Dynamic methods of identification of electromagnetic parameters of power transformers in non-stationary mode

. The report reflects the promising solutions for the creation of adaptive protection means and emergency automatics in the implementation of methods of identification of dynamic systems. Presented original research assigned at directly addressing the causes of the sensitivity lack of the transformer equipment protection – compensation for the magnetizing current of the power transformers by linearizing their transfer characteristics.


The problem of identification of mathematical models parameters of power equipment
Most of the numerical and analytical methods, used in practice for studying transients of electrical power equipment, are based on the use of its idealized (conservative) mathematical description and, as a result of this, inadequately reflect the real dynamic characteristics of the interconnected power systems. Initially, this caused by the methodological error due to the insufficient mathematical rigor of the formulation of the research problem, and in some cases in using of unreliable primary (initial) parameters of the mathematical model during the performance of numerical experiments. All this inevitably causes errors in solving the problem when performing computational studies and can lead to subsequent misinterpretation of the obtained results, which is ultimately reflected in the erroneous view of the operation reliability and stability of the power system [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].
As a rule, the idealization of mathematical models of electrical power equipment of the network is based on the use of concentrated complex equivalent circuits, whose linear parameters are calculated using known analytical expressions under the assumption of stationarity (single part) of the electromagnetic process. Moreover, in the conditions of a full-scale physical experiment with a new electrical power equipment, the idealization of adjacent elements of the power network is used. Quite often, power sources of the examined electrical equipment are idealized, thereby introducing methodological roughness and possible distortions of the experimental results. However, due to the unjustified complication of the model and the inability to meet the computational performance requirements of modern microcontrollers, the use of a rigorous mathematical description of electrical power equipment transients may not be acceptable when solving the problem of improving and expanding of the functionality of microprocessor protection means and automatics [11][12][13][14][15][16][17][18].
In connection with these technical difficulties, the task of creating adaptive parametric mathematical models with a relatively simple structure that adequately reflects the dynamic properties of electrical power equipment in non-stationary operating modes of the electrical network becomes crucial. Refinement of electrical parameters for the adopted (specified) structure of the mathematical model of electrical power equipment is carried out using the methods of identification of dynamic systems.

Identification of parameters of electrical power equipment of the electric power system in various applications
Consider the task of identifying the electrical parameters (R, L & C) of some electrical power equipment (load, transformers, etc.) of an electrical network connected to the busbars of the electric power system (EPS). To do this, we use the classical mathematical description of transients of electric installations using instantaneous values of external (measured) and internal (model) electrical signals. At the same time, to simplify the form of writing identification equations, symbolic operators of differentiation (p) and integration (1/p) of functions (electrical signals) in the time domain are used. Also for clarity, the adopted block diagrams of identification objects are given, and the determination of the RLC-parameters of a given model of electrical equipment is carried out as a result of solving a system of nonlinear differential and integral equations of its transient processes. Naturally, the mathematical description of the transients of the model corresponds to the actual connection scheme of the investigated electrical (electromagnetic) equivalent.
Identification of the parameters of electrical power equipment in such a formulation of the problem allows to perform the synthesis of adaptive, self-adjusting systems for monitoring and controlling the operating modes of the power system. At the same time, the control of electric power system can be performed as a function of change (deviation) of the identified RLC-parameters of the electrical network equivalent.
The structural scheme of the identification of electrical parameters of electrical power equipment is shown in Figure 1.
The scheme includes the main microprocessor modules of parametric identification. Naturally, its connection is made to its primary converters of electrical signals to measuring voltage transformers TVG 1 and TVG 2, as well as to measuring current transformers TA.1 and TA.2 and the corresponding secondary converters of electrical current signals (current sensors i/u, Figure 1) and voltages (voltage sensors u/u, Figure 1). Subsequently, the electrical signals of voltage and current are subjected to analog-to-digital conversion in the ADC modules (in Figure 1, ADC blocks). Digital voltage signals (u) and current (i), measured using an ADC, are fed to a processor (CPU, Figure 1), where integral and differential conversions of digital current and voltage signals are performed (1/p (n) and p (n) ) blocks in Figure 1) in accordance with the mathematical formulation of the problem of parametric identification. The solution of the system of identification equations E(X) with respect to the desired RLC -parameters is carried out using the numerical gradient method in the controller (CPU module, in Figure 1), the essence of which is described below.
-the matrix of sensitivity coefficients, p.u.; X  -the column vector of increments of the desired parameters, p.u.; ( , ) E t X -the column vector of the integral values of the objective function, defined by the expression, p.u.: where T -the averaging interval, which is assumed to be 20 ms.
In comparison with the objective function ( ), its integral values ( ) are signdefinite. Performing the transition according to (3), the final equations of the functions of the root-mean-square error ( ) and its sensitivity are obtained: The vector function • 1 2 ( ) for the transformer model after intermediate transformations will be determined by equation (4). In order to reduce the basic materials, only one objective function and one of the thirtysix sensitivity equations are given, that are corresponding to the first diagonal element of the matrix The presented above expression (5) is a power balance equation on the switching busbars of the primary winding of a two winding power transformer. In accordance with the Kontorovich theorem [3], the convergence of the numerical solution of the nonlinear system of equations (2)  Analysis of the results presented in Figure 2 showed a weak sensitivity of the identification equations to incorrectly specified initial conditions of the electrical parameters of the transformer primary winding. In particular, even with a 100% error in setting the starting values, the root-mean-square error is no more than 3% (Figure 2, a), which guarantees the speed of the algorithm, even if the initial conditions are zero.
At the same time, the algorithm hypersensitivity to the specification of zero initial values of the electrical parameters of identification of the transformer magnetization branch was revealed. Under these starting conditions, the error reaches a value of more than 250 p.u. (Figure 2, c), which can lead to incorrect operation of the identification algorithm. To eliminate this defect, it is recommended to take overestimated initial values of the identified resistances of the magnetizing branch. In this region of positive deviations of the identification parameters, the surface of the vector function of the root-mean-square error is almost linear; its field lines monotonously decrease with respect to maximum values of about 250 %. The nature of the dependence of the vector function of the mean square error from the error of inaccurately specifying the initial resistances of the secondary winding is almost linear -the value of the error is proportional to the error of the initial conditions in modulus over the entire range of values under study.
To accelerate the convergence of the problem of identification of electrical parameters, it is recommended to use the second-order sensitivity equations (equations of "fastest descent"), which are used in gradient methods for solving systems of nonlinear equations.

Testing of identification methods for power transformers under the physical experiments of the switching processes
Testing of the developed algorithm for the identification of electrical parameters of transformer equipment was carried out when performing physical experiments on the switching of three-phase transformers in idle mode. When performing full-scale experiments, solution of several tasks was simultaneously performing -testing of methods and assessment of the effectiveness of algorithms for identification of electrical parameters of transformer equipment in non-stationary modes of their operation. Full-scale tests for the identification of electrical parameters were performed for the following transformers: 1. block three-phase two-winding transformers of type TC-630 000/330, with the scheme and connection group Y/Δ-11, installed at 330 kV outdoor switchgear of the Leningradskaya NPP; 2. three-phase two-winding transformers of type TDC-80 000/110, TDC-125 000/110 and TD-40 000/110, with a scheme and connection group of windings Y/Δ-11 installed at 110 kV substations of the 110 kV distribution network of "JSC Lenenergo"; 3. three-phase two-winding transformer with dry insulation of type Trihal 2 500/10, with a scheme and a connection group of windings Δ/Y-0-11, installed in the Substation-14 of the distribution network of JSC Nissan Manufacturing Rus (St. Petersburg). In connection with a significant amount of physical experiments materials, the most typical results of transients are described below with reference to the two-winding transformers of the TDC brand installed at 110 kV substations of the 110 kV distribution network of JSC "Lenenergo". The error assessment of the electrical parameters identification with the use of above described method was performed fully with respect to all the specified transformers and autotransformers. Figure 3 shows digital oscillograms of phase currents of a power two-winding transformer TDC-80 000/110 (ΔPUnLoad = 85 kW; IХХ = 0.55 %; ΔPShortCircuit = 320 kW; UShortCicuit = 11 %) in the switching mode on idling. However, the highest current value is typical for phase A, its instantaneous value is about 6.25 p.u. (Figure 3). In the subsequent stage of the non-stationary mode for about 0.5 s (Figure 3), the phase currents are damped to amplitude values close to the rated current (381.72 A). In the process of switching on, the start of the longitudinal current differential protection of the transformer was detected, the tripping of which was blocked by the acting of current filters of double frequency (100 Hz).     Based on the above, an important practical conclusion follows that for efficient and high-quality control of the magnetization current in the sub-transient (initial) stages of nonstationary idling modes, a refined mathematical description of power transformer equipment is required, since its calculated equivalent circuit with passport characteristics does not meet high accuracy requirements. Considering this, the results of the study of adaptive compensation of magnetization currents of power transformers and displacement currents, which are typical for the extended intersystem (interstate) connections, are presented in [4][5][6][7][8][9][10]. Unlike with the currently existing stationary compensation solutions in [4][5][6][7][8][9][10][11][12][13][14][15][16][17], the use of additional software algorithms for identification of power equipment parameters is proposed. The proposed method allows to reduce the minimum tripping current to the level of 0.1-0.15 p.u. and to implement a dynamic correction of the tripping with a minimum level of brake signals (Кt = 0.1-0.15 p.u.).
The industrial introduction of such highly sensitive digital protection systems with a speed equal to the period of the industrial frequency will significantly improve the stability of the interconnected power systems and minimize the harm from damages of the protected equipment.

1.
A methodology and software modules for identification of electromagnetic parameters of power transformers have been developed. As a result of the assessment of the sensitivity of identification equations of the power equipment parameters, it is shown that the surface of the objective function always has a global minimum, which identifies the existence and uniqueness of the solution of the system in the space of identifications parameters.
2. It is established that the developed software algorithm is especially sensitive to the assignment of zero initial values of parameters Rμ, Lμ of the magnetization branch of power transformers. In the latter case, the root-mean-square error of the identification equations can reach critical (up to 250 p.u.) values in terms of speed and stable convergence of the numerical method. Positive non-zero initial conditions are characterized by monotonously decreasing vector function of the root-mean-square error E(X).
3. The weak sensitivity of the iterative process of identification the longitudinal parameters R and L to the assignment of them with zero initial values, at which the rootmean-square error is less than 3 %, is revealed. Similar indexes of the algorithm sensitivity were detected in assignment of zero initial values of the electrical parameters R1 and L1 σ of the primary winding of a two-winding transformer. Moreover, throughout the entire 4. As a result of full-scale experiments, the approbation of the developed algorithm for identification of electrical parameters of three-phase power two-winding transformers with a rated voltage of 330 kV, 110 kV and 10 kV in non-stationary operation modes was carried out. It is established that the calculated and averaged characteristics of the leakage inductances of the HV (L1 σ ) and LV (L2 σ ) windings in the sub-transient (initial) stage of the non-stationary switching mode of power three-phase transformers are almost linear.
The mutual inductance of the LV and HV windings (M21) of three-phase power transformers varies asymptotically in a fairly wide range up to ± 15 % regarding the steadystate values. In the final stage of the non-stationary mode, the mutual inductance of the transformer windings M21 tends to values inversely proportional to the current value of steady-state idle mode.
5. The impossibility of applicability the linear calculated equivalent circuits of transformer equipment, obtained based on its passport characteristics, is substantiated. It is shown, that in the sub-transient (initial) stages of non-stationary modes, the reduced value of the error in calculating the currents of power transformers with the use of their passport characteristics, can reach 10-15 %. For effective and high-quality reproduction of magnetizing currents of power equipment, a refined mathematical description is required taking into account the nonlinearity of its characteristics.