Abstract
Analysis of stability is essential for power systemplanning and stable operation. The results of transient andsteady-state stability studies depend on the appropriatechoice of load models. This study investigates modelingissues of the induction motors that are used as a dynamicpart of load models. Due to its superiority, we propose theapplication of the rotor speed frame based model ofinduction motors instead of a system frequency frame basedmodel in power system stability simulations. A comparativestudy of steady-state stability of a power system isperformed with different induction motor models. Theimplementation of lower-order models gives more optimisticresults. A larger proportion of induction motor loaddegrades the stability of power systems. It is shown that athird-order model implemented in a rotor speed framebased system is most suitable for stability studies.Index Terms-- Induction motors, load modeling, power systemstability, power system simulation, power system transientstability, power system steady state stability
I. INTRODUCTION
THE catastrophic consequences of blackouts in NorthAmerica, Europe and even East Asian countriesdemonstrate the urgency of proper security assessment andplanning for the stable operation of power system grids [1-3].An accurate security assessment requires accurate modeling ofpower system components. This is a great challenge for powerengineers as networks become increasingly complex andcomponents exhibit highly non-linear behavior [1, 4].Moreover, load modeling has become difficult due to therandom behavior of the load and the accumulation of largevolumes of measurement data[4].Modeling of the load involves two major issues: modelingand parameter identification [4-6]. The measurement data areused for the parameter identification of the composite load onthe load bus [4]. The dynamic part of the composite load issometimes represented by an induction motor[1, 4-6]. Powersystem stability is affected by the sum of the dynamic motorloads connected to the system and the line loading level, soload behavior differs under different system conditions [6].This means that choosing an accurate induction motor modelis important to the accuracy of system stability analysis.The third-order model is well preferred in stabilitysimulations [2, 4, 7] while a first-order model has beensometimes employed in some voltage stability studies[8]. Thefifth order model is used as a ‘full model’ to compare theperformance of lower order model. In [9], the singularlyperturbed lower-order models have been shown to havesimilar characteristics as the full model. The comparison ofthird and fifth order model response against the network faultis reported in [10]. The comparison of fifth and third ordermodel of DFIG response to a fault has been presented in [11].The responses of linear and nonlinear induction motor modelswith various inputs are compared in [12]. It is obvious that theimplementation of different models gives different results.This, in turn, affects the accuracy of the stability analysis.However, comparative stability assessment implementing thedifferent dynamic load models has not yet been sufficientlystudied.Induction motors are generally modeled in a frame rotatingsynchronously with system frequency [1-9, 12-15]. However,disturbance in the system sometimes causes a step change inphase angle which results in an unbounded frequency [15].This results in discontinuity of the state variables of inductionmotors, i.e. the stator and rotor fluxes. On the other hand, theinduction motor modeled in rotor speed based reference frame[16, 17] is free from above mentioned discontinuity problemwhich is discussed later.In this paper, we will show the superiority of the rotor-speedframe based models over the system frequency based framemodels. We will compare the responses of all the modelsunder same set of disturbances [10-12]. We will developinduction motor models of first and third order in a rotorspeed-based frame. Then, we will implement these models in aT2three-machine nine-bus power system. The small signalstability of a power system with induction motor models willbe investigated. Simulations are carried out usingMATLAB/Simulink software.
II. CONVENTIONAL INDUCTION MOTOR MODELS
A. Fifth, third and first order model in eframeThe dynamics of an induction motor can be represented bya set of five differential equations such as (1) [13]. Here,eqs v , evds , ' , ' e eqr dr v v represent the stator and rotor voltages,e , e , 'e , 'eqs ds qr dr represent the stator and rotor fluxes, , s r r r arestator and rotor resistances, e , e , 'e , 'eqs ds qr dr i i i i are stator and rotorcurrents, H is the inertia constant and , , r b e are the rotorspeed, base frequency and supply frequency respectively. em Tis the electromagnetic torque and mech T is the mechanicaltorque on the motor shaft. It is not common to implement thisfifth-order model in stability studies because of therequirement to balance the degree of approximation withsynchronous machines. However, the fifth-order model issometimes used as a benchmark to evaluate the accuracy oflower order models [9-11].The third order model can be obtained by ignoring thedynamics of the stator fluxes and the first order model can beobtained by ignoring the dynamics of both the stator and rotorfluxes. It can easily be shown that this first order model isequivalent to the well-known model expressed by phasorquantities [14 and m x are stator, rotor and mutualreactance respectively.III. PROPOSED INDUCTION MOTOR MODELA. A problem in eframe modelsIn the transient/steady-state stability simulation framework,voltages and currents are shown by phasors for the purpose ofreducing the computational requirements. This phasor-basedcomputation framework allows a time-step of 10 [ms] whileinstantaneous value-based computation requires a time-step ofless than 1 [ms].In the phasor-based framework, both slow and step changesin phasors are permitted. This means phasors can changediscontinuously, resulting in an unbounded frequency at themoment of discontinuity. This may result in a discontinuouschange in induction motor state variables or fluxes, asexplained in Appendix B. Fig.1 shows the discontinuity of qaxisrotor flux when phase angle is step-changed by 30degrees at 0.3 seconds. This is caused by an inappropriatecoordinate system, which results in the improper definition ofstate variables.

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