modelling and real-time simulations of electric propulsion
TRANSCRIPT
Modelling and Real-Time Simulations of Electric Propulsion Systems
Mario Porru, Alessandro Serpi, Andrea Floris, Alfonso Damiano Department of Electrical and Electronic Engineering
University of Cagliari Cagliari, Italy
Abstract—This paper presents an advanced modelling of an Electric Propulsion System (EPS) for light-duty Electric Vehicles (EVs). It is developed within the Matlab Simulink environment by taking into account suitable models of each EPS components, namely the energy storage system, the electrical drive and the transmission system, as well as the overall vehicle model. Particularly, a Li-ion battery is considered, which supplies a Permanent Magnet Synchronous Machine (PMSM) through a two-level DC/AC converter. The latter is driven by a PI-based control system, which assures adequate PMSM performances at any speed within its operating speed range. Vehicle modelling is introduced as well, i.e. the PMSM is coupled to EV wheels through a single-gear transmission system. The proposed modelling approach is validated through both conventional and real-time simulations, the latter being carried out by interfacing Matlab Simulink with an OPAL-RT device.
Keywords— Electric propulsion system, Electric vehicles, Drivetrain design, Real-Time simulations
I. INTRODUCTION
Electric Vehicles (EVs) are surely one of the most promising solutions for reducing emissions and the dependence from fossil fuels in the upcoming future. The most important EV component is the Electric Propulsion System (EPS), which consists mainly of an Energy Storage System (ESS), an electrical drive and a transmission system [1]. Particularly, the ESS should be characterized by high energy density in order to enable a wide EV driving range. However, adequate power density and dynamic performances should be guaranteed as well in order to cope with high and sudden power demand, as those occurring over EV acceleration and regenerative braking. The electrical drive represents the key element of the overall EPS. It generally consists of a traction inverter and an electrical machine, which can operate in either motoring or generating mode. The electrical machine is coupled to EV wheels through a transmission system, which is designed in accordance with electrical drive features and vehicle requirements [2].
Based on the previous considerations, it can be stated that overall EPS performances and efficiency depend on those of its main components. Consequently, these should be modeled accurately in order to enable a proper EPS assessment by means of numerical simulations [3]. However, advanced modelling of each EPS component may increase simulation time
dramatically, especially when power electronic devices are involved. These generally require stiff solvers with variable time steps in order to provide accurate results. As a result, only short time windows can be simulated successfully due to the very high computational effort. A viable solution consists of employing suitable real-time simulator, which enables advanced EPS modelling and rapid simulations simultaneously [4]–[6].
In this context, an advanced EPS modelling is presented in this paper. It is developed with the aim of accounting for each main EPS component, namely the ESS, the electrical drive and the transmission system, as well as for vehicle model. The proposed modelling approach has been validated within the Matlab Simulink environment, which provides suitable libraries for modelling each EPS component, i.e. SimPowerSystems and SimDriveline. Numerical simulations have been performed by following two different approaches; the first one consists of conventional stiff simulations with variable time steps and regard a short time window due to high computational effort. The second approach consists of employing a real-time simulator provided by OPAL-RT, which enables accurate EPS simulations over wider time periods. A comparison between these two approaches has been performed in order to highlight the most important differences, especially in terms of simulation times.
II. EPS OVERVIEW
An EPS consists of several components, the most important ones being highlighted in Fig. 1. It has to be designed in order to satisfy vehicle requirements, such as maximum acceleration, speed and gradeability. Reference can be thus made to the vehicle traction characteristic depicted in Fig. 2, in which Fw and Pw denote vehicle traction effort and power respectively, V being the vehicle speed. Particularly, at low vehicle speeds, Fw is generally upper bounded by tyre-ground contact issues, Pw
Fig. 1. The main EPS components.
Fig. 2. Vehicle traction effort (Fw, blue) and power (Pw, red) characteristics with vehicle speed, together with the overall resistance force (Fr, green).
being quite low. As the speed increases, Pw increases until vehicle base speed is reached (Vb). Then, Fw decreases while Pw is kept constant at its rated value. This occurs until vehicle maximum speed is reached (Vmax), in correspondence of which Fw equals the overall resistance force (Fr). It is worth noting that Vb is generally quite lower than Vmax in order to exploit full EPS power capability over a wide speed range. Consequently, a wide constant-power speed region must be assured by a suitable EPS design, especially regarding the electrical drive and the transmission system.
The ESS represents the on board energy source, which has to supply the electrical drive during motor operation, as well as to store some of the energy recovered during regenerative braking. The ESS is coupled with the electrical drive either directly or through appropriate DC-DC converters. Several kinds of ESS could be employed as EPS energy source, among which electrochemical batteries, ultra-capacitors and fuel cells. However, electrochemical batteries are the most popular solutions, especially those based on Lithium technologies.
Regarding the electrical drive, it consists mainly of the traction inverter and the propulsion motor. The former is generally represented by a two-level DC/AC power electronic converter, whereas the propulsion motor is generally either an Induction Machine or a Permanent Magnet Synchronous Machine (PMSM). The latter represents the most suitable choice in terms of efficiency, power and torque density [7]. However, PMSM suffers from high-speed operation issues due to permanent magnet excitation. These can be overcome by appropriate machine design and/or control algorithms in order to enable wide constant-power speed region [8].
The electrical drive is coupled with vehicle wheels through the transmission system, which can be chosen in accordance with electrical drive features and vehicle requirements. Particularly, single-gear transmission is surely the simplest configuration, it being well suited for electrical drives characterized by a wide constant-power speed region. Alternatively, multi-gear transmissions can be employed; these allows the electrical drive to operate within high-efficiency regions, but they introduce additional losses, weights, volumes and costs.
III. EPS MODELLING
In order to assess overall EPS performances and efficiency properly, advanced modelling of each EPS component is introduced. This accounts for specific component features and operating constraints, thus enabling an appropriate EPS performance assessment over any operating condition.
A. Energy Storage System
A Li-ion battery is considered in this paper as the EPS energy source, whose equivalent circuital representation is depicted in Fig. 3 [9]. Particularly, battery voltage can be computed in accordance with either charging or discharging mode by means of the following relationships:
*
0
*0 0.1
B itc
B itd
QE E K i it A e
Q it
Q QE E K i K it A e
it Q Q it
where E0 is the initial constant voltage, K is the polarization resistance, Q is the battery capacity, A and B being the exponential voltage and capacity respectively. Furthermore, i* and it denote the low frequency current component and the extracted battery capacity. Therefore, battery state-of-charge is computed as
%
1100 1 .
t
o
soc i t dtQ
Based on (1) and (2), it is thus possible to achieve appropriate battery charge and discharge characteristics, which are followed in accordance with EPS requirements.
B. Electrical Drive
A three-phase Surface-Mounted Permanent Magnet Synchronous Machine (SPM) is considered as the propulsion motor due to its numerous advantages, among which its relatively simple control system [10]. This is based on SPM equations in the dq synchronous reference frame, which can be expressed in terms of space vectors as
dqdq dq
div r j L i L j
dt
where r and L denote the phase resistance and the synchronous
Fig. 3. Equivalent circuit of Li-ion battery.
Fig. 4. The voltage follower PI-based control structure of the SPM, in which FFC denotes the feed-forward compensation block.
inductance, whereas vdq and idq are the phase voltage and current space vector respectively. These depend on the corresponding three-phase quantities as
2 4
j j j3 3dq a b c
2x x x e x e e , x v,i
3
in which {a,b,c} denote the three-phase terminals and ϑ is the electrical rotor position. In addition, still referring to (3), ω is the electrical rotor speed and Λ denotes the magnitude of the flux linkage due to permanent magnets. Hence, based on (3), SPM electromagnetic torque and power can be achieved as
e q m q
3 3T p i , P i
2 2
where p is the number of pole pairs. Whereas Joule losses can be determined as
2 2J d q
3P r i i .
2
Referring to both (3) and (5), it is worthy of note that these are achieved on condition of negligible magnetic anisotropy and saturation effects, as generally occurs for SPM. Hence, focusing on (5), it can be stated that SPM electromagnetic torque can be driven by means of iq only. As a result, for a given reference torque value, different id profiles can be chosen in accordance with several SPM control strategies [11]. Among these, the Maximum Torque Per Ampere (MTPA) is widely employed, especially for EPS. It consists of prioritizing the achievement of the reference torque value by minimizing current magnitude to the maximum extent.
In order to implement MTPA successfully, the voltage follower PI-based control structure depicted in Fig. 4 can be employed. It consists of two inner current control loops, which have to guarantee an appropriate tracking of the reference dq current components. The latter are determined in accordance with MTPA operation, i.e. based on voltage, torque and power requirements. Particularly, id* is held constant at zero unless motor voltage demand exceeds the rated voltage. In these cases, which generally occur over high-speed operation, id* is decreased appropriately in order to reduce motor voltage demand, thus keeping it within the corresponding operating boundaries. Whereas iq* is computed in accordance with motor torque and power requirements, as stated by (5).
In conclusion, it is worth noting that both id* and iq* must comply with current limitation constraint in order to prevent SPM failure. This means that constant-power operation may be assured up to a certain maximum speed value, above which reduced-power operation occurs [8].
C. Transmission System
Assuming the SPM characterized by a wide constant-power speed region, a single-gear transmission is considered in this paper, which is the simplest and most frequently used system for EVs. It presents a fixed gear ratio, which is defined as
m m w
w
r
V
in which ωm and ωw are the motor and wheel rotational speeds respectively, whilst rw is the wheel radius. Hence, based on (7), τ can be set in accordance with maximum SPM and vehicle speed. However, this choice has to be performed also in accordance with SPM torque and vehicle traction effort, as highlighted in Fig. 5. Consequently, the employment of a single gear transmission generally occurs jointly with electric motors characterized by wide constant-power speed regions in order to comply with the vehicle traction characteristic shown in Fig. 2.
D. Vehicle Modelling
Vehicle dynamics concerns a complex system of equations, which have to take into account all the forces acting on the vehicle during its motion. However, for energy analysis purposes, reference is generally made to a vehicle climbing a straight road [1], as shown in Fig. 6. Consequently, vehicle acceleration can be expressed as
w r
dVm F F
dt
in which m and δ is the vehicle mass and the mass factor respectively. Regarding Fr, it consists of several components as
r a g rollF F F F
where Fa is the aerodynamic drag, which accounts for the air resistance against the motion of the vehicle. This can be
Fig. 5. Wheel torque and motor speed characteristics with vehicle speed for a given output power and two single-gear ratios: τ(1) (reds) and τ(1) > τ(2) (blues)
Fig. 6. Vehicle model.
expressed as
21
2a f d wF A C V V
in which ρ is the air mass density, Af is the equivalent front area of the vehicle, Vw is the wind speed and Cd is the aerodynamic drag coefficient, which depends on the vehicle shape. Still referring to (9), Fg and Froll are the grading and rolling resistances respectively, which depend on vehicle mass and the slope θ as
singF mg
cosroll rollF mgC
where g is the gravitational constant, while Croll is the rolling resistance coefficient.
IV. SIMULATIONS
The EPS modelling introduced in the previous section is validated through a simulation study, which is carried out in the Matlab Simulink environment in accordance with the setup shown in Fig. 7. The Simscape library has been employed for modeling each EPS component, namely the Li-ion battery, the traction inverter, the SPM, the transmission system and the overall EV, as detailed in the following subsection.
A. Simulation Setup
First of all, reference has been made to the SimPowerSystems sub-library, which encloses suitable Li-ion battery, traction inverter and SPM blocks. The former is based on (1) and (2), whose parameters have been set in accordance with Table I. Whereas SPM parameters and rated values are summarized in Table II. Particularly, an SPM characterized by a
Fig. 7. Simulation set-up.
TABLE I LI-ION MAIN PARAMETERS
Symbol Unit Value
Nominal voltage Vn V 483
Rated capacity Q Ah 36
Initial state-of-charge soc0 - 0.8
TABLE II SPM PARAMETERS AND RATED VALUES
Symbol Unit Value
Rated Power Pn kW 40.3
Rated Torque Tn Nm 110
Rated Speed ωm,n rpm 3500
Maximum Speed ωm,max rpm 12000
Phase resistance r Ω 0.021
Synchronous inductance L mH 0.975
Pole pairs p - 2
PM flux Λ V∙s 0.1754
TABLE III VEHICLE MAIN PARAMETERS
Parameter Symbol Unit Value
Base speed Vb km/h 50
Design maximum speed Vmax km/h 171
Maximum gradeability (@ Vb) αmax ° 33
Vehicle mass m kg 1130
Frontal area Af m2 2.06
Drag coefficient Cd - 0.29
Friction coefficient Croll - 0.006
Wheel radius Rtyre m 0.28
wide constant-power speed region has been considered [12], which is fed by a conventional three-phase two-level DC/AC converter, whose switching frequency is set to about 10 kHz. Regarding the DC-Link, it is modeled by means of an equivalent capacitor, whose capacitance and rated voltage have been set to 820 µF and 560 V respectively. The DC-Link has been coupled with the Li-ion battery through an appropriate inductive filter in order to smooth battery current variation, but without impairing EPS dynamic performances. Referring to the SPM control system, it is implemented in accordance with Fig. 4 by means of the main Simulink library. Particularly, PI regulators have been set in order to achieve adequate dynamic performances, the sampling time interval being 102.4 µs.
Subsequently, reference is made to the SimDriveline sub-library, which enables both transmission system and vehicle modelling. Hence, a single-gear transmission has been employed, whose gear ratio has been set to 7.39 in order to assure the tracking of the vehicle traction characteristic. The main parameters of the EV model used in this paper are summarized in Table III. In this context, it is worth noting that simulations do not account for slope and wind speed, which have been set both equal to zero.
In conclusion, still referring to Fig. 7, the driver has been modeled by means of a driving cycle and a speed controller. Particularly, two different driving cycles have been considered; the first one is developed in order to compare conventional variable time-step and real-time simulations to each other. Whereas Artemis Extra-Urban driving cycle is considered because it reproduces real driving conditions accurately, which
Fig. 8. Torque (Nm) and speed (rpm) achieved by conventional (left) and real-time simulations (right): Te* (cyan), Te (blue), ωm* (red) and ωm (brown).
Fig. 9. Currents (A) and DC-link voltage (V) achieved by conventional (left) and real-time simulations (right): idq* (yellow & cyan) and idq (orange & blue).
Fig. 10. Battery voltage (V), current (A) and state-of-charge achieved by conventional (left) and real-time simulations (right).
are characterized by rapid acceleration and braking. The vehicle reference speed imposed by each driving cycle is thus tracked by means of a PI-based speed controller, which provides the reference torque to the EPS.
B. Simulation Results
Simulation results refer to both conventional and real-time simulations at first, the latter being performed by means of an OPAL-RT device. Particularly, the OP4500 is interfaced with the Matlab Simulink environment, thus enabling the real-time simulation of the proposed EPS. The corresponding results are shown from Fig. 8 to Fig. 10. Particularly, a rapid vehicle acceleration from rest to 50 km/h is considered. Subsequently, after few seconds, a quick stop is imposed. The comparison
between conventional and real-time simulation results reveal a very good agreement, although some differences occur on both DC-link voltage and battery current evolutions over the first stage of braking. Particularly, when the torque is reversed in order to brake the vehicle, DC-link voltage increases and forces battery current to reverse quickly in order to recover most of the energy coming back from the electrical drive. However, as soon as the battery current reaches its minimum allowable value (about –41 A), the braking resistor is activated in order to prevent battery overcurrent. As a result, some ripple on both DC-link voltage and battery current occurs over conventional simulations. This is more significant over real-time simulations due to the fixed time step, which is thus too large in order to simulate such a fast phenomenon properly. However, it is worth
Fig. 11. Torque (Nm) and speed (krpm) achieved by real-time simulations over the Artemis Extra-Urban Driving Cycle: Te (blue) and ωm (brown).
Fig. 12. Currents (A) and DC-link voltage (V) achieved by real-time simulations over the Artemis Extra-Urban Driving Cycle: idq* (yellow & cyan), idq (orange & blue) and Vdc (gray).
Fig. 13. Battery voltage (V), current (A) and state-of-charge achieved by real-time simulations over the Artemis Extra-Urban Driving Cycle.
noting that this mismatch has a very little effect on the overall EPS performances, which are thus well simulated by means of both conventional and real-time approach. It is also worthy of note that conventional simulations require about 5 hours to be carried out on a standard PC (quad-core intel i7, 2 GHz, RAM 12 GB), whereas real-time simulations are accomplished in just 9 s, as expected. This reveals conventional simulations unsuitable when long time horizons have to be considered, as needed for simulating EPS performances over real driving cycles.
In conclusion, simulations also refer to EPS operation over the Artemis Extra Urban driving cycle, whose corresponding results are depicted from Fig. 11 to Fig. 13. Particularly, only real-time simulations have been carried out because the driving cycle duration prevents conventional simulations to be accomplished successfully. Simulation results achieved in this case reveal a good tracking of the reference speed profile. In addition, appropriate flux-weakening is provided over high-speed operation, as revealed by the d current component evolutions depicted in Fig. 12
V. CONCLUSION
An advanced modelling of an Electric Propulsion System (EPS) has been presented in this paper. It is developed by taking into account suitable modelling of the main EPS components and enables the assessment of EPS performances by numerical simulations. It has been shown that real-time simulations allow the employment of long time horizons by achieving a accuracy similar to conventional simulations, which are characterized by excessive simulation times. The effectiveness of the proposed methods will be tested also by experiments, further investigations on which will be presented in future works.
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