Minimum Permeance Estimation of Variable Reluctance Machines by Using Neural Networks

A new approach was studied in tbis paper to calculate minimum permeance (Pmin) of variable reluctance machines (VRM). Finite element method (FEM) and neural network (NN) were employed together for estimation. The data collected by an electromagnetic finite element software (Flux 2D) were used to train NN. Trained NN was tested by another data set which are not in the training data set. Total estimation error in the test set was observed less than 2.5%. A similar study was performed with the data set collected using flux tube analysis (FT A). In this case, much larger data set was constructed by IT A since this method allows to generate larger data set. After training NN by this data set, it was tested by a test set generated by IT A. The total estimation error was observed less than 5%. I.Introduction The variable-reluctance machine (VRM) is a doubly salient synchronous machine used as aerospace motors, generators in wind energy systems and in other applications, ranging from fractional horse power up to several hundred kilowatts. The design of machine is complicated because of its strong spatial and magnetic nonlinearities, combined with a large number of degrees of freedom [I]. One of the minor problems in the design process is the calculation of minimum permeance (Pmin). In the literature, there are two methods commonly used. These are : finite element method (FEM) and flux tube analysis (FTA). The main differences between these methods are accuracy and calculation time. Although FEM can make an accurate estimatio~ it takes a long time. On the other hand, the calculation time can be made shorter by using FTA, but this causes loss of accuracy. The study in this paper aims to estimate Pmin in a significantly short time period without losing too much accuracy by using FEM and neural networks (NN) together. 2.VariableReluctance Machine(VRM) Fig.l illustrates the rudiments of a VRM and one of its driving circuits. The diagram illustrates eight stator poles and six rotor poles. In the case illustrated there are four separate circuits or 'phases.' VRM may also be designed, depending on the application, with one, two, three, four or even more phases. The salient poles on the stator carry concentrated windings of particularly simple form, but the salient poles of the rotor carry no windings of any kind. Both stator and rotor cores are constructed from laminated material, to reduce iron losses and …


I.Introduction
The variable-reluctance machine (VRM) is a doubly salient synchronous machine used as aerospace motors, generators in wind energy systems and in other applications, ranging from fractional horse power up to several hundred kilowatts.The design of machine is complicated because of its strong spatial and magnetic nonlinearities, combined with a large number of degrees of freedom [I].One of the minor problems in the design process is the calculation of minimum permeance (Pmin).In the literature, there are two methods commonly used.These are : finite element method (FEM) and flux tube analysis (FTA).The main differences between these methods are accuracy and calculation time.Although FEM can make an accurate estimatio~it takes a long time.On the other hand, the calculation time can be made shorter by using FTA, but this causes loss of accuracy.The study in this paper aims to estimate Pmin in a significantly short time period without losing too much accuracy by using FEM and neural networks (NN) together.

2.VariableReluctance Machine(VRM)
Fig. l illustrates the rudiments of a VRM and one of its driving circuits.The diagram illustrates eight stator poles and six rotor poles.In the case illustrated there are four separate circuits or 'phases.' VRM may also be designed, depending on the application, with one, two, three, four or even more phases.The salient poles on the stator carry concentrated windings of particularly simple form, but the salient poles of the rotor carry no windings of any kind.Both stator and rotor cores are constructed from laminated material, to reduce iron losses and for manufacturing convenience.As seen in Fig. I, diametrically opposite stator poles are excited simultaneously, and excitation of one pair of poles causes a pair of rotor poles to be attracted magnetically into alignment producing the basic torque of the device.The figure implies excitation of poles AA', and subsequently poles BB', are excited then the rotor poles bb' would move into alignment with them (with clockwise rotation).The switching sequence of the stator circuits is determined by the rotor position using some suitable transducer [2].l.Aligned position When any pair of rotor poles is exactly aligned with stator poles of any phase, that phase is said to be in the aligned position.

2.Unaligned position
When the interpolar axis of the rotor(the axis exactly in the middle of two subsequent rotor poles) is aligned with the poles of any phase, this phase is called in the "unaligned position".

Dimensions Effecting Pmin
Basic machine dimensions which effect Pmin has been shown in Fig. 3 There is a nonlinear relation between these dimensions and Pmin.Two approaches can be used to relate geometry and Pmin.The first one is to use dimensions directly Pmin=fl:0r, 0s, Rr, Rs, Lrph, ro, rsbi) The second one is to use proper ratios of dimensions as proposed in reference [5].
In addition to these dimensions, the length of the machine and the geometric shape of the poles are other important factors which need to be taken into consideration in some manner.In this study, since two dimensional finite element software has been used, the effect of length was not considered.
Moreover, poles are assumed in plain shape as shown in Fig. 3.

Minimum Permeance Calculation Methods.
As it is mentioned earlier, the uncertainty in the flux path of the magnetic circuit makes an accurate analytical calculation very difficult.Flux tube analysis is used to calculate Pmin analytically, which gives an approximate result.A comprehensive study about this method can be found in reference [4].On the other hand, an accurate calculation of Pmin is a very important step in the accurate calculation of motor torque and current.
In order to enhance the accuracy in the calculation, finite element method (FEM) is the most commonly used numerical analysis technique.The price of the accuracy is longer calculation time than that of flux tube analysis.If this calculation is a routine task performed in a motor design process many times, the application of FEM to each different geometry is obviously not feasible.Using FEM and interpolation techniques reduce calculation time, but this reduces the accuracy as well.Ui(h+ 1) = LWij(h+ 1) Yj(l) + 9i(h + 1) where, wij(h + 1)-weight between i th neuron of layer h + 1 andt neuron of layer h.9i(h+ 1)-threshold to the i th neuron in (h+ 1)th layer.uj(h+ 1)-input to the i th neuron in h+ 1thlayer The quantity 8Ep/Bwij(h)is calculated by the folowing expressions.
where OJ(H)= -(Ti -Yj ( H ) ) The above algorithm is commonly known as error back propagation.The constant 11 is the learning step while the constant v is the momentum gain.~wij(h) indicates the weight change in the previous iteration.
Weights are iteratively updated for all P training patterns.The training process may require many such sweeps.Suffcient learning is achieved when the total error function, Erota1=LA p=1,2,3 ... P summed over the set of all p training patterns goes below a preselected value E. [3] 4. The proposed method for calculation of Pmin.
The main objective in this study is to reduce the calculation time significantly while increasing the accuracy as much as possible.
In order to carry out this objective, FEM and NN will be employed together.Flux-2D (2 Dimensional Electromagnetic FEM package) has been used as a training data preparing tool.The accuracy of the proposed method has been tested by using the same tool.
Since FTA is an analytical way to calculate Pmin, data set generation using this method is very easy and allows us to generate much larger data set than FEM does.In order to see how this large but inaccurate data set effects generalization of NN, another data set was generated by using FTA, and then NN was trained by using these data.
In order to link real machine dimensions with their corresponding Pmin, reasonable machine dimensions have to be determined as a first step.After having determined reasonable machine dimensions by using criteria given in reference [4],corresponding Pmins were calculated and normalized between 0 and 1. Basically, three kinds of data set were generated.These can be summarized as follows; 1.Data Set Type.I.
2.Data set Type II.
Based on the relationship ofPmin= f{kl,k2), the data were generated by using Flux tube Analysis as proposed in reference [4).By randomly changing k 1 and k2 values within the same range as above, the corresponding Pmins were calculated analytically.

3.Data set Type ill
The Pmin=f{er, 0s, RI, Rs, Lrph, ro, rsbi) relation was used to produce this data set.It was generated at the same time with the data set Type II.Pmin was calculated by using FTA for each random dimension.

Fig. l
Fig.l Elements of 4-phase VRM showing one circuit.

Fig. 2 .
Fig.2.Flux distribution of a VRM which is in unaligned position.Minimum permeance is the permeance of a VRM magnetic circuit where the associated Fig.4 Topology of a 2-layer feedforward ANN.Artificial neural networks (ANNs) are computing systems whose structures are inspired by a simplified model of the human brain.A typical 2-layer feedforward ANN is given in Fig.4.It consists of an input layer, Testing of Neural Network.Data set type I consists of 35 data to use training.In order to make the most use the data available, following training method was used.4 patterns of 35 were held out as a test file each time.Remaining 31 patterns were used to train the NN by considering overtraining pOSSloilitycarefully. Whenever overtraining point was reached in terms of test file error, the training was stopped.Then 4 patterns were included to the training set, but other 4 patterns were held out as a test file.Then it was contiuned to train the NN until over training point.This process was repeated until all the patterns were used as a test pattern.