**2. Mathematical models used in isolated electrical grid**

The aim is to analyze the dynamics of the induction motors fed directly from the isolated electrical grid. For this purposes the mathematical model of isolated electrical grid has been develop consisting diesel electrical aggregate and unregulated induction motors.

Diesel generators are used as the main sources of electricity in cases of isolated systems. In many applications the diesel generator can suffer significant impacts loads that can produce disturbances in the isolated grid. However, the autonomous operation of the synchronous generator is characterized by a change in steady state which causes a change in voltage and frequency, which in turn affects the quality of electric power systems.

The model of diesel electrical aggregate considered in this study consists of: a diesel engine and a speed controller, a synchronous generator and a voltage controller, a mechanical connection between engine and electrical machine shaft.

The synchronous generator is presented as a machine with three armature windings, a magnetizing winding on the rotor and damper windings. One damper winding is located along direct-axis (D), and one along the quadrature-axis (Q). The basis of the mathematical model is a set of differential equations of the synchronous generator in the standard *dq*-axis form (Kundur, 1994). The voltage equations are written in generator (source) convention system, in which synchronous machines are usually represented:

$$-\mathbf{u}\_d = r \cdot \mathbf{i}\_d + \frac{d\psi\_d}{dt} - \omega \cdot \psi\_q \tag{1}$$

$$-u\_q = r \cdot i\_q + \frac{d\psi\_q}{dt} + \omega \cdot \psi\_d \tag{2}$$

$$E\_q = e\_q + \frac{x\_{1d}}{r\_1} \cdot \frac{d\psi\_1}{dt} \tag{3}$$

$$0 = r\_D \cdot i\_D + \frac{d\psi\_D}{dt} \tag{4}$$

$$0 = r\_Q \cdot i\_Q + \frac{d\psi\_Q}{dt} \tag{5}$$

where *u*, *i*, *r,* and denote voltage, current, resistance and flux respectively.

74 Induction Motors – Modelling and Control

term interruptions in the motor power supply.

electrical as well as mechanical systems.

cases, faulty contactor may produce multiple switching on and off. However, these interruptions will affect the dynamics of both electrical and mechanical variables. Therefore, it is interesting to analyze dynamics of the induction motor in case when it comes to short-

That's why the dynamic behavior of induction motors fed directly from isolated electrical grid, as well as dynamics of aggregate is in focus of researchers. Presently, advanced modeling and digital simulation techniques can be used to analyse the dynamics behavior of

The aim is to analyze the dynamics of the induction motors fed directly from the isolated electrical grid. For this purposes the mathematical model of isolated electrical grid has been

Diesel generators are used as the main sources of electricity in cases of isolated systems. In many applications the diesel generator can suffer significant impacts loads that can produce disturbances in the isolated grid. However, the autonomous operation of the synchronous generator is characterized by a change in steady state which causes a change in voltage and

The model of diesel electrical aggregate considered in this study consists of: a diesel engine and a speed controller, a synchronous generator and a voltage controller, a mechanical

The synchronous generator is presented as a machine with three armature windings, a magnetizing winding on the rotor and damper windings. One damper winding is located along direct-axis (D), and one along the quadrature-axis (Q). The basis of the mathematical model is a set of differential equations of the synchronous generator in the standard *dq*-axis form (Kundur, 1994). The voltage equations are written in generator (source) convention

> *d dd q*

> *q qq d*

> > 1 1 1 *d*

> > > *dt*

*dt*

*r dt*

0 *<sup>D</sup> D D <sup>d</sup><sup>ψ</sup> r i*

0 *<sup>Q</sup> Q Q <sup>d</sup><sup>ψ</sup> r i*

*q q <sup>x</sup> <sup>d</sup><sup>ψ</sup> E e*

*<sup>d</sup><sup>ψ</sup> u ri ω ψ dt* (1)

*<sup>d</sup><sup>ψ</sup> u ri ω ψ dt* (2)

(3)

(4)

(5)

develop consisting diesel electrical aggregate and unregulated induction motors.

**2. Mathematical models used in isolated electrical grid** 

frequency, which in turn affects the quality of electric power systems.

connection between engine and electrical machine shaft.

system, in which synchronous machines are usually represented:

The model of a synchronous generator is given in rotor reference frame ( is rotor electrical speed). The equations of excitation in motor (load) convention system are written. The voltage controller, modeled as PI, is implemented in the model of the synchronous generator.

The model of the prime mover - the diesel engine assumes that the engine torque is directly proportional to the fuel consumption. In order to describe the dynamic behavior of the diesel engine it is necessary to set up a system of differential equations which includes an equation of the engine, the turbocharger, the air collector, the exhaust system, and the speed controller. Taking into account these equations requires the knowledge of characteristics of diesel engines that require complex experimental measurements, according to (Krutov, 1978; Tolšin 1977). The studies carried out in (Erceg & et al. 1996) showed that the mentioned omissions do not affect significantly the results and that the proportionality of torque to the amount of injected fuel can be assumed. This simplification is allowed when it is of interest to observe dynamics of a synchronous generator as well as induction motors. The speed controller is modeled as PI and implemented in the model.

Sudden impact load on the diesel electrical aggregate is the most difficult transition regime for units due to electricity loads and also due to torsional strains in the shaft lines. A more significant disturbance, which is at the same time very common in practice, is the direct-online starting of induction motors to such a grid. Starting of induction motors will cause voltage dips and will reduce engine speed depending on the time of the starting of each motor. This will also cause torsional stresses in the shaft. Thus, the mechanical coupling of a diesel engine and a synchronous generator is considered to be a rotating system with two concentrated masses. Masses are connected by flexible coupling. The flexible coupling allows these masses to rotate at a different speed in transients.

The variable angle of rotation between these masses occurs during the transient, in period when mechanical balance between diesel engine and electric generator is disturbed., The torque which appears at coupling zone between two concentrated rotational masses allows thus the analysis of the torsional dynamics in the coupling.

Induction motor as an active consumer and its parameters were analyzed in (Maljkovic, 2001; Amezquita-Brook et al., 2009). According to (Jones, 1967; Kraus 1986) three phase squirrel-cage induction motors are represented with stators and rotors voltage equations:

$$
\mu\_{dIM} = R\_{sIM} \cdot i\_{dIM} + \frac{d\psi\_{dIM}}{dt} - \omega \cdot \psi\_{qIM} \tag{6}
$$

$$
\Delta u\_{qIM} = \mathbf{R}\_s \cdot i\_{qIM} + \frac{d\psi\_{qIM}}{dt} + \omega \cdot \psi\_{dIM} \tag{7}
$$

$$0 = R\_r \cdot i\_{DIM} + \frac{d\psi\_{DIM}}{dt} - \left(\omega - \omega\_{IM}\right) \cdot \psi\_{QIM} \tag{8}$$

$$0 = R\_r \cdot i\_{QIM} + \frac{d\psi\_{QIM}}{dt} + \left(\omega - \omega\_{IM}\right) \cdot \psi\_{DIM} \tag{9}$$

The Dynamics of Induction Motor Fed Directly from the Isolated Electrical Grid 77

Moment of switching/disconnecting on the network is controlled by means of the subsystem ON/OFF, also, the part of this subsystem is used to set the time of switching/disconnecting

All calculations were carried out by means of the "Variable-Step Kutta-Merson" method – an explicit method of the fourth order for solving the systems of differential equations, with

the load on the motor shaft.

variable integration increment.

**Figure 1.** Simulation of the applied mathematical model

**Figure 2.** Diesel engine and speed controller model

where: *u*, *i*, R, and denote voltage, current, resistance and flux respectively of an induction motor.

All winding currents, in the transient *dq* axis model of induction motors as well as in a synchronous generator model, are selected as state variables. The model is completed with an equation of the rotational mass motion (Vas, 1996). All variables and parameters are in per unit (p.u.). The motor's equation of motion involves electrical torque (*T*eIM), whereas (*T*lIM) represents load torque on the motor's shaft.

When the induction motor starts unloaded, then the torque *T*lIMn equals zero. Also, for this analysis the loading with a constant load was selected.

Loads, induction motors (index IM), are connected directly to a synchronous generator (index SG), what means that they are on the same voltage as the generator terminals: -*u*d=*u*dIM1=*u*dIM2, -*u*q=*u*qIM1=*u*qIM2. According to the Kirchhoff's law, the current relationship between supplying and receiving elements are: *i*d=*i*dIM1*+i*dIM2, *i*q=*i*qIM1+*i*qIM2.

The validity of the mathematical model of the generator-unit at impact load, direct-on-line starting of non-loaded induction motor, was checked in the previous work (Mirosevic, et al. 2002a, 2011b) by comparing the results of the simulation and the measurement on the generator-unit with a diesel engine of 46.4 kW, 1500 r/min and a synchronous generator of 40 kVA (3x400/231 V, cos =0.8; 57.7 A; 1500 r/min; 50 Hz), to which a motor drive of 7.5 kW ( 380 V, 14.7 A, 2905 r/min, cos =0.9) was connected. The results obtained by numerical calculation indicate that, the set mathematical model can be applied with sufficient certainty

The analysis of the dynamics of induction motors fed directly from the isolated electrical grid was performed by the application of program package "Matlab/Simulink". The block diagram of integral motor drives is presented in Figure 1 involves: a diesel engine (DM, SC), a three phase synchronous generator and voltage controller, their mechanical coupling and induction motors fed directly from the synchronous generator terminals.

Block diagram of Diesel engine and speed controller is presented in Figure 2 and represents subsystem of block naimed as DM SC in Fig. 1.

The Simulink is used to obtain a model of a diesel generator unit, as well as induction motors by means of basic function blocks that can be linked and edited to subsystem such as subsystems IM 1 and IM 2 in Figure 1 which representthe first and the second induction motors respectively. As one can see in Figure 3 components of the subsystem IM 1 that are used in the calculation of variables are presented.

Induction motors are connected to the network using power supply subsystem, while the load on the motor shaft is represented with subsystems: load IM1 and load IM2 in Figure 1. Moment of switching/disconnecting on the network is controlled by means of the subsystem ON/OFF, also, the part of this subsystem is used to set the time of switching/disconnecting the load on the motor shaft.

All calculations were carried out by means of the "Variable-Step Kutta-Merson" method – an explicit method of the fourth order for solving the systems of differential equations, with variable integration increment.

**Figure 1.** Simulation of the applied mathematical model

76 Induction Motors – Modelling and Control

(*T*lIM) represents load torque on the motor's shaft.

analysis the loading with a constant load was selected.

between supplying and receiving elements are: *i*d=*i*dIM1*+i*dIM2, *i*q=*i*qIM1+*i*qIM2.

induction motors fed directly from the synchronous generator terminals.

subsystem of block naimed as DM SC in Fig. 1.

used in the calculation of variables are presented.

where: *u*, *i*, R, and

sufficient certainty

motor.

<sup>0</sup> *QIM*

*r QIM IM DIM <sup>d</sup><sup>ψ</sup> R i ωω ψ dt*

All winding currents, in the transient *dq* axis model of induction motors as well as in a synchronous generator model, are selected as state variables. The model is completed with an equation of the rotational mass motion (Vas, 1996). All variables and parameters are in per unit (p.u.). The motor's equation of motion involves electrical torque (*T*eIM), whereas

When the induction motor starts unloaded, then the torque *T*lIMn equals zero. Also, for this

Loads, induction motors (index IM), are connected directly to a synchronous generator (index SG), what means that they are on the same voltage as the generator terminals: -*u*d=*u*dIM1=*u*dIM2, -*u*q=*u*qIM1=*u*qIM2. According to the Kirchhoff's law, the current relationship

The validity of the mathematical model of the generator-unit at impact load, direct-on-line starting of non-loaded induction motor, was checked in the previous work (Mirosevic, et al. 2002a, 2011b) by comparing the results of the simulation and the measurement on the generator-unit with a diesel engine of 46.4 kW, 1500 r/min and a synchronous generator of 40 kVA (3x400/231 V, cos =0.8; 57.7 A; 1500 r/min; 50 Hz), to which a motor drive of 7.5 kW ( 380 V, 14.7 A, 2905 r/min, cos =0.9) was connected. The results obtained by numerical calculation indicate that, the set mathematical model can be applied with

The analysis of the dynamics of induction motors fed directly from the isolated electrical grid was performed by the application of program package "Matlab/Simulink". The block diagram of integral motor drives is presented in Figure 1 involves: a diesel engine (DM, SC), a three phase synchronous generator and voltage controller, their mechanical coupling and

Block diagram of Diesel engine and speed controller is presented in Figure 2 and represents

The Simulink is used to obtain a model of a diesel generator unit, as well as induction motors by means of basic function blocks that can be linked and edited to subsystem such as subsystems IM 1 and IM 2 in Figure 1 which representthe first and the second induction motors respectively. As one can see in Figure 3 components of the subsystem IM 1 that are

Induction motors are connected to the network using power supply subsystem, while the load on the motor shaft is represented with subsystems: load IM1 and load IM2 in Figure 1.

(9)

denote voltage, current, resistance and flux respectively of an induction

**Figure 2.** Diesel engine and speed controller model

The Dynamics of Induction Motor Fed Directly from the Isolated Electrical Grid 79

maximum torque is achieved, the load of the synchronous generator reaches its maximum

At the instant of starting, as one can see in Fig. 4a, the air-gap torque is momentarily increased; reaches maximum value of 0.86 p.u. and change in it can be noticed during the whole start-up period of the first induction motor. The instantaneous torque oscillates about

**Figure 4.** Transients of: air-gap torque of induction motors (*T*eIM1, *T*eIM2) and speed transient of induction

(a) (b) *T*eIM1 *T*eIM2 *w*IM1 *w*IM2 *u w*SG *w*DM

The oscillations in the air-gap torque are caused by the interactions between the stator and rotor flux linkage. The negative oscillations in the electromagnetic torque of the induction motor are presented at the beginning of the start-up period. These are periods of momentary deceleration that occur during regeneration when the electromagnetic torque becomes negative. The rotor speed only increases when the torque is positive. The oscillations that are present in transient of air-gap torque of the first induction motor are damped at the end of start up period and finally the steady state condition is attained without oscillations.

The response of the air-gap torque is in accordance with the response of the motor currents. Transients of stator currents of induction motors and their components, for both cases, are

Under this condition the starting current is large. The starting current of an induction motor is several times larger than the rated current since the back emf induced by Faraday's law grows smaller as the rotor speed increases. However, a large starting current tend to cause the supply voltage to dip during start-up and can cause problems for the other equipment

SG) and diesel

IM2); terminal voltage (*u)*, speed transient of synchronous generator (

DM), during direct-on-line starting of: a) unloaded b) loaded induction motors

and then decreases rapidly.

positive average value.

motors (IM1, 

engine (

presented in Figure 5.

that is connected to the same grid.

**Figure 3.** Induction motor model
