sec) Fig. 39: Battery power, engine power, SOC, engine fueling rate and speed of vehicle for HWFET (Highway) Drive Cycle RESULT ANALYSIS Fuel economy values are very realistic and practical compared to the present vehicles. Load leveling concept was used to maximize fuel economy. As can be seen from the results, the values of design variables are considerably different from those obtained in the individual subsystem level optimization. This was expected because the different subsystems interact with each other and their performance is dependent on each other. Some variables reached upper and lower bounds during system optimization. Optimized BSFC value changed from 197 g/kw-h to 251.7g/kW-h from subsystem to system optimization. It was expected as the power demand from the engine has increased to due to auxiliary devices like pump and fan in system optimization. UDDS HWFET US06 Fuel economy 55.7mpg mpg 48.6 mpg 91 P a g e

92 5) REFERENCES [1] Couse Material, ME-438, Fall 2011, University of Michigan, Ann Arbor [2] Heywood JB. Internal combustion engines fundamentals. New York: McGrawHill, 1988 [3] Papalambros, P.Y., Wilde, D.J. Principles of Optimal Design Cambridge University Press [4] Incropera F. P., Dewitt D. P., Fundamentals of Heat and Mass Transfer, Fifth Edition. ISBN: [5] Hollister T., Weber R., Optimization Study of a Finned Heat Exchanger in an Orbiting Space Station, ME 555, Project Report: [6] Ng E. Y., Johnson P. W., and Watkins S., An analytical study on heat transfer performance of radiators with non-uniform airflow distribution, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering : 1451 [7] Designing a more Effective Car Radiator, MAPLESOFT tutorial [8] Tremblay, O., Dessaint, L., A Generic Battery Model for the Dynamic Simulation of Hybrid Electric Vehicles, Vehicle Power and Propulsion Conference, VPPC IEEE, pp ,12 Sept [9] Liu Jinming, Modeling, Configuration and control optimization of power-split hybrid vehicles, PhD dissertation, University of Michigan Ann Arbor. [10] R. Rynkiewicz, Discharge and charge modeling of lead acid batteries, in Proc. Appl. Power Electron. Conf. Expo., vol. 2, 1999, pp [11] Z. M. Salameh,M. A. Casacca, and W. A. Lynch, A mathematical model for lead-acid batteries, IEEE Trans. EnergyConvers., vol. 7, no. 1, pp , Mar [12] Pesaran, Ahmad A. "Battery Thermal Management in EVs and HEVs:." National Renewable Energy Laboratory. (2001): n. page. Print. [13] Keyser, Matthew, and Ahmad A. Pesaran."Thermal Characteristics of Selected EV and HEV Batteries."National Renewable Energy Laboratory. (2001): n. page. Print. [14] "Thermal Management."Technical Press Day (2009): n. page. Web. 5 Feb < [15] Pesaran, Ahmad A. "Thermal Evaluation of Toyota Prius Battery Pack." National Renewable Energy Laboratory. (2002): n. page. Print. [16] Park, Sungjin. "A Comprehensive Thermal Management System Model." (2011): Print. [17] Pesaran, Ahmad. "Cooling and Preheating of Batteries in Hybrid Electric Vehicles." 6th ASME- JSME Thermal Engineering Joint Conference. (2003): Print. 92 P a g e

93 VI) APPENDIX I: A) MATLAB CODE Engine Subsystem (PULKIT GUPTA) clearall clc %Inequality constraint A= [ ; ; ]; b= [ ]'; % Upper and lower bound lb= [ ]'; ub= [0.11 inf ]'; % Equality constraint Aeq= []; Beq= []; %Starting points, x= [B S CR k f] optimset('fmincon') % opts=optimset('display','iter', 'TolFun',1e-20,'Algorithm','active-set'); opts=optimset('display','iter','algorithm','interior-point'); [x, fval,exitflag, output,lambda,grad,hessian]= fmincon(@final,x, A, b, Aeq, Beq, lb, ub,@nonlincon,opts) %Objective function: function bsfc= Final(x) %% Parameters Pin= 100; % Intake Pressure Pex= 150; % Exhaust Pressure Ti = 300; % Intake Temperature Te = 450; % Exhaust Temperature Qloss = 1.5; % kj/kg Qlhv= 44000; % kj/kg R = 0.287; %kj/kg K g = 1.4; %for ideal air-fuel mixture AFR= 14.6; % stoichiometric ratio nc= 4; %4 cylinder nr= 2; %2 rev/cycle; 93 P a g e

94 Tcool= 353; % Coolant Temperature K= 0.06; % Conductivity v= 10^-4; % viscosity Tg= 1273; % Temp. Gas N= 3000; % RPM %% Variables list Initialization B= x(1); S= x(2); CR= x(3); k= x(4); f= x(5); %Non linear constraints: g6 = 2*S*N -900; g7 = 480-2*S*N; g15= T3-3100; g16= -bsfc; c=[g6 g7 g15 g16]; ceq=[]; CRs= CR*100; Up= 2*S*N/60; A= (2*(pi*B*B)/4) + (pi*b*s); Vd= (pi*b*b*s)/4; Vc= Vd/(CRs-1); V0= Vc+k*Vd; P0= Pin; V1=Vd+Vc; T0= (1-f)*Ti+f*Te*(1-((1-(Pin/Pex))*((g-1)/g))); m= P0*V0/(T0*R); T2= T0*((V0/Vc)^(g-1)); P2= P0*((V0/Vc)^g); mf= (m*(1-f))/(afr+1); Cv= R/(g-1); 94 P a g e

95 P= (P0+P2)/2; T= (T0+T2)/2; h= 3.26*(B^(-0.2))*(P^0.8)*((Tg)^(-0.55))*(Up^0.8); Qloss = A*h*((T- Tcool)/1000); T3=(((mf*Qlhv)- Qloss)/(m*Cv))+T2; P3= (P2*T3)/T2; T4= T3/((CRs)^(g-1)); P4= P3/((CRs)^g); P5=Pex; T5= (T4*P5)/P4; T6=T5; W12= m*cv*(t0-t2); W34= m*cv*(t3-t4); W10= Pin*(V0-V1); Wg= W12+ W34+ W10; Wpump= (Pin- Pex)*Vd; Wnet = Wg+ Wpump; IMEP = Wnet/Vd; FMEP = IMEP*(0.063+(0.008*Up)); BMEP= IMEP-FMEP; Wb= BMEP*Vd; bsfc= (3.6* *mf)/Wb end 95 P a g e

96 B) MATLAB CODE - Radiator Subsystem (VASU GOEL) Main Body of the code clearall;clc; formatlong; % Design Variables % x(1)-length of Radiator % x(2)-width of Radiator % x(3)-thickness of One Fin % x(4)-tube Height % x(5)-fin Height % x(6)-number of Fins per Tube % x(7)-number of Tubes % x(8)-gap between fins Optimal Design of Hybrid Electric Vehicle for Fuel Economy %% Different Starting Points % x0=[0.45; 0.35; ; ; ; 50; 41; ]; % %x0=[0.01; 0.31; ; ; 0.42; 0.01; 30; 0.003]; % %x0=[0.21; 0.41; ; ; 0.42; 100; 320; 0.003]; % %x0=[0.21; 0.41; ; ; 0.002; 590; 320; 0.10]; % %x0=[0.27; 0.38; ; ; 0.300; 990; 320; 0.02]; % %% Setting Lower Bounds, Upper Bounds, Options and FMINCON. lb=[ ;0.30; ;0.001; ; ;20;0.001]; ub=[0.50;0.45;0.50;0.45;0.45;inf;inf;0.50]; options=optimset('display','iter','algorithm','sqp','tolx',10^(-8),... 'TolCon',10^(-8),'TolFun',10^(-8),'MaxFunEvals',7000,... 'TypicalX',[0.50;0.45;0.0001;0.001;0.010;300;30;0.001],'PlotFcns',... {@optimplotx,@optimplotfval,@optimplotstepsize,@optimplotfirstorderopt}); options; [X,FVAL,EXITFLAG,OUTPUT,LAMBDA,GRAD,HESSIAN]=fmincon(@Heat_Dissipation_Final, x0,[],[],[],[],... lb,ub,@nonlinear_constraints_final,options) [NonLinear_Constraints,Ceq]=nonlinear_constraints_Final(X); EigenVal_Hessian=eig(HESSIAN) NonLinear_Constraints % END P a g e

97 Radiator Subsystem : Objective Function Calculation % FUNCTION TO CALCULATE THE HEAT DISSIPATION RATE % Design Variables % x(1)-length of Radiator % x(2)-width of Radiator % x(3)-thickness of One Fin % x(4)-tube Height % x(5)-fin Height % x(6)-number of Fins per Tube % x(7)-number of Tubes % x(8)-fin Gap %x0=[0.40; 0.30; ; ; ; 200; 20]; %x1=[0.46; 0.43; ; ; ; 280; 33]; Optimal Design of Hybrid Electric Vehicle for Fuel Economy function f=heat_dissipation_final(x) T_coolant=393.15;%(in Kelvin) Obtained from Engine Subsystem %% RADIATOR DIMENSIONS T_rad=0.025; %Radiator Thickness W_fin=T_rad; %Width of Fin W_tube=T_rad; %Width of Tube %% AIR PROPERTIES Ka=0.0266; %Thermal Conductivity (W/m.k) Ca= ; %Specific Heat Capacity (J/kg.K) Rho_a=1.137; %Density (kg/m3) Mu_a= ; %Dynamic Viscosity (Pa.s) VOL_a=1.1086; %Volumetric Flow Rate (m3/s) T_air=298.15; % Temperature of Air = Kelvin %% COOLANT PROPERTIES (50-50 Mixture of Water and Glycol) Kc=0.415; %Thermal Conductivity (W/m.k) Cc= ; %Specific Heat Capacity (J/kg.K) Rho_c= ; %Density (kg/m3) Mu_c= ; %Dynamic Viscosity (Pa.s) VOL_c= ; %Volumetric Flow Rate (m3/s) %% TUBE MATERIAL PROPERTIES K_Al=180; %Thermal Conductivity of Aluminium % K_Al=14; %Thermal Conductivity of Steel % K_Al=398; %Thermal Conductivity of Copper %% Calculating Area on Coolant Side (Ac) P_tube=2*(W_tube+x(4)); Ac_tube=W_tube*x(4); % Cross Section Area of Tube A=P_tube*x(1); %Surface Area of each tube 97 P a g e

98 Ac=x(7)*A; %Total Surface Area on Coolant Side for all tubes %% Calculating Area on Air Side (Aa) P_fin=2*(x(3)+W_fin); %Fin Perimeter Ac_fin=W_fin*x(3); %Cross Section Area of Fin A_fin=P_fin*x(5); %Surface Area of One Fin Ab_no_fin=((x(1)*T_rad)-(x(6)*Ac_fin))*2;%Base Area on each tube without fins A_t=(x(6)*A_fin+Ab_no_fin);%Total Surface area of one fin-tube assembly in contact with air Aa=x(7)*A_t; %Total Surface area in conact with air for all tubes %% Calculating Convective Heat Transfer Coefficient of Coolant Dh_tube=4*Ac_tube/P_tube; %Hydraulic Diameter V_coolant=VOL_c/(x(7)*Ac_tube); %Coolant Velocity inside Tubes Re_c=Rho_c*V_coolant*Dh_tube/Mu_c; %Reynolds No. of Coolant Flow Pr_c=Cc*Mu_c/Kc;%Prandtl No. of Coolant Nu_c=0.023*(Re_c^0.8)*(Pr_c^(1/3)); hc=nu_c*kc/dh_tube; %% Calculating Convective Hear Transfer Coefficient of Air Gap_fins=x(8); A_gap=x(5)*Gap_fins;% Cross Section Area of Air Gap Dh_gap=4*A_gap/(2*(x(5)+Gap_fins)); % Hydraulic Diameter of Air Gap N_gap=(x(6)-1)*(x(7)-1); % Total number of Air Gaps V_air=VOL_a/(N_gap*A_gap);% Air Velocity inside Air Gap Re_a=Rho_a*V_air*Dh_gap/Mu_a; %Reynold's Number of Air Flow Pr_a=Ca*Mu_a/Ka;%Prandtl Number of Air ha=580; m=sqrt((ha*p_fin)/(k_al*ac_fin)); eta_f=tanh (m*x(5))/(m*x(5)); eta_o=1-(1-eta_f)*a_fin*x(6)/(x(7)*a_t); Ef=sqrt((K_Al*P_fin)/(ha*Ac_fin)); ha=eta_o*ha; %% Calculating Net Heat Dissipation UA=1/((1/(hc*Ac))+(1/(ha*Aa))); C_air=Ca*Rho_a*VOL_a; C_coolant=Cc*Rho_c*VOL_c; if (C_air<C_coolant) Cmin=C_air; Cmax=C_coolant; else Cmin=C_coolant; Cmax=C_air; end Cr=Cmin/Cmax; ITD=T_coolant-T_air; % Initial Temperature Difference NTU=UA/Cmin; % Number of Transfer Units e=1-exp((1/cr)*(ntu^0.22)*(exp(-cr*(ntu^0.78))-1));% Effectiveness Q=e*Cmin*ITD;% Net Heat Dissipation Rate (Watt/s) f=-q/1000;% Returning the function value as negative of Q (in KW/s) end % END P a g e

99 Radiator Subsystem : Non-linear Constraint Function Code %% FUNCTION TO CALCULATE THE HEAT DISSIPATION RATE function [C,Ceq]=nonlinear_constraints_Final(x) %% Design Variables % x(1)-length of Radiator % x(2)-width of Radiator % x(3)-thickness of One Fin % x(4)-tube Height % x(5)-fin Height % x(6)-number of Fins per Tube % x(7)-number of Tubes % x(8)-fin Gap Optimal Design of Hybrid Electric Vehicle for Fuel Economy %% Temperature of Coolant T_coolant=393.15;%(in Kelvin) Obtained from Engine Subsystem %% RADIATOR DIMENSIONS T_rad=0.025; %Radiator Thickness W_fin=T_rad; %Width of Fin W_tube=T_rad; %Width of Tube %% AIR PROPERTIES Ka=0.0266; %Thermal Conductivity (W/m.k) Ca= ; %Specific Heat Capacity (J/kg.K) Rho_a=1.137; %Density (kg/m3) Mu_a= ; %Dynamic Viscosity (Pa.s) VOL_a=1.1086; %Volumetric Flow Rate (m3/s) T_air=298.15; % Temperature of Air = Kelvin %% COOLANT PROPERTIES (50-50 Mixture of Water and Glycol) Kc=0.415; %Thermal Conductivity (W/m.k) Cc= ; %Specific Heat Capacity (J/kg.K) Rho_c= ; %Density (kg/m3) Mu_c= ; %Dynamic Viscosity (Pa.s) VOL_c= ; %Volumetric Flow Rate (m3/s) %% TUBE MATERIAL PROPERTIES K_Al=180; %Thermal Conductivity of Aluminium % K_Al=14; %Thermal Conductivity of Steel % K_Al=398; %Thermal Conductivity of Copper %% Calculating Area on Coolant Side (Ac) P_tube=2*(W_tube+x(4)); Ac_tube=W_tube*x(4); % Cross Section Area of Tube A=P_tube*x(1); %Surface Area of each tube Ac=x(7)*A; %Total Surface Area on Coolant Side for all tubes %% Calculating Area on Air Side (Aa) P_fin=2*(x(3)+W_fin); %Fin Perimeter Ac_fin=W_fin*x(3); %Cross Section Area of Fin A_fin=P_fin*x(5); %Surface Area of One Fin Ab_no_fin=((x(1)*T_rad)-(x(6)*Ac_fin))*2;%Base Area on each tube without fins 99 P a g e

100 A_t=(x(6)*A_fin+Ab_no_fin);%Total Surface area of one fin-tube assembly in contact with air Aa=x(7)*A_t; %Total Surface area in conact with air for all tubes %% Calculating Convective Heat Transfer Coefficient of Coolant Dh_tube=4*Ac_tube/P_tube; %Hydraulic Diameter V_coolant=VOL_c/(x(7)*Ac_tube); %Coolant Velocity inside Tubes Re_c=Rho_c*V_coolant*Dh_tube/Mu_c; %Reynolds No. of Coolant Flow Pr_c=Cc*Mu_c/Kc;%Prandtl No. of Coolant Nu_c=0.023*(Re_c^0.8)*(Pr_c^(1/3)); hc=nu_c*kc/dh_tube; %% Calculating Convective Hear Transfer Coefficient of Air %% Calculating Net Heat Dissipation Gap_fins=x(8); A_gap=x(5)*Gap_fins;% Cross Section Area of Air Gap Dh_gap=4*A_gap/(2*(x(5)+Gap_fins)); % Hydraulic Diameter of Air Gap N_gap=(x(6)-1)*(x(7)-1); % Total number of Air Gaps V_air=VOL_a/(N_gap*A_gap);% Air Velocity inside Air Gap Re_a=Rho_a*V_air*Dh_gap/Mu_a; %Reynold's Number of Air Flow Pr_a=Ca*Mu_a/Ka;%Prandtl Number of Air ha=580; m=sqrt((ha*p_fin)/(k_al*ac_fin)); eta_f=tanh (m*x(5))/(m*x(5)); eta_o=1-(1-eta_f)*a_fin*x(6)/(x(7)*a_t); Ef=sqrt((K_Al*P_fin)/(ha*Ac_fin)); ha=eta_o*ha; %% Calculating Net Heat Dissipation UA=1/((1/(hc*Ac))+(1/(ha*Aa))); C_air=Ca*Rho_a*VOL_a; C_coolant=Cc*Rho_c*VOL_c; if (C_air<C_coolant) Cmin=C_air; Cmax=C_coolant; else Cmin=C_coolant; Cmax=C_air; end Cr=Cmin/Cmax; ITD=T_coolant-T_air; % Initial Temperature Difference NTU=UA/Cmin; % Number of Transfer Units e=1-exp((1/cr)*(ntu^0.22)*(exp(-cr*(ntu^0.78))-1));% Effectiveness Q=e*Cmin*ITD;%Net Heat Dissipation Rate (Watt/s) %% Calculating the Non-Linear Constraints g1=p_tube*(x(1)-x(6)*x(3))-x(6)*a_fin; g2=2.65-m*x(5); g3=2-ef; g4=0.25-ntu; g5=1.5*x(3)-x(8); g6=x(8)*(x(6)-1)+x(6)*x(3)-x(1); g7=x(7)*x(4)+(x(7)-1)*x(5)-x(2); C=[g1;g2;g3;g4;g5;g6;g7]; Ceq=[]; end % END P a g e

101 C) MATLAB CODE Battery Subsystem (ARCHIT RASTOGI) Battery Subsystem: Constraint function function [c, ceq] = confun(x) soc_min = 0.3; soc_max = 0.9; Vcell=9; Cmax=1; Q=5; m=50; Mmax = 40000; Cmin =- 1; Imin_ch=-2; Idis_max=5.5; rc = -8000; lb = [25,5, 0.3, 25000,-12000]; ub = [50,25,0.85,50000,-3000]; x = scaling(x,lb,ub,2); Ns=x(1); Np=x(2); soc=x(3); Pbat = x(4); Preg =x(5); Voc= Ns*2*( *soc *(soc^2) *(soc^3)); R = 0.01*Ns/Np*( *soc *(soc^2) *(soc^3)); if soc<=0.8 w = 1; elseif soc >0.8 && soc<=0.9 w = 9-10*soc; else w = 0; end % Nonlinear inequality constraints g7= Ns*Np*m-Mmax; g8=((voc-sqrt(voc^2-4*r*(pbat)))/2)-ns*vcell; g9 = Pbat-Ns*Vcell*Np*Cmax*Q; g10 = ((Voc-sqrt(Voc^2-4*R*(Pbat)))/(2*R))-Np*Idis_max; g11 = Np*Imin_ch-((Voc-sqrt(Voc^2-4*R*(Preg)))/(2*R)); g12 = ((Voc-sqrt(Voc^2-4*R*(Pbat)))/(2*R*Q*Ns))-Cmax; g13= Cmin-((Voc-sqrt(Voc^2-4*R*(Preg)))/(2*R*Q*Ns)); 101 P a g e

102 g14 = w*rc-preg; c = [g7,g8,g9,g10,g11,g12,g13,g14]; ceq = [];% Nonlinear equality constraints Objective function function f = objfun(x) lb = [25, 5, 0.3, 25000,-12000]; ub = [50,25,0.85,50000,-3000]; x = scaling(x,lb,ub,2); f= -x(4)-x(5); Main clc clearall salgo = 'interior-point'; %salgo = 'active-set'; %salgo = 'sqp' options = optimset( 'Display','iter', 'LargeScale','on', 'MaxIter', 1000, 'MaxFunEvals', 3000,... 'Algorithm',sAlgo,'TolFun',1e6,'FinDiffType','central','PlotFcns',@optimplotfval,'AlwaysHonorConstraints','bounds') ; lb = [25, 5, 0.3, 25000,-12000]; ub = [50,25,0.85,50000,-3000]; x0 = [ ,-10000];% Make a starting guess at the solution xoexcel= x0'; % scale inputs X0 = scaling(x0,lb,ub,1); LB = zeros(size(x0)); UB = ones(size(x0)); A = []; b= []; Aeq =[]; beq = []; [xopt1, fval1, exitflag1, output1, lambda1,grad1,hessian1] = fmincon(@objfun,x0,a,b,aeq,beq,lb,ub,@confun,options); DispResults(xopt1, exitflag1, output1, lb, ub); xoptim = (scaling(xopt1,lb,ub,2))'; Scaling Function 102 P a g e

103 function x_out = scaling(x,l,u,type) % scaling.m scales or unscales the vector x according to the bounds % specified by u and l. The flag type indicates whether to scale (1) or % unscale (2) x. Vectors must all have the same dimension. for i=1:1:5 if type == 1 % scale if x(i)>0 else end x_out(i) = (x(i)-l(i))/(u(i)-l(i)); x_out(i) = (2*x(i)-u(i)-l(i))/(u(i)-l(i)); elseif type == 2 % unscale if x(i) >0 else end end end x_out(i) = l(i) + x(i)*(u(i)-l(i)); x_out(i) = (l(i)+u(i) + x(i)*(u(i)-l(i)))/2; Display Function function DispResults(xopt, exitflag, output, lb, ub) global FCALL GCALL CALCCALL % unscale design variables xopt_unscale = scaling(xopt,lb,ub,2); disp (['algorithm disp (['x_opt : ' output.algorithm]); Optimal Design of Hybrid Electric Vehicle for Fuel Economy : (' num2str(xopt_unscale(1),'%.1f') ',' num2str(xopt_unscale(2),'%.1f') ',' num2str(xopt_unscale(3),'%.1f') ',' num2str(xopt_unscale(4),'%.1f') ',' num2str(xopt_unscale(5),'%.1f') ')' ]); disp (['exitflag disp (['iterations : ' num2str(exitflag)]); : ' num2str(output.iterations)]); disp (['firstorderopt : ' num2str(output.firstorderopt, '%11.4g')]); disp' ' end 103 P a g e

104 D) MATLAB CODE Battery Thermal Management Subsystem (MINGXUAN ZHANG) symsabmlpowereff_fanvair_fnuatp_1rtinqintqgrho_1cpbtbdhkbtm_baracremuprfrho_2 Vair_t %control c_h=1; %1 for cooling, 0 for heating %Constant R=286.9; cpb=700; P_1=10^5; %air constant %heat capacity of battery %inlet pressure %Dependent constant mu=184.6/10^7; kf=26.3/10^3; cpf=100; %air viscosity in tube %air conduction coeff %heat capacity of air %Parameter eff_fan=0.2;%guess P_2=P_1; rho_b=1811; %gap pressure %density of battery %System relative parameter Qg=500; Tb=40+273; %heat generate by battery %inital battery temperature %Varable Tin=25+273; At=2/10^2; l=0.1819; a=1/10^2; b=0.8/10^2; m=10; Tg=30+273; %inlet temperature %cross area of tube %length of the gap %width of the gap %hight of the gap %number of tunnal %target temperature %for first order plot %At=0:0.001:0.05;% m=1:10;% a=0:0.001:0.05;% b=0:0.001:0.02;% l=0:0.01:0.5; 104 P a g e

105 %Equation rho_1=p_1/r/tin; rho_2=rho_1; Ac=a*b; %cross area of gap Dh=4*Ac/(2*a+2*b); %specific diameter Pr=cpf*mu/kf; % for i=1:100 % Qint=Qg-rho_b*cpb*(Tb(i+1)-Tb(i)); % end Qint=Qg; %running at steady state Tm_bar=(Tin+Tb)/2; Nu=Dh./kf.*Qint./m./((a+b).*2.*l)./(Tb-Tm_bar); if Tin>Tb ch=0; else ch=1; end if c_h==1 Re=(Nu./0.023./Pr.^0.3).^(5/4); else Re=(Nu./0.023./Pr.^0.4).^(5/4); end Vair_t=Re.*mu./Dh./rho_2; Surf=((5*l+a.*m).*b+(5*l+a.*m).*(l+At.^0.5/2)+(l+At.^0.5/2).*b)*2; Vol=Surf*0.02; mass=vol*1.2/10^3*10^6 Vair_f=m.*Vair_t.*rho_2.*Ac./At./rho_1 Power=Vair_f.^2.*At.*rho_1./eff_fan 1. Optimal analysis code: a) 105 P a g e

106 clc clear CC=13/2/0.02/1.2/10^3; alpha=0.01; %Initial guess At=0.02; l=0.4465; x=[l;at]; Power=0.1057; a=1/10^2; b=0.8/10^2; %width of the gap %hight of the gap m=5; %calculate for m=1:10 for i=1:100 l=x(1); At=x(2); pf=(power/m^0.5)*[-2.5*at^-1*l^-3.5, -1*At^-2*l^-2.5]; ph=[10*l+a*m+10*b+2.5*at^0.5, 1.25*l*At^(-0.5)+0.25*(b+a*m)*At^(-0.5)]; P=eye(2)-ph'*(ph*ph')^-1*ph; x_temp=x-alpha*p*pf'; l=x_temp(1); At=x_temp(2); pf=(power/m^0.5)*[-2.5*at^-1*l^-3.5, -1*At^-2*l^-2.5]; ph=[10*l+a*m+10*b+2.5*at^0.5, 1.25*l*At^(-0.5)+0.25*(b+a*m)*At^(-0.5)]; h=5*l^2+(a*m+2*b)*l+2.5*l*at^0.5+(b+a*m)/2*at^0.5+a*m*b-cc; x=x_temp-ph'*(ph*ph')^-1*h; e=p*pf'; if abs(h)<0.001 if abs(e)< P a g e

107 break end end end st(m,1)= m st(m,2)=l st(m,3)=at st(m,4)=abs(5*l^2+(a*m+2*b)*l+2.5*l*at^0.5+(b+a*m)/2*at^0.5+a*m*b-cc) st(m,5)=abs(power/at/l^2.5/m^0.5) end b) clc clear Power=0.1057; a=1/10^2; b=0.8/10^2; At=0.02; CC=13/2/0.02/1.2/10^3; for m=1:20 aa=5; bb=(a*m+10*b)+2.5*at^0.5; cc=(b+a*m)/2*at^0.5+a*m*b-cc; l=(-bb+(bb^2-4*aa*cc)^0.5)/2/aa E=Power/At/l^2.5/m^0.5 g=5*l^2+(a*m+10*b)*l+2.5*l*at^0.5+(b+a*m)/2*at^0.5+a*m*b-cc Surf=((5*l+a.*m).*b+(5*l+a.*m).*(l+At.^0.5/2)+(l+At.^0.5/2).*b)*2; Vol=Surf*0.02; mass=vol*1.2/10^3*10^6 st(m,1)=m; st(m,2)=l; st(m,3)=g; st(m,4)=e; st(m,5)=mass; end ww=st; 107 P a g e

108 VII) APPENDIX II A) Radiator Subsystem: First Order Design of Experiments ariation of and constraints with radiator length. Fig 40: Radiator Subsystem: Variation of Q, g1, g2, g3, g4, g5, g6 and g7 with length of radiator keeping other variables fixed 108 P a g e

109 ariation of and constraints with fin thickness. Fig 41: Radiator Subsystem: Variation of Q, g1, g2, g3, g4, g5, g6 and g7 with fin thickness keeping other variables fixed 109 P a g e

110 ariation of and constraints with tube height. Fig 42: Radiator Subsystem: Variation of Q, g1, g2, g3, g4, g5, g6 and g7 with height keeping other variables fixed 110 P a g e

111 ariation of and constraints with fin height. Fig 43: Radiator Subsystem: Variation of Q, g1, g2, g3, g4, g5, g6 and g7 with fin height keeping other variables fixed 111 P a g e

112 ariation of and constraints with number of fins per tube. Fig 44: Radiator Subsystem: Variation of Q, g1, g2, g3, g4, g5, g6 and g7 with number of fins per tube keeping other variables fixed 112 P a g e

113 ariation of and constraints with number of tubes. Fig 45: Radiator Subsystem: Radiator Subsystem: Variation of Q, g1, g2, g3, g4, g5, g6 and g7 with number of tubes keeping other variables fixed 113 P a g e

114 B) Radiator Subsystem: Second Order Design of Experiments ariation of with length and width of radiator Fig 46: Radiator Subsystem: Variation of Q with length and width of radiator keeping other variables fixed ariation of with fin thic ness and gap between fins Fig 47: Radiator Subsystem: Variation of Q with fin thickness and gap between fins keeping other variables fixed 114 P a g e

115 ariation of with fin thic ness and tube height Fig. 48: Radiator Subsystem: Variation of Q with fin thickness and tube height keeping other variables fixed ariation of with radiator length and fin height Fig 49: Radiator Subsystem: Variation of Q with fin height and radiator length keeping other variables fixed 115 P a g e

116 ariation of with fin height and radiator width Fig 50: Radiator Subsystem: Variation of Q with fin height and radiator width keeping other variables fixed ariation of with fin height and fin thic ness Fig 51: Radiator Subsystem: Variation of Q with fin height and fin thickness keeping other variables fixed 116 P a g e

117 ariation of with fin height and tube height Fig 52: Radiator Subsystem: Variation of Q with fin height and tube height keeping other variables fixed VIII ariation of with number of fins per tube and fin thickness Fig 53: Radiator Subsystem: Variation of Q with number of fins per tube and fin thickness keeping other variables fixed 117 P a g e