These machine models are also valuable for developing, studying and evaluating the suitability and performance of connected power electronic circuits and their associated control and protection strategies.
The machine topology being considered in this work departs from the conventional machines in a number of ways. Firstly, it consists of a significantly large number of stator phases, up to 24 or even higher. Secondly, both even and odd number of stator phases are applicable to this machine topology. Thirdly, the way in which the machine stator phases are connected and linked to external circuits is different. Unlike conventional machines where the stator phases are connected in series or parallel and in either star/delta connections on the ac side, in this machine topology each individual machine stator phase winding terminals are connected to a dedicated power electronic phase module as discussed in chapter 3 and chapter 4.
The multiphase electronically commutated DC machine topology, its multiphase windings and the associated power electronics topology represents a significant depar- ture from the conventional machines and power circuit topologies currently in use such as, induction & synchronous machines with voltage and current source converters. As such, the generic machine simulation models developed so far and incorporated in commercial simulation packages do not lend themselves well to simulation studies of this machine/converter topology.
To enable transient, dynamic and steady state behaviour characterisation of this machine/converter topology and assessment of suitable power electronics and the protection strategies, suitable machine models have to be developed. This chapter details the machine modelling approach adopted for this machine & converter topology.
5.2
Modelling Approaches
A number of machine modelling methods have been proposed in literature [174– 178, 171] and some are in wide use today. In order to determine the most suitable approach for modelling this machine topology, a review of the modelling approaches used for machine simulation models will be briefly highlighted. The four most applicable machine modelling approaches used in simulation of machine and converter systems are; (a) Finite Element modelling, (b) Phase domain machine modeling, (c)
5.2 Modelling Approaches
DQ machine modelling and (d) Voltage behind reactance modelling. Each one of these methods has some key strengths and also some drawbacks depending on the intended purpose of the model.
Finite Element Modelling
Machine FE models are very accurate since they take into account the geometrical, material and operational details of electrical machine [179–181]. As a result, Finite Element methods are widely used in machine design stages. However, the intended purpose of the machine models in this work is for drive system dynamic behaviour prediction to facilitate the detailed design of the drive system, where not only the electric machine is included, but also the power electronics and control systems that interact with it. Owing to the need to calculate the machine parameters at each time step, the computational requirements of time stepping 2 & 3 dimensional FE simulations are very high. The computational burden coupled with the FE modelling complexity renders this approach less suitable for modelling such a drive system.
Phase Domain Modelling
In this formulation, the machine is represented in its natural form by lumped parameter coupled circuits in physical variables and abc phase coordinates. This modelling approach lends itself well to simulation of machines connected to power electronic circuits as machine circuits are directly inserted into the overall system circuit network equations, thus providing a simultaneous solution. Owing to this integration of the machine model with the rest of the power electronic system, the machine converter system behaviour during certain machine/converter failure modes such as open or short circuit faults can be readily simulated. Various academic publications [174, 182–185] have shown that this machine modelling approach improves numerical accuracy and stability of the system being simulated. In this formulation, the stator inductances are rotor position dependent and will have to be computed at each rotor position. Furthermore, the inversion of the inductance matrix is required at each time step during the numerical integration of the differential equations as the rotor changes position. This imposes a heavy computational burden during simulation particularly
5.2 Modelling Approaches
if the size of this matrix is large, as is the case with multi phase machine topology being considered. For machines with relatively low number of phases, this drawback can be mitigated by computing analytical expression of the inductance matrix inverse offline either by hand calculations or using symbolic maths software packages such as Mathematica and implementing the resultant simplified equations in the machine models. Work has also been reported in literature ([186], [187], [186]) where block matrix inversion techniques exploiting the circulant nature of the submatrices such that inductance matrix inversion is avoided. Instead, analytical expressions of submatrices of the inverse of the inductance matrix are used [188],[189]. However, all these techniques become tedious and error prone when significantly high number of phases are considered.
DQ Modelling
Thedq-machine modelling formulations have gained wide acceptance in many nodal
analysis-based electromagnetic transient programs (EMTP-type) and state variable- based simulation programs (e.g., Simulink, PLECS e.t.c.) as built-in standard library components that are extensively used both in industry and academia. This formulation is based on the transformation of machine variables into an imaginary reference frame,
the so calleddqPark transformations. Withdqtransformation, the sinusoidal machine
variables (volts, currents, fluxes) are transformed into DC quantities. This is quite attractive from a computational overhead viewpoint as it results in a constant steady state matrix of self and mutual inductances (i.e. inductances are independent of rotor position). Thus, in the machine model implementation the inductance matrix does not need to be recalculated at each simulation time step as the rotor changes position, making it numerically efficient. However, this method does not lend itself well to the simulation of the machine/converter topology proposed (machine and associated power electronics) for the following reasons. Prediction of a number of variables is required
to interface thedqcircuits of the model with the phase-domain representation of the
power electronic circuit being simulated. A number of methods are used to interface
thedqmachine model to the rest of the system circuits. These include; use of a Norton
5.2 Modelling Approaches
to become time-independent and the Norton current sources is computed from the
dqmodel. Alternatively, a compensation method in which the main machine stator
circuit is represented by a Thevenin equivalent circuit and interfaced to thedqcircuits
can be used. However this is limited to a number of topologies. A third approach
used is to interface the machinedq model with the stator circuit as a compensated
current source with special terminating resistance. The Norton current source that represents the machine is calculated using the previous time point terminal voltages of the machine. These interface methods often cause deterioration of numerical accuracy if larger time steps are used and often small time steps are required to keep interfacing errors within required tolerances. In some cases, this can lead to stability issues of the overall system. Often, terminating impedances / snubber circuits are needed at the machine terminals to aid numerical stability and facilitate integration with the rest of the power circuit. Furthermore, representation of the machine internal phenomenon such as short circuits and more importantly, the machine behaviour during electronic commutation process and system faults is somewhat difficult. This is particularly important in this drive topology where the power electronic modules are an integral part of the machine phases and can not be decoupled when studying the behaviour of this machine during normal and faulty operating conditions.
Voltage Behind Reactance Modelling
Recent studies have also proposed alternative approaches based on Voltage Behind
Reactance (VBR) modelling for electrical machines [190–197]. In this approach, the
machine equations are reformulated such that the machine rotor subsystem dynam- ics are decoupled from the stator subsystem dynamics. Here, the rotor subsystem
dynamics are represented using state variable equations in thedqcoordinate system
whereas the stator subsystem dynamics are represented in circuit form in theabcphase
coordinate system. Reformulating the machine equations in this way brings several advantages, particularly in drive system simulations where the machine is integrated as part of a power electronic system network.
Since the machine stator subsystem is represented in the physicalabcphase domain