Open access peer-reviewed chapter - ONLINE FIRST

MNLR and ANFIS Based Inductance Profile Estimation for Switched Reluctance Motor

By Susitra Dhanarajalu

Submitted: November 17th 2020Reviewed: February 10th 2021Published: March 3rd 2021

DOI: 10.5772/intechopen.96555

Downloaded: 88

Abstract

This chapter aims in presenting the methods for the accurate estimation of highly non linear phase inductance profile of a switched reluctance motor (SRM). The magnetization characteristics of a test SRM is derived from the SRDaS (Switched Reluctance Design and Simulation) simulation software. Statistical interpolation based regression analysis and Artificial Intelligence (AI) techniques are used for developing the computationally efficient inductance model. Multi Variate Non linear Regression (MVNLR) from the class of regression analysis and Adaptive Neuro Fuzzy Inference System (ANFIS) under the class of AI are implemented and tested on the simulated data. Non linear Inductance profile L(I,θ) of SRM is successfully estimated for the complete working range of phase currents (Iph). At each Iph, L(I,θ) values are estimated and presented for one cycle of rotor position (θ). Estimated inductance profile based on the two proposed methods is observed to be in excellent correlation with the true value of data.

Keywords

  • SRM
  • electromagnetic profile
  • multivariate non-linear regression
  • ANFIS

1. Introduction

Over past two decades, there has been noticeable increase in the research publications on switched reluctance machine (SRM). The machine can be operated as both generator and motor by suitable control techniques. It has been proved from research publications that SRM is a valid alternative to conventional machines in almost all industrial applications. The characteristics of electromagnetic parameters are highly non-linear due to the following reasons; i) highly saturated working zone ii) eddy current and hysteresis current effects iii) double salient structure and iv) non uniform air gap. All these effects makes the machine’s flux linkage and torque as the non-linear function of phase current (Iph) and rotor position(θ). Flux linkage of the machine depends on its phase inductance. For the analysis and control of the machine, it is important to establish the accurate nonlinear mapping between its phase inductance, phase current and rotor position. The design and electrical analysis of the machine is greatly dependent on its electromagnetic behavior. Linear mathematical models are proposed by many publications that are not applicable for real time control. Non-linear models are developed by few researchers based on the following techniques; a) Analytical model [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] in which Fourier series based expressions are derived. This method is highly time consuming and not suitable for real time control applications. b) magnetic model [11] c) finite element model (FEM) [12, 13] provides accurate results whereas it involves complicated mathematics computation. d) Artificial neural network models [14, 15, 16, 17, 18, 19, 20, 21, 22].

All these models have either fine precision or good computation speed. Both the benefits are not satisfied in any of these models. Few of the recent research publications have reported using the Adaptive Neuro fuzzy inference system (ANFIS) techniques for the computation of magnetic parameters [23, 24, 25] and estimation of rotor position [26]. On the overview of the publications on modeling of SRM, it is observed that none of the papers have paid attention in using Multivariate nonlinear regression technique (MVNLRT) for estimating its nonlinear inductance model. In this chapter, non-linear inductance model of SRM is developed based on the MVNLRT and ANFIS. Also the comparison between the results of ANFIS and MVNLRT inductance models are presented. This chapter is organized as follows; Section II discusses the mathematical model of SRM and the necessity of inductance modeling. The nonlinear inductance model based on MVNLRT and the one based on ANFIS are presented in section III and IV respectively. The comparative analysis of both the models is presented in section V and the inferential remarks are given in section VI.

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2. Mathematical model of SRG and the necessity of inductance modeling

SRM is a singly excited and doubly salient machine shown in Figure 1. The number of phases wound on the stator poles differs from machine to machine. There is certain number of suitable combinations of stator and rotor poles thus giving many machine configurations. The phase windings are excited sequentially for rotating the rotor in desired direction. For efficient control, the phase windings are excited with the current pulses based on the rotor position.

Figure 1.

Three phase 6/4 pole SRM configuration.

The electromagnetic characteristics of the machine vary depending on the relative rotor position with respect to stator pole axis and magnitude of current in the excited phase winding. An ideal machine is one with equal stator and rotor pole axis and saturation of the core and leakage fluxes are neglected. In such a ideal machine, the variation of phase inductance is triangular in shape. However, in a practical machine, the variation is highly nonlinear. When the stator and rotor poles are coincident (aligned position), the inductance is maximum and minimum inductance is experienced when the rotor inter polar axis coincides with the excited stator pole axis (unaligned position). The position of rotor between these two positions is called intermediate positions. Upon exciting a particular stator phase, its nearest pair of rotor poles tends to move towards the path of minimum reluctance along the excited stator poles by developing a required torque. The magnitude of torque developed depends on the magnitude of the excited current and rate of change of inductance with rotor position. Accurate rotor position sensors, power converter and a controller with good control technique are required for the successful operation and precise control of SRM [27].

Owing to the high non-linearity of the machine, a mathematical model describing the behavior of the machine is required. This model can be used as a base for simulation and verification of static and dynamic characteristics of SRM.

2.1 Energy balance equation

The conversion of electrical energy to mechanical energy and vice versa by an electromechanical energy converter is illustrated by the energy-balance scheme shown in Figure 2.

Figure 2.

Electromechanical energy conversion model.

A magnetic system couples the electrical and mechanical systems. The electromechanical energy converter illustrates the relationship between the electrical energy, mechanical energy, energy losses, energy stored and energies transferred between the electrical and mechanical systems. From Figure 2, and making an assumption that Wiron = 0 and Wleak = 0, the following equations are developed.

Wea=WeiWcuWleakE1
Wemec=WeaWfieldWironE2
Wmech=WemecWfriWinerE3
Energy input=WeaE4
Energy losses=Wcu+Wiron+WfriE5
Energy stored in the system=Wleak+Wfield+WinerE6
Energy output=WmechE7

Thus Energy output = Energy input-Energy losses-Energy stored.

In an ideal system, if the energy losses are neglected, the output energy is equal to the energy input into the system minus energy stored in the various components of the system. However, the stored energies can be recovered through regenerative process.

2.2 Equation of phase voltage

Electromagnetic characteristics of a 6/4 pole SRM is shown in Figure 3. The illustration shows the mapping among the flux linkage (ψ), current (i) in excited phase and rotor position (θ) of SRM. The high non-linearity behavior of the machine is clearly visible from the figure. The curve is linear for small values of excitation current. As the rotor rotates from unaligned to aligned position, the mapping among the motor parameters becomes highly nonlinear.

Figure 3.

Magnetization characteristics of three phase 6/4 pole SRM.

Figure 4 shows the equivalent circuit of an excited phase of the motor.

Figure 4.

Electrical equivalent circuit of the SRM.

According to Figure 4, the instantaneous voltage across the terminals of a phase of SRM winding related to the flux linked in the winding is given by,

Vpht=itphrph+phiph,θphdtE8
ψph=VphiphrphdtE9

Due to the double salient structure and effect of magnetic saturation, the flux linkage (ψph) and inductance of excited motor phase winding varies as a function of both rotor position (θph) and phase current (iph). Therefore (8) is written as given in (10).

Vph=irphph+dLphiph,θiphdtE10
Vph=irphph+Lphdiphdt+iphLphθdtE11
Vph=irphph+Lphdiphdt+12iphdLphω+12iphωdLphE12

2.3 Torque equation

When a phase of the SRM is excited by a voltage source, the instantaneous power is obtained by multiplying (12) by iph.

P=Vphiph=i2rphph+iphLphdiphdt+ωi2phdLphE13
Vphiph=i2rphph+ddt12Lphi2ph+12ωi2phdLphE14
Vphiph=i2phrph+dwfdt+TmωE15

where

Wf=12Lphiph2E16

is the energy stored in the magnetic field.

and

Tm=12i2phdLphE17

is the expression for instantaneous torque produced in the energized phase winding. In (15), The instantaneous electrical power supplied to the motor is given on the left hand side of the equation. On the right hand side of equation, the first term represents the power loss due to phase resistance, the derivative of stored energy in the magnetic field is represented in the second term and the mechanical power developed by the motor is represented in the third term.

The stator and rotor pole arc combinations are equal in SRM. The machine has triangular shaped inductance characteristics. The inductance value is maximum at aligned position and minimum at unaligned position. For a 6/4 pole SRM, an inductance profile for unequal stator and rotor pole arc combinations are shown in Figure 5. From the Eq. (17), it is clear that SRM develops positive torque when the phase is excited during the period of rising inductance, when dL>0. Negative torque is developed when the phase is excited during the period of the falling inductance, when dL<0and torque developed is zero when the phase is excited during the period of constant inductance, when dL=0. The machine can transform reversible modes of motoring and generating operations by suitable control of the switches in power converter. From this, it is understood that the operating mode of the machine greatly depends on its phase inductance value. Therefore inductance needs to be modeled accurately for real time applications.

Figure 5.

Variation of inductance for one rotor pole pitch.

The torque developed can be normally expressed in terms of the change of coenergy with respect to rotor position. The Ψicharacteristics of the SRM for a given rotor position is shown in Figure 6. These characteristics can either be obtained by conducting an experiment on the SRM or by performing magnetic field analysis. The area representing stored field energy (Wf) and co-energy (Wc) are marked in Figure 6(a) for a given phase current. The area below the ψi, characteristics can be written as.

Figure 6.

Magnetic characteristics of SRM. (a) Electromechanical energy conversion. (b) ψ i characteristics for a given rotor position.

Wc=0iψdiE18

Consider the movement of the rotor through an infinitesimal displacement Δθwhen the current is held constant. The ψicharacteristics of SRM for the given rotor position is shown in Figure 6(b). During the movement of the rotor, a definite amount of mechanical work is done by the motor. The electrical energy (ΔWe) fed to the motor is partially used to produce mechanical motion (ΔWm) and a part of the energy gets stored in the magnetic field (ΔWf). From Figure 6(b), the change in electrical energy input is,

ΔWe=area WXYZE19

and the change in stored field energy is

ΔWf=areaOXYareaOWZE20

and the mechanical work done is,

ΔWm=ΔWeΔWf=θ=Area WXYZ(areaOXYareaOWZ))=Area WXYZareaOXYareaOWZ=AreaOWXE21

The area OWX represents the change in the co-energy (ΔWc) and the torque can be calculated as.

T=ΔWcΔθE22

It can be observed that only a part of the energy input into the motor is utilized for torque production. If the ψicharacteristics were highly non-linear in the aligned position and linear in the unaligned position, then, the magnitude of the torque developed would be higher compared with the situation when the ψicharacteristics are linear for both the positions.

3. Inductance modeling of SRM using multivariate nonlinear regression technique

Regression analysis uses many techniques for modeling and analyzing parameters. It is useful in accurate mapping of dependent and independent variables even under unknown physical process. In non-linear regression model, the observed data are expressed as a nonlinear function of parameters and proved to be more accurate even for extrapolated data range. A nonlinear model is expressed as given in (23).

y=fXβ+εE23

where

  • y - Vector of the observed response variables (n × 1).

  • X - matrix determined by the predictor variables (n × p).

  • β - vector of unknown parameters to be estimated (p × 1).

  • f- function of Xand β.

  • ε- vector of independent, identically distributed random disturbances (n × 1).

Multivariate analysis encompasses the observation and analysis of more than one outcome variable and gives an excellent solution for nonlinear problems. This approach executes well in high dimensional problems especially when the available data set is of very small size. In this section MVNLR model is used for estimating the inductance parameters. Coefficients of predictor variables are estimated using least square method. The inductance values between an unaligned and aligned position from the magnetization characteristics of [28] has been used in exploring the non-linear relationships between the phase inductance L(i,θ), phase current(I) and rotor position(θ) and is expressed in terms of least square polynomial approximation of third order and the MVNLRT based inductance model of the three phase,6/4 SRG is expressed by the Eq. (24).

L1L2L3= 2.605×1069.645×1059.933×1042.631×1069.741×10510.032×1042.579×1069.549×1059.834×104I3I2I+5.518×1081.914×1073.752×1045.573×1081.933×1073.79×1045.463×1081.894×1073.714×104θ3θ2θ+5.825×1071.881×1072.745×1065.883×1071.91×1072.772×1065.767×1071.862×1072.718×106I2θIθ2+1.095×1031.106×1031.084×103E24

Comparison of true values Vs MVNLR estimated values is shown in Figure 7. From the analysis, it is clearly evident that the MVNLR predicted inductance values shows a strong agreement with the true data values used for developing the model.

Figure 7.

True data set vs MVNLR predicted data.

4. Inductance modeling of SRG using adaptive neuro -fuzzy inference system

The ANFIS is an adaptive neuro - fuzzy inference system (Jyh-Shing et al. 1993) [29]. The fuzzy inference system (FIS) has been developed for the inductance estimation of SR machine. FIS has two inputs and single output. Phase current (Iph) and rotor position (θ) are the input variables and phase inductance, L(I,θ) is the output variable. The input variables are fuzzified using 12 triangular membership functions. Fuzzy rule base has 12*12 (144) if-then rules. The ANFIS architecture is shown in Figure 8. There are 5 layers in the ANFIS network.

Figure 8.

ANFIS network structure.

Layer 1:Both the inputs are fuzzified with 12 triangular membership functions. The output of input membership function 1 is Ok1=μAkiand the output of input membership function 2 is, Ok2=μBkθ, i and θ are the input to nodes 1 and 2, respectively. Ak and Bk are the linguistic labels (mf1, mf2… mf12.) associated with the node functions.

The output of the input membership functions specifies the degree to which the given i and θ satisfies the quantifier Ak and Bk. Triangular shaped membership functions μAkiand μBkθare used with a maximum equal to 1 and a minimum equal to 0. The generalized triangular membership function of the current is given by

μAki=0iakbkakckickbk0iakakibkbkickckiE25

Similarly, the generalized triangular shaped membership function of the rotor position is given by

μBkθ=0θakbkakckθckbk0θakakθbkbkθckckθE26

where ak, bk and ck are the adaptable variables. The triangular shaped functions varies with the change in the parameter values.

Layer 2:Fuzzy AND operator is implemented in this layer.

Wk=μAki×μBkθE27

where k = 1, 2…12;

Layer 3:Scaling or normalizing of firing strengths takes place in this layer.

W¯k=Ok4=W¯kfk=W¯kmk+nkθ+rkE28

Layer 4:The output of the fourth layer comprises a linear combination of the inputs multiplied by the normalized firing strength. Output of this layer is given by.

Ok4=W¯kfk=W¯kmki+nkθ+rkE29

Output of layer 3 is W¯k.The modifiable variables mk, nk and rk are known as consequent parameters.

Layer 5:This layer is a simple summation of the outputs in the previous layer. The phase inductance L(i,θ) is obtained from the overall output.

Ok5=Wk¯fk=kWkfkkWkE30

The magnetic parameter values are considered during the period of unaligned to aligned rotor position for developing the inductance model. The range of the current and rotor position during this period are 0 ≤ I ≤ 25 A and 0 ≤ θ ≤ 45 deg.

The ANFIS information are as follows.

Total number of parameters: 504; Number of nonlinear parameters: 72

Number of linear parameters: 432; Number of fuzzy rules: 144

Number of training data pairs: 182; Number of checking data pairs: 0

Number of nodes: 341.

The triangular membership functions for the two inputs: current and rotor position are given in Figure 9(a) and (b) respectively.

Figure 9.

Fuzzy triangular membership functions for the input parameters. (a) Phase current. (b) Rotor position.

The 144 rules framed in ANFIS is shown in Figure 10. The fuzzy inference process is interpreted from this rule viewer. The AFIS surface mapping between input and output parameters is depicted in Figure 11. For low values of current, the variation in inductance is observed to be linear. After saturation, the variation is highly non linear. Figure 12 shows the comparison of the original test data with the ANFIS predicted data. From this comparison, it is clear that the inductance estimation from ANFIS is in very good concurrence with the test values of inductance. Figure 13 shows the MATLAB simulink model of ANFIS inductance predictor.

Figure 10.

Rule viewer - ANFIS.

Figure 11.

Mapping surface of L(i,ϴ).

Figure 12.

True data vs ANFIS predicted data.

Figure 13.

ANFIS inductance predictor from Simulink.

5. Comparison of MVNLRT and ANFIS based inductance models

Both the proposed inductance models based on MVNLRT and ANFIS uses the data from the magnetization characteristics of the machine. Figure 14 shows the comparison chart of the inductance values obtained from the ANFIS and MVNLRT models and Figure 15 illustrates the comparison of these two models along with the actual inductance values. For both the methods, the predicted and true values are compared at various phase current values and the same is shown in Figure 16. Root-mean-squared error (RMSE), Mean Absolute Error (MAE) and Maximum Absolute value Error (MAVE) for both the methods at different currents are given in Table 1. The RMSE for ANFIS and MVNLRT for full range of currents is shown in Figure 17. From the careful observation of comparison charts and error values, it is inferred that both ANFIS and MVNLRT models produces accurate mapping between input and output parameters. Among these two methods, ANFIS model has proved to have developed the model with higher precision and with least errors.

Figure 14.

MVNLR vs ANFIS models.

Figure 15.

True vs MVNLR vs ANFIS data.

Figure 16.

MVNLR VS ANFIS at various currents.

Current in ampsFor θ varying from 0 to 45 degrees at each phase current
ANFIS Estimator(Error in Henry)MVNRT Estimator(Error in Henry)
Labs maxLabs meanLRMSELabs maxLabs meanLRMSE
02.7E-033E-048E-041.1E-021.6E-034E-03
59E-046.5E-051.7E-043.3E-031.4E-043.6E-04
107.1E-041.6E-054.2E-051.3E-036.6E-051.7E-04
159.7E-041.6E-044.2E-041.2E-032.9E-047.6E-04
208.5E-041.3E-043.4E-047.5E-041.5E-044E-04
259.3E-048.1E-052.2E-042E-032.9E-047.7E-04

Table 1.

RMSE, MAE and MAVE at different currents and rotor position.

Figure 17.

L_RMSE - ANFIS vs MVNLR.

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6. Conclusion

This paper has presented two methods of computationally efficient modeling techniques for a 3 phase,6/4 switched reluctance generator for obtaining the nonlinear inductance model by means of adaptive neuro fuzzy inference system and Multivariate nonlinear regression technique. The developed inductance profile based on both these methods is trained with the same set of true data. Both the estimated models depicts the nonlinear relation between phase winding inductance at various operating currents and at various rotor positions at each current. MVNLR predicted model is in good concurrence with the true model. The ANFIS model shows an excellent concurrence with true data and reports only an insignificant error throughout the data range. From the observations and analysis of results obtained, it is inferred that both the techniques are proficient in estimation of SRM inductance parameter within acceptable accuracy limits. The methods are highly certain in presenting a superior performance when applied to modeling, prediction and control. Both are very simple, easy to implement and are sure to replace any computationally intensive numerical model of SRM. Of the two methods proposed in this chapter, ANFIS has high computational speed, simple structure, least training least epoch, faster learning, good convergence and presents a superior performance. The computational accuracy and easiness of estimation validate that ANFIS is highly competent tool for real time control of SR machine.

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Susitra Dhanarajalu (March 3rd 2021). MNLR and ANFIS Based Inductance Profile Estimation for Switched Reluctance Motor [Online First], IntechOpen, DOI: 10.5772/intechopen.96555. Available from:

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