HUNGARIAN JOURNAL OF INDUSTRY AND CHEMISTRY VESZPRÉM Vol. 42(2) pp. 109–113 (2014) STABILITY AND PARAMETER SENSITIVITY ANALYSES OF AN INDUCTION MOTOR ATTILA FODOR1 , ROLAND BÁLINTB1 , ATTILA MAGYAR1 , AND GÁBOR SZEDERKÉNYI2 1Department of Electrical Engineering and Information Systems, University of Pannonia, Egyetem u. 10., Veszprém, 8200, HUNGARY 2Pazmány Péter Catholic University, Práter u. 50/a, Budapest, 1083, HUNGARY BE-mail: balint.roland27@gmail.com A simple dynamical model of an induction motor is derived and analyzed in this paper based on engineering principles that describe the mechanical phenomena together with the electrical model. The used state space model consists of nonlinear state equations. The model has been verified under the usual controlled operating conditions when the speed is controlled. The effect of load on the controlled induction motor has been analyzed by simulation. The sensitivity analysis of the induction motor has been applied to determine the model parameters to be estimated. Keywords: induction motor, stability analysis, sensitivity analysis Introduction Induction motors (IM) are the most commonly used elec- trical rotating machines in several industrial applications. Irrespective of size and the application area, these motors share the most important dynamical properties, and their dynamical models have a similar structure. Because of the specialties and great practical impor- tance of IMs in industrial applications, their modelling for control purposes is well investigated in the literature. Besides basic textbooks [1-3], there are several papers that describe the modelling and use the developed mod- els for the design of different types of controllers: vec- tor control [1, 4], sensorless vector control [5] and direct torque control (DTC) [6]. The aim of this paper is to build a simple dynamical model of the IM and to perform its parameter sensitivity analysis. The results of this analy- sis will be the basis of the next step since the final aim of our study is to estimate the parameters of the IM and design a controller that can control the speed and torque of the IM. The state space model has been implemented in the Matlab/Simulink environment which enables us to analyze the parametric sensitivity based on simulation ex- periments. Nonlinear Model of an Induction Motor In this section a state space model of an induction mo- tor (IM) is presented. The model development is largely based on Refs.[1, 8-10]. For constructing the IM model, the following modelling assumptions are made: 1. a symmetrical triphase stator winding system is as- sumed, 2. the flux density is radial in the air gap, 3. the copper loss and the slots in the machine can be neglected, 4. the spatial distribution of the stator fluxes and aper- tures wave are considered to be sinusoidal, 5. stator and rotor permeability are assumed to be infi- nite with linear magnetic properties. According to the above modelling conditions the mathematical description of the IM is developed through the space-vector theory. If the voltage of the stator is pre- sumed to be the input excitation of the machine, then the spatial distribution along the stator of the x phase volt- age can be described by the complex vector vsx(t). We can determine the orientation of the voltage vector vs, the direction of the respective phase axis and the voltage po- larity. is(t) = 2 3 ( a0 · isa(t) + a1 · isb(t) + a2 · isc(t) ) = √ 2 · ieff(t) ·ejω0t+ π 2 +ϕi, (1) where a is the ej120 o vector and isa, isb and isc are the following: isa(t) = Re(a 0 · is(t)) = Re(is(t)), isb(t) = Re(a 2 · is(t)),and isc(t) = Re(a 1 · is(t)). In Eq.(1), 2/3 is the normalizing factor. The flux den- sity distribution can be obtained by integrating the cur- rent density wave along the cylinder of the stator. The flux linkage wave is a system variable, because it con- tains detailed information about the winding geometry. 110 Figure 1: The equivalent circuit of the IM The rotating flux density wave induces voltages in the in- dividual stator windings. Thus stator voltage vs(t) can be represented as the overall distributed voltages in all phase windings: vs(t) = 2 3 ( a0 ·vsa(t) + a1 ·vsb(t) + a2 ·vsc(t) ) = √ 2 ·veff(t) ·ejω0t+ π 2 +ϕu, (2) where a is the ej120 o vector and isa, isb and isc are the following: vsa(t) = Re(a 0 ·us(t)) = Re(us(t)), vsb(t) = Re(a 2 ·us(t)),and vsc(t) = Re(a 1 ·us(t)). Considering the stator of the IM as the primer side of the transformer, then using Kirchoff’s voltage law the follow- ing equation can be written (Fig.1): vs(t) = is(t) ·Rs + dφs(t) dt . (3) As for the secondary side of the transformer, it can be deduced that the same relationship is true for the rotor side space vectors: vr(t) = ir(t) ·Rr + dφr(t) dt = 0. (4) Eqs.(3) and (4) describe the electromagnetic interaction as the connection of first order dynamical subsystems: φs(t) = is(t) ·Ls + ir(t) ·Lm,and (5) φr(t) = is(t) ·Lm + ir(t) ·Lr. (6) Since four complex variables (is(t) , ir(t) , φs(t) , and φr(t)) are presented in Eqs.(5) and (6), flux equations are needed to complete the relationship between them. The mechanical power (Pmech(t)) of the IM can be defined as: Pmech(t) = Wmech(t) dt , (7) where the mechanical energy (Pmech(t)) in the rotating system can be given by the following expression: Pmech(t) = Wmech(t) dt = Tmech(t) ·ωr, (8) where Te(t) is the torque and ωr is the angular velocity of the IM. Afterwards the energy balance of the IM is as follows: We = Wmech + WR + WField, (9) Figure 2: The equivalent circuit of the d-axis of the IM Figure 3: The equivalent circuit of the q-axis of the IM where Pe = We(t) dt = 3 2 Re (us · is + ur · ir) (10) is the input electrical power, PR = WR(t) dt = 3 2 Re ( Rs · |is|2 + Rr · |ir|2 ) (11) represents the resistive power losses and PField = WField(t) dt = 3 2 Re ( dφs dt is + dφr dt ir ) (12) is the air gap power. Using Eqs.(8-12), Pmech(t) = Tmech(t) ·ωr = 3 2 · Lm Lr ·φs(t)× is(t),and (13) Te = 1.5p · (φdsiqs −φqsids). (14) The equivalent circuit of the IM can be decomposed to di- rect axis and quadratic axis components by Park’s trans- formation as shown in Figs.2 and 3. The actual terminal voltage v of the windings can be written in the form v = ± J∑ j=1 (Rj ij)± J∑ j=1 ( dφj dt ) , (15) where ij are the currents, Rj are the winding resistances, and φj are the flux linkages. The positive directions of the stator currents point out of the IM terminals. By composing the d and q axes of the IM the follow- ing equations can be written: vqs = Rs · iqs + φqs dt + ω ·φds, (16) vds = Rs · ids + φds dt −ω ·φqs, (17) vqr = R ′ r · i ′ ds + φ′dr dt + (ω −ωr) ·φ′dr,and (18) vdr = R ′ r · i ′ dr + φ′dr dt − (ω −ωr) ·φ′qr. (19) 111 Figure 4: Response to the step change of the speed-controlled IM dωm dt = 1 2H (Te −F ·ωm −Tmech) (20) where ω is the reference frame angular velocity, ωr is the electrical angular velocity, φqs = Ls · iqs + Lmi′qr, (21) φds = Ls · ids + Lmi′dr, (22) φqr = Lr · i′qr + Lmi ′ qs,and (23) φdr = Lr · i′dr + Lmi ′ ds. (24) The above model can be written in state space form by ex- pressing the time derivative of the fluxes and ω from the voltage and swing equations Eqs.(21-24). The nonlinear state space model of the IM is given by Eqs.(25-29): dφqs dt = −Rs Ls − L2m L′r ·φqs Rs ·Lm L′r(Ls − L2m L′r ) ·φ′qr − ω ·φds + vqs, (25) dφds dt = −Rs Ls − L2m L′r ·φds + Rs ·Lm L′r(Ls − L2m L′r ) ·φ′dr − ω ·φqs + vds, (26) dφ′qr dt = −R′r Lm − L′r·Ls Lm ·φqs + −R′r ·Ls Lm · (Lm − L′r·Ls Lm ) ·φ′qr − ω ·φ′dr + ωr ·φ ′ dr + vqr, (27) dφ′dr dt = −R′r Lm − L′r·Ls Lm ·φds + −R′r ·Ls Lm · (Lm − L′r·Ls Lm ) ·φ′dr + ω ·φ′qr −ωr ·φ ′ qr + vdr,and (28) dωr dt = 1 2H · 1.5 ·p ·Lm Lm · (Ls − L2m L′r ) ·φ′dr ·φqs − 1 2H · 1.5 ·p ·Lm Lm · (Ls − L2m L′r ) ·φ′qr ·φds − F 2H ·ωr − Tm 2H . (29) The state vector of the above model is x=[φqs, φds, φ′qr, φ ′ dr, ωr] T ∈ R5, and the input variables are organized in terms of the the input vector u=[vq, vd, −Tmech]T ∈ R3. It is assumed that all the state variables can be measured i.e. y = x. Model Validation The dynamical properties of the IM have been investi- gated. The response of the speed-controlled motor has been tested under step-like changes. The simulation re- sults are shown in Fig.4, where the fluxes (φqs , φds, φ′qr, and φ′dr) and the angular velocity (ω) are shown. Model Analysis The above model Eqs.(21-24) has been verified by simu- lation against engineering intuition. Local Stability Analysis As the final aim of our research is to estimate the param- eters of a particular Grundfos IM, first of all the resis- tances (Rs and Rr) of the IM were measured. Afterwards the values of the inductances (Lls, Llr and Lm) and the mechanical parameters (H and F ) of a similar IM with similar Rs and Rr found in the literature have been used. The parameters used during the model and the sensitivity analyses are the following: Rs = 0.196 Ohm Rr = 0.0191 Ohm Lls = 0.0397 H Llr = 0.0397 H Lm = 1.354 H H = 0.095 F = 0.0548 p = 1 (30) 112 Figure 5: The model state variables for a ±50% change of Lm It can easily be seen that the IM model is nonlinear since there are products of two state variables in the following equations: • Eq.(27): φ′dr is multiplied by ωr • Eq.(28): φ′qr is multiplied by ωr • Eq.(29): φ′dr is multiplied by φqs • Eq.(29): φ′qr is multiplied by φ′ds For the local stability analysis we have to calculate the eigenvalues of the system. The examined equilibrium point is φqs = −0.744 Vs φds = 1.0287 Vs φ′qr = −0.174 Vs (31) φ′dr = −0.087 Vs ωr = 48.477 s −1 The numerical value of the Jacobian of the nonlin- ear model (i.e. the state matrix of the locally linearized model) is as follows: −2.504 −50 2.4328 0 0 50 −2.504 0 2.4328 0 0.2370 0 −0.244 −1.522 0 0 0.2370 1.5222 −0.244 0 −8.617 17.077 0 0 −0.288 . (32) The eigenvalues of the state matrix of the linearized sys- tems are: λ1,2 = −2.504 ± j49.98, λ3,4 = −0.243 ± j1.534,and λ5 = −0.288 (33) Figure 6: The model state variables for a ±50% change in Rs It is apparent that the real parts of the eigenvalues are negative with small magnitudes. Parameter Sensitivity Analysis the sensitivity of the nonlinear model to the mutual induc- tance has been investigated. The steady state value of the system variables does not change (as is apparent in Fig.5) even for a considerably large change in Lm. Sensitivity analysis of the inductances Llr, Lls of the stator and ro- tor, resistances of the stator Rs and the damping constant F has also been investigated. In this investigation it can be seen that the values of the state variables were changed a bit, as shown in Fig.6. The analysis of the resistances of the rotor R′r and the inertia H of the rotor showed that every value of the state variables changed significantly, as shown in Fig.7. As a final result of the sensitivity analysis, we can de- fine the following groups of parameters: • Not sensitive: Mutual inductance Lm. Since the state space model of interest is insensitive in this respect, the values of this parameter cannot be reliably deter- mined from measurement data using any parameter estimation method. • Sensitive: These sensitive parameters are candidates for parameter estimation. – Less: inductances Llr and Lls, resistance of the stator Rs and the damping constant F . – More: resistances of the rotor R′r and the iner- tia H of the rotor. • Critically sensitive: none. 113 Figure 7: The model state variables for a ±50% change in H Conclusion The simple nonlinear dynamical model of an IM has been investigated in this paper. It has been shown that the model is locally asymptotically stable with regards to a physically meaningful equilibrium state. The effect of the controlled generator has been analyzed by simulation us- ing a traditional PI controller. It has been found that the controlled system is stable and can follow the set-point changes. Seven parameters of the model were selected for sensitivity analysis, and the sensitivity of the state vari- ables has been investigated. As a result, the parameters were partitioned into three groups. Based on the results presented here, a future aim is to estimate the parameters of the model for a real system from measurements. The sensitivity analysis enables us to select the candidates for estimation that are inductances Llr and Lls, resistances Rs and Rr, the damping constant F and the inertia H. An additional future aim is to develop a model for con- trol purposes and investigate different controllers before applying them on real systems. ACKNOWLEDGEMENT We acknowledge the financial support of this work by the Hungarian State and the European Union under the TAMOP-4.2.2.A-11/1/KONV-2012-0072 project. We ac- knowledge also the investigated IM of Grundfos Hungary Producer Ltd. 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