Parameter Optimal Identification of Dual Three Phase

Parameter Optimal Identification of Dual Three Phase Stator Winding Induction Machine Lucian Nicolae Tutelea1, Ion Boldea1, Sorin Ioan Deaconu2 1. Politehnica University of Timisoara, Electrical Engineering Department, Romania 2. Politehnica University of Timisoara, Electrical Engineering and Industrial Informatics Department, Romania [email protected], [email protected], [email protected] Abstract- The parameter identification of a dual three phase’s stator winding induction machine is approached in this paper by the genetic optimal algorithm. The estimated parameter are: the voltage ratio between main and auxiliary winding, the main winding resistance and leakage reactance, the cage rotor resistance and reactance, the coupling leakage reactance between main and auxiliary winding, the magnetization non saturated reactance, equivalent iron loss resistance and mechanical losses including their variation with speed. The parameter are calculated from standard no load and short circuit test performed on both stator winding, by minimization the sum of squared errors between measured and computed currents, active power and reactive power in several points. The proposed method reduces the measurement error influence on the estimated parameters and for leakage inductances that slightly depend on the current could be also considered the best constant values approximation.

I.

INTRODUCTION

The dual stator windings induction machine (DSWIM) is frequently present in the literature as an alternative in regenerative energies conversion [1-5]. Standard and advanced field-oriented vector controls of induction motors (IM) require accurate values of the motor parameters which are used in control algorithm. Methods of induction machine parameter identification from load tests [6-8] require generally a power converter and speed sensor and mechanical coupling of two electrical machines. Genetic algorithm is proposed in [9] to identify the induction machine parameter from no load start transients. The start current waveform comparison with the simulated current and also speed comparison are used to fit the machine parameters. The method required to record the speed, currents and voltage waveform during transients. Standstill, rotating machine and time domain parameter identification methods are presented in comparison [10] for multiphase induction machines parameters identification with reasonable good agreement. In this paper a new optimal method to identifies the parameters of the two stator winding induction machines [11] using only industrial measurement equipment (Voltmeter, Ampere-meter, Power –meter) from standard tests (no load and two phase short circuit) is proposed. This method does not need mechanical coupling of the induction machine with a load machine and does not need speed measurement. The method is able to deal with the measurements error.

978-1-4799-5183-3/14/$31.00 ' 2014 IEEE

R1

L1σ L12σ

R’2

L’2σ L’rσ

V1

V '2

Lm

Rm

R 'r s

Fig. 1. Equivalent phase circuit of the DSWIM.

The equivalent circuit of DSWIM, figure 1 contains parameters that will be founded: The no load and short circuit tests are performed on the two windings induction machine in order to identify the circuit parameters as: a) no load test with the main winding (noted with 1) supplied, the auxiliary winding (noted with 2) is open; b) no load test with the auxiliary winding supplied, main winding is open; c) two-phase short circuit test supplying the main winding with the auxiliary winding open; d) two-phase short circuit test supplying the main winding with the auxiliary winding in short circuit; e) two-phase short circuit test supplying the auxiliary with the main winding open; f) two-phase short circuit test supplying the auxiliary winding with the main winding in short circuit. For each of these tests the balance of the active and reactive powers may be considered, from which it results two equations for each test. Additionally, at the open circuit tests the voltages on the unsupplied winding are measured as well, two more equations popping up. Resistances R1 and R2 may be measured in dc. The iron losses and the mechanical losses, ' which are in relation with Rm and Rr are calculated through a s succession of open circuit tests and then more equations, are available. Finally, the equation system is over determined, but all the measurements are affected by errors. The relations between the machine parameters and the measured variables are not linear due to the products and then the equations system is practically impossible to solve classically, even using non linear methods. The paper is organized as follows: Section 2: proposed DSWIM computation of the parameters through optimal

231

methods (the circuit model, the measured errors reduction, the boundaries of the candidate parameters vector, the objective function), Section 3: parameters identification results, Section 4: conclusions. II.

COMPUTATION THE PARAMETERS THROUGH OPTIMAL METHODS

A. The Circuit Model An acceptable solution for the over determined system of equations, knowing that the measurements are affected by errors, is the minimization of the sum of the error squares for the active and reactive power, which equals the computation of the circuit parameters with an optimal problem. In conclusion, for the identification of the circuit parameters optimization algorithms may be used, having the circuit parameters as optimization variables, and having the sum of the error squares and the computed powers ( from the equivalent scheme and the candidate parameters) as the objective function. The domain of search for the parameters may be established considering a simplified circuit model (neglecting the influence of the rotor at the open circuit tests and neglecting Lm and Rm at the short circuit tests) and the maximal influence of the measurement errors. For tests a, b, c and e the system of equations which is solved is:

⎛ R + j ( X sσ + X 12σ ) + Z m ⎛V s ⎞ ⎜ s ⎜⎜ ⎟⎟ = ⎜ Zm ⎝ 0 ⎠ ⎜ ⎝ ⎛I ⎞ ⋅ ⎜⎜ s ⎟⎟, ⎝I r ⎠

⎞ ⎟ ' Rr ⎟⋅ ' + jX r + Z m ⎟ s ⎠ Zm

(1) where S is 1 for a and c and 2 for b and e. The slip is 1 for short circuit test and should be determinate from torque equation for no load test. In the parameter optimal identification approaches, the parameters are assumed to be know (given initial values or randomly generated in the finding interval) and than the stator and rotor currents are computed. Using the optimization algorithm the parameter is changed in order to minimization the sum of square error of currents, active and reactive power. The mechanical loss is also a parameter that should be determinate and consequently it is assumed to be known. More ever the mechanical losses variation with the speed could be determinate if the no load test is performed also at small voltage close to the stability limit. The no load tests starts with the larger voltage that is around the 60% (50%) of the rated voltage in order to consider the constant non saturate link inductance in the equation during to parameter identification. The link inductance variation due to the saturation is determinate separately considering the maximum supplied voltage larger than rated voltage but the saturation curve identification is not subject of this paper. The mechanical losses are considered through the friction torque which has three components: the constant component or

Columbian friction component T0, the viscous friction component, k1Ω, and the fan component, k2Ω2.

Tmec = T0 + k1Ω + k 2 Ω 2 =

2

, ⎛ω ⎞ ω = T0 + k1 1 (1 − s ) + k 2 ⎜⎜ 1 ⎟⎟ (1 − s )2 p ⎝ p ⎠

(2)

where ω1 is the angular frequency, p – number of pair’s poles and s -the rotor slip. The mechanical losses torque is equal with the electromagnetic torque:

Tmec =

Vs2

3p

ω1

R2 s

R Rs + C1 2 + j ( X sσ + X 12σ + C1 X 2σ ) s

2

, (3)

where C1 consider the magnetization current component and it is a complex number, usually close to unity with a small imaginary part that usually is not considered. In order to secure the algorithm accuracy for a large variety of induction machine, considering that the renewable energy application at very low speed implies large number of poles (ratio between leakage inductances and link inductance could be larger that for usual machines) we decide to consider the complex value of C1.

⎛ 1 j C1 = 1 + (R s + j ( X sσ + X 12σ )) ⋅ ⎜⎜ − ⎝ Rm X m

⎞ ⎟. ⎟ ⎠

(4)

The equation (3), considering the torque from equation (2) is polynomial equation of degree 4 which is solved numerically. The following rows will be present how to generate in a simple way the polynomial coefficients. At the first the equation that should be solved is:

T (s ) =

ct s ⇒ T (s )Pd (s ) = ct s ⇒ P(s ) = ct s . Pd (s )

(5)

In the computer code, the polynomial coefficients are stored in a vector and in order to find the equations roots, the ‘P’ vector should be computed. The ‘T’ vector coefficient in the descending degree of ‘s’ is coming directly from (2):

[

]

T = k 2 Ω12 , − k1Ω1 − 2k 2 Ω12 , T0 + k1Ω1 + 2k 2 Ω12 , where

Ω1 =

ω1 p

,

(6) (7)

In order to compute the denominator polynomial coefficient Pd, we introduce the P1, respectively P2 polynomials whose coefficients are linked to the real,

respectively imaginary part of the denominator from (3): (8) P1 = [R s − imag (C1 ) X rσ , real (C1 ) R r ] , P2 = [X sσ + X 12σ + real (C1 ) X rσ , imag (C1 ) R r ] , (9) Pd = P1 ∗ P1 + P2 ∗ P2 , (10) P = T ∗ Pd , (11)

232

where * means vector convolution product. The right side coefficient ct from (5) is:

ct =

3

Ω1

V s2 Rr ,

measurement is considered consistent if the relative error in the measured power satisfies the following inequality: ε m ≤ ε av + σ ε , (23)

(12)

A thrust weighing factor, w, could be computed for each The four degree polynomial equation (5) has four solutions. point based on the average error, standard deviation of the The good solution is only the smallest positive pure real series test and individual error: solution. No real solution of equation (5) means that an ε2 − 2 induction machine with the candidate parameters does not ε av +ε m2 +σ ε2 w = e . (24) have equilibrium point at given voltage. New candidate parameters should by tried. For the tests d and f we have the following equations: ⎞ ⎛ ⎟ ⎜ jX 12σ + Z m Zm ⎛ V 1 ⎞ ⎜ R1 + j ( X 1σ + X 12σ ) + Z m ⎟ ⎛⎜ I 1 ⎞⎟ (13) ⎜ ' ⎟ ' ' ⎟ ⋅ ⎜ I '2 ⎟ ⎜ V jX Z R j X X Z Z = + + + + ⎜ 2⎟ 12σ 2 2σ 12σ m m m ⎟ ⎜ ' ⎟ ⎜ 0 ⎟ ⎜ R 'r I ' ⎝ ⎠ ⎜ Zm Zm + jX 2σ + Z m ⎟⎟ ⎝ r ⎠ ⎜ s ⎠ ⎝

(

V − for test d , V 1 = ⎧⎨ s ⎩0 − for test f 0 − for test d V '2 = ⎧⎨V − for test f , ⎩ s

)

(14) (15)

The candidate parameter vector is: X=[ke R1 R2 Rr Rm Xσ X2σ X12σ Xrσ Xm T0 k1 k2] .

(16)

) The estimated stator current I s is computed at the test

voltage using the candidate parameters and equivalent scheme according to performed test (a, b, c, d, e, f). The active and reactive power are computed for no load test considering three phase current and for short circuit 2 phase current: ) ) P( a,b) = 3V( a,b) real I ( a,b) , (17)

(

(

)

)

) ) Q( a ,b ) = −3V( a,b ) imag I ( a,b) , ) ) P(c,d ,e, f ) = 2V(c,d ,e, f ) real I (c,d ,e, f ) , ) ) Q(c,d ,e, f ) = −2V(c,d ,e, f ) imag I (c,d ,e, f ) .

(

(

)

)

(18) (19) (20)

B. The Measured Errors Reduction The algorithm itself is filtering the measured error, but some errors are very large and then it is better to eliminate those points. Measuring the voltage, current, active and reactive power gives the possibility to estimate some error level in the measurements. The relative error for no load test and short circuit test are:

ε m( a ,b ) =

Ps2 + Q s2 −1 , 3V s I s

ε m( c ,d ,e , f ) =

Ps2 + Q s2 −1 . 2V s I s

(21)

(22)

The average error εav and standard deviation of the error σε could be computed for each test series. The inconsistent measurements points are eliminated from the parameter computation and also from graphic presentation. A

C. The Boundaries of the Candidate Parameters Vector The boundaries of the candidate parameter vector are computed from experimental data using classical approximation of the induction motor at no load test, respectively short-circuit and considering the maximum influence of the neglected part and also the measured accuracy. The auxiliary winding is considered in the stator frame of the main winding:

V '2 = k e ⋅ V 2 ,

(25)

where the ratio factor ke may be considered as the one used in designing, or may be calculated experimentally, as being part of the unknown parameters of the machine, and

V N kw ke = 1 = 1 1 , V 2 N 2 kw2

(26)

is the value of design. In case that ke is part of the unknown series, the computation of the search domain is needed: ⎛ V ⎞ ke max = max ⎜ 1 ⎟ (1 + 2ε v ) , (27) ⎝ V20 ⎠ where V1 is the supply voltage of the main winding, V20 is the voltage at the terminals of the auxiliary winding measured:

⎛V ⎞ ke min = ⎜ 10 ⎟ (1 − 2ε v ) , ⎝ V2 ⎠

(28)

and V10 is the voltage at the terminals of the main winding when the auxiliary winding is supplied and εv is the maximum value added in both voltmeters (a couple percents). If the algorithm of identification requires an initial value it may be considered that: V ⎞ 1⎛ V (29) k e0 = ⎜⎜ 1 + 10 ⎟⎟ . 2 ⎝ V 20 V 2 ⎠ The maximum and minimum values of the stator winding resistances R1, and R2 are computed considering theirs dc measured values and dc resistances measurement including possible resistance variation due to the windings temperature variation:

233

Rs max = Rsdc (1 + ε Rs ) ; Rs min = Rsdc (1 − ε Rs ) .

(30)

The rotor resistance boundaries are computing considering the short circuit resistances and the boundaries of the stator resistances: R'r max = max (R1sc1 ) − R1min ; R'r min = min(R1sc1 ) − R1max . (31) The minim rotor resistance should by positive, so the computed values will set to zero if the large measurement error produce a negative value. The short circuit resistance values are computed from the short circuit tests series c): P R1sc1 = msc1 . (32) 2 2 I msc 1 The iron losses resistance boundaries: 1 + 2ε v Rm max = , (33) ⎛P ⎞ min ⎜ m0 − R1min I m2 0 ⎟ 1 ⎝ 3 ⎠ − 2 max ( R10 ) max (V )

2 +ε 2 2 ε x2 = ε xIav xPav + ε xQav =

where indices x means test case a, b, c, d, e, and f. The current, active and reactive power errors εxI, εxP, εxQ are computed for each trust point:

ε xI = 1 −

2 = ε va

The rotor leakage boundaries are the same as the stator coupling leakage reactance X12σ. The magnetization reactance boundaries are computed from the no load tests on the main winding: X m max = max ( X10 )(1+ 2εv ) ; X m min = max ( X10 ) − X max . (39)

T0 max =

(

Ω1

);

T0 min = 0 ,

T k1max = 0 max ; k1min = 0 , Ω1 k2 max =

k0 max ; k2 min = 0 . Ω1

V21i

; ε xP

) ) Px Qx = 1− ; ε xQ = 1 − , Px Qx

2 = ε vb

∑ wbε vbi2 , ∑) wb

; ε vbi = 1 −

V12i V12i

(45)

(46)

.

(47)

PARAMETERS IDENTIFICATION RESULTS

The test was performed on the 11 kW, 8 poles, 630 V, and 50 Hz dual stator three phase induction machine with cage rotor. The main dimensions are: stator core outer diameter Dso = 267.8 mm, stator core inner diameter Dsi=180 mm, core length Lc = 230 mm, number of stator slots Ns = 72, with cross section Ss = 128 mm2, and rotor slots Nr = 58, with cross section Sr = 111 mm2. The first winding is a two layer winding with 8 teeth coil span and 0.94 mm2 cross section placed near air-gap while the second winding has a single layer winding with 1.73 mm2 cross section. Each winding has 96 turns per phase. A genetic algorithm with the population size of 200 and number of the generation 100 was used to find the machine parameter based on described no load and short circuit test series. The algorithm are transferred always the best machine code to the new generation and it has a mutation rate 0.5%. The parameter boundary and their founded values are given in table I. Variable

ke R1 (Ω) R2 (Ω) Rr (Ω) Rm (Ω) X1σ (Ω) X2σ (Ω) X12σ (Ω) Xrσ (Ω) Xm (Ω) T0 (Nm) k1 (Nm s) k2 (Nm s2) Pmec (W)

(40) (41) (42)

D. The Objective Function The objective function is sum of the squared current, active and reactive power error for each series tests plus the squared error on the opening winding for the no load tests. For each of the six tests the active and reactive power errors are calculated: 2 +ε2 , ε t = ε a2 + ε b2 + ε c2 + ε d2 + ε e2 + ε 2f + ε va vb

V21i

III.

The boundaries of the mechanical losses components are:

min Pm0 − 3R1min I m2 0

Ix

∑ waε vai2 ; ∑)wa

ε vai = 1 −

(34)

where R10 is the no load resistance computed from ‘test a)’ series: 3V 2 R10 = m0 . (35) Pm0 The stator winding leakage reactance’s boundaries are computed using the short circuit reactances: X X1σ max = min ( X1sc 2 ) ; X1σ min = 1max , (36) 10 X X 2σ max = min ( X 2 sc 2 ) ; X 2σ min = 2 max , (37) 10 X12σ max = max ( X1sc1 ) − min ( X1sc 2 ) ; X12σ min = 0 . (38)

) Ix

The voltage ratio error:

m0

Rm min = max ( R10 )(1 − 2ε v ) − R1max ,

∑wxεxI2 + ∑wxεxP2 + ∑wxεxQ2 , (44) ∑wx ∑wx ∑wx

Minimum 0.933 3.315 2.04 1.637 877.39 0.29 0.49 0 0 107.66 0 0 0

Table I Maximum 1.073 4.485 2.76 3.177 1239.26 2.91 4.9 9.219 9.219 122.19 0.516 6.567*10-3 84*10-6

Found value 0.992 4.212 2.491 2.243 1418.2 0.2989 3.9399 7.352 2.835 108.47 0.3043 1.494*10-3 51*10-6 58.027

The no load measured current, active power and reactive power are shown in figure 2 in comparison with the computed vales, based on parameter identifications.

(43)

234

No load current

Short circuit 1 - active power

2.6

800

2.4

700

2.2 600 500

1.8 Power (W)

Current (A)

2

1.6 1.4

400 300

1.2

main model main exp aux model aux exp

1 0.8 0.6

0

50

100

150 Voltage (V)

200

250

200 main model main exp aux model aux exp

100

300

0

0

20

40

a)

60 Voltage (V)

80

100

120

b)

No load - active power 300

Short circuit 1 - reactive power 1800

250

1600 1400 1200 Reactive (VAr)

Power (W)

200

150

1000 800 600


100

50

0

50

100

150 Voltage (V)

200

250

400


200

300

0

0

b) No load - reactive power

60 Voltage (V)

80

100

120

Fig. 3. The short circuit with one open winding comparison: a) current; b) active power; c) reactive power.

1800 1600 1400 Reactive (VAr)

40

c)

2000

1200 1000 800 600 400


200 0

20

0

50

100

150 Voltage (V)

200

250

300

The short circuit with an open winding measured current, active power and reactive power are shown in figure 3 in comparison with the computed vales. The short circuit with the one winding in short-circuits measured current, active power and reactive power is shown in figure 4 in comparison with the computed vales. The current, active and reactive power average errors (rms value) are presented in Table II for each test.

c)

Test type a b c d e f

Fig. 2. No load test and computation comparisons: a) current; b) active power; c) reactive power. Short circuit 1 - current 8 7

Table II. I (%) P (%) 5.4 6.0 3.8 5.8 4.9 10.4 1.9 2.5 3.2 6.0 1.3 1.3

Q (%) 9.9 6.1 3.0 5.8 3.5 2.4

6

Current (A)

5 4 3 main model main exp aux model aux exp

2 1 0

0

20

40

60 Voltage (V)

80

100

120

The overall average error is 5.3%. The larger error in the no load tests could be explained by measurements error and voltage and machine unbalance. The voltage unbalanced average on the main winding was 0.7% but at low voltage the unbalanced level was 1.7%. This is producing a 2.2% average unbalanced current with the maximum at 4%. When the auxiliary winding was supplied the unbalanced current level was 3.2% with some peaks at 4.4%.

a)

235

The short circuit reactance, fig. 6, is less affected by random error but the coupling leakage reactance of the stator windings and the rotor leakage reactance are affected by saturation and they are not constant in a real machine as in the model.

Short circuit 2 - current 11 10 9

Current (A)

8 7 6

16 5


2 0

10

20

30

40 Voltage (V)

50

60

70

12 Reactance (Ohm)

3

1

X1sc1 X2sc1 X1sc2 X2sc2

14

4

80

a)

10

Short circuit 2 - active power

8

6

1400

1200

4

1000

2

0

2

Power (W)

800

6 Voltage (V)

8

10

12

Fig. 6. Short circuit reactance.

600

The parameter variation and the measurement error are comparable with the average error between model current and power respectively measured current and power. The optimization algorithm features is presented through the objective function evolution, figure 7 and through induction machine parameter evolutions, figure 8.

400 main model main exp aux model aux exp

200

0

4

0

10

20

30

40 Voltage (V)

50

60

70

80

b) Short circuit 2 - reactive power 1200

0.25

cost min cost max cost med

1000

0.2

Cost function

Reactive (VAr)

800

600

0.15

400

0

0.1


200

0

10

20

30

40 Voltage (V)

50

60

70

0.05

80

20

c) Fig. 4. The short circuit type 2 comparisons: a) current; b) active power; c) reactive power.

40

60

80

100 120 Generation

140

160

180

Population evolution statistics.

Fig. 7.

Second windig voltage ratio 1.06

The variation of the short circuit resistance, fig. 5, versus test voltage shows also the measurements error level.

1.05 1.04

6.6 R1sc1 R2sc1 R1sc2 R2sc2

6.2

1.03 k e=V 1/V 2

6.4

Resistance (Ohm)

6

1.02 1.01

5.8

1

5.6

0.99

5.4

0.98

5.2

0.97

5

20

40

60

80

100 120 Generation

a)

4.8 4.6

0

0

2

4

6 Voltage (V)

8

10

12

Fig. 5. The short circuit resistance.

236

140

160

180

200

Windings resistance

Mechanical losses at synchronous speed

4.5

62 60

4

R1 R2 Rr

58 56 Pmec (W)

(Ohm)

3.5

3

2.5

54 52 50 48

2

46 1.5

0

20

40

60

80

100 120 Generation

140

160

180

44

200

8

X1 X2 X12 Xr

6

IV.

(Ohm)

5 4 3 2 1 0

0

20

40

60

80

100 120 Generation

140

160

180

200

160

180

200

c) Magnetizsation reactance 118 117 116

Xm (Ohm)

115 114 113 112 111 110 109 108

0

20

40

60

80

100 120 Generation

140

d) 1450

1400

Rm (Ohm)

40

60

80

100 120 Generation

140

160

180

200

CONCLUSIONS AND DISCUSIONS

This paper presents an optimal method for induction machine parameter identifications which could be allayed to classical rotor cage machine with a single three phase winding in the stator and also to a dual stator winding cage-rotor induction machine. The estimation is performed by using an equivalent scheme of the DSWIM and simple standard test without requiring mechanical coupling of two machines or a mechanical sensor. The model uses constant parameters as they are required in practical applications. The linkage inductance variation with the magnetization current could be added further using no load test points at larger voltage. During the parameter identification the machine speed is also estimate at each no-load test point. If the speed variation range is small then the mechanical losses distribution in Columbian friction, viscous friction and fan friction is poor in precision but the total mechanical losses (sum of the components) has a good precision. As a future work, a better thrust weighing factor could be computed considering the unbalanced current level and the short circuit resistance deviation from to the average value. ACKNOWLEDGMENT This work was partially supported by the strategic grant POSDRU 107/1.5/S/77265 (2010) of the Ministry of Labor, Family and Social Protection, Romania, co-financed by the European Social Fund – Investing in people. This project was developed through the Partnerships in priority areas program - PN II, with the support of ANCS, CNDI - UEFISCDI, project no. 36/2012.

Iron losses resistance

1350

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1300

[2]

1250

1200

20

f) Fig. 8. The induction machine parameter evolutions: a) voltages ratio; b) windings resistances; c) leakage reactance’s; d) magnetization reactance; e) iron losses resistance; f) mechanical losses at synchronous speed.

b) Leakage reactances

7

0

0

20

40

60

80

100 120 Generation

140

160

180

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