,
STOCKHOLM SWEDEN 2020
Modeling and control design of an
electric vehicle powertrain
DJANGO LE CLERRE MARAINE
KTH ROYAL INSTITUTE OF TECHNOLOGY
Sammanfattning
Detta examensarbete har utförts inom ramen för det så kallade Hybrid Innovative Power-train (HIP)-projectet vid företaget Altran. Syftet med arbetet har varit att kunna förse olika klienter med en modulär plattform för att kunna modellera och elfordon så noggrant och snabbt som möjligt. Huvudidén har varit att ta fram ett modellbibliotek i Simulink rep-resenterande nyckelkomponenterna i ett vägfordon. Plattformen möjliggör modellerandet av helelektriska fordon samt även plug-in hybridfordon och bränsecellsdrivna fordon. Inom ramen för detta examensarbete har elmotor, omriktare samt dc-dc-omvandlare modellerats. Modeller både inkluderande och exkluderande kraftelektronikens (antaget ideala) switchför-lopp har tagits fram. Modellen inkluderande switchförswitchför-lopp har även använts för att studera en metodik för att diagnostisera omriktarfel. Första delen av examensarbetet presenterar en litteraturundersökning med särskilt fokus på metoder för feldetektering. Matematiska modeller och regulatordesign beskrivs i arbetets andra del. Arbetets sista del beskriver mjukvaruimplementeringen samt analyserar de erhållna simuleringsresultaten.
Abstract
This degree project work has been carried out in the context of in the Hybrid Innovative Powertrain (HIP) project presently executed by the company Altran Technologies S.A.. Its purpose is to provide clients with a modular platform to model and simulate their vehicle as accurately and as fast as possible. The idea is to build a custom Simulink library with a block for each major component found in any road vehicle. The platform must enable clients to model battery electric vehicle as well as plug-in hybrid or fuel cell vehicles with different powertrain topologies. It must provides component blocks for each layer of control, from high-level energy efficiency algorithms embedded in high authority hardware to low-level component control solutions.
The work presented here contributes to the development of the modular powertrain elec-tric part with the elecelec-tric motor, the inverter and the chopper. It is composed of three main steps :
• Building an electric powertrain model with its proper control schemes in order to enable fast simulations. The model must be split into blocks that become part of the project library.
• Building a more realistic model that takes into account the discontinuous switching dynamics and the discrete nature of the actual controllers, in order to verify and validate the control effectiveness
• Designing a diagnostic method for inverter faults
The first part of this work is dedicated to a literature review of automotive electric drives solutions as well as the research in progress in the domain, notably in fault detection schemes. The mathematical models and controller designs used for the simulations are developed in a second part. The last part presents the software implementation aspects and the analysed simulation results.
Acknowledgments
I would like to thank my main supervisor Takwa Douiri for her patience and useful guidance during this internship. Thanks to Damien Clemençon and Niki Halttunen for their support and their constructive remarks. Above all, I would like to thank my university examiner and former teacher Oskar Wallmark for passing on to me his interest and enthusiasm about electric drives trough high quality courses and helpful Skype calls during the degree project.
Contents
1 Introduction 6
2 Literature review 7
2.1 Automotive electric drives . . . 7
2.2 Alternatives . . . 8
2.3 Electric drives faults . . . 9
2.4 Faults diagnosis techniques (FDI) . . . 10
2.4.1 Overview . . . 10
2.4.2 Application to electric drives . . . 11
3 Mathematical models 12 3.1 Electric motor . . . 12 3.1.1 Physical model . . . 12 3.1.2 Controller design . . . 14 3.2 DC/DC converter . . . 18 3.2.1 Physical model . . . 18 3.2.2 Control design. . . 20 3.3 Inverter . . . 22 3.3.1 Physical model . . . 22 3.3.2 Control design. . . 24 3.4 OC faults . . . 26
3.4.1 Faulty inverter model . . . 26
3.4.2 FDI schemes. . . 27
4 Matlab model implementation and results 29 4.1 General simulation set up . . . 29
4.2 DCDC converter . . . 32
4.3 PMSM with average inverter . . . 33
4.4 Whole electric powertrain . . . 39
4.5 Powertrain switched model . . . 45
4.6 Inverter with OC fault . . . 50
5 Prospects 58 6 Conclusion 59
List of Figures
2 Mono-directional current source inverter topology . . . 93 Z-source inverter topology . . . 9
4 Electric drive with open-end winding motor . . . 10
5 Different frames used for PMSM modelling . . . 13
7 DCDC converter . . . 18
8 Three phase AC motor with inverter . . . 22
9 Switching waveform and voltage reference . . . 24
10 2 level inverter space vector hexagon . . . 25
11 Motor with averaged inverter model - 1st test - motor currents and torque. . 34
12 Motor with averaged inverter model - 1st test - motor voltages . . . . 35
13 Motor with averaged inverter model - 1st test - vehicle speed and motor torque 35 14 Motor with averaged inverter model - 1st test - motor currents in Park frame 36 15 Motor with averaged inverter model - 2nd test - motor currents and torque . 36 16 Motor with averaged inverter model - 2nd test - vehicle speed and motor torque 37 17 Motor with averaged inverter model - 2nd test - motor currents in Park frame 37 18 Motor with averaged inverter model - 2nd test - slower FW - motor currents and torque . . . 38
19 Motor with averaged inverter model - 2nd test - slower FW - motor currents in Park frame . . . 38
20 Powertrain with averaged converters model - - DClink voltage and chopper inductor current . . . 39
21 Powertrain with averaged converters model - motor currents and torque . . . 40
22 Powertrain with averaged converters model - DClink voltage and chopper inductor current . . . 41
23 Powertrain with averaged converters model - powertrain power flow . . . 41
24 Powertrain with averaged converters model - motor currents in Park frame . 42 25 DClink request transient test - start at 410V . . . 43
26 DClink request transient test - start at 600V . . . 44
27 Matlab function delay problem . . . 46
28 Motor currents . . . 47
29 Motor currents - faster current controller . . . 47
30 DClink voltage and chopper current - faster current controller . . . 48
31 Motor currents - faster current controller and higher DClink capacitance . . 48
32 DClink voltage and chopper current - faster current controller and higher DClink capacitance . . . 49
33 Models comparison - motor currents. . . 49
34 OC fault impact - test3 . . . 51
35 OC fault impact - test2 . . . 51
36 OC fault impact - test1 . . . 52
37 FDI using reference currents - test 1. . . 53
38 FDI using an observer - test 1 . . . 53
39 FDI using reference currents - test 2. . . 54
40 FDI using an observer - test 2 . . . 54
41 FDI using reference currents - test 3. . . 55
42 FDI using an observer - test 3 . . . 55
43 FDI using reference currents - test 4. . . 56
44 FDI using an observer - test 4 . . . 56
45 switching signals and voltages . . . 68
1
Introduction
Energy and environmental issues currently represent major concerns at the international level. World population growth and the activity of the industrialized and emerging countries have led to an explosion in energy needs. The use of fossil resources is responsible for the majority of greenhouse gas emissions in the atmosphere, this pollution is at the root of the climatic upheavals we are facing today. In addition to the environmental impacts, fossil fuel reserves are limited and will not be able to meet global demand in the medium term. Of all the sectors implicated in these environmental problems, the the automobile is regularly mentioned; the number of vehicles on the road is constantly increasing. Thanks to government restrictions on pollution thresholds and technical innovations, manufacturers are moving towards technologies that emit fewer pollutants. The different cases and geopolitical or economic considerations are determining factors in the technological choices: types of electrical machines, arrangement of the battery pack or the battery cells, fuel cell technology and chemistry, cooling systems etc. The battery and fuel cell electric vehicles technologies offer a promising solution in this field and brings the zero emission propulsion target within reach.
As a result, manufacturers must bring forward cutting-edge powertrain technologies to provide customers with compact, energy efficient, long-range, environmental friendly cars at limited costs. This challenge requires a lot of research and development to design better and better technological solutions to fulfill the needs in low carbon emission transports necessary for slowing down global warming and improving cities air quality. The decrease in development costs highly relies on software simulation to accelerate the testing procedures and limit the use of expensive experimental set ups.
It this spirit, a project of modular platform with a set of blocks easy to assemble and ready for fast simulations was launched. It is thought to bring an exhaustive library of models to meet any needs in components and system parameter study, control validation or energetic optimization.
2
Literature review
2.1
Automotive electric drives
The majority of automotive electric drives are composed of a battery pack, a DC-link capac-itor, a two-level three phase voltage source inverter (VSI), a three phase AC machine, and optionally, a DC-DC two quadrant boost converter. An example of an electric powertrain topology for a hybrid vehicle with two electric motors/generators (MG) is shown on figure
1awhich is similar to the one of the 2010 Toyota Prius [1].
(a) electric powertrain topology for a hybrid electric bi-motor topology
(b) electric powertrain topology for a full electric vehicle application
Those two topologies are mainly used in hybrid and full electric vehicles because of their low cost and high efficiency [2]. The DC-link capacitor prevents too large voltage and current ripples that could damage both the battery and the converters. The optional two quadrant chopper is used to provide a variable DC voltage that can be fed to the inverter in order to improve torque and power or efficiency depending on the driving conditions. The AC machines gradually replaced DC machines since the development of digital signal processors (DSP) and variable speed drives (VSD) with the rise of computational power in the 1980’s. The AC machines are more efficient and reliable because they do not need mechanical commutation and then no brushes that wear out through time. Moreover, they offer more power density, which is crucial for designing compact electric and hybrid powertrains. The only drawback of AC machines is the more complex control schemes needed to provide good dynamics. The main types of AC machines used so far in widespread electric cars are the squirrel cage induction machine (IM) for example in the Tesla Roadster [3], a wound rotor synchronous machine for the Renault Zoé [4] and permanent magnets synchronous machines (PMSM) for most of the cars sold nowadays, notably Toyotas and GMs [5]. The PMSM
is widely used because of its very high power density and its superior efficiency. Most automotive electric drives uses vector control (VC), also called field oriented control (FOC), for the motor alongside with space vector modulation (SVM) for the inverter as both are easily implemented on recent DSPs and provide very good dynamics coupled with few torque harmonics. SVM is preferred to carrier-based pulse width modulation (PWM) because the latter need the injection of a homopolar component in the sinusoidal modulating signal to achieve similar harmonic performances and same AC/DC-link voltage ratio (AC phase to phase voltage fundamental component magnitude = DC voltage instead of√3/2 DC voltage for sinusoidal PMW) while staying in the linear modulation region [6]. In practice, the control algorithm is implemented on a DSP clocked at 100 MHz which uses two current sensors and a resolver or an encoder to measure the rotor position [2]. It is then termed a
sensored control in opposition to a sensored control which uses an observation technique to
estimate the rotor position, not used for automotive purposes for control performance and robustness reasons [3].
2.2
Alternatives
Some other converter topologies as well as other electric machines have been investigated for automotive applications although they are too expensive to implement on a product sold on a wide scale like electric cars. There are the current source inverters (CSI) [2,7] which enable both to step up the DC source voltage, which must be in series with an inductor, and to provide a quasi-sinusoidal three phased voltage to the motor, which in turn requires parallel capacitors to smooth out the distorted currents. But they need more active components than VSI to allow bidirectional current flow. CSIs are globally more expensive, more complex to control and less efficient than VSIs. Z-source inverters (ZSI) [2, 7] are an other alternative to VSI which can step up or down the DC-link voltage and feed a motor with three phase quasi-sinusoidal currents. They have a similar efficiency but they require more passive devices and need more complex control schemes. Finally, there are the multilevel VSIs [2, 7] like the neutral point clamped (NPC) topology which provide a lower harmonics/fundamental currents ratio as well as a lower voltage load on power devices compared to the basic two-level VSIs. Presently, their complexity, size and cost prevent them from being competitive, though.
There is also a way to achieve similar voltage pattern as a three level inverter by using two basic two-level inverters and plugging them on a machine with open-end windings [8]. This is equivalent to having each of the three phases fed by a H-bridge. This enables the usage of new pivot voltage space vectors for the nearest three vectors (NTV) SVM modulation [9]. This allows lower current harmonics content. However, as the machine phases are not connected to each other anymore, there is a homopolar current component that arises if an elimination technique like the 3D SVM is not used to cancel it [8]. Such a method is complex and results in more harmonic content than for the single inverter case. With this set up, the maximum phase voltage is multiplied by two or √3, without and with homopolar current cancellation respectively, compared to the single two-level inverter case.
Figure 2: Mono-directional current source inverter topology
Figure 3: Z-source inverter topology
2.3
Electric drives faults
AC motor drives components can suffer from failures as they wear out due to many factors such as overvoltage, high temperatures or mechanical vibrations. The failures can occur in the motor and in the static converter(s) and more specifically in the subcomponents like sensors, power semi-conductors, printed circuit boards (PCB), etc. Power modules represent 34% of power converters failures and power converters contribute to 80% of total electric drives failures [10, 11]. Moreover, the power devices gathered with their gate drivers and control circuits represent 50% of the industrial power converters failures [12]. Open-circuit (OC) and short-circuit (SC) are the two main types of failure occurring in converters. OC faults do not damage harshly the power devices even though they produce distorted currents and thus create fast electromagnetic torque oscillations which cause mechanical stress. On the other hand, SC faults results in very high currents which leads to overheating and failure of the power electronic devices within a very short time if they are not dealt with.
Mechanical and electrical failures can also occur in an electric machine. The rotor can suffer from eccentricity or bearing failures as well as broken rotor bars or end rings in the case of an IM [12]. The permanent magnet machine can experience the demagnetization of its permanent magnets in case of high currents. Both types of machine can experience
Figure 4: Electric drive with open-end winding motor
stator internal open-circuits or short-circuits which lead to high currents and require relays to open as soon as possible. The inter-turn fault that occurs when there is a isolation problem between stator coil windings, might lead to short circuits and is then also important.
2.4
Faults diagnosis techniques (FDI)
2.4.1 Overview
Diagnostic is done in three steps : detection, isolation and identification of the fault to re-spectively detect an unexpected behaviour, locate the faulty component and assess the type of fault and its severity. [13, 14]. there are model-based diagnostic techniques which uses mathematical models of the plant with one or more observers and compare its (their) out-put(s) with measured states of the plant. This method requires a good enough observability of the system and an accurate model which might be tough or even impossible to derive for complex systems.
An other possibility is to use signal-based algorithms to extract different patterns in the time, frequency or time-frequency domains with digital signal processes like histograms, fast fourier transforms (FFT), short-time fourier transforms (STFT) or wavelet transforms (WT).
An alternative to those two techniques is to use historical data of the plant to train a knowledge-based algorithm that is meant to recognize patterns in the plant monitored sig-nals. The major drawbacks of this technique are its high computational cost and its incapa-bility to recognize unknown faults that did not appear in the training data.
Active fault diagnosis is, contrary to the three others, an invasive technique that consists in adding a signal to the input of the system and extract its output to monitor the added input component impact.
The three passive techniques can be combined into a hybrid diagnosis technique to remove some drawbacks that they present alone.
2.4.2 Application to electric drives
SC faults are very fast and destructive, so they are tackled by inverters built-in devices that shut down the converter when an overcurrent is detected. On the other hand, OC faults might be longer to detect and don’t automatically lead to a shutdown of the drives. This is mainly why FDI techniques focus on OC [12].
All the FDI techniques outlined above can be applied to electric drives. The goal is to design a detection and isolation algorithm that is robust against false alarms and which does not depend on the operating conditions. Fault detection through monitoring of time signals can be done either with current or voltage signals even if voltage measurements require additional sensors, or at least less straightforward algorithms, and are then less common. The detection is either based on a complex pattern recognition algorithm or simply based on a specific value that dramatically changes when a fault occurs. Some methods rely on a park transform of the measured currents and the measure of the derivative of the space vector phase. Other robust techniques use the phase currents averaged and normalized values and compare them with a relevant threshold for detection and isolation. To compare reference and real currents is also a robust way to detect and isolate an OC fault [12].
Instead of just using measurements, a current observer can be used to compare its output with the measures. The resulting residual is compared to a threshold to perform detection and identification, like for the previous techniques. More complex techniques like artificial intelligence and neural networks were used also for FDI techniques related to electric drives [12, 15].
3
Mathematical models
3.1
Electric motor
3.1.1 Physical model
AC machines modeling is made simpler by reducing the three phase model in a two phase equivalent by using a tool called the Park transform. It transposes the three phase electrical values from the usual abc frame into a dq0 frame depending on an angle θ. The Park transform matrix preserving amplitude can be expressed as :
[P (θ)] = 2 3
cos θ cos θ − 2π3 cos θ − 4π3 sin θ sin θ − 2π3 sin θ − 4π3
1 2 1 2 1 2 (1)
The one preserving power is obtained by replacing the scalar coefficient 23 by q23 and the third row of the matrix with three √1
2. The one preserving amplitude will be used in the rest
of this thesis. By taking θ = 0 and θ = θr, the latter being the electrical angle between phase
a and rotor fluxes directions, one gets respectively the Clarke transform and the rotor-flux
fixed Park transform :
[Cαβ0] = 2 3 1 −12 −1 2 0 √ 2 3 − √ 2 3 1 2 1 2 1 2 ; [Pdq0] = 2 3
cos θr cos θr− 2π3 cos θr− 4π3
sin θr sin θr−2π3 sin θr−4π3
1 2 1 2 1 2 (2)
and their inverse :
[Cαβ0]−1 = 1 0 12 −1 2 √ 2 3 1 2 −1 2 − √ 2 3 1 2 ; [Pdq0]−1 = cos θr − sin θr 12 cos θr− 2π3 − sin θr− 2π3 12 cos θr− 4π3 − sin θr− 4π3 12 (3)
The zero component, also called homopolar component, of a three phase quantity is often of little importance as the current zero component does not contribute to build a magneto-motive force (MMF) [16] responsible for torque creation in an electric motor. Moreover, as long as a three phase load doesn’t have a grounded neutral point, the sum of the currents is inevitably zero. If, added to this, the load is balanced, then the voltage zero component is also zero [17]. As a result, the following matrix transforms are often used to reduce the model from a three-phased to a two-phased model :
[Cαβ] = 2 3 1 −1 2 − 1 2 0 √ 2 3 − √ 2 3 ; [Pdq] = 2 3
cos θr cos θr−2π3 cos θr−4π3
sin θr sin θr−2π3 sin θr−4π3
(4)
the model developed here is designed specifically for control purposes as it relies on several assumptions [16, 18] :
Figure 5: Different frames used for PMSM modelling
• Magnetic circuits are considered linear which means that all inductances are constant and saturation effects are neglected.
• Core losses, due to induced currents and hysteresis losses, are neglected
• The stator windings are supposed continuously and sinusoidally distributed, thus the MMF in the airgap is considered sinusoidal.
• The stator flux linkage due to the rotor magnets is assumed sinusoidal
Those approximations are not relevant for further analysis for instance for optimal control by considering core losses [18, 19] or for electric machines conception and optimization.
The model of the permanent magnet synchronous machine (PMSM) based on the previous assumptions is derived by applying Kirchhoff voltage law on the three phases and the energy conservation law for electromagnetic torque Tem derivation. Then, by applying the park
transform to the three obtained electric equations and adding the mechanical equation, one get the state-space equations for the PMSM in the rotor-fixed rotating frame [16, 18, 19] :
Ld did dt = vd− Rsid+ Lqiqwr (5) Lq diq dt = vq− Rsiq− Ldidwr− ψmwr (6) Jdwm dt = Tem− Tf − TL (7)
Where J is the total system inertia referred to the motor shaft, Ld and Lq are the motor
inductances in the d and q directions respectively, Rsis the stator phase resistance, and Tem,
Tf and TL are the electromagnetic, friction and load torques respectively. id and iq are the
currents creating magnetic flux in the d and q directions respectively, wr and wm are the
electrical and mechanical angular velocities respectively, linked by the number of pole pairs
np :
The electromagnetic torque and the friction torque formulas are :
Tem =
3
2np(ψmiq+ (Ld− Lq)idiq) (9)
Tf = blinwm+ bquw2m (10)
Where blin and bqu are the linear and quadratic friction coefficients respectively.
3.1.2 Controller design
For automotive purposes, one must design a controller that receives a torque request and outputs adapted voltage references that must be fed to the motor to meet the request. To do so, we can decompose the controller in two stages :
• An inner closed loop current controller that computes the voltages required by the plant to reach the currents request
• An outer feed-forward torque controller that receives a signal from the throttling action of a driver and computes current references based on a given torque request
The current controller usually reads the currents from two phases, since the sum of the three is 0, the rotor position, the DC-link voltage and the current references. It generates a voltage reference for the inverter. Because they can differ in practice, the real rotor angle θr
will be called differently from the measured one θ1. the controller have to compute the rotor
speed:
ω1 = dθ1
dt (11)
The controller applied here is a field oriented vector controller which uses the motor’s angle information to apply its control in a frame fixed to the rotor flux dynamics. In the case of a PMSM, the rotor flux fixed frame is the same as the rotor fixed frame since its magnets are responsible for the flux. The Park transform matrix 4 is used. When expressed in the Park frame, The abc sinusoidal quantities become constant (at steady-state), and then far easier to track for a linear PI controller. Besides, as mechanical dynamics are far slower than electrical dynamics, wr can be considered as a slowly varying parameter and then, the
electrical equations can be studied as linear ones for control purposes.
By replacing the derivatives with the Laplace operator p in5and6and putting the result into a compact matrix form, one gets :
id iq ! | {z } idq = " pLd+ Rs −Lqwr Ldwr pLq+ Rs #−1 | {z } G " vd vq # − " 0 ψwr #! | {z } vdq−E (12)
It is a system of two coupled first order ordinary differential equations (ODE). In order to use two PI controller for id∗ and iq∗ references, it is useful to decouple the equations thanks to an inner feedback loop, as proposed in [20, 21], with added active damping fictitious re-sistances for external disturbances and modeling errors rejection. The estimated parameters
used by the controller will have the hat notation within this thesis. The decoupling inner feedback and the PI controller are given by :
W = −Rˆ a,d − ˆLqwr Ldwr −Ra,q ! ; F = kp,d + ki,d/p 0 0 kp,q+ ki,q/p ! (13) The block diagram of the whole system is shown in figure 6with the kind permission of Oskar Wallmark.
Figure 6: Closed loop current control block diagram [19]
The internal feedback loop and the PI controller are designed so that the closed-loop system behaves like two decoupled one dimensional first-order systems with bandwidth αc :
Gcl= G0F(I + G0F)−1 = " αc p+αc 0 0 αc p+αc # ; G0 = G(I + GW)−1 (14)
If we consider no parameter estimation errors, firstly id and iq dynamics are decoupled,
and secondly the targeted dynamics can be achieved by pole placement by setting :
kp,d = αcLˆd ki,d = αc( ˆRs+ Ra,d) (15)
kp,q= αcLˆq ki,q = αc( ˆRs+ Ra,q) (16)
Finally, the active resistance are chosen in such a way that the disturbance and modeling errors rejection dynamics are as fast as the closed loop dynamics [20] :
Ra,d= αcLˆd− ˆRs Ra,q = αcLˆq− ˆRs (17)
One can choose the bandwidth by using its link to the rise time of a first order system :
αc =
ln(9)
tr,c
(18) The current controller outputs a voltage reference :
As explained later for the inverter and its control, the voltage reference magnitude must be limited to a value Vmax. When the reference magnitude exceeds Vmax, the controller reacts
: vd,sat∗ = Vmaxv ∗ d |v∗ dq| ; vq,sat∗ = Vmaxv ∗ q |v∗ dq| (20) Because of this control structure with an internal saturation, anti wind-up must be im-plemented for both the d and q PI controllers to counteract an irrelevant rise of the PI integrators that could occur when |vdq∗| is above V
max and that leads to overshoots and
globally poorer current dynamics. An anti wind-up implementation, called back-calculation, proposed in [21] is applied. The references generated by the PI controllers can be expressed as :
vd,P I∗ = kp,d((i∗d− id)) + ki,d((i∗d− id) +
1
kp,d
(v∗d,sat− vd∗))/p (21)
v∗q,P I = kp,q((i∗q − iq)) + ki,q((i∗q− iq) +
1
kp,q
(vq,sat∗ − vq∗))/p (22) The part added in the integrator only prevent the integrator output from rising when the voltage reference is saturated, but is zero otherwise and then does not affect the unsaturated behaviour.
The current controller is now fully defined. The outer torque controller that outputs the current references need to be tackled. The design presented in this thesis is based on the maximum torque per ampere (MTPA) computation and is directly inspired by what can be found in [22,18]. MTPA equation is derived by calculating the maximum torque per current magnitude : ∂Tem ∂id i s=cste = 0 ⇒= iq2− id2− ψ Ld− Lq id = 0 (23) id≤0 ==⇒ id = ψ 2(Ld− Lq) − v u u t ψ 2(Ld− Lq) !2 + iq2 (24)
The output signal sent by the throttle can also be interpreted by the torque controller as a current magnitude Inorm. By putting it in the previous equation, one gets :
Inorm2− 2i∗d 2− ψˆ ˆ Ld− ˆLq i∗d = 0 i ∗ d≤0 ==⇒ i∗d= ˆ ψ 2( ˆLd− ˆLq) − v u u u t ˆ ψ 4( ˆLd− ˆLq) 2 +I 2 norm 2 (25) And the q component reference computation is straightforward :
i∗q =qI2
norm− i∗d2 (26)
Thus, the throttle signal is equivalent to the torque request :
Tem∗ = 3 2np(( ˆψm+ ( ˆLd− ˆLq)i ∗ d) q I2 norm− i∗d2) (27)
But, this won’t be exact for high speeds conditions for which a closed-loop FW control must be implemented alongside the torque controller. By looking at equation 7, one can see that at steady-state and no load conditions :
Vq = ψmωr
| {z }
EM F
(28)
Which means that in order to keep iq to a constant value, the motor must be fed by a
q component voltage equal to the electromotive force (EMF) which is proportional to the
rotor’s electrical angular speed. However, the maximum voltage Vmax available depends on
the DC-link voltage and the inverter control scheme, as explained further in this section, where the inverter model is tackled. The base or nominal speed ωbase can be defined as the
threshold value above which the EMF goes beyond Vmax. Then we can use the contribution
of the term Ldidwr in equation 7 to counter the magnets flux by decreasing id to allow for
torque creation above base speed.
This can be achieved thanks to a second closed loop control which uses the voltage reference Vdq∗ magnitude to compute the i∗d,f w component that must be added to i∗d (which will now be called i∗d,1 to avoid notation problems) to compensate for the difference between the EMF and Vmax when the latter is the lowest :
i∗d,f w=
Z
γ(Vmax2 − vd∗2− v∗q2) ; γ = αf w
2max(ωr, ωbase) ˆLdVmax
(29)
id,min− i∗d,1 ≤ i
∗
d,f w ≤ 0 (30)
Here, id,min is a value that can be implemented in order to avoid permanent magnets
demag-netization which might occur because of too negative id currents. As the field weakening
impacts the currents references, its dynamics interacts with the currents dynamics so the two cascaded loops must have separated bandwidth. αf w ≤ αc/10 seems to be a good choice
because then FW is also much faster than the mechanical dynamics ∼ blin/J . For further
analysis and comprehension, a comprehensive stability study of this scheme is proposed in [19].
Now, to determine the iq component, one must study how the torque might evolve for
a constant throttling action when field weakening is triggered. The optimal choice in term of convenience for the driver is to find a way to keep the same torque for a given throttling action. To do so, the following scheme can be adopted :
i∗q = T ∗ em 3 2np( ˆψm+ ( ˆLd− ˆLq)i ∗ d) ; i∗d= i∗d,1 + i∗d,f w (31)
Finally, the last limitation concerns current. Indeed the motor stator windings heat up due to copper and core losses. The amount of heat that can be evacuated by the cooling system, has a higher limit that fixes a value for the maximum current Imax, for obvious safety
and longevity reasons. Then :
i∗q ≤qI2
3.2
DC/DC converter
The DC/DC converter model also relies on assumptions :
• As for the motor, the inductors as well as the resistances and capacitors are considered linear and invariant.
• The diodes and transistors are considered as perfect uncontrolled and controlled switches respectively, which means that conducting and switching losses are neglected and the switchings are supposed instantaneous.
3.2.1 Physical model
The DC/DC converter used in hybrid/electric cars is a basic chopper. It is used to control the DC voltage available for the electric motor(s). It is then more advantageous for performances to choose a converter that outputs, for a given voltage input, a high voltage output. This can be done by a boost converter. Using a buck converter would be irrelevant because the battery would then be oversized for the traction motor, which is a waste of resources and money. An other key feature expected from such an inverter is to enable the current to flow both from and into the battery, for motoring and regenerative braking scenarios respectively. This is achieved by using two transistors instead of one transistor and a diode used in a mono-directional chopper. A two quadrant or bi-directional boost chopper is then used. In an automotive application, the chopper is supposed to maintain a desired output
Figure 7: DCDC converter
voltage V2 for an input battery voltage V1, while the motor, which is modeled as a constant
power load (CPL) since they behave alike for steady-state conditions i.e when the throttling action and the vehicle speed are constant, draws a power P . The converter is controlled by a pulse width modulation (PWM) method that outputs a square signal S1 of switching period Ts and duty cycle d1.1 − d1 can be interpreted as the proportion of the switching period spent
connecting the input and output voltage sources. when it is 1, the input and output are always connected and the output voltage converges to the input one while current flows from or to the load. when it is 0, the input current rises as long as the transistor does not switch,
and the DC-link capacitor is isolated with the load. If the load is a motor, the capacitor charges during regenerative braking mode, and discharges when the motor request power. In between it is a combination of both and the input/output voltage relation at steady state, for an ideal converter as supposed here, is V2 = 1−DV1 :
d1 = 1 Ts Z t0+Ts t0 S1dt (33)
S2 is its complementary and its duty ratio is then d2 = 1 − d1. It is renamed d for simplicity
because it will be used as the control signal below. The capacitor has a capacitance C and the inductor has an inductance L. The converter is a two states system, with states iL and
v2, that can be modeled with the non-linear equations : LdiL dt = v1− S1v2 (34) Cdv2 dt = S1iL− P v2 (35) As outlined in [23], this is the exact or switched model. A large signal averaged model can also be obtained by applying to the plant a moving average on a small enough time window considering the system dynamics. Supposing that the PWM switching period is small enough, one gets [24] :
LdhiLiTs dt = hv1iTs − (1 − d)hv2iTs (36) Cdhv2iTs dt = (1 − d)hiLiTs − P hv2iTs (37) It can be noticed that it is not an exact moving averaged of the switched model, mathemat-ically speaking, since the average of two switched variables product is the product of the averaged variables. However this model performs an accurate approximation of the switched model behaviour, as will be shown through simulations further in this thesis, while reducing the computational load by removing the fast switching dynamics. The next step, necessary to design a linear controller, is to linearize the plant around an operating by defining steady state voltages, inductor current and duty cycle V1, V2, I, D and introduce small perturbations
around those pointsv∼1 = v1− V1 ; ∼
v2 = v2 − V2 ; ∼
iL= iL− I ;
∼
d = d − D. The small signal
averaged model is obtained by computing the steady state relations and then the first order Taylor series expansion of the large signal averaged model [23, 24] :
Ld ∼ iL dt = ∼ v1+ ∼ dV2− (1 − D) ∼ v2 (38) Cd ∼ v2 dt = (1 − D) ∼ iL− ∼ dP V1 + P ∼ v2 V2 2 (39) Which can be written in state-space form :
˙ ∼ iL ˙ ∼ v2 | {z } ˙ X = 0 D−1L 1−D C P CV2 2 | {z } A ∼ iL ∼ v2 | {z } X + 1 L V2 L 0 CVP 1 | {z } B ∼ v1 ∼ d | {z } U (40)
From which the two following transfer functions are deduced : Gid= ∼ iL ∼ d = V2 Lp p2− P CV2 2 p + (1−D)LC 2 (41) Gvi= ∼ v2 ∼ iL = −LP p + V 2 1 CV1V2p (42)
One can notice that this system is unstable for positive values of P because it forces the poles of Gid into the right-half complex plane. Moreover, the voltage transfer function
presents a right half plane zero which might become a problem as explained further during control design.
3.2.2 Control design
Now that a linear model of the DC/DC converter has been obtained, it can be used to design a linear controller by pole placement. The controller reads the DC-link voltage and the inductor current as well as their corresponding reference value. It generates a duty cycle value for the PWM scheme which drives the transistors. A cascaded control scheme composed of two PI control loops, an inner one for the current and an outer one for the voltage, is chosen, as proposed in [23, 24]. Let’s use the notations kp,iL, ki,iL, kp,v, ki,v for the PI coefficients and wi, wv for the closed loop bandwidth targeted. The PI controllers are
given by the formulas :
Fi = kp,iL+ ki,iL/p (43)
Fv = kp,v+ ki,v/p (44)
The closed loop transfer functions can be computed :
Gcl,iL = V2 L(kp,iLp + ki,iL) p2+ (V2 Lkp,iL− P CV2 2 )p + V2 Lki,iL+ (1−D)2 LC (45) Gcl,v = −LP p+V2 1 CV1V2 (kp,vp + ki,v) (1 − CVLP 1V2kp,v)p 2+ ( V1 CV 2kp,v − LP CV1V2ki,v)p + V1 CV 2ki,v (46)
From which the constraints that must fulfill the PI controllers to ensure closed loop stability are deduced : kp,iL > LP CV3 2 (47) kp,v < CV1V2 LP (48) ki,v < V2 1 LPkp,v (49) (50)
One also gets the PI coefficients values that provide the closed loop system with the desired pole locations :
kp,iL = L V2 P CV2 2 + 2wi ! (51) ki,iL = L V2 w2i − (1 − D) 2 LC ! (52) kp,v = CV1V2(LP w2v+ 2wvV12) (V2 1 + LP wv)2 (53) ki,v = CV3 1V2w2v (V2 1 + LP wv)2 (54) Which gives the closed loop transfer functions :
Gcl,iL = V2(kp,iLp + ki,iL) L(p + wi)2 (55) Gcl,v = (−LP p + V12)(kp,vp + ki,v) CV1V2(p + wv)2 (56) The current controller generates a duty cycle signal that is of course kept between 0 and 1 by a saturation. Therefore, an anti wind-up technique equivalent to the one used in the current controller is implemented for the inductor current PI control loop. The current signal from the voltage loop is not limited here but it might be in practice.
The system dynamics are fixed for a given operating point with given pole locations targets. However, the PI controller should allow the closed loop system to remain stable and as effective as possible for the whole range of operating points that might be reached in driving conditions. By taking a look at equations 46 to 49, it appears that the higher the bandwidth, the less the PI coefficients sensibility to operating point variations. It means that the changes in performances of the closed loop system, with PI controllers designed for a given OP, would be minimized in case of OP variations if the bandwidths were taken as high as the hardware allows.
The operating point chosen for the control design is also of high importance since it must be selected in order for the system to be stable in every situations. Thus, a worst scenario OP should be selected regarding the stability constraints. One can see that the three stability margins decrease when P increases as well as when V1and V2decreases. The worst scenario is
when the battery voltage and the DC-link voltages are at their respective minimums while the motor is drawing the maximum power. Those minimums and maximums must be specified before the controller design and then used as parameters. To cope with any disturbance or modelling error that might lead to instability, one should even take more extreme values than the minimum voltages and maximum power that the powertrain is supposed to reach. The bandwidth can be calculated with the same formula as for the AC motor current controller and by choosing the rise time. It would be preferable to have a DC-link current loop at least as fast as the motor current loop to cope with the sudden changes in power demand which are equivalent to sudden OP shifts.
wi = log 9 tr,i ; wv = log 9 tr,v ; tr,i ≤ tr,c (57)
The last thing that must be taken into account is the right half plane zero of the voltage transfer function that might interfere with the current dynamics. The current pole location must be well chosen to preserve dynamics separation as much as possible [23]. The zero tends towards infinity as the motor power demand P decreases, so its dynamics will eventually cross the current dynamics as the load decreases like in many driving situations.
In terms of sampling time choice for the discrete controller, a simple experimental rule of thumb used in control classes states that the sampling bandwidth should be chosen between 10wmax and 30wmax, with wmax the maximum bandwidth among open-loop and closed-loop
plant dynamics as well as PI controller dynamics. The highest security margin is considered here which results in the recommendation :
Ts,DC ≤
2π
30 max(ki,iL/kp,iL, wi, pole(Gcl,iL))
(58)
3.3
Inverter
3.3.1 Physical model
The three phase inverter converts the DC-link DC voltage into a three phase AC voltage that can be fed to an AC motor.
Figure 8: Three phase AC motor with inverter
Figure 8 displays the motor/inverter set up that will be modeled in this thesis. The switched model takes into account the voltage harmonics i.e the motor phases are fed with a non-sinusoidal switched voltage waveform. As stated before, the machine is considered balanced and Y-connected with an ungrounded neutral point. The transistors of a same leg are controlled with complementary switching signals Sa,b,c. The voltages between a phase
defines the set up :
ia+ ib+ ic= 0 (59)
van+ vbn+ vcn = 0 (60)
vaM,bM,cM = Sa,b,cVdc (61)
idc= idca + idc,b+ idc,c= Saia+ Sbib+ Scic (62)
By using these equations and linear algebra one gets [17]:
van vbn vcn = Vdc 3 2 −1 −1 −1 2 −1 −1 −1 2 Sa Sb Sc (63)
This equation is true if the voltage drops across the semi-conductors are neglected, which is assumed here. The matrix is of rank two, which means that there are infinite possibilities of switching patterns to obtain a given set of phase voltages. It leaves a degree of freedom which Figure 8brings forward :
va0 vb0 vc0 = van vbn vcn + vn0 vn0 vn0 = Vdc Sa Sb Sc − Vdc 2 1 1 1 (64)
Vn0 is the voltage between the neutral point and the fictitious middle point of the
DC-link. This is the zero component that appears clearly when Clarke transform is applied to the phase voltages :
[Cαβ0] van vbn vcn = vα vβ 0 ; [Cαβ0] va0 vb0 vc0 = vα vβ vn0 (65)
This characteristic of the inverter is used for control purposes as explained in the control design part.
For the averaged model of the powertrain, an averaged model of the inverter is needed. It can be modeled as an inverter which generates a constant phase voltage, on a switching period, equal to the average of the waveform generated by the switched model during this same period. And its switching period can be considered infinitesimal so it produces exactly the requested voltage at any time. It is exactly the same as just neglecting the voltage harmonics and considering that the motor receives only the fundamental of the switched waveform. The averaged inverter model does not have any dynamics contrary to the DCDC converter, since the inverter does not have elements that can accumulate energy, hence no states that varies dynamically. The switching signals are reduced to their equivalent duty cycle. For a phase i:
di = 1 Ts Z t0+Ts t0 Sidt (66)
Finally, the equation linking the DC current with the phase currents can be rewritten for the averaged model :
3.3.2 Control design
The switching signals are generated by a control scheme in order to replicate the voltage reference vdq∗sat on average on a sampling period Ts by switching at the right moment
between two reference samplings. The switching period Tsw is defined as the time needed
for all the transistors to do the adequate switchings and come back to their previous state and is not necessarily equal to Ts. Some switching patterns called discontinuous allow the
transistors of a leg not to switch during a period. In this thesis a continuous modulation method is used and all the transistors switch twice in a period. A synchronous sampling is used, which means that the measured signals received by the motor current controller are sampled and transistors switching times are computed at a multiple of the switching rate. It can be at the same rate but here an asymmetrical sampling is used. It samples the incoming signals and computes the voltage references and switching times at twice the switching rate. The switching instants are not exactly symmetrical from the first half switching period to the second, hence its name.
Figure 9: Switching waveform and voltage reference
Figure 9shows this asymmetry with the voltage reference that changes in the middle of the switching period Tsw. However the difference from one sample to the next is very low
in practice due to a very high switching frequency compared to the motor current dynamics bandwidth.
the controller must find a set of switching instants that provides the adequate duty ratios to meet the voltage request on average on Ts. Equation 63can be rewritten in terms
of requested voltage and duty cycles : [17]:
v∗an vbn∗ vcn∗ = Vdc 3 2 −1 −1 −1 2 −1 −1 −1 2 da db dc (68)
The simplest way to compute the duty cycles is to compare a triangular wave of period
added to the phase voltage reference to improve the linear modulation range or to achieve discontinuous switching patterns to decrease switching losses [6]. This is fit for analog control implementation. Space vector modulation is an other way to achieve the same purpose and is suitable for digital implementation. It consists in reading the voltage references in the Park frame instead of the normal one. only the Vd and Vq components are provided by the
current controller. The rotation matrix of rotor electrical angle θr is applied to express the
voltage reference in the Clarke frame as a 2D space vector, as displayed by figure 10.
Figure 10: 2 level inverter space vector hexagon
This is an hexagon divided in six sectors separated by the voltage vectors that the in-verter can actually generate. The ones and zeros represent the states of the inin-verter’s upper switches. When they all share the same state, the space vector value is zero. This is again a way to show that the zero component does not appear in the 2D Clarke/Park frame and then does not play a role in the motor current dynamics. The current controller limits the voltage reference magnitude in advance so it does not go beyond the biggest inscribed circle of the hexagon. Its radius, represented by Vbase on the figure, and called Vmax in this thesis,
is equal to : Vmax = cos (π/6) 2Vdc 3 = Vdc √ 3 (69)
This circle represents the linear modulation region. To go beyond, some overmodulation techniques exist notably for allowing more power during transients[19] but at the cost of poorer harmonic content leading to higher torque oscillations.
The SVM algorithm calculates the space vector angle, deduces the sector he belongs to, projects it onto the vectors surrounding its position, calculates the ratio between each pro-jection and the maximum value it could take and then deduces the duty cycle corresponding to each non-zero (or active) vector. these duty cycles multiplied by the sampling period gives the time during which the transistors are supposed to be in the position generating
the corresponding space vector. The degree of freedom remaining is the placement of the zero vectors in the sampling period. It is the mathematical equivalent of choosing the zero component [6]. In this thesis, both zero vectors duty cycles are taken equal :
d0,1 = d0,2 =
1 − dactive,1− dactive,2
2 (70)
Now that the duration of each switching position is known, and that the switches are in a zero vector position at the beginning of each sampling period, the best pattern is to switch one transistor at a time. For instance, the two possible paths to generate an averaged space vector located in the first sector are :
(0, 0, 0) − (1, 0, 0) − (1, 1, 0) − (1, 1, 1) (1, 1, 1) − (1, 1, 0) − (1, 0, 0) − (0, 0, 0)
When the voltage reference is near zero, an issue arises. A very small active vector is demanded, then a very short impulse is required. If it is shorter than the switching times computation time, the switching will occur too late, impacting the performances, as outlined in [21]. Even if the computation delay is not modeled in this thesis, the issue is tackled to prepare for the next model improvements. the solution is to generate the switching times for the next sampling period, which causes a delay equal to Ts between the controller demand
and the actuator response. Added to this, the average delay between the sampling time and the switching time of a leg is almost equal to Ts/2. The overall delay is :
Td=
3
2Ts (71)
First, it must be taken into account during voltage space vector reference computation, when applying the rotation matrix :
v∗α,sat vβ,sat∗ ! = cos (θr+ w1Td) − sin (θr+ w1Td) sin (θr+ w1Td) cos (θr+ w1Td) ! v∗d,sat vq,sat∗ ! (72) Furthermore, the delay may lead to the plant instability. By modeling the switching actuators as a delay added to the linear plant transfer function, a minimum sampling bandwidth of 25αc
is recommended for comfortable gain and phase margins [21]. The resulting recommendation on sampling time is :
Ts≤
2π 25αc
(73) The same reasoning could suit the DCDC converter controller. However the rule of thumb was used for simplicity.
3.4
OC faults
3.4.1 Faulty inverter model
A simple way to adapt the inverter model for open circuit fault is found in [25]. The reasoning presented here is similar. In case of a fault from the transistor or its control circuit, the states
S1=0 ; S2=1 S1=1 ; S2=0 S1=0 ; S2=0
i>0 idc=0 ; VM=0 idc=i ; VM=VDC idc=0 ; VM=0
i<0 idc=0 ; VM=0 idc=i ; VM=VDC idc=i ; VM=VDC
Table 1: Boolean table of an inverter leg
of the two transistors of a leg are not correlated anymore. By building the Boolean table of a leg, one can find the way to adapt equations 61,62. One can refer to figure 8 for notations. Let’s note the upper and lower transistors states S1 and S2. The table gives the
new equations true for each phase ph
vphM = (S1+ S2(iph< 0))Vdc (74)
idc,ph = (S1+ S2(iph< 0))iph (75)
3.4.2 FDI schemes
The FDI schemes presented here focus on OC faults detection by manipulating time signals and focusing on current monitoring. they require more lines of code in the controller and then maybe more computational power but do not need more sensors than the two current ones already available. The first technique is inspired by the work done in [26]. The signals needed are the current references i∗ph and the measured ones iph. Their difference eph can
be averaged on an electrical period, to remove the current ripples influence, and normalized by being divided by the averaged measured current.
hiphi = w1 2π Z t0+w12π t0 iphdt (76) eph = i∗ph− iph (77) kph = hephi h|iph|i (78) This obtained value is then independent of the load conditions since it does not depend on the current magnitude. It equals 0 in healthy conditions because the current follows the reference. It can be compared to a fixed threshold to detect and isolate the fault. The method relies on the hypothesis that the current waveform is exactly half sinusoidal when an OC fault occurs in a leg. The value of kph is then either 1 or -1 if the upper or lower
transistor are respectively faulty [26].
The second technique relies on an observer that imitates the motor behaviour by using the current state equations. It uses a constant gain which must be tuned to allow for robust detection for different operating conditions.
d ˆid dt d ˆiq dt = −Rs Ld Lq Ldwr −Ld Lqwr − Rs Lq "ˆ id ˆ iq # + " 1 Ld 0 0 L1 q # " vd vq # − " 0 ψwr #! + " K1 K3 K4 K2 # "ˆ id ˆ iq # − " id iq #! (79)
For simplicity of gain tuning, K1=K2 and K3=K4=0. Form factors are calculated for
Both are calculated with the same equation used for previous technique and RMS is obtained just by squaring the signal before averaging. Then the difference of the form factors f fph
and ˆf fph gives residuals rph that can be compared to a threshold that must be found for
robust detection purposes.
hiphiRM S = s w1 2π Z t0+w12π t0 i2 phdt (80) f fph= h|iph|iRM S h|iph|i (81) rph= |f fph− ˆf fph| (82)
In healthy conditions, the form factors both equal almost 1.11 and the residual of each phase is very close to 0. The work of [27] provide active thresholds that happened to be effective in their case but ineffective in the case of the PMSM modeled in this thesis, because they were always higher than the residuals even in faulty conditions, and then no fault was ever detected. Hence, they are not used in this thesis.
The technique proposed allows for faulty leg detection. To detect the faulty switch, one must also look at hiphi sign. If it is positive, then the faulty transistor is the lower one, and
4
Matlab model implementation and results
4.1
General simulation set up
The models simulated in this thesis are based both on the assumptions outlined in the mathematical model chapter and on what follows:
• All the sensors are supposed perfects which means that the measured values used for control purposes and the real physical values are supposed perfectly identical. Then, one must acknowledge that the PMSM indirect field oriented control is done with a perfectly measured rotor angle.
• The estimated parameters used in the controllers are supposed to match the real ones perfectly unless the contrary is clearly indicated
• the parameters are time-invariant since temperature dependency and skin effect are neglected
• Mechanical parts like shafts and wheels are supposed to be infinitely stiff (no shaft torsion, no torque transmission delay and no wheel slip) and the internal frictions are neglected. Thus, the friction torque Tf which appears in equation 7 is zero like the
motor internal friction coefficients blin and bqu
• the battery is considered as a perfect voltage source
The environmental parameters as well as the basic car parameters that were used in the platform models are given in this table :
parameter value
gravity acceleration g 9.81 m/s2
air density ρ 1.225
SCx 0.75 m2
rolling resistance coefficient Crr 0.006
vehicle mass M 1468 kg
wheel radius R 0.3 m
Table 2: environment and car mechanical parameters
Those figures give access to the load torque, which is due to rolling resistance and drag, and the total inertia, both referred to the motor shaft (sp is the speed of the car):
TL= RFL r = R(12ρSCxsp2+ CrrM g) r (83) Jtot = Jmot+ Jdrivetrain r2 + M R2 r2 (84)
The load torque referred to the motor shaft varies from 3 to 23 N.m for standstill and highway at 130 km/h respectively. Two performance indicators are now chosen arbitrarily to size the electric motor :
parameter value acceleration time from 0 to 100 km/h t0−100 8 s
nominal speed spn 70 km/h
Table 3: performance targets
Then some mechanical and design characteristics are chosen, arbitrarily again but rele-vantly according to the literature that proposes PMSM motors with 4 or 8 poles [19,1] and gear ratios of 3 to 4 for small motors for hybrid powertrains [1], up to 10 for full electric vehicles [4, 3]:
parameter value
fixed gear ratio r 9 number of pole pairs np 4
Table 4: motor mechanical parameters
The literature proposes battery voltages from 200 to 400 V and DC-link voltages from 200 to 650 V. Battery and the DC-link voltages are then chosen [8, 1] :
parameter value
Vbattery 400 V
VDC,min 400 V
VDC,max 600 V
Table 5: battery and DC-link operating voltages Those values permit to compute the motor nominal characteristics :
characteristic formula value
wm,n rspRn 5567 rpm wr,n npwm,n 2π*371 rad/s Tn 1.1MR∗100/3.6t 0−100r 187 N.m Pn wm,nTn 109 kW Vn Vmax = VDC,min√ 3 231 V In 23 ∗ PVnn 315 A
Table 6: motor nominal characteristics
The nominal mechanical rotational speed is here defined as the rotor shaft rotational speed when the car is going at the nominal speed outlined in the performance arbitrary specifications. 1.1 is used as a correction coefficient to roughly consider air and road re-sistant forces during acceleration when computing the nominal torque. It is defined as the torque that must output the motor to reach the acceleration target specified in the arbitrary
performance specifications. Then the obtained nominal mechanical power is considered equal to the nominal electric power because the losses can be neglected on high power operating point. The nominal voltage is taken equal to Vmax which (roughly, because of the MTPA
control) represent the maximum average voltage per phase enabled by both the DC-link and the SVM modulation in linear range, as explained in the mathematical model chapter. One might notice that here, the motor nominal voltage is defined at the minimum voltage that can output the DC/DC converter. This is also arbitrary because it cannot be clearly speci-fied without a complete book of specifications written by a car manufacturer. The nominal current, which is (roughly, because of the MTPA control) the current needed to develop the nominal torque, is deduced from the electrical power and voltage.
Finally, the values chosen for the stator resistance and inductances as well as the PM flux linkage are based on [21] in per unit and converted in physical values by multiplying them with the corresponding base values for three phase systems. One can verify that the physical values obtained are consistent with what is found in [19] :
base value formula
Zbase Vn/In
Lbase Zbase/wr,n
ψbase Vn/wr,n
Table 7: base values
Parameter per unit physical value
Rs 0.02 14.7 mΩ
Ld 0.6 1.89e−4 H
Lq 1 3.15e−4 H
ψm 1 99 mWb
Table 8: motor parameters
A rough sizing is done for the chopper passive components. The values found in [1], for choppers with rated power between 20 and 36 kW, are between 0.9 and 2.6 mF and 0.2 and 0.4 mH for capacitors and inductors respectively. The passive components sizing does not seem to be correlated with the converter and motor rated powers in the case of the Prius 2004 and 2010, whereas it seems to be the case for the American cars. By supposing that these components are sized only to fulfill the specifications about DC-link voltage and inductor current ripples limitations, one could use some formulas outlined in [28] for instance. However those equations requires performance targets and must be used for specific operating points with given switching frequency, duty cycles and motor power demand, so many things that are not chosen very accurately here. Thus, some consistent values are taken with regard to the literature. Besides, The switching frequency of the converters is often between 5 and 10 kHz and can even reach 20 kHz sometimes [1, 29, 4]. Then both the chopper and inverter are given a relevant and common switching frequency : One might notice that the switching
parameter value
CDC 1.5 mF
LDC 0.5 mH
fsw 10 kHz
Table 9: converters parameters
looking at the nominal values, it appears that even if the car was supposed to go at 140 km/h, twice the nominal speed, the fundamental frequency of the stator currents would be 742 Hz. The ratio between the switching frequency and the fundamental of requested currents, also called pulse number, would then be equal to 13.5 which is good enough to provide motor currents without too much harmonics.
All the models are Simulink block diagrams using parameters generated by Matlab scripts. No specific toolboxes were used. The solver chosen for all simulations is of variable step time in order to get the fastest simulations possible :
parameter value
simulation time 30 s
solver VariableStepAuto (ode23t)
throttling signal triggering 0.1 s Table 10: simulation parameters
The automatic choice is done by Matlab which selects the variable step solver that best suits the model being simulated. The only parameter fixed by hand is the relative tolerance which defines the numerical computation accuracy. It is scarce but sometimes necessary to tweak it in order to get the best transients computations. All the other solver parameters are automatically set during compilation and simulation steps. The start of the throttling action is slightly delayed to get better plots. The torque reference appearing on the plots is the one calculated with equation 31.
4.2
DCDC converter
The controller developed in the mathematical model part is linear and receives signals that are perturbations around the OP chosen for the design. The measured signals are absolute and then must be compared to their corresponding OP value to obtain the difference that can be used by the controller. The output signals from the controller must then be transformed back to absolute values by adding them their corresponding OP value. Besides, this DC converter receives its DClink voltage request from a decision algorithm with a higher level of authority that is not modeled in the modular platform. With this information lacking, it is all right to suppose that the requests are steps. However steps are too sharp requests that might lead to huge problems of powertrain effectiveness and stability. A voltage request rate limitation is therefore introduced. Moreover, an operating point is chosen for the controller design.
