power plant, and battery and hybrid electric vehicles. Research in this field is tremendously going on, particularly from electric power generation point of view. Available literatures are showing a general feasibility of multiphase system [4, 5]. On generation side, the concept of multiphase (more than three-phase) machine was initially originated in late 1920s when larger power generation got hampered due to the limitations in circuit breaker interrupting capability. To overcome this situation, attention of scientists was diverted to the machine having double windings embedded in its stator [6]. It was after this time that research continued in the field of multiphase machine with steady, but in slower way. Utilization of multiphase synchronous generator was used in 1980 [7] for power generation in electric railway coaches. A few mathematical analysis of alternator with two three-phase winding was carried out by using orthogonal transformation for the elimination of time-dependent coefficient from system differential equations [8]. Six-phase synchronous generator in conjugation with three-to six-phase conversion transformer was analyzed for harmonic content [9]. In six-phase synchronous machine, mutual coupling effect between two sets of balanced three-phase stator winding is considered in [10], and under steady-state ac-dc stator connection [11] has been also presented and analyzed by using average-value modeling with line commutated converter [12]. A detailed mathematical modeling of six-phase synchronous generator using Park’s variable has been carried out in [13] under different working conditions at stand-alone mode, where an enhanced power handling capability by 173% was achieved when compared with its three-phase equivalent. A detailed experimental investigation of six-phase synchronous generator in stand-alone mode was carried out for renewable power generation in conjugation with hydropower plant [14]. Considering the suitability in generating mode, this chapter presents a mathematical modeling of grid connected six-phase synchronous generator applicable for wind power generating system, followed by the dynamic response under load variation.
Being an integral component in wind power generation, an operational stability of six-phase synchronous generator under steady state (i.e., small signal stability) is also of prime importance. Although, the small-signal stability analysis of three-phase synchronous machine is available in few available literature [15] using root locus [16] and Nyquist criteria [17]. But, for multiphase (i.e., six-phase) synchronous machine, a very limited literature is available for small-signal stability analysis. An introductory analysis of synchronous machine was reported in [18] followed by the determination of stability limits under parametric variation and different working conditions [19] when compared with its three-phase counterpart [20]. With the aim to access the suitability and applicability in wind power generating system, this chapter is dedicated to present a small-signal stability analysis of grid connected six-phase synchronous generator showing a comparison with its three-phase equivalent. For this purpose, a linearized version of six-phase synchronous generator model has been derived and used to evaluate the system eigenvalue. Eigenvalue criteria are used for small signal stability analysis under different machine parametric variation. A comparative analysis, from stability view point, is also presented using Park’s (dq0) variable for both grid connected three- and six-phase synchronous generator.
1.2 Analytical Modeling of Six-Phase Synchronous Machine
To design a six-phase machine, it is a common strategy to split the stator winding into two through phase belt splitting namely, abc and xyz having the angular displacement of α = 30°, to have asymmetrical winding [4, 18, 19]. Rotor of the machine remains same having the field winding fr and damper windings kd and kq along d-q axes, respectively. While going onward for the mathematical modeling, some of the important simplifying assumptions are considered [21, 22]:
Both the three-phase stator windings (abc and xyz) are symmetrical balanced having a perfectly sinusoidal distribution in the air-gap.
Flux and mmfs are sinusoidal with no space harmonics.
Saturation and hysteresis effects are ignored.
No skin effect, i.e., winding resistance, is not dependent on frequency.
Although, voltage and electromagnetic torque can be mathematically expressed in terms machine variables, which results in non-linear differential equations [22]. The non-linearity is due to the time varying inductance term. For simplicity, with constant inductance terms, concept of reference frame theory is used, and equations are preferably written in rotor reference frame using Park’s equation. Mathematically, voltages and flux linkage per second of a six-phase synchronous machine using Park’s variables are as follows [21–23]:
1.2.1 Voltage Equation
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)
where p shows the differentiation function w.r.t. time.
1.2.2 Equations of Flux Linkage Per Second
(1.9)
(1.10)
(1.11)
(1.12)
(1.13)
where
(1.15)
(1.16)
The parameters of rotor circuit are referred to the stator winding set abc. These voltage and flux linkage equations suggest the equivalent circuit, as shown in Figure 1.1, wherein Llm and Lldq represent leakage inductance of common mutual and cross mutual coupling between d and q-axis of stator circuit, respectively:
(1.17)