by the traction motor is split between the engine and generator (often referred to as auxiliary power unit,. APU), and the battery. A simple strategy that proved.
2005-01-1163
Control-Oriented Modeling and Fuel Optimal Control of a Series Hybrid Bus Michele Anatone, Roberto Cipollone, Andrea Donati, Antonio Sciarretta University of L’Aquila, Department of E nergetica, DOE - L’Aquila, Italy
Copyright © 2004 SAE International
ABSTRACT
larger vehicles such as city buses.
The paper describes the derivation of a real -time controller for the energy management of a series hybrid city bus. The cont roller is based on Optimal Cont rol theory and on a control -oriented model of the propulsion system. The model is of the quasistationary, backward type, and it is derived from tabulated data of the single components provided by the manufacturers and basic, first-principle equations. The fuel cons umption obtained with the optimal controller is compared with that yielded by a conventional controller tracking the battery state-ofcharge.
The performance and fuel economy of a series hybrid vehicle depend heavily on the applied energy management, i.e., how the electric power required by the traction motor is split between the engine and generator (often referred to as auxiliary power unit, APU), and the battery. A simple strategy that proved to be robust is the ”on/off” or ”thermostat” cont rol [1]. The basic idea is to turn on or off the APU based on the battery state-of-c harge (SOC). This concept was extended in various rule-based power split controllers, often referred to as ”continuous” controllers [2].
INTRODUCTION
A decise improvement with respect of such strategies is achieved with optimal controllers. These require the mathematical definition of an optimization c riterion, usually the minimization of engine fuel consumption, submitted to various constraints among which the most important is that the battery state-of-charge must remain within an acceptable band (self-sustainability). If the driving conditions along a given route are known in advance (this is the case, e.g., of urban buses), mathematics techniques such as Dynamic Programming (DP) can be applied to derive an optimal controller [3]. In ot her applications where the driving conditions vary, some degree of predictivity is needed. The future driving conditions could be forecasted, in detail (e.g., using static route mapping and a telemetry system [4]) or a dopting some global parameter over a given time horiz on [5].
Hybrid electric propulsion emerged in the last years as a reasonable mid-t erm solution to a number of challenges related to part-load efficiency and pollution of conventional, internal combustion engine powered vehicles. Hybrid powertrains are characterized by two or more prime movers, usually an engine and an electroc hemical battery pack. Series hybrid powertrains use the internal combustion engine (ICE) to extend the driving range of a purely electric vehicle. The engine output is converted into electricity using a generator and then it can either directly feed an electric traction motor or charge the battery. Regenerative braking is possible using the traction motor as a generator and storing the electricity in the battery. The ICE operation is not related to the requirements of the vehicle, thus the ICE can be operated with optimal efficiency and emissions. A disadvantage is that this configuration needs three machines, and at least one of them has to be sized for the vehicle maximum power, thus limiting t he possibility of downsizing. A consequence of this fact is that series hybrid propulsion has been mainly adopted for
A much simpler and more applicable approach consists of sub-optimal, real-time c ontrollers, which are based on the minimization of a properly defined cost function which depends only on t he actual driving conditions. A controller of this type has been
proposed by one of the aut hors for parallel hybrid vehicles [6] starting from a heuristic analysis of DP results [7]. In this paper a sub -optimal, real -time controller is derived from the methods of O ptimal Cont rol and Pontryagin’s Minimum Principle in particular [8]. The c ontroller is applied to a series hybrid bus under testing at the facilities of the authors’ institution [9] and it is bas ed on a c ontroloriented model of the system.
HYBRID VEHICLE MODEL The aim of this section is to illustrate a detailed but structurally simple mathematical model of the hybrid bus powertrain. The system considered here consists of: (i) an electrochemical battery, (ii) an auxiliary power unit (APU) wit h a spark ignited, natural gas fed, engine and an electric generator, (iii) a AC traction motor equipped wit h an DC/AC converter, (iv) a transmission consisting of a fixed gear reductor and a final drive, (v) the wheels and bus body [9]. The model derived belongs to the class of backward-facing models. Each power converter is represented by a submodel, whose input variable is the power at the output stage of the component. The power chain is simulated going backward from the vehicle/road interaction, once the power required at the wheels is known from the route characteristics (velocity profile, grade).
added to the vehicle and propulsion system mass) and thus it is evaluated as follows:
m'v m f m p 1.03(mv m pr )
(2)
The relevant model data are listed in table 1.
mv cr cD Af Rt ηt
4436 kg 0.015 0.40 4.7 m2 0.0259 m 0.9
Table 1: Vehicle relevant data TRANSMISS ION MODE L The torque and speed at the output stage of the traction motor are calc ulated, according to t he backward c ausality adopt ed, from the speed and force required for traction. The wheels’ radius, the final drive transmission ratio and the fixed gear transmission ratio are lumped in an overall transmission coefficient Rt , such that:
Tm Fv Rtt1
m
60 vv 2 Rt
(3)
(4)
VEHICLE MODEL When the vehicle speed profile is known for a given route, this submodel calculates the force required for the t raction. The total t raction force is the s um of five contributes due to the rolling resistance, t he aerodynamic drag, the vehicle acceleration or deceleration, the road slope, and the friction braking, respectively:
Also a transmission efficiency t which is mainly refered to the fixed gear has been considered. The sign -1 is validfor traction ( Fv 0 ), ths sign +1 for braking ( Fv 0 ) The values of t and Rt are listed in table 1.
TRACTION MOTOR MODE L
dv 1 Fv mv gcr a c D A f vv2 m' v v mv g sin Fbr 2 dt (1) In this equation c r is the rolling resistance coefficient, c D the aerodynamic drag coefficent, mv the total mass, A f the frontal area, a the external air density, the road slope. The latter is tabulat ed as a function of the distance covered for a given route, while the acceleration is calculated from the speed profile. The total mass mv is the sum of vehicle mass, propulsion system mass, mass of fuel and payload. The inertial mass m' v takes into account the equivalent rotational masses (a 3% is
The traction motor is modelled through its quasi stationary efficiency map. The motor efficiency is provided by the manufacturer as a function of input (electric) power and shaft speed. From t hese data, the input power is calculated as a function of speed and torque:
Pm f m m , Tm The function f m m , Tm is shown in figure 1.
(5)
d cb,0 b cb,1 b Pb cb, 2 b Pb2 dt cb,i cb,i 0 cb,i1 b
Figure 4 clearly shows are linear cb,i b , i 0,1,2
i 0,1,2 how with
(8) (9)
the curves very good
approximation.
Figure 1: Traction motor efficiency as a function of speed and electric power (t op). Mot or electric power as a function of speed and torque (down)
TRACTION BA TTE RY MODE L According to the backward causality adopt ed t he battery power is the input variable of the submodel. In principle, the battery current and volt age may be evaluated from a power balance by using information on the battery polarization curve. Here a static map obtained from the manufacturer is used, which provides the battery current as a function of the required power and the battery depth-ofdischarge:
I b f b Pb ,
Figure 2: Battery functions f I b (top) and f b Pb , (bottom)
(6)
The variation of t he battery state-of-charge is calculated from the battery current and power using a second tabulated function:
Pb , d f (I b ) dt Pb ,
Pb 0 Pb 0
(8)
Figure 3: Battery depth-of-discharge variation as a function of power and depth-of-discharge, tabulated (solid curves) and parameterized (circles)
The t wo functions f b Pb , and f I b are shown in
MODEL OF THE APU
figure 2. The combination of equations (6) and (7) gives the variation of as a function of the battery
The input variable of this submodel is the electrical power Papu required at the AP U output stage. The
power. This dependency is shown in figure 3 for positive Pb (battery depleting) together with a parameterization which is very useful as a cont rol oriented model:
operating point of the engine is related only to the power and not on its single factors, voltage and current. Thus the operating point is selected in such a way to maximize the APU efficiency at every power request. The efficiency is a tabulated function of the engine torque and speed:
apu f apu , Tapu The
map
f apu , Tapu
provided
(10) by
the
manufacturer is illustrated in figure 5. The engine speed that maximizes the APU’s efficiency as a function of the APU power is shown in figure 6. Figure 6: Engine speed as a function of the AP U power
Figure 4: Curves of cb,i b , i 0,1,2 as a function of the battery depth-of-discharge, from equation (8) (solid curves), linearly fitted as in equation (9) (curves with markers)
Figure 7: Fuel power as a function of the APU power
OPTIMAL CONTROL THEORY
Figure 5: Contour plot of the APU efficiency as a function of engine speed and torque
The backward model derived in the previous Section is characterized by a single state variable – the battery state of charge, equation (8) – and by a single output – the fuel consumption mass flow rate, equation (11) –. The power from the two paths are balanced at the DC bus level:
Pb Papu Pm A lower limit of 2000 rpm has been introduced to avoid instability. Combining the two informations, the input power, i.e., the chemical power associated with the fuel, is evaluated as a function of the ouptut power. This dependency is shown in figure 7 together with a linear approximation of the same curve. The agreement bet ween t he t wo curves clearly leads to a control -oriented model for this component of the Willans type [10]:
Pf Pf , 0
Papu
apu
(11)
where Pf ,0 22.5 kW is the external power loss and
(12)
A control variable u is defined as t he ratio bet ween the power provided/absorbed by the battery path and the total power:
Pb uPm , Papu (1 u) Pm
(13)
The optimization problem can be stated in rigorous mathematical terms as follows: find the control law u(t ), t 0, t fin that minimizes the fuel consumption
t fin
0
m f dt with a constraint over the battery state-of-
charge, (0) (t fin ) .
apu 0.3286 the efficiency. The fuel consumption
mass flow rate is m f Pf H f , being H f the fuel lower heating value.
Using t he results of the previous sections, the problem is formally described in the framework of optimal control theory [8]. The Hamiltonian function is constructed as follows:
H ( , u, Pm ) m f
(14)
being a Lagrange multiplier. The corres ponding Euler-Lagrange equation is:
H
(15)
The condition for u (t ) to be optimal is that it minimzes the Hamiltonian. Notice that since t he system is not explicitly time-dependent, the Hamiltonian function is constant during time.
The derivative in equation (15) may be calculated using the parameterization of equation 9. A further simplification consists in neglecting the
dependency of from which is justified if varies slightly around its initial level. Since this is actually the goal of t he control, suc h a simplification will be used in the following, implying that:
done for . This means that equation (17) can be rearranged as:
u 0 (t ) arg min Pf u, Pm 0 0 , u, Pm u
(16)
The value is that corresponding to t he fulfillment (0) (t fin ) . Under this of the condition
An alternative approach may use the controloriented model derived in the previous Section to calculate the Hamiltonian as an analytical function. After minimization with respect to u the optimal control law then follows. The operating limits of u are given as follows:
1
Papu,max Pm
u 1, Pm 0
Papu,max
1 u 1
Pm
0
assumption, a suboptimal cont rol law is found generally as:
u 0 (t ) arg minm f u, Pm 0 0 , u, Pm (17) u The latter equation corresponds to the minimization of a cost function which depends only on the current driving conditions (via Pm ) and on 0 . Therefore it my be deemed as a real-time controller, provided that the optimal value of 0 is known. This in turn depends on the whole driving cycle but it can be reasonably estimated using, e.g., the t echniques proposed in [6, 11] for parallel hybrid vehicles. It is interesting to notice that an alternative way to apply equation (17) is in terms of power flows. The
first part of the right end term m f
can be easily
converted into fuel chemical power aft er multiplication by the fuel lower heating value. The
term can be also converted into electrochemical power after multiplication by the battery open-circuit voltage. This is slightly dependent on the state-ofcharge but, if the latter is kept reasonably constant, this dependency can be neglected as it has been
(18)
The search for the optimal values u 0 (t ) may be done with a simple inspection procedure, e. g., several values of u (t ) are tested at eac h time and the respective values of the cost function are compared to find the minimum. The accuracy of such method increases with the dimension of the test set.
0, 0 constant
, Pm 0
(19)
(20)
since the motor power is always lesser than t he maximum power available from the battery. To calculate analytically the optimal cont rol law various cases should be considered. For Pm 0 the limits of u given by equations (19) and (20) are both positive. Thus the battery power is negative and the Hamiltonian (see equation (7)) is calculat ed as follows:
H u, Pm
Pf , 0 Hf
1 u Pm apu H f
uPm
(21)
The Hamiltonian varies linearly with u and it exhibits a minimum for u 1 or u 1 Papu,max Pm , according to the sign of the derivative of equation (21) with respect to u :
1 apu H f 1, u0 1 Papu,max Pm , 1 apu H f
(22)
For 0 Pm Papu,max the limits of u have opposite signs. In the range
1 Papu,max Pm u 0
the
Hamiltonian varies linearly with u . In u 0 there is
a discontinuity, being H 0 H 0 . For 0 u 1
the Hamiltonian may have a local minimum, given by the derivative:
Pm H cb,1 Pm 2cb, 2 uPm2 0 u apu H f
(23)
38, 72, and 129 s), leading to a substantial underestimation of the total traction power. Nevertheless, during the rest of the route, the agreement bet ween measured and predicted dat a is quite satisfactory.
Resulting in:
1 u*
apuH f
cb,1 (24)
2 Pm cb, 2
if 0 u* 1 . Summarizing, the optimal control variable is in this case:
1 apu H f 1 Papu,max Pm , (25) u0 * arg min H 0 , H 0 , H u , 1 apu H f u
Figure 8: Speed profile for the model validation
For Pm Papu,max again the limits of u are both positive, thus the battery power is also positive. A possible stationary point for the Hamiltonian is found using equation (24). It is optimal if it respects the constraints over u , otherwis e the optimal control law lays over a ”constrained arc”:
u 0 (t ) arg min H u , H u * , H 1 u
(26)
Figure 9: Altitude profile, mapped data (circles ) and polynomial fitting curve (solid)
being u 1 Papu,max P m .
OPTIMAL CONTROL RESULTS
MODEL VALIDATION
The policy given by E q. 17 has been tested along a given route from t he city of L’Aquila, to the Faculty of Engineering located on a hill around 950 m high. The road slope x has been calculated from
The model validation has been carried out accordi to the following procedure. During a bus trip along a flat piece of route, the motor speed m and the motor power Pm were recorded. The vehicle speed is calculated using equation (4) and it is shown in figure 8. The vehicle acceleration is calculated from the filtered speed. The friction braking force was not recorded, thus it represents a strong source of uncertainty. The road slope, though always less than 2-3% has a strong influence on the traction power, and thus it has been estimated from altitude data. The reconstructed altitude profile is shown in figure 9, together with the polynomial fitting curve used to calculate the grade profile. The motor power predicted with the system model previously presented is compared with t he recorded trace in figure 10. The figure clearly shows the contribut e of the under estimated friction braking force during heavy decelerations (located at about
topographic data, while the speed profile vv t has been recorded t hrough an on-road measurement. The electric power required at the input stage of t he traction mot or is calculated with the model validat ed in the previous Section and it is shown in figure 11. The sub-optimal controller adopted minimiz es the Hamiltonian function with a search procedure. The test set of the control variable includes the values u 1,0.5,......1 plus the lower and upper values dictated by the APU operating limits. Moreover, the optimal control law depends on the value of . There is only one optimal value which fulfills the condition (0) (t fin ) (continuous curve in figure 12). Higher values of tend t o penalize too much the use of the battery as a prime mover, thus the final state-of-charge is too high (dash-dot curve in Figure 12). Lower values of tend t o overfavour
the use of the battery, thus the final state-of-charge is too low (dashed curve in figure 12).
Figure 10: Motor power, recorded (solid) and predicted (dashed)
Figure 11: Electric power required (positive) or provided (negative) by the traction motor as a function of time
intermediate levels of SOC the AP U status is kept equal to that at the previous time step. The improvement in fuel economy obtained with Optimal Control with respect to a conventional controller is evident from the figures. Due to its poorly efficient on/off nature, the thermostatic controller exhibits a fuel mass consumption which is 29% higher t han the global optimum calculat ed with Dynamic P rogramming. The figures also confirm that the control laws calculated with the minimization of the Hamiltonian are nearly optimal. Only a difference of less than 1% arises bet ween the fuel mass consumption obt ained with Dynamic Programming and the minimization of the Hamiltonian. The difference slightly increas es to 5% if the analytical minimization is considered, mainly because of the approximation errors inherent to the control-oriented model, and despite the fact that t he analytical minimization may find optimal values of which are not included in the test set of the search procedure. However, analytical minimization exhibits the decisive advantage of a lesser computing time required.
Figure 12: Trajectories of battery state-of-charge ( 1 ) obt ained with 7.5 108 (dash-dot),
6.7 108 (continuous), and 6.0 108 (das hed) The sub-optimal state-of-charge trajectory is compared in Figure 13a with thos e calculated using: (i) Dynamic P rogramming, (ii) a thermostatic-type controller, (iii) the procedure described a previous Section 3 to find the minimum of the Hamiltonian analytically. Figure 13b also shows a comparison of the fuel mass consumed. The Bellmann’ algorithm of Dynamic Programming [7] was implemeted with a time step of 1 s and a state-ofc harge step size of 5.0 10 5 . The test set of u is the same as for t he sub-optimal controller. The thermostatic-type controller uses two limits for t he SOC low . When state-of-charge, SOC high and
SOC SOClow the APU output is set to 30 kW. When SOC SOC high the APU is turned off. For
Figure 13: Battery state-of-charge trajectory (a) and mass of fuel consumed (b): Dynamic Programming (bold), sub-optimal controller (continuous), suboptimal controller with analytical minimization (dashdot), on/off ”thermostat” controller (dashed) The behavior of the various control strategies compared is better illustrated by figure14, which shows the power from the APU as a function of time. The on/off nature of the thermostatic-type controller is evident in the bottom figure. The figure relative to the sub-optimal controller with analytical minimization shows a higher number of APU starts
and stops, when c ompared with the optimal and t he sub-optimal trajectories. This behavior could be avoided by properly penalizing any engine start with an additional fuel mass consumption.
agreement bet ween predicted and meas ured dat a is quite satisfactory for the purpos es of this study. The sub-optimal controller has been tested for a route characterized by high variations of altitude. The trajectories obtained have been compared with those calculated with a global optimizing tool, the Bellmann’s algorit hm of Dy namic Programming, exhibiting no significant deterioration of t he performance criterion. The improvement obtained with respect to a conventional, on/off regulator is indeed about 29%. A slightly higher (5% ) fuel consumption has been exhibited by a variant of the sub-optimal controller that uses the control -orient ed model for an analytical minimization, instead of a trial-and-error search procedure. Despite this poorer performance, such variant has the promising advantage of a computing time 5 to 10 times lower.
ACKNOWLEDGEMENTS Figure 14: Power from the APU, top to d own: Dynamic Programming, sub-optimal controller, suboptimal controller with analytical minimization, on/ off ”thermostat” controller
CONCLUSIONS An energy management controller for a series hybrid vehicle has been developed. The controller has been synthesized using Optimal Cont rol theory with the aim of minimizing fuel consumption while guaranteeing the battery self-sustainability. Due to some approximations aimed at simplifying the coupled E uler-Lagrange equation, the controller should be deemed as sub-optimal. Its more interesting feature is the circumstance that the value of the control variable at each time depends explicitly only on the driving conditions at that time, thus it can be applied as a real-time controller also when the vehicle driving profile is not known in detail. The sole piece of predictive information needed is the constant value of the Lagrange multiplier. This plays the role of a weight factor between a term related to fuel consumption and a term related to state-of-charge deviation, which are summed to build the cost function to be minimized. The synthesis of the c ontroller required t he development of a mathematical model of t he system. A simple structure was chosen and t he model was built using polynomial fitting of components’ maps and basic firstprinciple relationships. The traction model has been validat ed with data recorded along a test route wit h very low altitude variations. Despite some uncertainties due to grade profile and friction braking events, the
This work has been developed under the project “Improvement of the Scientific and Technological Network”, funded by the It alian Ministry for Education, University and Research (MIUR), years 2000-2003.
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DEFINITIONS, ACRONYMS, ABBREVIATION SIMBOLS c b,i Coefficients c r Rolling resistance coefficient c D Aerodinamic drag coefficient t Time u Cont rol variable v Speed x Vehicle travelling distance Af Vehicle front al area F Force Hf Lower heating value I Electric current M Mass P Power Rt Overall transmission coefficient T Torque Road slope angle Efficiency Battery charging constant Lagrange multiplier Density Rotational speed Battery depth of discharge SUBSCRIP TS 0 Initial a Air apu Auxiliary power unit
b br f fin m p pr v
Battery Brake Fuel Final Motor Payload Propulsion system Vehicle
