Abstract
This paper describes a new method to identify the equivalent circuit parameters from finite element modelling and optimisation tool. The optimisation strategy is interesting one, because it allows to find a set of equivalent circuit parameters with only 4 known results instead of solving a polynomial of degree six (or more) and then looking for more known variable and results. The proposed method has three highlight points. The first point is that this new method will make it possible to split stator and rotor leakage inductances. The second one is that the equivalent circuit parameters are depending on working point saturation state. The last one is that the used quantities can be extracted either from modelling or from measurements.
Introduction
The induction machines are more and more used in variable speed drives, like in electric cars (Tesla and Renault Twizi). In the industry as in France, 70% of the electric energy consumption is due to rotating machines [1], it highlights the importance of the speed variation to reduce losses and to adapt the machine to the load. Today, the electric machine is not considered alone, but as a part of a complex system like an electric car for instance. Co-simulation between system software and finite element software exists, but it can be time consuming. Co-simulation between system software using tables offers a good compromise between time and precision, but this method is not adapted for induction machines. Equivalent circuit is more suited for this kind of study. The classical equivalent circuit with grouped leakage inductances can be determined with no-load and short circuit tests. Since the saturation is changing for each working load [2,3], this circuit can be used only for the linear part of the curve which gives the torque variation versus speed. For the starting working point and the maximum torque, it is necessary to adapt the equivalent circuit parameters to the magnetic state of the machine for a given working load.
Method using FEM to extract equivalent parameters is already used in [2,6] but, in some cases, the method seems to be laborious to setup and it allows to extract values only for one working point. The proposed method has two points of originality. First, the used quantities can be extracted either from modelling or from measurements. The second interesting point is the splitting of stator and rotor leakage inductances. This separation is also used in [2,6] using finite element computation and measurement but the method are heavy to apply.
In the following, the proposed method is described, as well as the used machine and the FEM model. The results obtained with the proposed method will be compared to the FEM results.
Used machine and finite elements model informations
The results presented in the paper concern to a 4 kW induction machine. Its characteristics are: 230/400 V–15.3/8:8 A–1435 rpm - p = 2 pole pairs – 36 stator slots and 28 rotor bars. The induction machine is modelled using the commercial software Flux® 2D. The computations are done in steady state AC application. As shown in Fig. 1, only one pole is taken into account in order to reduce time computation. The stator outer and inner radius are respectively 86 mm and 54.3 mm. The outer rotor radius is 53.85 mm. The used mesh contains 26735 nodes. It is a 2nd order mesh.
Meshed 2D geometry & electric circuit.
The used materials are M800-100A for stator and rotor lamination, copper for stator winding and aluminium for rotor bars. There are 78 turns per stator phase whose resistance is 1.6 Ω, the stator end winding inductance is computed analytically [7] and it is equal to 1.5 mH. Rotor end ring short circuit portion is computed analytically with Trickey formulas [7,8], the temperature is not variable according to the slip. The supply voltage is 230 V rms and the used circuit is given in Fig. 1.
Classical method
The determination of the equivalent circuit scheme 1 is shown in Fig. 2, it is based on no-load and short circuit tests. Table 1 gives measurement and modelling results of these two tests.
Scheme 1 – equivalent circuit.
∼
Various reasons can explain the differences between measurements and modelling results: the analytical computation of end winding inductance and end ring impedance may be be approximate, the iron losses are not taken into account during the solving process of Flux® 2D. The finite element results will be taken into account as reference in this paper.
∼
The extracted results from FEM are the electromagnetic torque 𝛤e, ref, the total consumed reactive power Qtot, ref, and the stator phase current Is, ref. The magnetic flux computed from the integral of the normal flux density component on a path in the air gap is called 𝜙 ref .
The used equivalent circuit scheme 2 is shown in Fig. 3. This scheme allows us to split the stator and rotor leakage inductances. Since, in Flux® 2D, the iron losses are not taken into account, the resistance Rμ is not represented in this circuit. The magnetisation voltage E
ref
is computed from the magnetic flux 𝜙
ref
according to expression (1). The factor k is computed from the no load test.
Scheme 2 - Used equivalent circuit for parameter identification.
Equivalent circuit parameters variation versus slip.
Comparison between reference results and analytical.
The parameters x
s
,
Altair HyperStudy® multi-disciplinary design exploration and optimization tool has been used to answer this identification problem by optimization. The optimization problem is to identify the parameters values allowing to get the best fitting for each one of the four objectives. The constraints on the parameter variations are:
The optimization is done for each slip value individually, which leads, in our case, to solve a set of 15 optimizations. The Global Response Search Method (GRSM) optimization method [9] has been used to solve each individual optimization, while master-slave setup has been used to run automatically the set of optimizations. For each optimization, we compare analytical computations with the scheme 2 to FEM computation of I s , Q tot , 𝛤 e and E.
Figure 4 shows the variation of the equivalent circuit parameters versus slip. The equivalent circuit parameters are depending on state of load of the machine, so the saturation can be taken into account. The unexpected result is the variation shape of the magnetisation reactance Xμ, the value of this reactance has been decreased by 80% for slip values between 40% and 60%. The other parameters seem not to be impacted in this interval, as well as the results. As the model results are fitting well with FEM results it is difficult to say if this behaviour is physical or due to the model itself since lot of set of solutions can be possible, this is the limitation of this method. This last one can be improved by adding more results to fit.
Figure 5 shows the comparison between reference results from FEM computation and analytical computation with optimized equivalent circuit parameters obtained by the identification method.
Perspective and conclusion
In this paper, a new method to identify equivalent circuit parameters circuit of induction machine has been presented. This method allows to split leakage reactances and to take into account magnetic state of the machine with sufficient precision. This circuit can be used for system analysis or for studying transient behaviour of induction machines, especially high power machines. Future work will consist to use this method to extract the parameters versus frequency and voltage supply, by the way a 3D matrices to cover all the working loads can be obtained.