Open access peer-reviewed chapter

# Analysis of Compensated Six-Phase Self-Excited Induction Generator Using Double Mixed State-Space Variable Dynamic Model

By Kiran Singh

Submitted: September 30th 2018Reviewed: October 30th 2018Published: May 15th 2019

DOI: 10.5772/intechopen.82323

## Abstract

In this article, a mixed current-flux d-q modeling of a saturated compensated six-phase self-excited induction generator (SP-SEIG) is adopted during the analysis. Modeling equations include two independent variables namely stator current and magnetizing flux rather than single independent variables either current or flux. Mixed modeling with stator current and magnetizing flux is simple by having only four saturation elements and beneficial in study of both stator and rotor parameters. Performance equations for the given machine utilize the steady-state saturated magnetizing inductance (Lm) and dynamic inductance (L). Validation of the analytical approach was in good agreement along with three-phase resistive or resistive-inductive loading and also determined the relevant improvement in voltage regulation of machine using series capacitor compensation schemes.

### Keywords

• mixed double state space variables
• self-excitation
• six-phase
• compensation
• induction generator
• non-conventional energy

## 2. Modeling description

Concerns about mathematical modeling of SP-SEIG, short-shunt series compensation capacitors and static resistive ‘R’ and balanced three-phase reactive ‘R-L’ loads as previously discussed in [1, 3] are not described in this section, only briefly summarized along with newly addition of long-shunt series compensation capacitors.

### 2.1 SP-SEIG model

A basic two-pole, six-phase induction machine is schematically described by its stator and rotor axis [1]. In which, six stator phases of both sets, a, b, c and x, y, z (set I and II, respectively) are arranged to form two sets of uniformly distributed star configuration, displaced by an arbitrary angle of 30 electrical degree gravitate asymmetrical winding structure. The distribution of 6-phases in 36 slots of 6 pole induction machine is also shown in Figure 1. Previously, voltage equations using single state-space variable namely flux linkage were used in the expanded form for six-phase induction machine.

After simplification of voltage equations using double mixed state-space variables namely stator currents and magnetizing fluxes in the machine model following form occurs from Eqs. (1)(39) of [1]:

Vdq=HdXdq/dt+JXdqE1

where [Vdq] = [Vd1 Vq1 Vd2 Vq2 0 0]t, [Xdq] = [id1 iq1 id2 iq2 ψdm ψqm]t and matrices [H] and [J] are given by Eqs. (2) and (3), respectively.

7Lσ1000100Lσ1000100Lσ2010000Lσ201Lσr0Lσr01+LσrLddLσrLdq0Lσr0LσrLσrLdq1+LσrLqqE2
r1wLσ1000wwLσ1r100w000r2wLσ20w00wLσ2r2w0rrwwrLσrrrwwrLσrrrLmwwr1+LσrLmwwrLσrrrwwrLσrrrwwr1+LσrLmrrLmE3

The nonlinear equations of voltage and current across the shunt excitation capacitor and series compensation capacitors (short-shunt and long-shunt) can be transformed into d-q axis by using reference frame theory, i.e. Park’s (dq0) transformation [4], are given by Section 2.2 of [1] and (Section 2.3 of [1] and by following Section 2.2), respectively. Modeling of static loads is also given in Section 3 of [1].

### 2.2 Modeling of long-shunt capacitors

Current through series capacitors Cls1 and Cls2 (in case of long shunt), connected in series with winding set I and II, respectively, is same as the machine current. The machine current along with series capacitance determine the voltage across series long-shunt capacitor and when transformed in to d-q axis by using Park’s transformation is given in Eqs. (4) and (5) [5, 6, 7].

ρVq1ls=iq1/Cls1ρVd1ls=id1/Cls1ρVq2ls=iq2/Cls2ρVd2ls=id2/Cls2E4
and the load terminal voltage is expressed as
VLq1=Vq1+Vq1lsVLd1=Vd1+Vd1lsVLq2=Vq2+Vq2lsVLd2=Vd2+Vd2lsE5

The remaining symbols of machine model have their usual meanings from Ref. [1] and Table 1.

 ψdm, ψqm, 1/Ldd, 1/Lqq, 1/Ldq d- and q-axis magnetizing flux linkages, saturation dependent coefficients of system matrix A p, σ, P, J, θr, ω, ωb, ωr differentiation w. r. t. time, index for constant, number of pole pairs, moment of inertia, electrical angular displacement of the rotor, reference frame speed, base speed and rotor speed Lm, L, Ldq, cosμ, sinμ steady-state saturated magnetizing inductance, dynamic inductance, cross-saturation coupling between the d- and q-axis of stator, angular displacements of the magnetizing current space vector with respect to the d-axis of the common reference frame Cls1 Cls2 long-shunt capacitors across the stator winding set I and II

### Table 1.

Machine model symbols.

## 3. Methodology

In this section, a numerical method is introduced to the solution of Eqs. (1)(3); where double mixed current flux state space model is discussed by [1]. The ordinary linear differential equations can be solved by the analytical technique rather than approximation method. Eq. (1) is non-linear differential and cannot be solved exactly with high expectations, only approximations are estimated numerically by computer technique using 4th order Runge-Kutta method or classical Runge-Kutta method or often referred as “RK4” as so commonly used [8]. The analytical response of compensated SP-SEIG in only single operating mode is carried out under significant configuration using RK4 subroutine implemented in Matlab M-file. The dynamic performances were determined under no load, R load and R-L loading condition in only the single mode of excitation capacitor bank, and in both modes of compensating series capacitor bank. The following analytical dynamic responses of series compensated SP-SEIG is considered for the validity of proposed approaches in this chapter.

Both analytical responses are well detailed in Section 4. The analytical study of compensated SP-SEIG is given in Section 4 by using an explicit MATLAB program incorporates the RK4 method. An Algorithm of RK4 method for the analysis of compensated SP-SEIG is also shown in Figure 2. The parameters of studied machine and saturation dependent coefficients of system matrix [H] are also reported by [1] for further dynamic analysis of saturated compensated SP-SEIG using double mixed state space variable model under constant rotor speed along with appropriate initial estimated variables values which are also responsible in the development of rated machine terminal voltage and it depends on other machine variables.

## 4. Analytical response

### 4.1 Short-shunt series compensated SP-SEIG

Performance of short-shunt SP-SEIG in only its single mode of operation using the values of excitation and series capacitor banks of 38.5 and 108 μF per phase, respectively, has been predicted from the built explicit MATLAB program using RK4 subroutine. Computed waveforms are shown in Figures 3 and 4. Application of short-shunt scheme results in overvoltage across the generator terminals as shown in Figure 3a, the per phase voltage level is more than the voltage level of Figure 5a and it is illustrated in Figure 3c.

#### 4.1.1 When both three-phase winding sets are connected in short-shunt configuration with independent R loading

The analytical d-q waveform of voltage, current, magnetizing flux and magnetizing current during no-load and sudden switching of R load of 200 Ω at t = 2 s are shown in corresponding Figure 3a, b, d and g. In addition, combined amplitude waveforms of magnetizing flux, steady-state saturated magnetizing inductance and dynamic (tangent slope) inductance is shown in Figure 3f. The angular displacements of the magnetizing current space vector with respect to the d-axis of the common reference frame are also shown in Figures 3j and k. The combined d-q axis voltage drop across the series short-shunt capacitor is given in Figure 3c and combined amplitude waveform of d-axis stator current and load current during no-load and sudden switching of R load of 200 Ω at t = 2 s is given in Figure 3e. The d- and q-axis load currents are also depicted in Figure 3h and i, respectively, during sudden switching of R load of 200 Ω at t = 2 s.

#### 4.1.2 When both three-phase winding sets are connected in short-shunt configuration with independent R-L loading

In the same order of figure numbers, all computed waveforms are shown in Figure 4 with sudden switching of R-L load (200 Ω resistance in series with 500 mH inductor) at t = 2 s. Analytical generated RMS steady state voltage and corresponding current at rated speed of 1000 RPM are shown in Figure 4a and b. Drops in d-q-axis generating and lagging load currents are shown in Figure 4b, h and i.

### 4.2 Long-shunt series compensated SP-SEIG

It is also seen that like short-shunt compensation, long-shunt compensation is also self- regulating in nature. In the same manner, analysis of long-shunt SP-SEIG along with a single mode of excitation and the series capacitor banks of 38.5 and 350 F respectively, have also been computed and predicted by using the RK4 subroutine and illustrated in Figures 5 and 6. Here, the value of series capacitor is more than the twice of short-shunt series capacitor.

#### 4.2.1 When both three-phase winding sets are connected in long-shunt configuration with independent R loading

The analytical waveform of voltage, current, magnetizing flux and magnetizing current during no-load and sudden switching of R load of 200 Ω at t = 2 s with long-shunt compensation along both three-phase winding sets are respectively shown in Figure 5a, b, d and g. Combined amplitude waveforms of magnetizing flux, steady-state saturated magnetizing inductance and dynamic (tangent, slope) inductance is shown in Figure 5f. The angular displacements of the magnetizing current space vector with respect to the d-axis of the common reference frame are also shown in Figure 5j and k. The application of the long-shunt scheme results in less overvoltage or reduced terminal voltage across the generator terminals. As it is shown in Figure 5a, the per phase voltage level is less than the voltage level of Figure 3a. It gives evidence that long-shunt SP-SEIG is able to deliver output power at reduced terminal voltages, as shown in Figure 5c. The combined d-q axis voltage drop across the series short-shunt capacitor is given in Figure 5c and combined amplitude waveform of d-axis stator current and load current during no-load and sudden switching of resistive load of 200 Ω at t = 2 s is given in Figure 5e. The d- and q-axis load currents are also depicted in Figure 5h and i respectively, during sudden switching of resistive load of 200 Ω at t = 2 s. The steady state no-load voltage is generated at rated speed of 1000 RPM.

#### 4.2.2 When both three-phase winding sets are connected in long-shunt configuration with independent R-L loading

In the same order of figure numbers, all computed waveforms are also shown in Figure 6 with sudden switching of R-L load (200 Ω resistance in series with 500 mH inductor) at t = 2 s. The RMS analytical value of steady state voltage is generated at rated speed of 1000 RPM. Same like as Section 4.1, the d-q axis drops in generating and lagging load currents are also given in Figure 6b, h and i.

## 6. Conclusion

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© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kiran Singh (May 15th 2019). Analysis of Compensated Six-Phase Self-Excited Induction Generator Using Double Mixed State-Space Variable Dynamic Model, Electric Power Conversion, Marian Găiceanu, IntechOpen, DOI: 10.5772/intechopen.82323. Available from:

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