Recent Developments of Electrical Drives - Part 10 ppsx

At the same time the difference between the energy value calculated from stator mesh and the energy value calculated from rotor mesh was computed Fig.. A compound drive simulator is pres

Figure 3 Generated meshes for the shit larger than one third of the air gap width.

Taking into account the stator node with number j neighboring with nodes i and i+ 1

of the rotor mesh, f the linear interpolation the equation for this node can be described as

follows (5):

V J θ i − Vi(γ i)− Vi+1(θ i − γi)= 0 (5) Whereθ i is the angle between nodes with the numbers i and i + 1, and γi is the angle

between the stator mesh node j and the rotor mesh node i

j+n+1

i+k+2

j+n

j i

i+k

j+n-i i+k-1 j+n-2

j-1 j-2

i-1 i-2 i+2 i+1

Figure 4 Part of the one level (z= const.) of the mesh with the overlapping region (parts of electrodes dashed, dimensions enlarge)—symmetrical air gap

i+k+2

j+n

j

i

i+k j+n-i

i+k-1 j+n-2

j-1 j-2

i-1 i-2 i+2 i+1

Figure 5 Part of the one level (z= const.) of the mesh with the overlapping region (parts of electrodes dashed, dimensions enlarge)—nonsymmetrical air gap

In the nonsymmetrical model nodes of the stator and the rotor meshes for overlapping bounds have different angleθ and radius r (Fig 5) The stator mesh node with number j neighbors with four nodes i , i + 1, i + k, and i + k + 1 of the rotor mesh.

The equation describing the value of the potential in the j node can be written down

using bilinear interpolation function in the following form (6):

V j( γ i , r i)= a0γ i r i + a1γ i + a2r i + a3 (6) Using equation (6) for each node at both boundaries (outer for the rotor mesh and inner

for the stator mesh) one obtains sub matrix of main matrix [M] containing five nonzero

elements for each row As a result one gets nonsymmetrical system of linear equations, which is solved using LDU decomposition method with permutation matrix (7)

Presented algorithm was implemented in a numerical program, which allows determining

a field distribution for every angular position of the rotor and every possible movement of its rotation axis

Integral parameters

The application allows calculating integral parameters for every position of the rotor—in particular the system energy that can be written down in general in form (8):



 w d  =





 E

0

D dE



(9) Using explicit choose shape functionsλ i in formula (8), one can calculate the total system co-energy as the sum of the energy accumulated in each of the mesh elements (10)

e



 e



ε



 e

grad2V i λ i d  e



(10)

Proceeding in the same way with the general expression (9) leads to formulas describing

force components in relation to surface S, which consists of the sum of elementary surfaces

S i in the single mesh elements As the result one obtains the normal force component in form (11):

F n = ε0

2



i



S i

(grad n2V i λ i − grad2

t V i λ i ) d S i (11)

By analogy the tangent force components can be written down as follows (12):

Ft =

i



S i

ε0(grad n V i λ i · gradt V i λ i)

Numerical verification

The basis of the verification of the presented model was the numerical experiment Air gap energy was calculated in the part common for both the rotor and the stator meshes and obtained results were compared The quality of energy calculation was determined on the basis of numerical testing of the convergence of the solutions from both meshes (Figs 6 and 8) The influence of mesh density on the value of energy accumulated in the air gap was analyzed for different positions of rotor (rotation and shift) It allows determining minimal mesh density for given accuracy of computations A clear tendency of both curves to reach the same value was observed It means a convergence of energy value and exact value The convergence was observed irrespective of the rotor’s location However, the slope of the curve changes, which results from different energy values for different locations of the rotor

At the same time the difference between the energy value calculated from stator mesh and the energy value calculated from rotor mesh was computed (Fig 7) Convergence to zero of the above difference was observed Like before the tendency appears irrespective

of the rotor’s position

Convergence of solutions determined on the basis of the values of potentials in the nodes of both rotor and stator meshes confirm the thesis that the implemented model is correct

Figure 6 Influence of the mesh density on the value of air gap energy for rotor position 0◦.

Calculating the changes of energy value for different rotor angular position (Fig 9) allows determining static torque as follows (13)

As a matter of fact, the approach based on Maxwell’s tensor is used (11, 12), whereas the above formula (13) is only a method of confirming the correctness of the results

Figure 7 Influence of the mesh density on the value of air gap energy for rotor position 30◦

Figure 8 Ratio of air gap energy calculated from stator and rotor meshes.

Figure 9 Dependence of air gap energy on the rotation angle of the rotor (about 25,000 mesh

elements)

Conclusions

Another step in developing the model will be extending it to the analysis of microacutators with leant rotation axis It will require interpolation by three variable function and not one variable function (symmetrical model) or two variable function (model with shifted rotation axis) as so far

The most important conclusion resulting from the carried out studies on the three-dimensional model for the analysis of the electrostatic micromotors is that it allows effective analysis for any position of the rotor—both rotation and rotation axis shift

Presented algorithm allows correct and exact representation of the changing width air gap in the model Since significant part of the main matrix rows is calculated only once and it’s only recalculated fragment is the one representing the air gap, it is possible to reduce computation time

The results of the numerical tests confirm the thesis about the correctness of the model Short computation times are obtained even with quite big number of mesh elements

References

coilguns, IEEE Trans Magn., Vol 31, No 3, pp 2052–2055, 1995

microfabricated variable capacitance motors, Sens Actuators, A21–A23, pp 173–179, 1990

Naukowe, Z 111 2002

movement, IEEE Trans Magn., Vol 31, No 3, pp 1920–1923, 1995

Vol 28, No 5, pp 2247–2249, 1992

S Kanerva1, C Stulz2, B Gerhard3, H Burzanowska2, J J¨arvinen3

HUT, Finland

sami.kanerva@hut.fi, slavomir.seman@hut.fi

christian.stulz@ch.abb.com, halina.burzanowska@ch.abb.com

bilal.gerhard@ch.abb.com, jukka.jarvinen@fi.abb.com

Abstract A compound drive simulator is presented, where a finite element method (FEM) model

of the electric motor is coupled with a frequency converter model and a closed-loop control system The method is implemented for SIMULINK and applied on a 2-MW asynchronous machine drive The results are validated by measurements and the performance is compared with an analytical motor model It is shown that simulation with the FEM model provides very good results and gives much better insight in the motor behavior than the analytical model

Introduction

As the demands for performance of electric drive systems increase, also the simulation software must follow the requirements Designers of frequency converters and electric motors rarely work in the same location, but they must be able to model both parts of the drive as accurately as possible Naturally, different expertise is required to model electrical machines or power electronics, but the key issue is to couple these models together in a way that experts in both fields can profit from each other by using the most advanced simulation models in their design

Accurate modeling of digital control systems requires simulation in multiple timescales, because different sampling times are used for measurement, filtering, estimation, and mod-ulation By including all detailed functions and sample times, it is possible to create very accurate simulation models of the converter control In such a case, however, also a detailed electrical machine model is needed in order to get the maximum advantage of the drive model

Finite element method (FEM) is a widely known method to model electrical machines with high accuracy For standard-type machines, two-dimensional field solution coupled

S Wiak, M Dems, K Kom˛eza (eds.), Recent Developments of Electrical Drives, 83–92.

2006 Springer.

with simple circuit equations of the windings is usually accurate enough, when the cross-section geometry and material properties are known [1]

The most problematic in the drive simulation is to couple the FEM computation with the converter model Most obvious method would be to couple the converter model in the FEM code and solve all the equations simultaneously with uniform time steps [2,3] However, such an approach is hardly applicable to a detailed converter model with digital closed-loop control because of the amount of programming, and the demand for common time step length would make the simulation too heavy with respect to existing computing facilities Reference [4] presents an indirect method for coupling time-stepping FEM simulation with SIMULINK using multiple sample times for different parts of the system model The method was applied to a cage induction motor and a frequency converter with direct torque control (DTC) The model of the control system was developed in order to investigate control-related topics and verified for steady-state and transient operation of the drive In its original state, it was using a motor model that was based on analytical equations

In this paper, the same method is applied to an asynchronous motor drive with DTC The frequency converter model is based on a real application, comprising a detailed model of the digital control system The frequency converter model is implemented in SIMULINK and

it is coupled with a two-dimensional FEM model of the asynchronous motor The system

is simulated in steady-state and transient operation, and the simulation results are validated

by a comparison with the measured results

Compound model of inverter-fed electrical drive

The general structure of the compound drive simulator is shown in Fig 1 The model

is implemented in SIMULINK but the execution of the model is controlled from within

Script file:

runA6ka.m

Input data

- Environment

- Model

- Operating

- Starting

conditions

Calculation of

initial conditions

Setup of

SIMULINK

Run simulation

Output results

(Plots, )

Motor

Inverter

Plot files

DC circuit

3-Phase 3-Level

Motor

Process

Measurements Inverter control

Model

Overall

DC voltage

Phase voltages

Torque Speed

Phase currents

Half DC voltages

DC currents

Speed reference

Torque reference Flux reference

Inverter control

Measurements

Flux reference

Speed control Torque reference

Load model

Speed reference

conditions

Figure 1 General structure of the drive simulator.

to motor

Figure 2 Simulation model of the three-phase three-level inverter.

MATLAB This allows to specify the plant parameters, operating and starting conditions very easily Based on the selected operating conditions, the initial conditions for continuous and discrete time states are determined This allows to start the simulation in a reasonably stable operating point The machine model in this drive simulator can be selected to be the simple analytical or the precise FEM-based model

The main components of the simulated plant are DC circuit, inverter, motor, process, and control Two basic control schemes can be selected: torque or speed control

Inverter and DC link The three-level inverter is modeled as a set of ideal switches, which can connect the phase voltages to either plus, neutral, or minus potential of the DC links Fig 2 gives a rough overview The switching pattern is given by the drive control The status of the switches together with the phase currents determines the currents in the DC bus bars of the DC link The current in the neutral bus bar is used to calculate the potential of the neutral point of the DC link The phase voltages transferred to the motor terminals are defined by DC link voltages and switching pattern

Analytical motor model and load The analytical motor model is used for simulations that will be compared to the FEM-based motor model It is based on the well-known space vector representation of the asynchronous machine It uses both the stator and the rotor fluxes as state variables The following features are present in the model:

rconstant air-gap and sinusoidal flux distribution along the air-gap

rno iron losses

rresistances and inductances are independent of frequency and temperature

rthe magnetizing inductance can saturate with increasing main flux

The model needs phase voltages and speed as inputs and produces phase currents and air-gap torque as outputs

The driven process is described by the differential equation of motion A single inertia is used The load torque may follow several functions of the speed (constant, linear, quadratic,

or mixed) The mechanical mass is driven by the electromagnetic torque of the motor and gives the speed as output

v3_vs v3_s

A6ka

VECTOR v3_s

Vector -> Switching

me

t_load n

Mechanical System

n_rot t_load Load Model

Inverter

Induction Machine Model (analytical

or FEM)

Current Meas.

Speed Meas.

v3_s

v3_is

vdc1_t2

vdc2_t2

n_rot VECTOR

Control_dtc6000_AD

vdc_id

Voltage Meas.

Figure 3 Model of the drive system implemented in SIMULINK.

Control The control model describes speed/torque control using a DTC algorithm The main func-tions of the ACS6000 drive are implemented as discrete funcfunc-tions on different time levels to appropriately represent the behavior of the real drive The detailed description of the DTC control cannot be in the scope of this paper

The top level of the SIMULINK environment is shown in Fig 3

Model of the asynchronous motor

Modeling by finite element method (FEM) The FEM model of the motor is based on two-dimensional finite element method and circuit equations of the windings [1] The magnetic field in the core region is calculated using magnetic vector potential formulation, in which the vector potential and current density

have only z-axis components.

The phase windings in the stator or rotor are modeled as filamentary conductors with uniformly distributed current flowing through all the coils that belong to the same phase The rotor bars are modeled as solid conductors, in which the current density varies according

to eddy currents The sources of the magnetic field are the phase currents, the voltage drop

in the rotor bars and the magnetic force of the permanent magnets, depending on the type and construction of the machine

The relations between voltage and current are determined in the circuit equations of the stator and rotor windings, which also include the end-winding impedances and the short-circuit rings As a result, only phase voltages are needed as an electrical input for the FEM model The electromagnetic torque is calculated by virtual work principle, and the movement of the rotor is determined from the equation of motion At each time step, new position is calculated for the rotor and the air-gap mesh is refined

FEM block for SIMULINK The FEM computation is implemented as a functional block in SIMULINK using dy-namically linked program code (S-function), as illustrated in Fig 4 The stator voltage