Mathematical model for calculating transients of a single-pole synchronous machine of longitudinally transverse excitation

. The article presents the results of studies of synchronous machines of longitudinally transverse excitation operating in the drives of rolling mills with a sharply alternating shock load, provides an algorithm for compiling a mathematical model using a system of differential equations of the Gorev Park, taking into account the parameters of longitudinally transverse excitation windings, provides a comparative analysis of the calculation results with the results of experiments on a physical model.


Introduction
Sharply variable loads of the metallurgical industry cause vibrations of the rotor and other operating parameters of synchronous motors of uniaxial excitation [1][2][3][4][5][6]. The use of automatic control systems /12,15/ even "strong" regulation is not able to provide an aperiodic transition to another mode of synchronous motors of uniaxial excitation. Due to the presence of longitudinal-transverse excitation and a powerful damping system distributed in the massive pole cores /18,19/ of the synchronous motor, it provides an aperiodic transition to another mode. The same circumstance leads to a significant complication in determining the parameters of substitution schemes.
The article presents the results of computational and experimental studies of transients of synchronous motors of longitudinally transverse excitation [1,2] in characteristic modes, such as the determination of oscillations of active and reactive power, the angle of the rotor run-out, dynamic stability /7/ with sudden surges and load drops, based on analytical and experimental data, an algorithm for calculating parameters is developed. Experiments conducted on a physical model of a longitudinally transverse synchronous motor based on an MCA 72/4 synchronous motor show that the proposed refinement of substitution schemes significantly reduces errors in the calculation of electromagnetic quantities for various transients.
The relevance of the problem of determining parameters and electromagnetic quantities is due to the fact that the success of the application of mathematical modeling of electric power systems ultimately depends on the quality of its solution.
Simulation of transient processes of synchronous machines of longitudinally transverse excitation can be carried out in various coordinate systems, the choice of which is determined by the specific conditions of the problem being solved. Synchronous motors of longitudinally transverse excitation are the most complex elements of electric power systems.
In a mathematical model, a detailed account of all elements of a system with synchronous machines using modern computing tools is impossible, therefore, when choosing mathematical models, one has to strive for the maximum possible simplifications, but in such a way that they correctly reflect the main processes of the engine under load over the entire range of rotation speed of synchronous machines /10,11/ even with asynchronous modes.

Methods
When calculating electromechanical and electromagnetic transients in a huge number of cases for systems with synchronous machines of longitudinally transverse excitation /8/ operating with sharply variable load, equations written in the coordinate axes d, q, rigidly connected to the rotor, or ds, qc, synchronously rotating are used for detailed accounting [2,3]. At the same time, mathematical modeling of transients can be carried out both according to the well-known complete or simplified Park-Gorev equations, and using substitution schemes represented in the coordinate axes d, q, o. The equations of machines in the axes d and q, which are a universal standard model, can be obtained from the original system of equations written in the system of phase coordinates a, b, c, by transformation. There are several forms of representation of the equations of the standard model, differing from each other only by the signs before the terms in both parts of the equations, which depends on the choice of positive directions of currents, voltages, coordinate axes, their rotation and rotation of the rotor.
Taking into account the fact that the d axis is ahead of the q axis, it is possible to write the equations of a typical universal mathematical model of a synchronous machine in the form: u d = ri d − pѰ d − ω s (1 + s)Ѱ q u q = ri q − pѰ q + ω s (1 + s)Ѱ d u fd = r fd i fd + pѰ fd u fq = r fq i fq + pѰ fq 0 = r к i kd + pѰ kd 0 = r kq i kq + pѰ kq T = Jω s p s + 1.5 (Ѱ d i q − Ѱ q i d ) Where ud, uq -longitudinal and transverse components of the voltage at the stator terminals of the machine; Ѱ d , Ѱ q -flow coupling of the stator windings along the longitudinal and transverse axes; Ѱ fd Ѱ fq -flow coupling of the longitudinal and transverse excitation winding; Ѱ k Ѱ kq -flow coupling of longitudinal and transverse damping massive contours; ufd and ufq voltage of the longitudinal and transverse excitation winding; id , iq -longitudinal and transverse components of the stator current; ifd, ifq,currents of the longitudinal and transverse excitation winding; iкd, iкq-currents of longitudinal and transverse calming circuits of massive poles;r-active resistance of the stator winding, r fd , r fq , -active resistance of the longitudinal and transverse excitation winding r kd r kq ω = 2πf synchronous angular frequency; rotor sliding relative to synchronous axis  (2) is the electromagnetic moment of the synchronous machine. In order to avoid easily occurring errors in modeling, the slip sign should be taken the same as the sign of the derivative of the angle θ, formed by the transverse axis of the rotor and synchronously rotating axis, t.u. s = pδ/ωс.
Then the sign of acceleration will coincide with the sign of the second derivative of the angle and, consequently, with the sign of the moment acting on the rotor of the machine. For motors of longitudinally transverse excitation (power flow from the network) -the torque on the shaft will be negative, and the electromagnetic torque will be positive.
Consequently, both acceleration in the direction of braking of the machine and sliding at a rotor speed less than synchronous will be positive.
To compile a mathematical model of a single-pole synchronous machine of longitudinally transverse excitation, all parameters are characterized in a system of relative units. In the engineering practice of analysis and calculation of transients in electric machines, various systems of relative units (OE) are used. The most common in the tasks of the electric power industry is a mutual system, or «system Х а », the transition to which does not change the form of the record of the original system (1), presented above in named units (n.u.) /4/. «SystemХ а » it differs from other modifications. Neglecting the subtleties of the transition to systems r.u. /9/ leads to significant errors in the study of transients. Therefore, this article presents an algorithm for the transition to the r.u. system in the formХ а . It includes the following steps: 1. The basic basic parameters are selected: Ib, Ub,, ωb, equal, respectively, to the amplitude values of the phase currents and stator voltages Im, Um and synchronous angular frequency ωs .
2. The basic values of other stator variables and moment are calculated using the formulas: where the base power Sb is assumed to be equal to the rated power of the machine: 3. The basic capacities of all rotor circuits are assumed to be the same in magnitude with the basic power of the stator: 4. The basic currents and voltages of the rotor circuits are found. At the same time, they proceed from the condition that the base current of the excitation winding creates the same first harmonic field in the air gap of the machine as the longitudinal reaction of the armature at its base current. It determines the ratio between the basic excitation current Ifb and the idle excitation current ifxx. To find this ratio, two equations are used representing the synchronous EMF of the generator in r.u: Where ifxx -the excitation current at idle; c -is a coefficient that takes into account the discrepancy between the actual idling characteristic /21/ and the straightened unsaturated initial part, xad -the reaction resistance of the stator.
From the equations for the excitation currents, equating their right parts, we find the basic current of the excitation winding: i f(oe) x a = = c i fq /x fxx Similarly, it is necessary to determine the basic currents of the damper windings, which will differ from the basic excitation current, since the damper windings have a different number of turns and a different winding coefficient. However, such a path is impossible for synchronous machines with massive rotor poles. Therefore, more often they proceed from time constants of contours determined experimentally or estimated by the design dimensions of the machine. Let us assume here that the required basic currents of the damper windings are obtained by any of the methods -IlDb, IkGb, then, based on the formulas, it is not difficult to determine the basic stresses of all rotor circuits. 5. Other parameters of the rotary circuits are calculated: Ψfb =Ufb /ωs; Zfb = Sb /I 2 fb; ZDb = Sb /I 2 lDb; ZkQb = Sb /I 2 kQb.
Naturally, the active resistance of the rotor excitation winding can be found using the time constant Т of this winding when all other circuits of the machine are open: The relative resistances of the damper windings are calculated similarly. The fundamental difficulties in determining the parameters are not related to the system of relative units entered, but to the impossibility of often directly using the factory parameters /16/ given in catalogs and experimental data. Therefore, it is necessary to resort to various simplified schemes of machine substitution in the analysis. Detailed algorithms for calculating the parameters and characteristics of synchronous uniaxial excitation motors are given in [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], the substitution scheme of which is characterized by seven parameters, which are defined below. At the same time, each of the parameters included in equations (1) was divided into the corresponding basic values and, without introducing special designations for variables in r.u., the following system of equations of a longitudinally transverse synchronous motor was obtained in relative units (1 + s)Ψ d − pΨ q ω s − ri q = u q pΨ f ω s + r fd i fd = u fd pΨ fq ω s + r fq i fq = u fq pΨ ID ω s + r ID i ID = 0 pΨ kQ ω s + r kQ i kQ = 0 М = T J ps + (Ψ d I q − Ψ q i d ) (12) where in the last equation of system (3) it is taken into account that TJ = J ωs 2 /Sb All values in the system of equations (3) are dimensionless, and time and the inertial time constant are written in seconds. In this case, the angle δ= ωs/p when integrating the slide is obtained in radians.
Considering that in the r.u. system inductance and reactance are numerically equal, we can write:  In these equations (4) for the flow couplings of the corresponding contours written in r.u.:хd , xq-synchronous inductive resistances along the longitudinal and transverse axes;хad, xaq-inductive resistances of the stator reaction along the longitudinal and transverse axes; хfd, хfq xlD , xkQ -reactive resistances of the excitation windings, l-th longitudinal and k-th transverse damping circuits. The systems of equations (3) and (4) represent a universal, complete model for studying transients of operating modes of synchronous machines of longitudinally transverse excitation. The mathematical model of a synchronous motor of longitudinally transverse excitation, taking into account the saturation of the magnetic system, takes into account the dynamic parameters of the synchronous motor.
At the same time, the system of complete Park-Gorev differential equations compiled for the study of transients should include parameters corresponding to this operating mode SM.
The structure of the Park-Gorev differential equations remains the same as for an unsaturated machine, however, the electromagnetic parameters of the machine do not remain constant, but are in complex dependence on the parameters of the mode and ultimately on the nature of the distribution of the magnetic field in the air gap. /21/ Table 1 shows quantitative comparisons of experimental and calculated values, deviations of the operating parameters of a longitudinally transverse synchronous motor from the steady-state values: electromagnetic moment ∆ Э ах of the displacement angle is ∆ , and the reactive power is ∆ .As you can see, the calculated values, taking into account saturation, are very close to the experimental ones.  The oscillogram (Fig. 1) of the transient process under shock load shows the forcing of the current in the transverse excitation winding from = 0 to =0.48 and at constant current in the longitudinal excitation winding to = . The analysis of the oscillograms of the transient process of charging and resetting the shock load on the shaft of a longitudinally transverse synchronous motor shows the effectiveness of applying and regulating the current in the transverse excitation winding, which provides an aperiodic transition of all operating parameters. Fluctuations in the operating parameters are almost absent. At the same time, the system of complete Park-Gorev differential equations compiled for the study of transients should include parameters corresponding to this operating mode.

Results
The structure of the Park-Gorev differential equations remains the same as for an unsaturated machine, however, the electromagnetic parameters of the machine do not remain constant, but are in a complex dependence on the parameters of the mode and ultimately on the nature of the distribution of the magnetic field in the air gap /17,19/.
Experimental studies were carried out on a physical model of a longitudinally transverse synchronous motor, manufactured on the basis of a synchronous motor of the МСА72/4, type, rated power S=15 кVА and voltage Uн=380 V, rated speed nn = 1500 rp/m with various modifications of the rotor with a sedative winding and with mass-washing poles, equipped with a transverse excitation winding.

Conclusion
A comparison of the calculation results on a mathematical model and the waveforms obtained under direct load showed that the difference in the oscillation amplitudes of the regime parameters from the established values is 7.2%.
The advantages of using a mathematical model and comparing the results obtained with the results of experiments conducted on a physical model are especially pronounced when analyzing processes in energy systems with clearly pole machines of longitudinally transverse excitation, as well as with machines with biaxial excitation with a symmetrical rotor, taking into account many damping circuits of a massive rotor.