Matrix structure of unified mathematical model of electric AC machines at control

. The matrix structure of the equations of a generalized electric alternating current machine is proposed, which, based on the Parke equations, is written in the coordinate axes of the machines rotating with the rotor speed. In the matrix structure, the column matrices of the derivatives of the stator, excitation and rotor windings are equal to the product of diagonal matrices consisting of the machine parameters and the column matrices of the flux links themselves and the sum of the matrix columns of the control parameters which are the matrix columns of the stator voltage, excitation voltage, and rotor voltage. It is shown that the matrix structure of a generalized controlled AC machine is transformed into mathematical models of almost all encountered AC electric machines, namely, into a synchronous machine with two excitation windings - a longitudinal and a transverse one; in a synchronous machine with a longitudinal field winding (classic); in an asynchronous machine with a squirrel-cage rotor; into an asynchronous machine with a phase rotor. It has been shown that the matrix structure includes the controls of these machines both from the stator and from the rotor. On the stator side for synchronous machines, it is a frequency control that regulates both the amplitude and frequency of the applied voltage, and on the rotor side, a constant voltage control is supplied to the longitudinal and transverse windings. For asynchronous machines, the stator and rotor are frequency-controlled. The following are examples of frequency control of an asynchronous machine both from the stator and from the rotor.


Introduction
When designing complex electromechanical devices for such, for example, systems as "wind-power engineering", "small hydropower engineering", etc., which can contain controlled electric machines of different types, it is required not only optimally join electric machine with mechanical one, but in a number of cases to optimize the choice and the type itself of the controlled electric machine.
The equations of a generalized electric machine are well known, which are given, for example, in [1]. It should be noted that these equations more reveal the principle of construction of electromagnetic and electrical connections in electric machine, for practical use they must be transformed taking into account the type of electric machine and the form of writing of their equations.

Materials and methods
The above circumstances predetermine the creation of a universal mathematical model of a controlled alternating current (AC) electric machine, which, remaining structurally unchanged, would allow for studying all modes of operation of AC electric machines used in practice: synchronous machines with electromagnetic excitation and permanent magnets, including frequencycontrolled ones; asynchronous machines with shortcircuited and phase-wound rotor, including frequencycontrolled ones.
Here, Park's equations, written in axes rotating at the rotor speed of ωr, were taken as the basis of mathematical model of electric AC machines, and the machine has two stator windings and four rotor windings. In contrast with these equations, not angle θ between the rotor axis, rotating at the rotor speed of ωr, and the synchronous axis, rotating at the synchronous speed of ωs, but the angle between the rotor axis and fixed axis, hereinafter signed as angle α, is selected as a power angle. Thus, the rotor speed is equal to ωr, where p -differentiation symbol, τ -synchronous time equal to τ=ωs·t=314·t. In addition, the equations of machines are written in flux linkages.
In this case the equations of controlled electric AC machines can be written in cell wise-matrix form, which is represented in the form.
Column matrices are the essence of a vector with projections on d, q axes: -derivatives of flux linkages of stator, field and rotor windings: -voltages of stator, field and rotor windings (control actions): First matrix As1 can be represented in the form: Since ωr -rotor speed of AC machine is a scalar value, then As1ω can be represented in the form: where J-matrix orthogonal to unity matrix E, i.e. (J 2 = -E), by analogy with complex unit j 2 = -1.
Besides equations (1), it is necessary to take into account equations of motion with motive moment mt and electromagnetic moment mem:   (3) It should be noted that the mentioned equations are written in the per unit system, for basic expressions the same units are taken as in the equations, which, for example, are given in [9; 10; 11].
They are easily defined from the matrix equality (direct and inverse matrices with inductive parameters of machine). In expression (4) the right part is a matrix of machine parameters in relative units, but it is necessary to have in mind that they are equal to the inductances of the machine in relative units, i.e. currents and flux linkages are connected by the values of the corresponding inductances, which in relative units are equal to the values of the passport parameters of the machine. Structure of mathematical model of synchronous machine with excitation along the longitudinal and transverse axes d and q. In this case the matrix form , since for such machines the rotor has damper windings that are short-circuited.
Mathematical model of "classical" synchronous machine (having one field winding, located along the axis d), here besides the equality to zero Ur=0, the following column-matrices will change: In addition, also diagonal matrices will change, which will occur in the form: As a result of equality kqsf=0 and kqfr=0, As2 and Cr2, will also be transformed, which will take the form: If the synchronous machine is made with permanent magnets, i.e. permanent magnets act as the exciter, then additionally the derivative matrix of flux linkages in expression (1) turns into the following matrix: And the voltage Udf, on which the flux linkage Ѱdf, depends, should be interpreted as a value that determines the coercive force of permanent magnets, or more exactly the magnetic energy value of permanent magnets, referred to the unit volume of permanent magnets [12].
Expression for the moment of "classical" synchronous machine will take the form: resistance matrix will appear in the form: The structure of asynchronous machine model in this case: in matrix (1) the 3 rd row disappears and in diagonal matrix the 2 nd column and equation (1) transform into form: Since the asynchronous machine is symmetrical in magnetic and electrical relation, then the parameters kds=kqs; kdsr=kqsr; kdr=kqr and rdr=rqr. Taking this into account, the submatrices are transformed into expressions, Substituting (7) for (6), one can obtain in expanded vector form an expression for derivative flux linkages of stator and rotor loops Ψs and Ψr: It should be noted that expressions (8) by form coincide with the equations of asynchronous machine mentioned in [13].
Expression for electromagnetic moment of asynchronous machine: As it was already mentioned, in the model structure of AC machines as control parameters can be used Us, fsvector and frequency of the rotor winding voltage. Since the structure of the mathematical model of AC machines is based on the writing in the axes d, q rotating at the rotor speed ωr, then there are no problems of modeling of voltages of field and rotor windings, as they are directly supplied from regulating devices (e.g., from the output of automatic excitation regulator of synchronous machine, either the output of frequency converter feeding the rotor winding for asynchronous machine).
This problem exists for stator winding. It is necessary to represent a vector of voltages, which supply stator winding of AC machine, whose components Uds and Uqs are, naturally, written in axes d, q, rotating with rotor speed ωr, in such a form so that to be able to join control system with the electric AC machine (e.g. frequency converter). That is, it is necessary that they reflect the E3S Web of Conferences 209, 02023 (2020) ENERGY-21 https://doi.org/10.1051/e3sconf/202020902023 change (regulation) of the amplitude and frequency of the voltage supplying the stator winding. For this purpose the following transformations must be performed [12].
Angle between axes d, q and fixed axes α0, β0 we designate as α, which is equal to α=ωr·τ and finally, angle θ=α+αs -angle between axes d,q and axes αs, βs, which is called power angle.
If the stator voltage vector to place in the initial state at an angle π/4 radians to the axes αs, βs, then its projections on these axes will naturally be the same and equal to Usα0=Usβ0=0.707·Us (in relative units.).   (11) General projections of these components on the axes d, q according to Fig.1 will be in the form: Substituting expressions from (11) for (12) and taking into account that the vector Us just like the axes αs, βs rotates at the speed ωs, then Usα=Usβ=Usα0=Usβ0=0.707·Us.  By means of simple transformations, expression (13) can be represented in a more convenient form: The control of the AC electric machine from the stator side is represented in (14). And for synchronous and asynchronous machines, this expression allows for taking into account the change and control of both the amplitude kus of the voltage supplied to the machine and its frequency kfs. Moreover, all other equations of AC machines remain written in the axes rotating at the rotor speed.
It should also be noted that in the absence of frequency converter in the stator circuit of AC electric machine, i.e. when the latter is connected directly to the electrical network kus=k us=1.
As it was said, Uf and Ur vectors can also act as control coordinates in the generalized cellular-matrix equations (1). As for the excitation voltage vector Uf, in synchronous machine its components Udf and Uqf along the axes d, q assume the presence of longitudinal and transverse field windings and when presenting the equations written in the axes d, q, rotating at the rotor speed, in the structure of the equations they are constant and, if necessary, controlled values.
Regarding the voltage vector Ur. For synchronous and asynchronous machines with short-circuited rotor it is equal to zero Ur=0. For double-way feed asynchronous machines: at supply both from the stator side and the rotor side its value, naturally, is not equal to zero and in general case its components are written in the form [12]:

Resuls
The control of a controlled synchronous machine with one field winding along the d axis has been investigated [12]. Just so a frequency-controlled synchronous machine with 2 windings along the axes d, q respectively, can be studied [14]. Here study of controlled asynchronous machine has been conducted. In this case, the structure of the equations in vector form is represented by the expression (6), and the components of the stator voltage are modeled by the expression (14), and the rotary expression (15).
Algorithm of solution of equations of asynchronous machine in general form is as follows:  Fig.2 (a, b, c, d) at kus=kfs=1 (i.e. nominal values of amplitude and frequency of stator voltage) at the first stage at short-circuited rotor Ur=0 in the range from 0 to 10 3 radians the starting of generator with the value mt=-0.3, is implemented and is set in the value ωr=1.01 (Fig.2,а). At 10 3 radians the voltage is supplied to the rotor winding with steady-state value kus=kfs=-0.15 (asynchronous machine with short-circuited rotor is transformed into double-way feed asynchronous machine). From 10 3 to 2000 radians the rotating frequency becomes equal to ωr=1.15 (Fig.2,a). Electromagnetic moment is determined by the value mt and remains constant mem=-0.3 (Fig.2,b) ("minus" sign corresponds to generator mode). Curve of general active power, equal to sum of powers of stator and rotor circuits, is shown in the (Fig.2,c): its steady-state value varies from value pw=-0.3 at short-circuited loop in the range from 0 to 1000 radians to value pw=-0.34 at τ>1000 radians, when the rotor speed is increased to ωr=1.15 by means of regulation of frequency of current in the rotor winding of double-way feed machine. Reactive power (Fig.2,d) varies from value qw=0.24 (consumes from network) to value qw=-0.218 (delivers to network). After returning to source mode from 2000 radians to 3000 radians (i.e. when kus=0), is supplied. At control kus=kfs=0.15 at 3000 radians the control equal to ωr=0.85 (Fig.2,a), the active power decreases from pw=-0.3 to pw=-0.25, respectively, and the reactive power, remaining in consumption mode, increases from qw=0.24 to qw=0.49, the moment mem is unchanged and equal to mem=-0. 3 Fig.3. Fluctograms of change of operating parameters of an double-way feed asynchronous machine when regulating voltage and frequency both from the stator side kus=kfs=0.7 and from the rotor side with kur=kfr=-0.15 and kur=kfr=+0.15.
Fluctograms of change of regime parameters of double-way feed asynchronous machine at control of voltage and frequency both from the stator side and from the rotor side are represented in Fig.3 (a, b, c, d). Here kus=kfs=0.7, kus=kfs=-0.15 at the first stage and mt=-0.3. After starting the steady-state value is ωr=0.7 then at kus=kfs=-0.15 (after 1000 radians) the ωr becomes equal to ωr=0.85 (Fig.3,a), the mem remains unchanged and equal to mem=-0.3 (Fig.3,b). The active and reactive powers vary from pw=-0.21 to pw=-0.246, respectively, and qw=0.17 (consumes from network) to qw=-0.166 (delivers to network) (Fig. 3, c and d).
There after source mode at 3000 radians at unchanged kus=kfs=0.7 the value kus=kfs, is regulated, which was set equal to kus=kfs=-0.15. At that the rotor speed ωr decreases to ωr=0.55. The active power accepts value pem=-0.16, and the reactive power in the consumption mode increases almost 2 times (as compared with the source mode) and becomes equal to q=0.34.
The given example of calculation confirms the efficiency of the universal mathematical model structure of controlled electric AC machines. It is known that main power control means for electromechanical AC transformers are frequency converters, which at present are implemented on completely controlled IGBTtransistors or GTO-thyristors with PWM control. The presented structure allows for taking into account in the mathematical model both amplitude and frequency of voltages supplying the stator and rotor circuits of AC machine.
If it is necessary to take into account a saturation, then this can be very simply done by existing methods: to operate by either saturated parameter values, or famous methods of their functional dependence [12]. As regards the consideration of harmonic composition of voltage at the outputs of frequency converters, their consideration also doesn't cause insurmountable difficulties -they can be taken into account in output voltage by means of harmonics factorized to Fourier series.