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Розглянуто використання методу повних рiвнянь складових миттевог потужностi для задач визначення електромагттних параметрiв асинхронного двигуна. Отримано залежно-стi, що описують складовi миттевог потужностi на нелтшному активному опорi та нелтшнш тдуктив-ностi ротора в залежностi вгд струму ротора. Показана ефективтсть використання методу повних рiвнянь складових миттевог потужностi при iдентифiкацiг нелтшних параметрiв асинхронного двигуна
Ключовi слова: асинхронний двигун, енергетичний метод, схема замщен-ня, нелтшт електромагттт пара-метри
Рассмотрено использование метода полных уравнений составляющих мгновенной мощности для задач определения электромагнитных параметров асинхронного двигателя. Получены зависимости, описывающие составляющие мгновенной мощности на нелинейном активном сопротивлении и нелинейной индуктивности ротора в зависимости от тока ротора. Показана эффективность использования метода полных уравнений составляющих мгновенной мощности при идентификации нелинейных параметров асинхронного двигателя
Ключевые слова: асинхронный двигатель, энергетический метод, схема замещения, нелинейные электромагнитные параметры
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UDC 621.313.333
|DOI: 10.15587/1729-4061.2017.104146]
REFINED CALCULATION OF INDUCTION MOTOR EQUIVALENT CIRCUIT NONLINEAR PARAMETERS BY AN ENERGY METHOD
M. Zagirnyak
Doctor of Technical Sciences, Professor, Rector Department of Electric Machines and Devices* E-mail: mzagirn@kdu.edu.ua D. Rod ' ki n
Doctor of Technical Sciences, Professor, Head of Department Department of Systems of Automatic Control and Electric Drive*
E-mail: sau@kdu.edu.ua Iu. Romashykhi n PhD, Associate Professor Department of Systems of Automatic Control and Electric Drive*
E-mail: romashykhin.iurii@gmail.com Zh. Romashykhina
PhD
Department of Electric Machines and Devices* E-mail: romashykhina.zhanna@gmail.com A. Nikolenko PhD, Associate Professor, Head of Department** Gagarina ave., 4, Dnipro, Ukraine, 49600 V. Kuznetsov PhD, Associate Professor** *Kremenchuk Mykhailo Ostrohradskyi National University Pershotravneva str., 20, Kremenchuk, Ukraine, 39600 **Department of the electrical engineering and electromechanic National metallurgical academy of Ukraine Gagarina ave., 4, Dnipro, Ukraine, 49600
1. Introduction
At the stage of improvement of mathematical description of electric machines, the importance of accurate determination of their characteristics with the use of various methods and means of implementation grows. At present, the processes of energy conversion taking into account the induction motor (IM) particular nonlinear parameters are the object of the research. There is also a necessity for the research of electrical energy processes regulation and its conversion into mechanical one when IM operates at constant values of voltage and frequency [1, 2]. As a rule, IM nonlinear parameters represent physical properties of structural materials and phenomena in them. These phenomena occur under the impact of electromagnetic actions that can be observed in
the process of transformation of the electrical energy into the mechanical one [3, 4].
In a certain way, availability of electric machines nonlinear parameters influences operating characteristics - the values of the starting and critical torques, admissible mechanical load, etc. In this connection, there appears a necessity for determination of real characteristics taking into consideration the peculiar features of the electric machine.
Knowledge of the parameters of induction machine non-linearities is required in two typical cases.
1. In design of new-generation machines meant for variable-frequency electric drive systems. It is necessary for creation of systems providing maximum starting torque, maximum overload capacity, minimum operating temperature at the assigned load, etc.
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2. In determination of IM complete electromagnetic parameters (EMP) in the process of production test after repair or during the current monitoring of parameters of the electrical equipment in service. It is necessary for determination of the real values of the starting torque, overloading and loading capacity of the electric motors.
Thus, development of the method for identification of IM parameters, taking into account their nonlinearity, is a topical task.
2. Literature review and problem statement
features of their use for IM EMP determination. Improvement of the energy method is shown in [15] in the form of introduction of the rotor nonlinear resistive impedance into the equivalent circuit; it takes into account the effect of current displacement in the rotor.
The analysis reveals that there are some factors due to which the interest in the considered problem shifts towards the IM characteristics detailing. Here the accent is based on the known propositions concerning IM nonlinear properties. This means that the possibilities of the energy method can be extended due to refinement of the expressions for the nonlinear parameters of the equivalent circuit.
The interest in the problem of determination of induction motors electromagnetic parameters essentially grew in connection with the development of static frequency converters for the purposes of alternating current controlled electric drive [5]. At that time, there appeared a great number of papers dealing with determination of resistive impedance and inductance concerning simplified and detailed equivalent circuits [6]. The simplest solutions were obtained due to the use of L-shaped equivalent circuits with the magnetizing circuit taken to the terminals. More accurate results were obtained when the T-shaped equivalent circuit was used.
The analysis of the equivalent circuit with linear (constant) values of its components and resistances with taking into account the effect of current displacement in the rotor is common for the papers related to the considered problem [7]. The problem of taking into account the current displacement effect in the rotor is further developed by division of the rotor into several layers [8]. However, the drawback of this approach consists in impossibility to know the IM real geometric parameters and low accuracy of the obtained results. Elimination of the drawbacks concerning the knowledge of the geometric parameters is described in the paper [9], where the parameters of rotor resistance and leakage inductance are changed in order to take into account the effect of current displacement. The method of IM parameters identification [10] based on the integral calculation in the quiescent state for the admitted assumptions concerning IM parameters constancy is rather accurate. However, constancy of parameters in [9] and [10] can only be taken with possible big errors.
In [11], it is shown that creation of IM mathematical model is one of the ways of obtaining information about it. However, the information of the real state of the motor and current displacement effect influence on the rotor parameters is not taken into consideration in [11]. The use of genetic algorithms is rather topical but quite a lot of experiments are required to create a genetic model [12]. The paper [13] contains description of a method developed for the online assessment of the stator and rotor resistance with the use of artificial neural networks. In the mentioned approach, the assessment of the rotor resistance is insensitive to the change of the stator resistance, which is inadmissible.
Thus, there are a great number of the IM parameter determination methods based on the use of mathematical models or neural networks. However, they do not enable the assessment of the nonlinear characteristics and properties of the electric machines of this class.
The energy method is one of the modern methods for IM EMP determination [14]. The paper [14] contains the structure of creation of identification equations and special
3. The aim and objectives of the study
The aim of the research consists in the development of the methods for taking into account IM equivalent circuit nonlinear parameters when the energy method is used.
To achieve this aim, the following tasks of the research were solved:
- refinement of the expressions describing nonlinear inductance in the function of the rotor current;
- determination of the instantaneous power components for nonlinear inductance;
- assessment of the efficiency of the induction motor equivalent circuit parameter identification taking into account the refined expressions for the calculation of nonlinear resistive impedance and inductance with the use of the energy method.
4. The materials and methods of the research
Taking into account IM nonlinear properties refers to the sphere of their parameters identification during post-repair tests and current monitoring of the electric machines condition in the process of their operation.
Literature [1, 15] review revealed that the rotor resistive impedance is characterized by nonlinear variation due to the influence of current displacement effect and can be approximated by an even-power polynomial. In this case, the polynomial for the description of the rotor nonlinear resistive impedance can be limited to the second power with sufficient accuracy [15, 16]:
r2 (i2)=r20+kR(i2 (t))2, (1)
where R20 - the rotor resistive impedance without taking into consideration its variation due to the displacement effect; kR - the coefficient of approximation of nonlinear resistive impedance; i2(t) - rotor current flowing through the nonlinear resistive impedance.
In the general form, the rotor current can be presented by the dependence:
M
i2 (t) = £ I2m cos (mWt - y m ) =
m=0
M M
= £ I2am cos (mWt)+ £ I2bmcos (mWt), (2)
m=0 m=0
where m - the number of the current harmonic; M -the quantity of the analyzed current harmonics; I2m -
the value of the rotor current of the m-th harmonic; Q - the circular frequency of the network; ym - the angle of current harmonics phases shift; I2am=I2mcosym - the cosine component of the rotor current of the m-th harmonic; I2bm= =I2msinym - the sine component of the rotor current of the m-th harmonic.
The effect of current displacement in the rotor at the start also influences the value of inductive reactance that can also be expressed by a nonlinear dependence on current [14]:
L2 (I2 ) = L20 - kL (i2
ps(t) =É pj(t),
j=1
Ri
L
^L2(I2)
pR2 (t) = i2 (t)R2 (I2 ) = i2 (t)(R20 + kR (i2
(i2(t))2);
pL2 (t) = i2 (t)
d(L2 (I2)i2 (t))
dt
= L,
(i. ) (t) ^ ii (t) ^Lt1
= ( L20 - kL
di2(t)
i2(t)-
(i2 (t))2)i2 (t) dt (i2(t))2)
d ( L20 - kL
dt
(5)
(3)
where L20 - rotor inductance without taking into account its variation caused by the displacement effect; kL - the coefficient of nonlinear inductance approximation.
IM EMP determination taking into consideration dependences (2) and (3) is possible with the use of the energy method. This method is based on the use of complete equations of balance of instantaneous power harmonics components of the power supply and all the consumers [15, 17].
When the energy method is used, the system of equations is based on equality of the components of instantaneous power p(t) of the power supply and the sums of powers of all the consumers [14, 15]:
Instantaneous power components for the rotor nonlinear inductance:
P = T A ;
0L2 20 0
PkaL2 = L20kLAk +
1
-kLWb(-2kÀ0Bk -EE^ - ni)A„iB„2
2
N N
hXX(ni - n2)A„iBn2 -
n1=1n2=1 N N
-EE (n + n2 )A„1B„2 - 2£E n1A„1B„2 );
(6)
PkbL2 = L20kLBk +
(4)
where j - the index of the corresponding consumer; G - the number of the consumers.
In relation to IM, each element of the equivalent circuit is considered to be a consumer (Fig. 1), i.e. consumers include stator resistive impedance R1 and inductance L1, rotor resistive impedance R2(I2) and inductance L2(I2) and magnetization circuit inductance LH [6].
+- kLWb(-2kA0Ak + EE(n ± n2)A„1A„2 +
2 n1=1n2=1
+ ^ E (n - n2)B„1B„2 - E E (n + n2 )B„1B„2 -
n1=1n2=1 n1=1n2=1
-2 E En^AL - BBjn 2 )),
Fig. 1. ^shaped induction motor equivalent circuit with nonlinear resistive impedance and nonlinear inductance of the rotor
The method for calculation of instantaneous power components for linear elements of the equivalent circuit is described in [14, 17]. Determination of instantaneous power components for nonlinear resistive impedance is given in [15]. That is why it is necessary to additionally obtain instantaneous power components for the rotor nonlinear inductance.
Instantaneous powers for nonlinear elements of the equivalent circuit are determined as follows:
where k - the number of the instantaneous power harmonic; P0L2, PkaL2, PkbL2 - constant, cosine and sine components of instantaneous power for nonlinear inductance;
- in the calculation of PkaL2, the following is taken into consideration:
- for the third summand k=n2-n1 at n1<n2;
- for the fourth summand k=n1-n2 at n1>n2;
- for the fifth summand k=n1+n2;
- for the sixth summand k=2n1 at n1=n2;
- in the calculation of PkbL2, the following is taken into consideration:
- for the third summand k=|n1±n2|;
- for the fourth summand k=n1-n2 at n1>n2;
- for the fifth summand k=n1+n2;
- for the sixth summand k=2n1 at n1=n2;
1 M M
A0 = 2 E E (I2am1 - I2bm1 + 2I2am1I2bm2 )'
1 f M M An = ^
E E 2I2am1I2am2 E E 2I2bm1I2
V m1=1m2=1 m1=1m2=1
M M
E E 2I2bm1I2bm2 + E (I2am1 1
2 ) 2bm1
(7)
1 f M M Bn = ^
E E 2I2am1I2bm2 + E E 2I2bm1I2am2
V m1=1m2=1 m1=1m2=1
M M
E E 2I2bm1I2am2 + E I2am1I2
+
+
where n - the number of the cosine or sine harmonic component of the square of the rotor current i22(t); I2am1, I2am2 - cosine components of the rotor current for harmonics m1, m2; I2bm1, I2bm2 - sine components of the rotor current for harmonics m1, m2; in the calculation of A0, m1=m2 is taken into account; in the calculation of An, Bn, the following is taken into account:
- for the first summand n=m2±m1 at m1<m2;
- for the second summand n=m1+m2;
- for the third summand n=m2-m1 at m1<m2;
- for the fourth summand n=m1.
For a T-shaped equivalent circuit (Fig. 1), the resistive impedances and inductances of the stator, rotor and magnetization circuit are the unknown parameters. The rotor and magnetization circuit current harmonics are not known either. The total number of the unknown parameters is 17.
To get the required number of identification equations, it is necessary to analyze three voltage and current harmonics at a time. Additional equations are obtained due to the use of equations created according to the first Kirchhoff's law. The specific features of the choice of voltage and current harmonics and generation of instantaneous power harmonic components are given in [14].
Thus, with the use of three harmonics of voltage and current (the first, the third and the fifth harmonics), the equations of the balance of instantaneous power harmonic components will be of the form:
P = P + P •
0s 0R1 0R2 '
P2as = P + 2aR1 -P + 2aR2 P2aL1+ -P + P • 2aL2
P2bs = P 2bR1 + P 2bR2 + P2bL1 + P2bL^ + P2bL2
P 4as = P + 4aR1 n ^ P4aR2 ^ P4aL1 + -P + -P • 4aL2
P 4bs = P _ 4bR1 + P4bR2 + P4bL1 + P4bL^ + P4bL2
P6as — P + 6aR1 -P + 6aR2 -P + 6aL1 P6aL^ + P • 6aL2
P6bs = P 6bR1 + P 6bR2 + P6bL1 + P6bL^ + P6bL2
P 8as = P + 8aR1 hP + 8aR2 ^ P8aL1 + -P + -P • 8aL2
P 8bs = P 8bR1 + P 8bR2 + P8bL1 + P8bL^ + P8bL2
P = P + P + P + P + P •
c 10as 10aR1 10aR2 r10aL1T r10aL^ T r10aL2 '
P = P + P + P + P + P •
c 10bs _ 10bR1 10bR2 10bL1 r10bL^ T r10bL2 '
the equations created according to the first Kirchhoff's law for the IM T-shaped equivalent circuit (Fig. 1):
I1a1 = I(ia1 + I2a1; I1b1 = I^b1 + I2b1; ' I1a3 = I^a3 + I2a3; I1b3 = I^b3 + I2b3; (9)
I1a5 = I(ia5 + I2a5; I1b5 = I^b5 + I2b5'
where P0s, P0Ri, P0R2 - constant components of instantaneous power of the voltage source, stator and rotor resistive impedances;
Pkas, PkaR1, PkaR2, PkaL1, PkaLp PkaL2 - cosine components of instantaneous power of the voltage source, stator and rotor resistive impedances, inductances of the stator, magnetization circuit and rotor;
Pkbs, PkbR1, PkbR2, PkbL1, PkbLja, PkbL2 - sine components of instantaneous power of the voltage source, stator and rotor resistive impedances, inductances of the stator, magnetization circuit and rotor;
I1a1, I1a3, I1a5, I1b1, I1b3, Iib5 - cosine and sine components of the first, third and fifth harmonics of the stator current;
W, W W W, IMb3, I^b5 - cosine and sine components of the first, third and fifth harmonics of the magnetization current;
I2a1, I2a3, I2a5, I2b1, I2b3, I2b5 - cosine and sine components of the first, third and fifth harmonics of the rotor current.
To confirm the obtained theoretical provisions, an experimental research was carried out.
5. The results of the experimental research of the induction motor parameter identification taking into account the nonlinear elements
The experimental research was carried out for IM switched to a transformer via a block of voltage and current sensors. The experiments were performed for the following voltage levels: 220 V, 180 V, 120 V, 40 V, recorded by the voltage and current sensors. Then the required levels of voltage and current harmonics were singled out.
During the experiments, 4AP100L4U3 IM was used:
- rated power of 4 kW;
- rated voltage of 220 V;
- rated current of 8.7 A;
- stator resistive impedance of Ri=1.35 Ohm;
- stator inductance of Lj=0.00676 Hn;
- magnetization circuit inductance of L^=0.246 Hn;
- rotor resistive impedance of R2=1.38 Ohm;
- rotor inductance of L2=0.00678 Hn.
The equivalent circuit parameters were calculated with the use of the system of identification equations (8), (9) the measured values of voltage and current, as well as instantaneous power harmonic components for the elements. Tables 1, 2 contain the measured values of the stator current harmonics for the following voltage levels 220 V, 180 V, 120 V, 40 V, and also the calculated components of the rotor current harmonics and the magnetization circuit. Table 2 contains the calculated electromagnetic parameters of IM equivalent circuit taking into consideration their nonlinearity. Table 2 also contains the coefficients of approximation of the rotor nonlinear resistive impedance and nonlinear inductance. Besides, these Tables contain the rotor current cosine and sine components I2a1, I2a3, I2a5, I2b1, I2b3, I2b5, resistive impedance R2o, inductance L2o and coefficients kR and kL describing the rotor nonlinear resistive impedance and nonlinear inductance.
As a result of the analysis of the obtained data (Tables 1, 2), errors of determination of IM equivalent circuit EMP are calculated; they are shown in Table 3. Maximum error of determination of the parameters of the analyzed motor is 4.07 %. Verification based on comparison of the stator current experimental and calculated curves (Fig. 2) revealed the adequacy of the obtained results as the determination coefficient R2 equals 0.995.
Measured and calculated values of induction motor currents
Tablel
Measured values of currents (A)
Parameter Harmonic number
cosine components sine components
1 3 5 1 3 5
Stator current harmonics I1 at the stator voltage 220 V 27.514 -0.687 0.052 25.96 -1.11 0.072
Stator current harmonics I1 at the stator voltage 180 V 24.48 -0.569 0.034 21.133 -0.603 0.021
Stator current harmonics I1 at the stator voltage 120 V 18.28 -0.259 7.0M0-3 13.60 -0.146 -1.9840-3
Stator current harmonics I1 at the stator voltage 40 V 6.70 -0.013 -2.2940-4 4.33 -4.2-10-3 -9.14-10-4
Calculated values of induction motor currents (A)
Magnetization current harmonics Ip at the stator voltage 220 V 1.807 -0.312 0.014 1.24 0.029 -0.01
Magnetization current harmonics Ip at the stator voltage 180 V 0.994 -0.179 -2.2340-3 0.107 -0.162 -3.0840-3
Magnetization current harmonics Ip at the stator voltage 120 V -0.597 0.083 7.75340-4 3.359 -0.057 8.7M0-4
Magnetization current harmonics Ip at the stator voltage 40 V -0.284 -4-10-3 -4.2540-4 0.551 -1.1-10-4 1.65-10-4
Rotor current harmonics I2 at the stator voltage 220 V 25.707 -0.375 0.038 24.718 -1.139 0.082
Rotor current harmonics I2 at the stator voltage 180 V 23.489 -0.39 0.036 21.026 -0.441 0.024
Rotor current harmonics I2 at the stator voltage 120 V 18.878 -0.342 6.23640-3 10.336 -0.089 -2.85-10-3
Rotor current harmonics I2 at the stator voltage 40 V 6.98 -8.940-3 -1.8740-4 3.778 -4.1-10-3 -1.0840-3
Table 2
Results of determination of induction motor parameters taking into account the equivalent circuit nonlinear elements
Parameter Identified parameters
Stator voltage value, V 220 180 120 40
Magnetization circuit inductance Lp Hn 0.248 0.248 0.248 0.246
Stator inductance L1, Hn 6.8-10-3 6.7-10-3 6.8-10-3 6.75-10-3
Rotor inductance L20, Hn 5.435-10-3 5.79540-3 6.28640-3 6.70740-3
Rotor resistive impedance R20, Ohm 3.235 2.74 2.06 1.463
Coefficient kR, Ohm/A2 1.359 10-3 1.31510-3 1.42-10-3 1.31440-3
Coefficient kL, Hn/A2 0.985340-6 0.9525-10-6 1.03-10-6 1.162-10-6
Table 3 Errors of induction motor parameter determination taking into account the nonlinear elements of the equivalent circuit
Parameter Errors of the identified parameters
Stator voltage value, V 220 180 120 40
Error of determination of magnetization circuit inductance ALp, % 0.813 0.813 0.813 0
Error of determination of the stator inductance AL1, % 0.591 0.887 0.592 0.148
Error of determination of the rotor inductance AL2, % 3.087 3.232 3.741 4.070
Error of determination of the rotor resistive impedance AR2, % 2.467 2.847 3.448 3.911
i, A
20
10
1 / 2
0.005 t, s
Fig. 2. The stator current curves: 1 — experimental; 2 — calculated
For practical purposes, it is convenient to use resistive impedance and inductance dependences on supply voltage for refinement of the start and energy characteristics, creation of models, etc. Dependences on the supply voltage of the efficient value of the rotor current first harmonic, rotor resistive impedance and rotor inductance are shown in Fig. 3-5. It can be seen from the analysis of the mentioned dependences that when the supply voltage changes, the rotor current values change (Fig. 3), the values of nonlinear resistive impedance (Fig. 4) and the rotor nonlinear inductance (Fig. 5) change accordingly.
I2, A
30
10
0
100
200
U, V
Fig. 3. The rotor current first harmonic effective value dependence on supply voltage
0
R2, Ohm 3 2.5 2 1.5
50
100 150 200 U, V Fig. 4. The rotor resistive impedance dependence on supply voltage
L2, Hn 0.0065
0.006
0.0055
0.005
0 50 100 150 200 U, V Fig. 5. The rotor inductance dependence on supply voltage
Thus, the analysis of the obtained results revealed that the energy method allows determination of IM parameters taking into account their nonlinearity in equivalent circuits. Results shown in Tables 1, 2 make it possible to come to the conclusion as to the expediency of the use of the energy method for identification of nonlinearities parameters.
The performed research made it possible to specify the calculation of IM parameters due to the use of the energy method. The essential advantage consists in additional introduction of nonlinear resistive impedance and inductance of the rotor into the equivalent circuit; they take into account the effect of current displacement in the rotor.
The performed research and the analysis of the obtained results revealed the efficiency and expediency of the use of the proposed method for the calculation of the nonlinear character of resistive impedance and inductance by the energy method.
The proposed method of induction motor nonlinear parameter determination with the use of the energy method is recommended for application at electrical-repair enterprises for identification of the parameters of motors with defects or breakages. The described method can be practically realized in creation of the software for determination of induction motor parameters in the problems of outgoing inspection at electrical-repair enterprises and enterprises-manufacturers.
In the future, the proposed method can be used for the analysis of the processes in IM as a nonlinear electromechanical converter. In this case, the number of the identified parameters in the equivalent circuit can be extended.
6. Discussion of the results of identification of induction motor electromagnetic parameters taking into account their nonlinearities
The expressions for taking into account the nonlinear character of the resistive impedance and inductance of the rotor made it possible to improve the energy method for identification of the induction motor electromagnetic parameters. The performed experimental research and the analysis of the obtained data revealed the efficiency and expediency of the use of the proposed method for identification of the induction motor electromagnetic parameters.
The improvement of the energy method consists in introduction of additional components into the system of identification equations. These components take into account the nonlinear character of the resistive impedance and inductance of the rotor. This allowed enhancement of the accuracy of identification of the induction motor electromagnetic parameters when the energy method is used.
In the future, the proposed method can be used to improve the accuracy of calculation of the induction motor electromagnetic parameters and published data in the problems of outgoing inspection at electrical-repair enterprises and enterprises-manufacturers.
7. Conclusions
1. Expressions enabling improvement of the method for the calculation of the induction motor electromagnetic parameters due to the introduction of nonlinear elements into the equivalent circuit to take into account the effect of current displacement in the rotor have been obtained.
2. Expressions for the instantaneous power components at the nonlinear resistive impedance and inductance, making it possible to take into consideration the effect of current displacement in the rotor during identification of the electromagnetic parameters by the energy method, have been obtained.
3. The use of the proposed method enabled improvement of the efficiency of the energy method application taking into account the effect of current displacement in the rotor, in particular, it allowed the decrease of the error of the induction motor electromagnetic parameters identification to 4.07 %. The adequacy of the obtained results has been assessed by the determination coefficient that makes 0.995.
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