Switched Reluctance Motor Srm Matlab

Arxiv:1601.03768V1 [Cs.Sy] 14 Jan 2016

Optimal Current Waveforms for Switched-Reluctance Motors

Abstract

In this paper, we address the problem of finding current waveforms for a switched reluctance motor that minimize a user-defined combination of torque ripple and RMS current. The motor model we use is fairly general, and includes magnetic saturation, voltage and current limits, and highly coupled magnetics (and therefore, unconventional geometries and winding patterns).

We solve this problem by approximating it as a mixed-integer convex program, which we solve globally using branch and bound. We demonstrate our approach on an experimentally verified model of a fully pitched switched reluctance motor, for which we find the globally optimal waveforms, even for high rotor speeds.

Introduction

We consider the problem of choosing optimal current waveforms for a switched reluctance motor (SRM). Traditionally, the shape of the current waveforms is determined in an ad-hoc manner (e.g., by fixing the turn-on angles for each phase winding as a function of rotor position; see [TK12]).

Although creating these waveforms does not require a detailed motor model, and implementing them is simple, waveforms produced in this manner rarely produce smooth output torque, and often decrease motor efficiency and exacerbate mechanical vibration and acoustic noise issues.

Also, because of inverter voltage limits, such waveforms may not even be realizable at high rotor speeds. We therefore propose to use optimization to find current waveforms that acheive a desired average torque, while minimizing a combination of resistive power loss and RMS torque ripple. We consider supply voltage limits, as well as current limits in each phase winding. Our model includes a detailed magnetic circuit, which can account for magnetic coupling between phases, and can be used to model motors with unconventional geometries and winding patterns, such as those with fully pitched windings.

We propose to solve this optimization problem by approximating it as a mixed-integer convex program (MICP). This MICP reformulation approach has several inherent advantages over more conventional methods, such as sequential quadratic programming. The most prominent, for our purposes, is that it can be solved globally by generic methods such as branch and bound, often in a reasonable amount of time. This has two benefits: first, it allows us to achieve the best performance possible for a given motor; second, it provides a metric against which other, suboptimal methods

1

can be compared. We note that although in general global optimization methods for solving MICPs can have very high (exponential) runtime, we find that global solutions can typically be found in a reasonable amount of time (1-5 minutes) for the problems we encounter. These waveforms can be computed and stored in a lookup table, indexed by desired torque and rotor speed. Such a table with hundreds or thousands of entries could be computed overnight on a standard multi-core computer.

Another advantage of an MICP formulation is that, if suboptimal solutions are acceptable, first-order methods exist that can produce a good solution very quickly especially when initialized with a decent initial guess. (see, i.e., [TMBB15]) This opens the possibility that (nearly) optimal waveforms can be produced by a smart control scheme even as motor parameters vary over the life of the motor.

This is especially attractive when combined with a lookup table containing precomputed, globally optimal waveforms: the precomputed waveforms can be used as an initial guess for an online optimization method when new waveforms (corresponding to updated motor parameters) are required.

We demonstrate our approach numerically on the experimentally validated motor model of [MWC01], which describes an SRM with fully pitched phases.

Previous Work

Current optimization for SRMs. Several authors have considered optimization of SRM cur- rent waveforms. The most similar work to our own is a series of papers by Lovatt and Stephenson [LS94], [LS97a], [LS97b], that use local optimization methods to find minimum RMS current wave- forms that acheive a given (average or pointwise) desired torque, subject to voltage and current constraints. Stankovic et al. derive optimal waveforms for a simple SRM, under several restrictive assumptions (e.g., sufficient drive voltage, no more than two simultaneously conducting phases); under these assumptions, it is only necessary to discretize the waveforms during commutation.

Kaiserseder et al. [KSAS03] also seek optimal current waveforms that produce smooth torque out- put and can be chosen to either minimize current RMS values, or minimize vibration resulting from radial forces. The method of optimization is not described. A similar optimization problem is posed by Chapman and Sudhoff[CS02] in the frequency domain; sequential quadratic programming is used to (approximately) solve it. Many other works optimize over current waveforms parametrized by only a few free variables, such as the firing angle or the corner locations of a trapezoidal wave- form. This of course requires predetermined current waveform shapes, which are not optimal in general. For some examples of this approach, see [MK03], [CMH93], and [CKKP02].

We also note that our approach can be viewed as an extension of the authors’ previous work on optimal current waveform design for permanent magnet motors; see [MB15]. Hybrid control and MPC for SRMs.

Here we list some other optimization-based techniques applied to control SRMs. Peyrl et al. propose a finite-set model predictive control approach, which involves online optimization directly over future inverter switching states. Due to the computa- tionally demanding nature of this technique, only very short prediction horizons (i.e., up to three steps) are considered. The work by Vasak et al. [VZP+07], uses a piecewise-affine model of the torque characteristic (as we do) to derive a feedback controller that guarantees a torque ripple.

However, their proposed model is relatively low fidelity: the dynamics are modelled as a first-order linear system, and the proposed torque characteristic (our gk(Fk, θ)) has only nine regions (For comparison, in §6 we use over one hundred regions.)

Micp.

Convex optimization problems can be solved efficiently and reliably using standard tech- niques [BV04] (and additionally, specialized modelling software, such as CVX [GB14], enables rapid development of convex optimization applications). An optimization problem with some integer vari- ables, but which is otherwise convex, is called a mixed integer convex program (MICP). Because of the presence of integer variables, MICPs are nonconvex optimization problems, and are difficult to solve (globally) in general (i.e., these problems are are NP-hard; see [KT06]). Indeed, all known global solution techniques for MICPs (such as the branch-and-bound algorithm), have exponential worst-case runtime. Nevertheless, many of these algorithms are effective in practice, and we found them to work well for the the optimization problem we formulate in this paper. In addition to global solution algorithms, many approaches exist to (approximately) solve MICPs more quickly; see [TMBB15] and references therein.

Our formulation is based on approximating the nonlinear constraint functions (the torque char- acteristics and the magnetic flux characteristics) by piecewise affine functions. These constraints can then be represented as a combination of integer and linear constraints using disjunctive pro- gramming. For details on disjunctive programming consult Balas [Bal79] and Ceria and Soares [CS99]. Disjunctive programming has found many applications in the past decade or so, such as process engineering [GT13], facility location, unit commitment and portfolio management [GL12], and optimal control [?].

Contribution

Our reformulation of the torque control problem as a MICP opens the door for two interesting possiblilites. The first is that global solution methods can be used to find the optimal waveforms, typically in a reasonable amount of time. This is of course advantageous in its own right, as it allows provably optimal waveforms to be implemented. It is also useful to verify the limits of performance of motors, and as a benchmark for comparing heuristic methods. The second possibility is that fast first-order methods, which are simple enough to run on embedded platforms, can be used as a heuristic to update the globally optimal waveforms as parameters vary over the life of the motor. Our proposed model is also much more general than the optimization models considered in previous works, and can therefore be used to capture more of the characteristc features of switched reluctance motors, such as magnetic coupling. We hope this generality will be useful for researchers investigating novel switched reluctance motor topologies, by giving them a practical method for optimal waveform generation, and by characterizing the theoretical performance of their designs.

Motor Model

We consider an abstract, lumped parameter model of a switched reluctance motor. The rotor, which does not contain any windings or magnetic elements, has angular position θ and angular velocity ω; we assume ω is constant.

The stator contains n electrical circuit branches, called windings. The winding currents are i ∈Rn, the winding voltages are v ∈Rn, and the magnetic flux linkages through the windings are λ ∈Rn. The stator also contains m magnetic elements, with magnetomotive force (MMF) vector F ∈Rm, and magnetic flux vector ψ ∈Rm.

We will assume that i, v, λ, F, and ψ are 2π-periodic functions of θ.

We Use A Prime (′)

to denote differentiation of these functions with respect to θ. To lighten notation, we often drop explicit dependence on θ.

3

Electrical dynamics.

Where R ∈Sn

++ is the (diagonal) resistance matrix. Note that ωλ′ is the time derivative of λ. Magnetic circuit. We assume the magnetic elements are connected by a (planar) magnetic circuit, which we describe in terms of mesh analysis (for an introduction to mesh analysis, see [DK84]). In particular, we assume that there are l circuit meshes (not including the outer mesh) each with an associated reference direction. The (reduced) mesh matrix M ∈Rl×m is such that

1

if magnetic element k is in mesh j, with coinciding reference directions

−1

if magnetic element k is in mesh j, with opposite reference directions

0

if magnetic element k is not in mesh j. Any flux vector ψ consistent with the magnetic circuit topology must be a linear combination of

(2)

for some φ(θ) ∈Rl, which we call the mesh magnetic flux vector. (Because M has full row rank in

General, There Is A Unique Φ For Any Such Ψ.)

In addition, the total MMF around each mesh is the sum of the MMFs of the magnetic elements that make up the mesh (accounting for reference direction), so the vector of mesh MMFs is given by the vector MF.

Electro-magnetic geometry. We define the electro-magnetic geometry matrix C ∈Rl×n such that Cjk gives the amount of current passing through mesh j per unit of current in winding k. The total current passing through each of the l meshes is therefore given by the vector Ci, which is related to the total MMF around the meshes by Amp`ere’s law: MF = Ci.

(3)

Similarly, the flux linkage is related to the mesh magnetic flux vector by λ = CT φ.

(4)

Magnetic characteristic. The magnetic flux and the MMF of the k-th magnetic element are

(5)

where the magnetic characteristic fk is a monotonically increasing function in its first argument. As a special case, if the functions fk are affine in Fk for each θ, with constant linear term (so that ψ(θ) = AF(θ) + b(θ), with A diagonal and positive definite), as in the case of a permanent magnet motor, then we have λ(θ) = Li(θ) + k(θ), where L = CT (MA−1MT )−1C is the inductance matrix and k(θ) = CT(MA−1MT )−1A−1Mb(θ) is the back-emf constant.

4

Torque.

0

fk(x, θ) dx.

0

fk(x, θ) dx. By introducing a phase torque function gk such that

.

(6)

Voltage limits.

|Vk(Θ)| ≤Vmax,

k = 1, . . , n.

(7)

Torque ripple.

0

τ(θ) dθ.

Τ(Θ) −Τ

2 dθ. Power loss. The power loss is the average resistive loss from the phase currents over one cycle:

0

i(θ)T Ri(θ) dθ.

Optimal Torque Control

The optimal torque control problem is to choose the phase voltages, phase currents, and eddy currents to achieve a desired average torque while minimizing the average power loss and torque

Τ = Τ Des,

equations (1), (2), (3), (4), (5), (6), and (7).

(8)

The parameters are the trade-offparameter α ≥0, the rotor angular velocity ω, the desired average torque τ des, the resistance matrix R, the mesh matrix M, the electro-magnetic geometry matrix

5

C, and the magnetic characteristic functions fk, for k = 1, . . , m. The problem variables are the 2π-periodic functions i, v, λ, F, ψ, and φ. Problem (8) is an infinite-dimensional optimization problem. The problem is nonconvex due to the magnetic characteristic (5) and the torque relation (6), and therefore is expected to be difficult to solve globally.

One approach is to find a locally optimal solution, using common local optimization meth- ods such as sequential quadratic programming Software that implements these methods is readily available; see [NW06].

In this paper, we pursue a different approach, and will instead show how to approach (8) by discretizing the variables, and (approximately) reformulating the problem as a mixed-integer convex program (MICP), a problem class for which efficient algorithms are available to find a good (or even globally optimal) solution.

We note that if the magnetic characteristic is affine, with only the offset depending on rotor position, the torque control problem reduces to a version of the formulation given in [MB15]. Our current problem can therefore be interpreted as an extension of that formulation to cover magnetic nonlinearities and reluctance torque.

Conversion To Micp

In this section we show how to convert (8) to an (infinite-dimensional) mixed-integer convex pro- gram. To do this, we use piecewise-affine approximations of the nonlinear equality constraints.

Approximation Of Magnetic Characteristic

Here we approximate the equation magnetic characteristic (5) by a set of linear and integer con- straints. We replace the constraint ψk = fk(Fk, θ) with the constraint

(9)

where ˜fk is a piecewise affine approximation of fk. In particular, we have

K

...

Are

the boundaries of the affine regions, so that we have

≤X ≤˜Fj

k.

K(Θ) And Sj

k(θ), for j = 1, . . , N, and k = 1, . , m, the approximate magnetic characteristic constraint (9) can be written as

Sj

k(θ) ∈{0, 1}.

Approximation Of Torque Function

In the same way, we can approximate the torque constraint (6) using a set of linear and integer constraints. To do this, we first approximate each torque function gk, for k = 1, . . , m, as a

K

...

K(Θ), . . . , Dn

k (θ) parametrize the affine functions. Then the approximate

(11)

to the constraints (10) above.

Improving The Micp Formulation

By converting the nonlinear constraints to linear and integer constraints, we have succeeded in our goal of making (8) into an MICP. However, the runtime of a global MICP solver often depends crucially on the problem formulation. Here we give an additional reformulation of the objective of (8) (specifically, of the the power loss) that may improve the performance of a MICP solver compared with the basic formulation.

Assuming that C has full column rank, we can use (3) to express the winding current as i = C†MF, where C† is the pseudo-inverse of C (or any other left inverse). Then the power loss

F(Θ)T Qf(Θ) Dθ,

where Q = MT (C†)T RC†M. For any feasible set of variables, and for any diagonal matrix D, this is equivalent to

!

dθ. In particular, if we choose D with nonnegative diagonal elements, such that Q −D is positive semidefinite, then the reformulated power loss function is convex in all its variables. Furthermore, the larger the elements of D are, the tighter the convex relaxation will be. Finding a suitable D can be cast as a small convex optimization problem; for details on this type of reformulation, see

Deployment In Out-Of-Position Situations

D. Bendjaballah1, A. Bouchoucha1, M. L. Sahli1,2* and J-C. Gelin2

Abstract

Side-impact collisions represent the second greatest cause of fatality in motor vehicle accidents. Side-impact airbags have been installed in recent model year vehicle due to its effectiveness in reducing passengers’ injuries and fatality rates. In meeting these requirements, simulations of folding and deploying airbags are very useful and are widely used. The paper presents a simulation method for the deploying airbags using three materials in different working conditions. Finite element analysis is primarily used to evaluate this concept. In these simulations, the gas flow is described by the conservation laws of mass, momentum, and energy. The numerical results indicate that the FE method in this paper is capable of capturing airbag deploying process accurately.

Keywords: Airbag simulations, Out-of-position, Crash, Modeling, Out-of-position

Background

The passive safety of cars has become a very high prior- ity issue for the automotive industry. Today, there are not only one or two airbags in a car; certain models have ten times more than that. With the increasing usage of airbags, the number of accidents where the airbag itself can cause an injury to the occupant also increases

(Augenstein Et Al. 2003; Gabauer And Gabler 2010;

Audrey et al. 2011). As is well known, safety belts are also now devices designed to provide protection to the users of vehicles during crash events, minimizing the loads necessary to adapt their movement to the move- ment of the car (Freesmeier and Butler 1999; Schmitt et al. 1997). In general, the seat belt is designed to restrain the occupant in the vehicle and prevent the

Occupant From Having Harsh Contacts With Interior

surfaces of the vehicles. The airbag acts to cushion any impact with vehicle structure and has positive internal pressure, which can exert distributed restraining forces over the head and face. As a safety component of auto- mobile, an airbag decreases occupants’ injury likelihood effectively in case of an accident (Ruff et al. 2007). These safety elements can reduce the death rates on the roads, and its protection effects have been widely approved (Crandall et al. 2001; Teru and Ishikawa 2003). With computational tools such as finite element methods designed for dynamic contact problems, crashworthiness simulations can now be used with reliable accuracy to evaluate occupant protection in various collision condi- tions with safety metric/parameters such as acceleration, head injury criteria, intrusion distance, intrusion vel- ocity, and neck forces (neck injury risk or whiplash).

Thus, new types of airbag products are being developed to handle different collision scenarios.

Become Standard Equipment On Most New Passenger

vehicles (Braver and Kyrychenko 2004; Teng et al. 2007; Yoganandan et al. 2007). The airbag cushion is com- posed of a woven fabric which is rapidly inflated during a car crash. The airbag dissipates the passenger’s kinetic energy thereby reducing injury through biaxial stretching of the fabric bag and escaping gas through vents. There- fore, the performance of the airbag is greatly influenced by the mechanical properties of the fabric. Generally, air bags are designed to deploy in a crash that is equivalent to a vehicle crashing into a solid wall at 8 to 14 mph.

Air bags most often deploy when a vehicle collides with another vehicle or with a solid object like a tree. There are various types of airbags: frontal, side-impact, and curtain airbags. In general, the passenger side airbags are usually larger than the driver airbags (see Fig. 1).

Besançon, France

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Bendjaballah et al. International Journal of Mechanical

Doi 10.1186/S40712-016-0070-2

Extensive studies have shown that the airbag deploy- ment in load cases consists of two occupant loading phases: a punch-out effect where the airbag bursts out of its container with the airbag and airbag module cover accelerating towards the occupant and a second loading phase during which the airbag is taking on its deployed shape and volume (membrane-loading effect). Bankdak et al. (2002) developed an experimental airbag test system to study airbag-occupant interactions during close proximity deployment. The results provided insight for simulating the effect of inflation energy and mass flow on target response. Bedard et al. (2002) found that while left-side (driver-side) impacts accounted for only 13.5% of all crashes, the fatality rate among these

Crashes Was 68.3% In Comparison To Front Impact

(48.3%), right-side impact (31.3%), and rear impact (38.4%). These studies underscore the importance of oc- cupant safety during side-impact collisions. In the last years, the current market requested to reduce the time and cost airbag development. In order to achieve this result, virtual simulations play an important role since they allow to minimize the number of experimental tests (Pei et al. 2013; Cao et al. 2014). Several simulation models of airbag were established (Wang et al. 2007). It is feasible to optimize the parameters of airbag deploy- ment using simulation technology. Experimental and numerical studies have quantified injury risks to close- proximity occupants from deploying side airbags. These studies have focused on the prevention of the most ad- verse effects of airbag deployment (Duma et al. 2003).

Other studies have proposed airbag characteristics to minimize particular biomechanical responses (Haland and Pipkorn 1996). In a more recent study, Marklund and Nilsson (2003) compared deformation patterns with experimental data as well as the computational costs associated with three different airbag deployment simu- lation methods; they concluded that the SPH method is relatively inexpensive and produces incremental deform- ation patterns that compare most closely to the experi- mental results. The process of inflation of an airbag is one of the determining factors in saving lives. The duration from the initial impact of the crash to the full inflation of an airbag is about 40 ms, and during this time, the airbag goes from being in a folded state to a fully inflated state, with a high internal pressure. After achieving this state, the airbag begins to deflate, thus providing a nice cushion for the body impacting it.

Ideally, the person in the crash should come into contact with the airbag at this time. In the present study, a large volume passenger side airbag model is developed to handle different collision scenarios. The main aim is evaluate the performance of deploying of passenger side airbag using finite element methods (FEM).

Materials

The tensile specimens were made in different airbags (P: Peugeot, R: Renault, and VW: Volkswagen) with a length of 200 mm long and a width of 40 mm. Table 1 shows the mechanical properties of the airbag.

Tensile Tests

To determine the mechanical properties of the material of airbag used in the test pieces, tensile tests were performed on Lloyd EZ20 universal testing machine in Constantine. These tests were conducted using rect- angular samples. The axial force and axial displacement acquired during a test are converted into stress and the strain in order to be used for the fabric material model.

The continuous recording of the stress-strain data was performed during both the load and unload phases. A minimum of five samples were made in order to check the repeatability of the measurements. All the data was collected by using a PC-based data acquisition system and analyzed by commercial software. The picture frame test device that is made for this study is shown in Fig. 2.

Fig. 1 a Frontal and side airbags. b Oblique view of facet occupant model in sitting posture following airbag deployment (Lim et al. 2014)

0.150

Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

Page 2 Of 9

Figure 3 shows the stress-strain relationship of the airbag sample under axial tensile loads. The results are showing a linear increase in extension with the increas- ing stresses. This is an expected output and it confirms with the theoretical behavior of a sample subjected to tensile stress. The rupture strain values for different airbags (R/P/VW) were 0.322, 0.441, and 0.472, respect- ively. The measured elastic parameters (i.e., Young’s modulus E and initial yield strength) and Poisson’s ratio are summarized in Table 2. The tensile tests of the woven fabrics can show differences on mechanical prop- erties because woven fabrics can resist in-plane shear loads once the yarn lock-up angle has been reached. The differences of material property on material direction can affect the shape of fully deployed bag (see Fig. 3b).

Theoretical Background

Numerical simulations of airbags use very complex and techniques such as an orthotropic model to identify the mechanical behaviors during the airbag inflation and the fluid mechanics (gas flow) to describe the inflator gas flow (pressure gradient) and improve the representation of the pressures within the airbag. To model the airbag as an orthotropic model, three material constants have to be provided. Assuming a plane stress condition, the

Ð1Þ

where σ is the normal stress and τ is the shear stress, the subscript refers to the principal material directions, i.e., the fill and warp directions. Also, ε and γ are the strain components. The material elastic constants Qij are

Ð2Þ

where E1 and E2 are the Young’s modulus in the fill and wrap directions and G12 is the shear modulus of the fabric material. νij is the Poisson ratio of the material.

The gas exerts a pressure load on the airbag causing it to expand. This expansion puts the airbag under tensile stress lowering the expansion rate. In this study, heat conduction and heat transfer is not taken into account.

Fig. 2 A photograph of Lloyd EZ20 universal testing Fig. 3 Stress versus strain using Lloyd EZ20 machine for a three different airbags at 0° and 90° and b VW airbag test specimens at

Different Angles

Table 2 Physical and mechanical properties of the airbag

Page 3 Of 9

In the deployment of an airbag, an inflator supplies high velocity gas into an airbag causing it to expand rapidly. The gas inside the airbag is assumed to be ideal, to be of constant entropy, and to satisfy the equation of state:

Ð3Þ

Here p, ρ, and e are respectively the pressure, density, and specific internal energy, and γ is the ratio of the heat capacities of the gas. The gas flow is described by the conservation laws for mass, momentum, and energy that

Ð4Þ

here, V is a volume, A is the boundary of this volume,

N Is The Normal Vector Along The Surface A, And U

denotes the velocity vector in the volume. Applying Bernoulli’s equation in the case of an ideal gas with

Ð5Þ

Here, the subscript ex denotes quantities at the throat of the tube. Furthermore u, p, and ρ denote the quan- tities inside that part of the tube that is supplying mass.

Materials And Boundary Conditions

The airbag system mainly consists of three parts: the airbag itself, the inflator unit, and the crash sensor or diagnostic unit. Thus, to study the behavior of the airbag using FE simulations, we need to have an FE model of the airbag in the folded position. A FE model of the airbag was used to simulate the test condition as shown in Fig. 5. LS-DYNA® material model FABRIC (MAT_34) is used to simulate the airbag material. It is a variation of the layered orthotropic material model. Additionally, in the LS-DYNA® material model, fabric leakage can be accounted for. However, for this CAB material, the leak- age is almost negligible and therefore no leakage is specified. The mechanical properties can be determined from the physical test. Typical material properties for airbag fabrics are taken as given in Chawla et al. (2004a) (Table 3). These properties are used to simulate inflation process of airbag (see Table 1). The car dashboard is modeled as the rectangular thin plate using a MAT_RI-

Gid Material, And The Degrees Of Freedom Are Con-

strained in all the directions. The similar properties of thermoplastic polymer are assigned for contact purposes. The porosity of the fabric is assumed zero. The nitro- gen gas is taken for inflating the airbag. Properties of nitrogen gas and initial bag conditions are shown in Table 4. The example on which we perform the study is a typical passenger side airbag. The geometric de- tails have been measured from a commercially avail- able airbag. The initial state of the airbag is a closed rectangular whose sides are to be finished to 482 × 635 mm2 and is shown in Fig. 4.

Table 3 Material properties of airbag and rigid plate used in FE

–

Table 4 Initial values used for FE simulation of the swelling of

3.33 × 10−4

Fig. 4 The initial airbag geometry in the form of a rectangular Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

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