RO131666A2 - Method for estimating parameters of brushless dc motors - Google Patents

Abstract

The invention relates to a method for estimating the parameters of brushless DC motors. The claimed method consists in estimating the parameters as an optimization problem solved by means of a PSO (Particle Swarm Optimization) algorithm, which is carried out as follows: a supply voltage Uis applied and gradually modified and the system statuses xi=[iiiω Θ] are measured in the sampling moments q*T, where i, i, iare the intensities of phase currents, ω is the rotor angular speed, Θis the rotor electric angle, Tis the sampling period and q is the number of samples; the initialization of a population of particles with aleatory positions within the searching space, the maximum iteration number Kof the algorithm and a maximum value of the objective function, noted with V; the objective function is evaluated for each particle, then the value of the objective function is compared with the value corresponding to the best position of a previous particle, followed by replacing it or not with the current position, as the case may be; the best position particle within the group is determined, and its position is alloted to the variable, afterwards modifying the position and speed of each particle according to pre-established relations, then passing to the next iterration or, if a stop condition is fulfilled, the algorithm is stopped.

Description

The invention relates to a method for estimating the parameters of DC motors without brushes. The method according to the invention consists of estimating the parameters in the form of an optimization problem, solved by means of a PSO (Party Swarm Optimization) algorithm which runs as follows: a supply voltage is applied which is modified in steps, and the system states are measured ξ = [ i _a i _B i _c ω Θ] at the sampling moments q * T _s , where i _a , ί _Β , and _c represent the intensities of the phase currents, ω is the angular velocity of the rotor, θθ is the electric angle of the rotor, T _s is the sampling period, and q is the number of samples; initialize a population of particles with random positions in the search space, the maximum number of iterations of the algorithm and a maximum value of the objective function, denoted by V _max ; for each particle the objective function is evaluated, then the value of the objective function is compared with the value corresponding to the best position of a previous particle and, if appropriate, it is replaced by the present position; determine the particle in the group with the best position, and assign its position to the variable; change the position and velocity of each particle, according to predetermined relations, then move to the next iteration, or, if a stop condition is met, the algorithm stops.

A; 1

Figures: 3

Starting from the date of publication of the patent application, the application provides, provisionally, to the applicant, the protection granted according to the provisions of art: 32 of Law no: 64/1991, except in cases where the patent application has been rejected, withdrawn or considered as withdrawn, the extent of the protection conferred by the patent application is determined by the claims contained in the application published in accordance with article 23 paragraphs) - (3).

IUI

III

METHOD OF ESTIMATING THE PARAMETERS OF THE ENGINE PARTS

C.C. WITHOUT PERU oncfoi. STATE OF INVENTION MARK Patent application

Nr. .......

Date of deposit · ,, | 3.ΗΒ8.2015 · »; _

The numerical methods of heuristic optimization have been particularly successful lately due to the very good results obtained in solving difficult optimization problems, as well as due to the simplicity of their implementation. The emergence of heuristic optimization methods made it possible to solve some global optimization problems that were considered difficult or impossible to solve. The development of heuristic optimization methods was favored by the increased computing power of microcontrollers and microprocessors as well as the fact that they are easy to implement.

Of the most used heuristic methods used in optimizations, we mention; Simulation of metal shafts (sirnulaled anneuling). TABU search (Tabu search), GRASP (Greedy Random Adaptive Search Procedure) method. Evolutionary calculus (genetic algorithms, evolution strategies, genetic programming ACO (Ant Colotiy Opiimization). WBM (Wasp Behavioral Model) etc. These techniques are based on concepts from biological evolution theory, intelligent problem solving, mathematical and physical, statistical mechanics. In general, these methods are difficult to implement in practice, and the major disadvantage of these methods is the large computing resources (memory and time) required.

The heuristic optimization technique used in this invention is of the Swarm Opiimization (PSO) type. PSO is a heuristic optimization algorithm of "population type" first introduced by Kennedy and Eberhart 11] and known as the intelligence technique of the cluster. The algorithm is based on the resemblance to the behavior of certain species such as bird flocks, fish banks or insect swarms. In this algorithm the population is called swartnl an element of the group is called 'parties' (particle) and the trajectory of each particle in the search space is controlled by a term called 'vetodty' (particle velocity), which is determined by their own experience, such as and from the experience of the other members of the group.

The invention relates to the formulation of the problem of estimation (identification) of the parameters of an e.c. without brushes as an optimization problem, which is then solved using a PSO lip algorithm. This method eliminates the disadvantage of large computing resources and time, solving the optimization problem much faster.

1. The mathematical model of the c.c. brushless used in the identification algorithm χίυ- 2015-- 0D3700 2 -06- 2015

The DC motor runs the brushes, referred to in the English literature as the BLDC (BrushLess Direct Current) motor ”, is an electric motor of the synchronous type, again brushes, usually three-phase, having the rotor built with permanent magnets and the stator windings. concentrated or evenly distributed. Specific is the electronic switching, made by switching the semiconductor elements in the structure of the three-phase power supply inverter, which is synchronous with the position of the rotor. The position of the rotor must be known at certain angles in order to align the applied voltage to the electromotive voltage. Usually, to obtain information on the position of the rotor, three Hali effect transducers are encapsulated in the stator.

The mathematical modeling of the drive system can be made based on the equivalent principle diagram of Fig. 1.1.

in the phase voltage equations, respectively [1], [2];

dt dt dl, dt 'dt of ^Kl dt

di - !, J- fi Mon ri (1.1) di dt (1.2) _r di —-ae 'rih ^{c 7} (1.3)

t Z.

where:

ui ,, u, ~ instantaneous values of phase voltages;

e _a , ei, _t e, ~ - instantaneous values of phase electromotive voltages:

i _a . ib, ic - instantaneous values of phase currents:

R _s , Li - stator phase resistance and inductance, assumed the same on all phases:

L _m - mutual inductance.

Given the star connection of the stator windings, the condition fi-, ⁺ fi + fi = θ (1 -4) allows the rewriting of equations (1) - (3) in the form;

« _A = ^ îfi + (ri ~ rii} —f ~ ^{+ e} « '(î -5) dt' 2 Ο ί 5- - 0 0 3 7 0 0 2 -06- 2015

Mo

u _h = RL + _(/.%-Fi ,,, + C: di (1.6) u = Rj + ('fi.-fij - ^ ..... t-e,. (1.7) Phase electromotive voltages depend on the position of the rotor and build a system symmetrical three-phase, I waveform imposed, usually trapezoidal. So they can be expressed as [2], [3]: e _a '= ίύ · K _r · f (fi,), fi.8) and „= io-K _s (1.9) e _r = ιο · _Κ: · / (θ ,. - · 4π / 3), (1.10)

where Θ is the electric angle of the rotor, ω is the angular velocity of the rotor. K _t . is the constant of the electromotive voltage (V / (rad / sec)). and the reference function / (0, -,) for the electromotive voltages is alternately trapezoidal, with amplitude 1 (Fig. 1.2),

The phase currents are alternatively rectangular, ie the alternations for the duration of Ξπ / 3, centered on those of the electromotive voltages (Fig. 1.2).

Considering the origin of the phase it in Fig. 1.2, the function ((O <.) Defined on intervals has the expression:

and i 1 for G _t , e - 2nd i : () i L 3 J 1 . 2-1 pemru Θ. 2π 1 i 1 ——ΐ θ ----- :, î π) ⁵ 3 3 '“j (0, ) = - L, .a (1.11) \ -1 for 0 _t ¹ 5t β π.— s' 3 J i _ 6 (5π ( for 0 ,. 5n Ί i π ( θ -. 2π ' . 3 'j

The equations of motion are added to the stress equations, m = - ^ - ΐ-, β-ω + ζη, (1.12), where m and m _s are the electromagnetic torque developed by the motor, respectively the static torque ,, / is the moment of total inertia (motor plus load), and B is the coefficient of friction.

Λ- 'Π 15 - (| ο 3 ₇ υ. Β * \

2 -Q6-2O15

The electromagnetic torque developed by the motor is the sum of the torques corresponding to the phases and depends on the phase currents. angular velocity and electromotive voltages and has the expression [1 j:

Κ · Ι _ν / '(θ f ί θ,.

- / 3) + ν / (θ ........ 4- / 3) ”(ΐ.13)

The torque is mainly influenced by the waveforms of the electromotor voltages induced in the stator due to the rotor motion, ideally, the electromotor voltages of both trapezoidal waveforms and the stator currents are rectangular (Fig. 1.2), resulting in a constant torque. In practice, there are torque pulses due to the design imperfections that lead to the removal of the electromotive voltages from the perfectly trapezoidal shape. current ripples resulting from PWM control or through hysteresis and switching.

Taking into account also the connection relation between the electric angle 6 _f . and the mechanical angle by the number of pairs of poles p.

6 .. = / + 0 ,,, - (, i-î4) and by the fact that ω = (Fio) di 'the operating equations (5) - (7), (12) and fl5) can be written under the form of the state equations, respectively:

dt _a _ A Κ ./ 'ίθ,) 1 __ ______ u' du R. K, f (e, -2-'3)) —- = _.-L, --—--- _{ω +} --- _u · dt L - l '- IL l di, _ R. KJ (Q, .- 4π / 3)) of ”” ω + _lt _ (1.16) (1.17) (1.18) c / ω _ ÂȚ / (e ,.) , _; Κ, / (θ ~ 2π / 3), dt J ^K * J '' / c J (6,, - 4π / 3), β i - o —_ / -: —-_ i ------- - _(!) - _m 'J .1' (1.19) ί / θ _? dt .20) ^ '2 0 1 5 - 0 0 3 7 0 0 2 -06- 2015

In the matrix form, the equation of state is;

£ = Α · „- B u;

wherein the vectors of the state (ς) and input (u) variables are:

ς = [(, /. ș ω (.

u = k <a, u. m

• .22) (1.23) and the matrices A and B have the form:

Γ R

/. ~ L v r:

O o

-IT"

A

L ~ L

KJ (Q _e ) ~ 2π / 3) KJ (Q, .- 4ni3)

L - L "

Κ, / (θ "-2π.'3)

Α2 / (θ, -4π'3)

Β where

J ii (J o

I) o <

o (i.25) / - ^L , o

We consider the DC motor. described by a set of differential equations «-dimensional;

<(Q of (2.1)

- ς e R ⁿ is the state vector of the system;

-re R ™ is the vector of unknown parameters: ^τ ~ ~

- f is a given function.

s

^ - 2 0 1 5 - 0 0 3 7 0 0 2 -06- 2015

The following system is defined for estimating unknown parameters:

of (2.2) where

- ξ e R ⁿ is the estimated state vector:

- faith 7? ^m is the vector of the estimated parameters.

The determination of the unknown parameters is done by minimizing an objective function obtained defined as the average of the squared errors between the measured values and the estimated ones of the state variables for a number of N samples (measurements):

A '· P; ΣΣΑ-ζ) ::, 3) where /' is the number of measured state variables. (V is the number of samples (measurements), and ξ ' ^! . And ξ are the measured values and. Respectively, estimated for state j at time point q * Ts (q number of sample, Ts sampling period).

This optimization problem is solved using the PSO algorithm described below. Considering the Af-dimensional search space (M · - number of unknown parameters), position and velocity of particle i are defined by the vectors ^ / - dimensional x, '= (xy, x, 2 ..... χ; χ) and respectively v, - = (v, 7, v, z ..... Vî.wJ. Each particle is potentially a solution to the optimization problem. The best position of a particle (that value of the position for which the objective function has the smallest value) is represented as pi = (p> hp, j ...... p ^), and will henceforth be referred to as pbest, and the best position of a particle in the whole group will be represented by _s = {p _g i, p _g 2,., .. p _4M ), called ghest.

At each step of the algorithm the position and velocity of each particle is modified according to the formula:

v * ' ¹ -Λ · [ν (+ c, r, (pbesp -xț) + c ₂ r, (gbest) -x *)] where h represents a constraint factor defined as follows:

- ^ - 2 0 1 5 - 0 0 3 7 0 0 2 -06- 2015

II (2.5)

- a ..... \ ί a -...... 4a a = c> * c 7> 4 (2.6) where k represents the number of the iteration, C'i and o are two coefficients that control the relative velocity of the particle. to the best local and global positions (called acceleration coefficients), and r \ and!> are two random numbers in the range [0, 1 j.

For the acceleration coefficients we propose a variable (time) evolution of them, as follows:

<'l - (%; ........ ^c

(2.7), \ ^{, k} max ·, c ₂ - (c _{2 /} -c _2i -) · - ----------- re2 / ^Λ τη: v where Ăv ,, _aA is the maximum number of iterations of the algorithm, C \ -, and c-χ are the initial values of the coefficients C; and cp; and c ^ r are the final values of the coefficients m and, respectively, c _d . For these coefficients we propose the values: Ci, = 2.5; (¾ = 0.5; c /: f = 0.5 and C2f = 2.5.

The algorithm for implementing the invention on a computer system is given below. The algorithm can be implemented and run in different programming environments, for example Matiab version 2014a can be used. which also has powerful libraries for optimization techniques.

The estimation algorithm using PSO.

Step 1. Apply a supply voltage LJ that is modified in steps and measure the system states (2 = [ζ i _fl i _v 6) #,]) at the sampled times q * Ts.

The voltage U _d (Fig. 1.1) is changed in steps of 20% of the maximum value corresponding to the tested engine, the intensities of the current currents 1. ,, /., 1. sc is measured with current transformers and a data acquisition system. The rotational speed is measured by a tachogenerator and a data acquisition system. The angle (U, is calculated on the basis of the rotational speed measurements using the relations ith.14). (1.15) or (1.20). 7's this

2015- - 00370 'ϊ '06 - 2015 sampling period, ten times less than the electric time constants of the engine tested, and q is a positive integer representing the number of samples and taking values from 0 to a maximum value / V. Example: if the identification experiment is conducted over a time interval of 100 seconds and Ά = 0.01 seconds, then q takes values between 0 and N ~ 10000.

Step 2. Initialize the population of particles with random positions in the search space (size 5), the maximum number of iterations k _mux of the algorithm and a maximum value of the objective function (2.3). rated V _max .

Example: Initialize a number of 50 particles (denoted pl. P2 ..... p50) with random values in the range [0, 10000] (an example of a particle is the vector pl = [3; 50; ll; 200; i000j), & _maA = 100, υ, _{Ιί) Α} = ϋ.Ι. Considering that 11) 000 samples were measured, the objective function for the pi particle is calculated with the relation:

V.

• J sOoOi'i _: __ V V {In i - ´iy

50000 (2.8) where i is the current _measured at the sampling time T $ Q * and _b is the current measured at the time of sampling tf'Ts ^ξ! : is the current i _c measured at the sampling time q * Ts cf is the rotor speed of the rotor ω measured at the sampling time (fTs df is the electric angle of the motor θ _ί: measured at the sampling time q '^ Ts ζ ^!! the same sizes presented above, but estimated by numerical integration of the mathematical model (1.16) - (1.20) with the model parameters given by the particle pl.

Step 3. For each particle, evaluate the objective function V _PJ , V _P \ .... using the relation (2.8). We will give to the variable V the smallest value of V „ _h V ,, \ ··, V _pSi >.

Step 4. For each particle, compare the value of the objective function with the value corresponding to the previous asbestos position. If the current value is better, replace phest with the current position.

JL -2 o 1 5 - 0 0 3 7 0 '

2-06-2015

Step 5. Determine the particle in the group with the best position and assign its position to the gem variable. A particle p _m has a better position than the particle p = {m * j) if

1 'pm * -1 ft ·

Step 6. Change the position and velocity of each particle according to the relations (2.4) - (2, /).

Step 7. If a stop condition is satisfied (V <V _{w. T. The} or k> k _MTA and go to step 8 Work.

otherwise go to Step 3.

Step 8. If (V <V _m f then the parameter values are equal to the ghest values.

STOP otherwise the result is not validated and the algorithm is resumed from Step 2.

The logic scheme of the algorithm is shown in FIG. 2.1.

Bibliography:

Jl] Kennedy, J. and Eberhart, R. C. Party swarm oplimization. Proceedings of IEEE International Conference on Neural Networks. Piscataway, NJ. pp. 1942-1948. 1995.

[2] V. Fedâk, T. Balogh and P. Zâskalicky, Dynamic Simulation of Electric Machinery and Drive Systems Using MATLAB GUI- Λ Fundamental Tool for Scientific Compuiing and Engineering Applications - Volume 1, Praf. Vasilios Katsikis (Ed.), ISBN: 978-953-510750-7. InTech, 2012.

[3] G. Prasad, N. Sree Ramya, P.V.N. Prasad, and G, Tulasi Ram Das, Modeling and Simulation Analysis of the Brushless DC Motor by using MATLAB. International Journal of Innovative Technology and Exploring Engineering, Vol. 1, Issuc 5. Oct. 2012.

A - 2 O 15 - - 0 0 3 7 0

O 2 -06- 20) 5

Claims (2)

Method of estimating the parameters of the DC motors. brush country characterized by using the numerical method of heuristic optimization Swarni Optimization Parties (PSO). % ν - ^ - 2 0 1 5 - 0 0 3 7 0 8 2-00-7815 Fig. I. de Schematic principle of the actuation system in the continuous field molar brush country A- - 2 0 1 5 - 0 0 3 7 0 0 2 -06- 2015 F / g 1.2 The electromotor voltages, the phase irons of irradiators and irradiators of liatl position JL '20 1 5-- 00370O 2 -06- 2015 Measure state variables £ = [ζ i _h i, 6) $.] Initialization of particle positions, V _mM and k. k = l Objective function rating ¹ for each particle replaces asbestos with current knowledge replaces asbestos with best asbestos Calculates the velocity for each particle Calculates the new position for each particle Fig. 2.1 Logical scheme for identifying the engine parameters