# Modeling magnetic saturation and saliency effects via Euler-Lagrange by brm24619

VIEWS: 0 PAGES: 9

• pg 1
```									    Modeling magnetic saturation and
saliency effects via Euler-Lagrange
models with complex currents for
three-phase permanent magnet machines

Pierre Rouchon, in collaboration with
Duro Basic, François Malrait from Schneider Electric, STIE.

Mines ParisTech
Centre Automatique et Systèmes
Mathématiques et Systèmes
pierre.rouchon@mines-paristech.fr
http://cas.ensmp.fr/~rouchon/index.html

IFAC Workshop E-COSM’09
Engine and Powertrain Control, Simulation and Modeling
IFP, Rueil-Malmaison, France
30 November – 2 December 2009
Outline

Motivations

Usual (α, β) model with saliency

The (α, β) model with magnetic saturation

A ﬁrst experimental validation

Concluding remarks
Modeling and control of electrical machines in future cars

Speciﬁc machines with unusual geometry (e.g. wheel-mounted
electrical machine), operating close to magnetic saturation,
exploring large speed domains.
Control of the electro-magnetic torque:
precision, bandwidth and robustness
minimizing energy losses
deceleration versus braking, coupling with ABS.
ESP and more generally advanced traction control system
by monitoring each wheel separately.

Need for dynamic modelling compatible with real-time applications
but including effects that are not usually taken into account in
standard (α, β) models.
Goal: how to base such modeling on variational principles underlying
Maxwell/Lorentz equations with magnetic energies directly expressed
with usual complex currents, ﬂuxes and voltage variables.
Focus: permanent magnet machines with saliency and magnetic
saturation.
Permanent-magnet machines: usual (α, β) models
In the (α, β) frame the dynamic equations read1 :

 d      d                                               ∗
      J θ = np          λıs + φenp θ − µı∗ e2np θ
¯        s                 ıs − τL
dt     dt

 d
λıs + φenp θ − µı∗ e2np θ = us − Rs ıs .
¯

s

dt
where
∗ (resp. ) stands for complex-conjugation (resp. imaginary
√
part),  = −1 and np is the number of pairs of poles.
θ is the rotor mechanical angle, J and τL are the inertia and
ıs = ısα + ısβ (resp us = usα + usβ ) is the stator current
(resp. voltage): complex quantities.
L +L       L −L
λ = d 2 q , µ = q 2 d with inductances Ld , Lq > 0.
¯                         ¯
The stator ﬂux is φs = λıs + φenp θ − µı∗ e2np θ with φ > 0
s
the constant ﬂux due to permanent magnets.
1
See, e.g., J. Chiasson: Modeling and High Performance Control of
Electric Machines, Wiley-IEEE Press, 2005.
Hamiltonian dynamical models with complex variables
The magnetic energy of the usual (α, β) model reads:
1                         2        1                    2
Hm (φs , φ∗ , θ) =
s                 (φs e−np θ ) − φ
¯       +         (φs e−np θ )
2Ld                                2Lq
where φs is the stator ﬂux. Magnetic saturations just mean that
Ld and Lq are not constant but depend also on φs .
Any three-phase permanent magnet machine with saliency and
magnetic saturation can be described by a dynamic model with
the following structure:2
d        d           ∂Hm           d                                  ∂Hm
J      θ   =−       − τL ,       φs = us − Rs ıs ,      ıs = 2
dt       dt           ∂θ           dt                                 ∂φ∗
s

where the magnetic energy Hm is considered as a function of
the rotor angle θ, the stator ﬂux φs and φ∗ .
s
When inductances Ld and Lq are constant, we recover the
usual (α, β) model (with saliency when Ld = Lq ).
2
D. Basic, F. Malrait, P. Rouchon: Euler-Lagrange models with complex
currents of three-phase electrical machines and observability issues. To
appear in IEEE Transactions on Automatic Control.
Experimental validations based on high frequency voltage injection
Rotor blocked at θ = 0 and high frequency voltage injection:

r    a           d        r    a               ∂Hm
us (t) = us + us f (Ωt),      φs = us + us f (Ωt) − Rs 2
dt                            ∂φ∗
s
ıs

r      a
where the complex quantities us and us are constant, the
function γ → f (γ) is 2π periodic with a zero mean .
a
For Ω large enough(typically ∼ 1kHz), φs = φr + us F (Ωt) where
s  Ω
F is the primitive of f , dF = f with a zero mean,
dγ
2π
0  F (γ) dγ = 0.
Then, ıs = ır + δıs where current ripples δıs result from ﬂux
s
a
ripples δφs = us F (Ωt):
Ω

r
us    ∂Hm r                               ∂ 2 Hm        ∂ 2 Hm
ır =
s        =2     (φ , (φr )∗ , 0),    δıs = 2            δφs +2 2 ∗ δφ∗ .
Rs    ∂φ∗ s s
s                                ∂φs ∂φ∗s       ∂ φs s
Magnet saturation: current ripples depend on current offsets

When Ld and Lq depend on φs , the magnetic energy

1                        2        1                          2
Hm (φs , φ∗ , θ) =
s                (φs e−np θ ) − φ
¯       +            (φs e−np θ )
2Ld                               2Lq

is no longer quadratic in (φs , φ∗ ).
s
a    r
Voltages us = us + us F (Ωt) generate current ripples δıs
∗
∂ 2 Hm        a
us F (Ωt)      ∂ 2 Hm            a
us F (Ωt)
δıs = 2                            +2 2 ∗                             .
∂φs ∂φ∗s          Ω          ∂ φs                 Ω

depending on ﬂux level φr given by ır = 2 ∂H∗ (φr , (φr )∗ , 0).
s           s     ∂φ
m
s     ss
u                   r
Thus amplitudes of δıs are related to current offsets ır = Rs .
s     s
Magnetic saturation implies that Ld , Lq are decreasing versus
2            ∂ 2 Hm
|φs |. Thus ∂φsHm∗ and
∂
∂φs          ∂ 2 φ∗
are increasing and amplitudes of
s
δıs increase too.
Experimental data on a 1.2 kW permanent magnet machine

The blue curve corresponds to the amplitudes of the
current-ripples (obtained via a simple PLL-ﬁlter), the red curve
to the current measures ıs (d-axis aligned with the rotor θ = 0).
Increasing dependance of the ripples versus the current offset
ır : inductances are decreasing functions of |φs |.
s
Concluding remarks

The saturation models for permanent-magnet machine are
based on variational principles. Experimental data provide
a ﬁrst validation of such modelling procedures that
preserve the physical insight while maintaining a synthetic
view without describing all the technological and material
details.
More complete validations could be done: at non zero rotor
velocity, the proposed computations of the current ripples
are still possible since they are obtained via usual
perturbations/averaging techniques. Such saturation
models can be used for control purposes: adaptation of
usual control schemes to take into account saturation
effects are under study.
Modeling of interconnected machines and generators can
also be developed with similar variational principles relying
on complex currents, ﬂuxes and voltages.

```
To top