Fuel Consumption Reduction with a Starter-Alternator using an MPC-based Optimisation

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1 Consumption Reduction with a Starter-Alternator using an PC-based Optimisation ark iert Ford Forschungszentrum Aachen, Germany eywords: model predictive control, optimization, receding horizon, hybrid drivetrain, starter-alternator Abstract In order to minimize the uel consumption and emissions o a sot hybrid vehicle while maintaining the SOC o the battery in a speciied window, an optimising control strategy is uired. An optimization developed or this purpose is introduced here. It employs a cost unction that is minimized o-line to reduce the calculation power uired or its implementation.. Introduction Spurned by the increasing electrical energy consumption in modern vehicles, the implementation o the starter-alternator is being developed by the automotive industry to provide power to the coming 4 power distibution network. It has been realized that the starter-alternator not only could be used to generate and start but also to provide boost to the combustion engine. With a starter-alternator integrated in its drivetrain, a vehicle could be classiied as a sot hybrid, and through the use o an optimising strategy, it could be controlled to reduce uel consumption and emissions as well as to keep the battery charged. In this paper, an optimisation based on model predictive control is introduced. It minimizes a cost unction over a sliding time rame and could be easily implemented in a real-time application without previous knowledge o the driving cycle.. Overview o Drivetrain with Starter-Alternator A vehicle that contains a starter-alternator integrated in its drivetrain can be treated as a parallel-hybrid-vehicle since the propulsion power could be provided by both the combustion engine and an electric machine. In drivetrains that are being developed or this technology, the starter-alternator may be placed inside the bell housing between the engine and gearbox. In this position, the starter-alternator can take the place o the engine's lywheel.. odes o Operation with a Starter-Alternator A starter-alternator is an electric machine that can unction as a motor or generator, and as such it can add a positive or negative torque to the crankshat o the combustion engine. When the electric machine is operated as a motor, it is said to boost. In this case, the electrical torque produced by the starter-alternator has the same sign as the torque produced by the combustion engine C. When the electric machine is operated as a generator, it produces an additional drag on the crankshat, and the electrical torque is negative with respect to the torque produced by the engine. he uired crankshat torque that is necessary to propel a vehicle at a certain speed and with a certain acceleration and gear is a unction o the rolling riction, aerodynamic drag, climbing resistance and inertia o the vehicle. It ollows that the uired torque is a unction o the wishes o the driver or o a prescribed driving cycle. he electrical and combustion-engine torques are added at the crankshat, and the sum must equal the uested torque at any given time: () hereore a starter-alternator can be used to shit the operating point o the combustion engine by choosing an electrical torque or a given uested crankshat torque. A shit in the engine torque against the backdrop o a map o speciic consumption b e or a compression-ignition engine is illustrated in igure. With the electric machine unctioning as a generator, the operating point o the combustion engine is shited to higher torques with lower speciic uel consumption values. he opposite occurs when the electric machine is operated as a motor. In this case the operating point o the combustion engine sinks to a lower torque and higher speciic consumption. A map o speciic uel consumption or a sparkignition engine is similar to its compression-ignition counterpart in that b e sinks with rising loads, although the minimum speciic consumption is not located on the maximum torque curve, as is the case with a diesel engine. he product o the speciic uel consumption and the speciic caloriic value o the uel C is inversely proportional to the indicated thermal eiciency o the combustion energy th, which deines the raction o the chemical energy in a mass o uel that is changed to kinetic energy []: It ollows that a shit to a higher speciic uel consumption is a shit to a lower thermal eiciency, and conversely a shit to a lower speciic consumption is simultaneously a shit to a higher, better eiciency. his ability to improve the operating point o the combustion engine with a starter-alternator may C th ( be C ) ()

2 be exploited by an optimising control system in order to improve the global eiciency o a drive train over a driving cycle. Such a control system must take into account that every shit in operating point brings about a change in the state o charge o the battery. hereore the state o charge must be weighed against an improvement in operating eiciency by an optimising control system. he eiciency actor describes the total eiciency o the electric machine and its drive circuit (inverter) and is a unction o the rotational speed and mechanical torque o the device. For the operation as a generator, the electrical power produced by the machine is deined positive and can be expressed as ollows: P, gen Pech,, gen and Similarly, the electrical power used when the electric machine is operated as a motor is deined as being negative and can be expressed as ollows: P P, mech, and, mot, mot (4) (5) In the simulation, the eiciencies,mot and,gen are obtained rom maps. Figure : Shiting Operating Point o Combustion ngine with lectric orque. ehicle and Combustion ngine odel A Ford Focus with a.5 liter, three cylinder spark-ignition engine and a starter-alternator was built up or testing purposes, and this vehicle is used as the basis or the simulations. he combustion engine produces a maximum mechanical power o 56 kw, while the starter-alternator can produce a maximum o 3 kw continuous mechanical power. In generator mode, it can produce a maximum o 5 kw electrical power. he starter-alternator is an asynchronous machine designed to operate on a 4 power distribution network. In the test vehicle, a 36, 7 Ah AG (absorbent glass matt technology) battery is used as an energy storage device. he vehicle is modeled in Simulink. he dynamic behavior o the spark-ignition engine is modeled by a dierential equation o the irst order, and the uel consumption is calculated using a uel mass-low map..3 Starter-Alternator odel Like the combustion engine, the dynamic behavior o the electric machine is also modeled by a dierential equation o the irst order. However, its time constant is approximately one hundred times aster than that o the combustion engine. he starter-alternator changes electrical to mechanical power as a motor and mechanical to electrical power as a generator. he positive or negative electrical power that is generated or used by the electric machine P can be expressed as a product o the mechanical power P ech and an eiciency actor : P P ech (3).4 attery odel lectrical power charging a battery is transormed into chemical energy and stored. When a battery is discharged, the stored chemical energy is transormed back into electrical energy. y every transormation there occurs an energy loss that is a unction o the internal ohmic resistance o the battery and the square o the charging or discharging current [5]. A schematic diagram o an external electrical power source or sink connected to a battery is illustrated in igure. U ext Source/Sink attery i R Di R Ci - - Figure : Schematic Diagram o attery and external oltage Source or Sink U int Analog to the electric power, the current i is deined as positive when it is charging and negative when it is discharging. he internal ohmic resistance during charging is represented in the igure by R Ci, and R Di represents the internal ohmic resistance during discharging. he battery eiciency is deined as the useable power that is stored or discharged divided by the sum o this power and the power dissipated in the internal battery resistance [5]. In the case that the battery is being charged, the battery eiciency C can be ormulated as ollows: C U R Ci i int, i (6)

3 he battery eiciency during a discharge can be expressed as ollows: D R Di i, i < U As in the case o the electric machine, the battery eiciency can be expressed by an eiciency actor describing the eiciency o an energy transormation. While the eiciency actor o the electric machine describes the eiciency o the transormation between mechanical and electrical energy, the eiciency actor o the battery,, describes the eiciency o the transormation between electrical and chemical energy. he change in the energy stored in the battery can be expressed with a simple irst-order dierential equation: When the battery is being charged and the input power is positive, the dierential equation has the orm: C Conversely, when the input power is negative and the battery is being discharged, the equation has the ollowing orm: he nominal state o charge is deined to be less than the maximum charge in order to allow or the storage o cheaply won energy such as that rom regenerative braking [3]. In the optimisation, the deviation in the nominal state o charge is to be minimized. It is deined as the quotient o the energy stored in the battery at to the energy stored at the nominal state o charge at,nom : he battery model used in the simulations was created using impedance spectroscopy. With this technique, the complex impedance o the battery over a range o state o charge values and currents is determined experimentally on a test rig and used to create a parameterized equivalent circuit. With this circuit, both the ohmic and imaginary components o the battery's internal resistance are calculated and used to predict the energy losses and terminal voltage. 3 Derivation o an PC-ased Optimisation An optimisation is uired that reduces the uel consumption o a vehicle whose drivetrain includes a starter-alternator without previous knowledge o the speed and gears chosen during a drive. A urther goal may be to reduce the exhaust gas emissions, and in the ollowing derivation, the inclusion o individual exhaust gas types is included in the optimisation. ecause these goals are achieved by shiting the operating point o the combustion engine with the starter-alternator, which eects the energy stored in the battery, it is necessary int P P, D P, SOC at at, nom C % D (7) (8) (9) () () to balance an improvement in the engine s perormance with a change in the state o charge o the battery. he uel consumption, exhaust gas production and state o charge could be used to deine a cost unction, and the goal o the optimisation can be stated as minimizing the cost unction over the driving cycle. ecause the driving cycle is unknown, it is necessary to make a prognosis o the vehicle s operation during a limited time rame. his optimisation horizon is chosen to be longer than the sampling time o the vehicle's powertrain controller, and a new optimisation over the horizon is calculated ater each sampling interval. ecause the optimisation is updated at a much shorter interval than the horizon length, the inluences o the length o the horizon and unexpectedly changing conditions are minimized [6]. he optimisation presented here is based on the basic theory o model predictive control in which a prediction o vehicle states is calculated over a sliding horizon using a plant model and predicted inputs []. In the optimisation it is assumed that the vehicle speed remains constant in the time rame. his assumption works well with the NDC test cycle, which consists o intervals o constant acceleration ollowed by intervals o constant speed, but or use in actual driving conditions, better results may be achieved by taking the acceleration o the vehicle into account. Simulations using optimisation horizons with several lengths were carried out, and a horizon o 8 seconds brought the best results. 3. Cost Function Components he state o charge SOC can be expressed as a state variable, and the change in the state o charge with respect to the nominal value SOC over the optimisation horizon is to be minimized. Using equations (3), (8) and (), the change in the state o charge can be expressed as an integrated unction o the rotational speed n, the torque and eiciency actors o the electric machine and, the eiciency actor o the battery, and the nominal charge att,nom : SOC H ( ( ( att, nom () he limits o the integration are the start time and the sum o the start time and the optimisation horizon length H. In order to carry out an optimisation, the perormance index should always be positive. hereore the absolute value o the change in the state o charge is used in the cost unction. he masses o uel consumed and exhaust gases produced are also actors that are to be minimized over the optimisation horizon. his can be accomplished by maximizing the average combustion eiciency and similar eiciencies expressing the energy production with respect to the exhaust gas masses that result, or by minimizing the reciprocals o the average eiciencies over the time horizon. A weighted sum o the reciprocals o these eiciencies can be expressed as a single value that is to be minimized. Such an expression, which characterizes the perormance o the system over the

4 optimisation horizon, is reerred to in the literature as a Lagrange index [4]. he combustion eiciency deined in equation () is inversely proportional to the product o the speciic uel consumption b e and the speciic heating value o the uel C, which is a constant. aximizing the average combustion eiciency can be achieved by minimizing the average speciic uel consumption. his component o a Lagrange perormance index can be expressed as ollows: L H C b H e ( (, ) R (3) he oset R and the weighting actor do not aect the minimum but can be used to set the range o the Lagrange uel component between and, which corresponds to and %. hey will be used later to assign weights to the minimization o individual exhaust gasses or uel consumption. Similarly, the speciic production o an exhaust gas in g/kwh is inversely proportional to an exhaust gas eiciency expressing the ratio o kinetic energy released to the mass o gas produced. As with the consumption eiciency, the average o a given exhaust gas eiciency can be maximized by minimizing the average speciic gas production, and the range o possible values o each Lagrange exhaust gas component L G can be set using a weighting actor G : L G H CG G b H G, e (4) Components or the uel and exhaust gases,, HC and NO X are added together to orm the complete Lagrange index L that is to be minimized: L L (5) he engine maps or the speciic consumption and exhaust gas production can be seen as describing suraces with the coordinates torque and rotational speed n. ecause the high and low areas o these suraces don t always match, it is sometimes necessary to make compromises in the minimization in order to meet emissions or uel consumption goals or a certain driving cycle. Compromises could be made by emphasizing certain components through their weighting constants, NOx. he individual components o the Lagrange perormance index could be expressed as the sum o a single integral and a constant: ( (, ) HC R L L L L H H C be ( (, b, e H H R R... H C b e C ( (, C b, R R... H H e NO X G P( (, Q ( (, ( (,... (6) he dependant variable P( (, is obtained by adding the products o the individual engine maps and their weights together point by point and can be reerred to as a perormance index map or surace. A sample perormance index map used in simulations to optimise a drivetrain with a spark-ignition engine is illustrated in Figure 3. Figure 3: Perormance Index ap or Spark-Ignition ngine he total cost unction is made up o the sum o the SOC and Lagrange components: H J H ( ( ( Q P( (, Q (7) he actors and are weighting actors that can used to emphasize the battery s state o charge or engine perormance in the total cost unction. An optimising control system is sought that minimizes the cost unction at each sampling time in order to ind the best compromise between the two sides. Finding a solution by solving a variation problem deined by the system describing the drivetrain and the combined perormance index is diicult because o the non linearities represented by the engine maps and the eiciency characterisctics o the electric machine and battery. However, by implementing a number o simpliications, it is possible to ind a closed solution to the minimizing problem that can be used by a control system to optimise the operation o a drivetrain in real time. 3. Simpliication through Restriction o System Dynamics and ariables he state equations o the system describing a hybridized drivetrain are made up o the equations o motion o the combustion engine, electrical machine and vehicle (8), (9) and () and the energy state equation o the battery (). Here ρ represents the air density, C W A represents the product o the vehicle's air drag coeicient and rontal area, v at

5 represents the vehicle speed, g represents the earth's acceleration, m eh represents the vehicle's mass and m Inirtia represents an equivalent additional mass that is eectively added to the vehicle mass during acceleration. C C C _ C _ In C In (8) (9) m eh ρcw Av v g Rmeh meh minertia RWheel meh () 5 v () PLoads 3 π RWheel i quation (), which deines the uested torque as the sum o the torques rom the combustion engine C and electrical machine, is a boundary condition to the system o equations. With the restriction that the torques C and remain constant in the interval between the sampling times o the driving cycle or uested torque, the cost unction can be simpliied. his is done by neglecting their rise-times, which can be done, because the dynamics o the combustion engine and electric machine are much quicker than the dynamics o the input signal describing the drive cycle. A urther simpliication can be achieved through the use o constant battery and electric machine eiciency actors and, and a inal simpliication is the assumption that the vehicle speed remains constant over the optimisation horizon. As a result, the rotational speed o the engine N becomes a constant in the cost unction. his o course produces deviations rom an optimal solution during accelerations, but the deviation is minimized by the uent updating o the optimisation. Using these simpliications, the integrals in the perormance index equation (7) could be replaced by simpler expressions. ecause Q in (7) is a constant, it can be eliminated rom the equation. he simpliied cost unction can be expressed as ollows: ( N For a given uired torque, the combustion engine torque C corresponding to can be calculated using the condition deined by equation (): C A deviation in the electrical torque causes a reciprocal deviation in the torque produced by the combustion engine according to equation (5): C he cost unction () can be rewritten in terms o the electrical and combustion engine torques and torque derivations and simpliied by using the relationship between and e ' as expressed in equation (3): 3.3 Closed Solution to Optimisation (3) (4) (5) (6) In order to obtain a closed control law that perorms the optimisation, it is necessary to estimate the average electric machine eiciency actor (,N) and treat it as a constant in the calculation o that is carried out in equation (3) and in the cost unction. In this case the cost unction can be expressed as ollows: (7) he minimization o the cost unction expressed by equation (7) can be carried out o-line on each element o the perormance index map described by P( C,N), and the result is an optimisation map that gives an optimised * combustion engine torque C or a given C. An optimisation map is illustrated in Figure 4. H N H J P( C ( N H J P( C J N H P( ( () he energy is the dierence between the nominal energy capacity o the battery and the amount o energy stored in the battery at the sampling time. he SOC- (irst-) component o the cost unction is minimized to zero when the electrical energy added to or taken rom the battery equals the negative o. his is accomplished by boosting or generating with the electrical torque, which is deined as ollows: Figure 4: Optimisation ap o Spark-Ignition ngine

6 In order to implement the map in a control system, the electrical torque and the corresponding combustion engine torque C are calculated with equations (3) and (4) or a given uired torque and state o charge. he optimised combustion engine torque C * or the measured engine speed N is determined rom the optimisation map, and the corresponding optimised torque to be produced by the starter-alternator is calculated using equation (8): 4 Simulations with Optimisation ethod in Closed Form Simulations using the optimisation method outlined above were carried out with and without regenerative braking or electrical loads between W and kw and compared with voltage control in order to judge whether improvements in uel consumption occur. he relative improvements using the closed solution optimisation over voltage control are shown in chart. he reerence strategy is voltage control, which is used in conventional vehicles on the road today. oltage control is accomplished by controlling the generator to maintain a constant voltage on the power distribution network. his indirectly causes the generator to provide all electrical load power, and so voltage control could be reerred to as a loadollowing mode. In the case o the 4 power distribution system, the generator is simply regulated to maintain 4. Regenerative braking was implemented by controlling the starter-alternator to provide as much negative torque as possible when deceleration was uested by the driver. he negative torque was limited by the maximum torque line o the electric machine and a maximum allowable voltage o 45 on the power distribution network. With these limitations it was ound that approximately 3 Wh or an average power o approximately 9 W over the cycle could be generated with regenerative braking. Percent Improvement C PCRegen Load Power in Watts PC Strategy (8) to the ability o the drivetrain to decouple the energy production schedule rom the engine operating points prescribed by the drive cycle and use the battery as an energy buer. As the load power rises, this ability to decouple production rom uest decreases, because the starteralternator is orced to constantly generate in order to keep up with demand. 5 Conclusion An optimisation based on model predictive control was introduced in this paper. It uses no previous knowledge o the driving cycle but employs a simple prognosis o the vehicle s speed in a sliding horizon to minimize a cost unction that balances improvements in uel consumption and emissions with a change in the state o charge o the battery. In order to create a closed orm o the optimisation that could easily be implemented in real time, the system equations had to be simpliied by assuming constant eiciencies o the battery and electrical machine. he optimisation was examined in simulations alone and in combination with regenerative braking. Despite its simplicity it was ound to perorm well with both applications, although the degree o improvement was dependant on the loading o the power distribution network. Reerences [] Pro. Dr. ir. on C. P.. ackx and Dr. ir. an den oom. odel Predictive Control, script o course taught by the Dutch Institute o Systems and Control, Delt, pp. 7-7, (). [] Gordon P. lair. Design and Simulation o Four-Stroke ngines, Society o Automotive ngineers, Warrendale, Pennsylvania, p. 7 (999). [3] Dr.-Ing. H. runner and Dipl.-Ing. A. Winger. Intelligente Regelstrategien ür Hybridantriebe, DI erichte Nr. 48, (998). [4] Otto Föllinger and Günter Roppenecker. Optimierung dynamischer Systeme, script o a course taught at the FernUniversität Hagen, (98). [5] L. Guzzella, L. and A. Amstutz. Fahrzeugantriebssysteme, script o a course taught at the idgenössichen echnischen Hochschule Zürich, pp (999). [6] Dr.-Ing. arkos Papageorgiou. Optimierung-Statische, dynamische stochastische erahren ür die Anwendung, Oldenbourg erlag, ünchen, pp (996). Chart : Improvement in Consumption with respect to oltage-control Strategy As might be expected, it could be seen that the combination o the optimisation and regenerative braking produces the best results. he relative improvement decreases as the load power increases, because the improvement rom the strategy is due