An induction motor parameter estimation method

Electrical Power and Energy Systems 23 (2001) 251±262

www.elsevier.com/locate/ijepes

An induction motor parameter estimation method D. Lindenmeyer a,*, H.W. Dommel a, A. Moshref b, P. Kundur b a

Department of Electrical and Computer Engineering, The University of British Columbia, 2356 Main Mall, Vancouver, BC, Canada V6T 1Z4 b Powertech Labs Inc., 12388 Ð 88th Avenue, Surrey, BC, Canada V3W 7R7 Received 16 May 2000; revised 14 July 2000; accepted 8 August 2000

Abstract The auxiliary systems of large thermal power plants are mainly driven by large induction motors. Their accurate modeling is essential for the analysis of black or emergency start procedures during power system restoration, where auxiliaries are started from relatively small diesel, hydro, or gas turbines, and where large voltage and frequency deviations occur. This paper proposes a new parameter estimation method for induction motors which allows to build induction motor models from manufacturer data such as nameplate data, and motor performance characteristics. The models can be used for black and emergency start studies with the Electromagnetic Transients Program (EMTP) or with stability programs. The method is based on a non-linear optimization routine. Its objective function, boundaries, and constraints are described in this paper. The method is ®rst formulated neglecting saturation and then extended to allow for the saturation of the motor leakage reactances. It is tested to ensure that it is suf®ciently robust and that its application gives the best agreement between model and manufacturer data. The effectiveness of the method is demonstrated for a 600-HP pump motor, such as typically used in large thermal power plants, and by a practical emergency start case study. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Motor modeling; Electromagnetic transients; Black start

1. Introduction After a widespread blackout, the power of large thermal generating units has to be restored quickly and reliably in order to be able to bring major loads back into service as soon as possible. If no assistance of other utilities is available, a restoration path from a relatively small hydro generator or gas turbine to a large thermal power station has to be built in order to start up the thermal power plant's auxiliaries. These auxiliaries are mostly driven by large induction motors, drawing high currents and requiring long starting times. In order to ensure a successful start of a thermal power plant, it is crucial for any restoration procedure to investigate the feasibility of such auxiliary motor start-ups [4]. This is usually accomplished using time-domain simulation programs such as the Electromagnetic Transients Program (EMTP) [7], or stability programs [13]. This paper focuses on the modeling of motors based on experience gathered during an actual emergency start study. The implementation of the induction motor differential equations in most simulation programs can be considered as mature, and in most cases no further improvement is required [4]. However, this still leaves the question open * Corresponding author. Tel.: 11-604-605-8830; fax: 11-604-822-5949. E-mail address: [email protected] (D. Lindenmeyer).

on how motor parameters can be ef®ciently and accurately determined from different sets of manufacturer data. Generally, induction motor parameter estimation methods can be classi®ed into ®ve different categories, depending on what data is available, and what the data is used for: 1. Parameter calculation from motor construction data. This method requires a detailed knowledge of the machine's construction, such as geometry and material parameters. On the one hand, it is the most accurate procedure, since it is most closely related to the physical reality. On the other hand, it is the most costly one since it is based on ®eld calculation methods, such as the ®nite element method [2,6]. This method is mainly applied in induction motor design. 2. Parameter estimation based on steady-state motor models. The methods in this category use iterative solutions based on induction motor steady-state network equations and given manufacturer data [9,10,15,17]. This is the most common type of parameter estimation for system studies since the data needed for it is usually available. 3. Frequency-domain parameter estimation. The stand-still frequency response (SSFR) method is based on measurements that are performed at standstill. The motor parameters are estimated from the resulting transfer function

0142-0615/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0142-061 5(00)00060-0

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D. Lindenmeyer et al. / Electrical Power and Energy Systems 23 (2001) 251±262

I1 R1

2. Input data

I2 X1

X2 X21

V1

XM I21

X22 I22

R21/s

R22/s

Fig. 1. Induction motor model without saturation.

[18]. The major advantage of this method is its accuracy. However, stand-still tests are not common industry practice, and this method can therefore not be used very often. 4. Time-domain parameter estimation. For this type of method, time-domain motor measurements are performed and model parameters are adjusted to match the measurements [5,11,14]. Since not all parameters can be observed using measurable quantities, the motor models need to be simpli®ed [11]. The method is costly, and the required data is usually not available. 5. Real-time parameter estimation. This type of parameter estimation is used to tune the controllers of induction motor drive systems. This requires real-time parameter estimation techniques, using simpli®ed induction motor models, that are fast enough to continuously update the motor parameters and therefore prevent the ªdetuningº of induction machine controllers [8,16]. The motor parameter estimation method proposed in this paper belongs to the second group of methods. It is suitable for system studies since suf®cient data is usually available to determine a motor model of suf®cient accuracy. Most methods in this category have the disadvantage that they only accept nameplate data [9,15], even when more data, such as performance characteristics, are available. Furthermore, they often ignore constraints on the machine parameters or saturation effects [10,17]. This paper proposes a new approach to overcome those drawbacks and to make it possible to ¯exibly determine motor models for any given combination of manufacturer data. The new motor parameter estimation routine can be considered as a generalization and combination of the methods introduced in Refs. [10,15]. In the following, the input data to the motor parameter estimation routine is described. Then, a parameter estimation algorithm neglecting saturation is developed, tested and extended to allow for saturation effects. The results for a typical 600-HP pump motor, such as commonly used in large thermal power plants, are given. Finally, the application of the method to a practical emergency start case study is demonstrated.

The proposed parameter estimation routine allows the input of different kinds of data, depending on their availability. The ®rst type of data is the one given on the motor nameplate, where subscript ªFLº refers to full load, and ªLRº to locked rotor test: ² ² ² ² ² ² ² ²

Rated power PFL; Rated voltage Vrated; Ef®ciency h FL; Power factor pfFL; Rated slip sFL; Starting current ILR; Starting torque TLR; Breakdown torque Tmax.

This set of data is not always suf®cient to obtain a motor model, which behaves accurately over the entire speed range. The new parameter estimation routine can therefore use extra motor performance data such as: ² current-slip characteristic Im.f.(s); ² torque-slip characteristic Tm.f.(s); ² power factor-slip characteristic pfm.f.(s); where ªm.f.º stands for ªmanufacturer dataº. Additional data, such as rotational losses Prot FL, can also be taken into account, which create constraints and boundaries for the non-linear optimization method of the following section. 3. Non-linear optimization procedure Our algorithm is based on the non-linear optimization routine Solnp [19,20], which solves non-linear programming problems of the general form: minimize F…X†

…1†

subject to : G…X† ˆ 0

…2†

Lh # H…X† # Uh

…3†

Lx # X # Ux n

…4† n

n

m1

where X [ R ; F…X† : R ! R; G…X† : R ! R ; H…X† : Rn ! Rm2 ; Lh ; Uh [ Rm2 ; Lh , Uh ; Lx ; Ux [ Rn and Lx , Ux ; Lx , X0 , Ux ; X [ Rn : The function F(X) represents the objective function. Eq. (2) stands for the equality constraints, Eq. (3) for the inequality constraints, and Eq. (4) represents the boundaries on the variables. X0 is the initial estimate for the solution. The underlying mathematical algorithm uses major iterations, in which a linearly constrained optimization problem is solved based on a Lagrangian objective function. Within each major iteration, minor iterations are carried out, based on linear and quadratic programming. Functions (1)±(4), as

D. Lindenmeyer et al. / Electrical Power and Energy Systems 23 (2001) 251±262

used for our parameter estimation routine, will be de®ned in the following sections.

which is de®ned as a quadratic error function: F…X† ˆ WI

4. Motor model without saturation

253

nI X iˆ1

D2I …si † 1 WT

nT X iˆ1

D2T …si † 1 Wpf

npf X iˆ1

D2pf …si † …16†

where

4.1. Objective function

X ˆ ‰R1 ; X1 ; XM ; X2 ; R21 ; X21 ; R22 ; K; mŠ

The objective function F(X) is developed starting from the steady-state equations for a double-cage induction motor. These equations can also be used for induction motors with deep-bar rotors and with a single-cage [13]. For the derivation of the steady-state equations the motor equivalent circuit of Fig. 1 is used [7,13]. All parameters and equations are given in per unit. The impedances for the two rotor circuits in Fig. 1 are

…18†

with K and m de®ned in Section 4.3, and DI …si † ˆ

Ic …si † 2 Im:f: …si † Im:f: …si †

DTc …si † ˆ D pfc …si † ˆ

T…si † 2 Tm:f: …si † Tm:f:

i ˆ 1; ¼; nI i ˆ 1; ¼; nT

pf…si † 2 pf m:f: …si † pf m:f: …si †

i ˆ 1; ¼; npf

…19† …20† …21†

R Zr1 ˆ 21 1 jX21 s

…5†

R22 1 jX22 s

…6†

The quantity si represents discrete values for the induction motor slip, and nI, nT, npf are the total number of data points available for current, torque, and power factor, respectively. The factors WI, WT, and Wpf are weighting factors described next.

…7†

4.2. Weighting factors

Zr2 ˆ

This results in the total rotor impedance Zr ˆ Rr 1 jXr ˆ jX2 1

Zr1 Zr2 Zr1 1 Zr2

I21 ˆ

Zr2 I Zr1 1 Zr2 2

…11†

The choice of the weighting factors for the objective function F(X) is of great importance for the optimization process. It is desirable to give each of the torque-slip, power factor-slip, or current-slip characteristics equal weight. Otherwise, if e.g. a large number of data points for the torque-slip characteristic and only a small number of points for the current-slip characteristic is provided, the latter would only have a negligible in¯uence on the value of the objective function if each point had equal weight. This would lead to signi®cant differences between manufacturer and calculated values in the current-slip characteristic. Generally, nameplate parameters are more reliable than motor performance characteristics. The latter do not always give exact numbers, but are rather indication of the generic performance behavior [1]. Based on these observations, we de®ne the following weighting factors: WI ˆ nT 1 npf

…22†

I22 ˆ

Zr1 I Zr1 1 Zr2 2

…12†

WT ˆ nI 1 npf

…23†

From the above equations we ®nally obtain the equations for current magnitude, power factor and torque:

Wpf ˆ nI 1 nT

…24†

and the total motor impedance as seen from the motor terminals Zmot ˆ R1 1 jX1 1

jXm Zr jXm 1 Zr

…8†

The motor current as a function of the slip follows as I1 ˆ

V1 Zmot

…9†

where V1 represents the motor terminal voltage. Using current divider equations we then get I2 ˆ

jXm I jXm 1 Zr 1

…10†

Ic …s† ˆ uI1 u

…13†

The importance of nameplate parameters is re¯ected by multiplying its weighting factors by an additional factor

pf c …s† ˆ cos /…Zmot †

…14†

WNP ˆ

…15†

4.3. Constraints and boundaries

Tc …s† ˆ

R21 R 2 2 uI u 1 22 uI22 u s 21 s

where the index ªcº stands for ªcalculatedº. Eqs. (13)±(15) form the basis for the objective function F(X) in Eq. (1),

nI 1 nT 1 npf 3

…25†

If neither constraints nor boundaries are used for the optimization, the result vector X may contain ªnon-physicalº

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D. Lindenmeyer et al. / Electrical Power and Energy Systems 23 (2001) 251±262

I2

I1 Rstat

Xstat

X2 X22

X21 Vstat

I21

Zr R21/s

Stator

I22

R22/s

Rotor

Fig. 2. Induction motor model with stator part replaced by Thevenin equivalent circuit.

values, such as negative values for resistances. Consequently, we de®ne the following boundaries: R1 ; X1 ; XM ; X2 ; R21 ; X21 ; R22 $ 0

…26†

We further de®ne a boundary condition for the design factor m [15]: R21 1 R22 X21 1 X22

…27†

0:4 # m # 1:1

…28†

mˆ

The maximum or breakdown torque Tmax is reached when the power ¯ow from stator to rotor is a maximum, i.e. when the following condition is ful®lled: q Rr …smax † ˆ …R2stat 1 …Xstat 1 Xr …smax ††2 † …36† smax The values for Rstat and Xstat are obtained from the Thevenin equivalent impedance that replaces the stator part of the induction motor (see Fig. 2): Zstat ˆ Rstat 1 jXstat ˆ

jXM …R1 1 jX1 † R1 1 j…X1 1 XM †

…37†

when neither the rotor type nor m are known. In cases where the rotor type is known, the boundaries in Eq. (28) are adjusted. The following boundary conditions are chosen, allowing for a ^10% variation of the values given in [15]:

From Eq. (7) we can further calculate the resistance Rr and reactance Xr of the rotor as:

0:45 # m # 0:65 for deep-bar rotors

…29†

Rr ˆ

2 2 1 R21 R22 …R21 1 R22 † 1 s2 …R21 X22 1 R22 X21 † s …R21 1 R22 †2 1 s2 …X21 1 X22 †2

0:90 # m # 1:10 for double-cage rotors

…30† Xr ˆ

1 R221 X22 1 R222 X21 1 s2 X21 X22 …X21 1 X22 † 1 X2 s …R21 1 R22 †2 1 s2 …X21 1 X22 †2

If the rotor is of type single-cage, the constraint for m is removed since a design factor in this case is not de®ned. Based on nameplate data, four constraints G1 …X†; ¼; G4 …X† are de®ned:

…38†

…39†

X2 2 KX1 ˆ 0

…31†

Eq. (36) can be solved analytically for smax and the maximum torque Tmax is then determined from Eq. (15).

Xr 2 X1 ˆ 0

…32†

4.4. Initialization

Tmax m:f: 2 Tmax …X† ˆ 0

…33†

2 2 PFL 2 …R1 I12 FL 1 R21 I21 FL 1 R22 I22 FL 1 Prot FL † 2 hFL ˆ 0 PFL …34†

The factor K in Eq. (31) is subject to another boundary, whose value is chosen according to [1,3,15]: 0 # K # 0:4

It is important to start the optimization process with initial estimates X0 as close as possible to the values that lead to a minimum for the objective function F(X). This is accomplished using approximate equations based on [15]:

h 0FL ˆ 0:25 1 0:75hFL R1 ˆ pf FL

…35†

Eq. (32) comes from the assumption that the rotor and stator reactances are equal [12,13].

RFL ˆ sFL

h 0FL 12 1 2 sFL h 0FL pf 1 2 sFL FL

…40† ! …41†

…42†

D. Lindenmeyer et al. / Electrical Power and Energy Systems 23 (2001) 251±262 Table 1 Induction motor nameplate data

Table 2 Induction motor characteristics

PFL (HP) Vrated (kV) Synchr. speed (RPM)

600 4 900

TFL (N m) IFL (A) pfFL h FL sFL (%)

4812 75.5 0.914 0.936 1.333

TLR (N m) ILR (A) pfLR

4861 480.8 0.29

Tmax (N m)

13 810

h 0FL XM ˆ …1 2 sFL † sin fFL RLR ˆ TFL h 0FL pf FL =

2 ILR 1 2 sFL

…43†

…44†

m ˆ 0:7

…45†

R21 ˆ RLR …1 1 m2 † 2 RFL m2

…46†

R22 ˆ R21

RFL R1 2 RFL

X21 ˆ 0

…47† …48†

R 1 R22 2 X21 ˆ 21 m

…49†

 s  1:0 2 2…R1 1 RLR †2 Xl ˆ ILR

…50†

X22

X1 ˆ

Xl 2

X2 ˆ X1 2 RFL Kˆ

255

…51† R21 m R22 m2 1 1

X2 X1

…52† …53†

4.5. Veri®cation of algorithm In case a set of adverse initial estimates is chosen, the optimization routine may converge to a local minimum for the objective function that does not represent the best possible solution. In order to ensure with a high probability that the parameter estimation algorithm leads to an absolute minimum, the initial estimates are varied randomly between 0 and 200% of the values calculated according to Eqs. (40)±

Speed (rpm)

I (A)

pf

0.0 150 180 400 405 600 630 750 765 850 888

480.8 ± ± ±

0.290 ± 0.306 ± 0.339 ± 0.416 ± 0.570 0.813 0.914

± ± ± ± ± 75.5

T (N m) 4861.0 5288.0 ± 6454.0 8243.0 ± 12202.0 ± 12609.0 4812.0

(53) except for the parameters m and K (Eqs. (29) and (35)), which are randomly varied between their boundaries. If the initial estimates are chosen close enough to the set of parameters that leads to the absolute minimum of the objective function, the objective function value should always be equal to or smaller than the values we obtain for random initial estimates. Different rotor parameter sets might lead to the same value for the objective function. In order to allow for a direct comparison between the results for different initial estimates, one of the rotor parameters is ®xed for this test. Otherwise the objective function may lead to the same value although the motor parameters are different. Therefore, as suggested in Refs. [13,15], the rotor reactance X21 is limited to a small value: 0 , X21 , 1023

…54†

For the test, we use the motor data of a 600-HP pump motor, which is currently in use in a large thermal power plant. Its nameplate data is listed in Table 1 and the torque-slip, current-slip, and power factor-slip characteristics are displayed in Table 2. For 100 different sets of initial estimates, the algorithm converged in 81 cases to the same objective function values and delivered the same motor parameters. The values for the other cases were signi®cantly higher and therefore belong to local minima that do not represent the best possible solution. The objective function values with F…X† , 0:2 are Table 3 Induction motor parameters (p.u.) X

Å (p.u.) X

DX (%)

R1 X1 R21 X21 R22 X2 XM m K

0.024111 0.76779 0.27005 0.00099999 0.27389 0.070127 3.2316 0.60206 0.91336

0.0005 0.0006 0.0257 0.0416 0.0271 0.0415 0.0293 0.0996 0.5426

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D. Lindenmeyer et al. / Electrical Power and Energy Systems 23 (2001) 251±262

0.0618

0.0616

Objective function F(X)

0.0614

0.0612

0.061

0.0608

0.0606

0.0604

10

20

30

40 50 Set of initial estimates

60

70

80

Fig. 3. Values for objective function.

displayed in Fig. 3 and the number of major iterations in Fig. 4. The ®rst set of initial estimates is the one calculated by Eqs. (40)±(53). In Table 3 the mean of the motor  and the percentage deviation DX from the parameters X mean are given. The deviations are negligible, despite the variation of the number of major iterations between 3 and 10. As a second test, the algorithm's robustness is investigated. ªCorrectº current-, torque-, and power factor-slip characteristics are created from the motor parameters in Table 3. These are then fed back into the algorithm. The difference between the parameters obtained from the ªcorrectº characteristics and from the manufacturer data, given in Tables 1 and 2, is below 1%. 5. Motor model with saturation In this section, we discuss how saturation effects can be taken into account. Only the saturation of the stator leakage reactance X1 and the rotor mutual leakage reactance X2 are considered, which is a good approximation for most motor start-up studies [15]. The equivalent circuit for a motor including saturation effects is depicted in Fig. 5 [13]. In the following, only the changes needed for the saturation effects are outlined.

5.1. Objective function To represent saturation of the leakage reactances, we use the ªdescribing functionº DF de®ned in Ref. [15]: DF ˆ 1 DF ˆ

for g . 1

2 …a 1 0:5 sin …2a†† p

…55† for g # 1

…56†

Isat I

…57†

a ˆ arcsin …g†

…58†

gˆ

where Isat stands for the saturation threshold current and I for the current ¯owing through the reactance. It is assumed that both stator and rotor leakage reactances have the same percentage part saturated, and de®ne the modi®ed reactances as: X1 ˆ X1u 1 DF1 X1s ˆ …1 2 SAT†X10 1 DF1 SAT X10 …59† X2 ˆ X2u 1 DF2 X2s ˆ …1 2 SAT†X20 1 DF2 SAT X20 …60† The factor SAT represents the per unit value of the saturable parts of the leakage reactances X1s and X2s with respect to the

D. Lindenmeyer et al. / Electrical Power and Energy Systems 23 (2001) 251±262

257

10

9

Number of iterations

8

7

6

5

4

3

10

20

30

40 50 Set of initial estimates

60

70

80

Fig. 4. Number of iterations.

tion, with two more variables added:

unsaturated leakage reactances X10 and X20: SAT ˆ

X1s X ˆ 2s X10 X20

X ˆ ‰R1 ; X1 ; XM ; X2 ; R21 ; X21 ; R22 ; K; m; Isat ; SATŠ

…61†

If detailed saturation data is available, the saturation threshold current can be ®xed, introducing an additional boundary.

The values of the currents and the describing functions are now determined iteratively, since the currents depend on the values of the reactances, and vice versa. This is shown in the diagram in Fig. 6, where the superscript ªnº stands for the nth iteration. The objective function F(X) for the saturated induction motor model has the same form as the one without satura-

5.2. Constraints, boundaries, weighting factors, and initialization In addition to the boundaries described earlier, we add [15]: 0:2 # SAT # 0:8

V1

…63†

I2

I1 R1

X1u

DF . X1s

…62†

X2u XM

DF . X2s

X21

I21

X22 I22

R21/s

Fig. 5. Induction motor model with saturation.

R22/s

258

D. Lindenmeyer et al. / Electrical Power and Energy Systems 23 (2001) 251±262

X1n = X10 ; X2n = X20

I1n ; I2n

DF1 ; DF2

X1 ; X2 I1n = I 1n+1 ; I2n = I 2n+1 I1n+1 ; I2n+1

| I1n+1 - I1n | < ε

No

| I2n+1 - I2n | < ε ? Yes Ic ; pfc ; Tc Fig. 6. Calculation of currents and describing functions.

1:5 # Isat # 3:0

…64†

This allows a deviation of ^60% from the value SAT ˆ 0:5 suggested in Ref. [15]. The breakdown torque for the saturated case can no longer be calculated analytically since, as outlined earlier, the currents have to be determined iteratively. Instead, it is found with an additional maximum ®nding routine. Since the torque function might have another (local) maximum, the range where the maximum torque is located is limited to: 0:0 # smax # 0:2

…65†

For the initialization procedure, the values calculated in Eqs. (40)±(53) remain the same, and two initial values for the new variables are added [15]: Isat ˆ 2:0

…66†

SAT ˆ 0:5

…67†

5.3. Veri®cation of algorithm For the veri®cation of the algorithm that includes saturation of leakage reactances, the same motor is investigated

D. Lindenmeyer et al. / Electrical Power and Energy Systems 23 (2001) 251±262

259

500

450

400

350

Current [A]

300

250

200

150

100

50

0

0

100

200

300

400

500 600 Speed [RPM]

700

800

900

Fig. 7. Current characteristic: `Ð', no saturation; `- ´ -', saturation; `W', manufacturer data.

and the same procedures are carried out as for the case without saturation. For 100 different initial estimates, the algorithm converged in 78 cases to the same objective function values. The objective function values and the number of iterations as a function of the set of initial estimates look similar to Figs. 3 and 4. The percentage deviation of the motor parameters from their mean is below 1%. 6. Results Figs. 7±9 show the results obtained from the algorithm for the 600-HP motor. A good agreement between the manufacturer and model characteristics can be observed. The match for the model with saturation is slightly better than the one for the model without saturation. This difference is not signi®cant for this particular motor. It can be explained by the additional degree of freedom that is introduced by the inclusion of saturation, which leads to a better overall ®tting. The results also show that close to the rated voltage, the model without saturation gives a reasonably good ®t, and can therefore be used for cases where voltage variations are small. The parameter estimation without saturation takes typically 20 s and with saturation typically 2.5 min on an IBM-compatible PC ®tted with a 200-MHz processor.

7. Emergency start case study The induction motor parameter estimation method is applied to the modeling of 20 induction motors of different size that drive the auxiliaries of a large thermal power station. Fig. 10 shows the simulated and measured generator currents during an emergency start procedure where power is supplied by a hydro power station. Simulation and measurement results are in good agreement. 8. Summary and conclusions This paper describes a new induction motor parameter estimation technique, based on induction motor steadystate analysis. Its core is a non-linear optimization routine that allows to ¯exibly take into account different types of data, such as nameplate data and motor performance characteristics in the form of boundary conditions and constraints. Models without saturation and with saturation representation are developed. Several tests, including a practical emergency start case study, show the usefulness and robustness of the algorithm and con®rm that the models developed with the routine agree well with manufacturer data. The new algorithm provides a simple and fast motor parameter estimation when motor start-ups have to be simulated in black start and emergency start studies.

260

D. Lindenmeyer et al. / Electrical Power and Energy Systems 23 (2001) 251±262 1

0.9

0.8

0.7

Powerfactor

0.6

0.5

0.4

0.3

0.2

0.1

0 0

100

200

300

400

500

600

700

800

900

Speed [RPM]

Fig. 8. Powerfactor characteristic: `Ð', no saturation; `- ´ -', saturation; `W', manufacturer data.

14000 12000

Torque [Nm]

10000 8000 6000 4000 2000 0

0

100

200

300

400

500 Speed [RPM]

600

700

800

Fig. 9. Torque characteristic: `Ð', no saturation; `- ´ -', saturation; `W', manufacturer data; `´ ´´', load.

900

D. Lindenmeyer et al. / Electrical Power and Energy Systems 23 (2001) 251±262

261

1800

1600

1400

Current [kA]

1200

1000

800

600

400

200

0

0

5

10

15

20 Time [S]

25

30

35

40

Fig. 10. Emergency start: `Ð', simulation; `- ´ -', measurement.

Acknowledgements [7]

The authors gratefully acknowledge the ®nancial assistance of the Natural Science and Engineering Research Council (NSERC) of Canada, and of B.C. Hydro and Power Authority, through funding provided for the NSERC±B.C. Hydro Industrial Chair in Advanced Techniques for Electric Power Systems Analysis, Simulation and Control.

[10]

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