On convergence of nonlinear topological algorithms for calculation of steady modes of electric networks

. The paper presents the results of a study of topological algorithms for the formation of a steady state of a complex electrical network, on the convergence of the iterative process. For a power transmission with distributed parameters, consisting of 5 nodes with a voltage of 500 kV, operating on a system of unlimited power, it is shown that there are two solutions and the need to introduce restrictions to obtain real solutions to nonlinear systems of equations. As the regime becomes heavier, the topological method ensures convergence and slow growth of the iterative process compared to the results obtained using the RASTR software package.


Introduction
The use of distribution coefficients of master currents as parameters of electrical networks makes it possible to simplify the performance of some complex calculations [1].Therefore, the use of current distribution coefficients from the perspective of the topological theory of electrical networks is of particular interest.
Problems associated with the analytical method for determining the Z matrix of nodal resistances can be solved based on the following.With a known matrix C, it is always possible to find a unique correspondence to the response of the circuit of the electrical network under study to disturbances of the driving currents [2].Current distribution coefficients, together with branch resistances, quite fully characterize the properties of electrical network circuits, which indicates a certain universality of their application.The complexity of the existing method for determining current distribution coefficients lies in determining the numerators of topological expressions by dividing the network into two parts in order to find two graph trees [3].
For the first time, the exact analytical dependence of the matrix of nodal resistances as a function of the matrix of current distribution coefficients was established by transforming the known matrix equations of the network state in [4].This paper presents the results of an analysis of the convergence of topological algorithms for the formation of steady-state modes, developed on the basis of the properties of possible graph trees, without dividing the network into two parts [5,6].

Materials and methods
The analytical method for constructing the topological matrix of the electrical network of the power system was chosen as the research method.This method is described in detail in [7].
The topological method is applied to a well-known circuit, studied in detail in [2], IEEE test circuits http://energy.komisc.ru/dev/test_cases.
The conducted studies showed that the elements of the matrix of current distribution coefficients of a circuit of arbitrary complexity can be determined based on the natural parameters of the network according to the formula [9]: is specific tree containing the i-th branch relative to the j-th node; ∑  is the arithmetic sum of possible trees of the graph.
The steady state of complex networks is described by the matrix equation developed in [6]: where  is the matrix of current distribution coefficients;  -diagonal matrix of branch resistances;  -diagonal matrix of nodal conjugate stresses; matrix-column of conjugate powers of node loads and generators.
The matrix-column of conjugate powers of nodal loads and generators  depends on the nodal voltages: where  -are given fixed values of generation power (load), coefficients  are expressed through the given values of capacitive conductivities , in mSm and  , in mSm -the conductivities of shunt reactors.

Results and discussions
For a comparative assessment of the convergence of iterative processes for calculating a steady state in a topological model, let us consider the example discussed in Chapter 3 [2]: Equivalent circuit of the system under study (resistance in Ohm, conductivity in cm) A comparison of the results of calculating the steady state performed by the method of simple iteration of the topological model and the Seidel method of the equation of nodal voltages in the form of a current balance is given in table1.It should be noted that the execution time of the proposed program was 10-2 seconds on a regular laptop.First of all, this is due to the fact that in this model there is no need to invert the conductivity matrix, which significantly reduces the number of computational operations.Comparative table of calculations: 1-Seidel method for equations of nodal voltages in the form of current balance, 2-method of simple iteration of the topological model, Δ-relative difference in percentage.
Let us now consider the 5-node test circuit rucase_5_4.txt.The equivalent circuit has the form shown in Fig. 2.  The calculations performed using both methods coincided exactly to the third decimal place.It should be noted that in this case there are several solutions to the vector equation ( 2), so it is necessary to take into account technical restrictions such as | | 550.
Moreover, the execution time and number of iterations, depending on the accuracy of the solution, are as follows: The required number of iterations for a 5-node scheme turned out to be greater than expected.This is because in this case we are considering a case close to the maximum power transmission power, in which additional "imaginary" solutions of the nonlinear system arise, to which the iteration converges very quickly, almost regardless of the required accuracy of the solution (no more than 5-6 iterations).As noted above, to exclude these imaginary solutions, additional conditions are imposed on the solution to limit the voltage module to 525 kV.Therefore, the number of iterations increases.If we reduce the power transfer by 20%, that is, instead of 5400 MW we take, for example, 4400 MW, then the iteration converges to the desired solution quite quickly.Below are the calculation results for a generation power of 4400 MW.As can be seen from the graphs presented in Figure 3, calculations by the Seidel method become significantly more complicated with increasing accuracy.At the same time, the topological model method proposed by the authors shows a slight increase in the complexity of calculations with increasing accuracy, which is an extremely promising method for calculating complex circuits with high accuracy.Figure 4 shows the general dependence of the increase in calculation duration with an increase in the number of iterations for the 5-node circuit presented in Figure 2. As can be seen from the figure, with an increase in the number of iterations, the calculation time increases according to a power law, which is shown by the solid line.
The equation for the approximating curve was obtained according to the recommendations [10]: 30 1.05 10  2970 .01 This type of dependence allows us to describe the ratio of the time of the number of iterations with the coefficient of determination  0.999.It should be noted that using the standard "Power" trend line in MS Excel gives the coefficient of determination value  0.88 , and for the "Exponential" trend line  0.93, which is significantly lower than the value we obtained.
The vertical dashes on the solid line in Figure 3 show the standard error of approximation.As can be seen from the figure, it does not exceed 5%, which also confirms the correctness of the chosen form of the equation.
Thus, it is shown that the topological model makes it possible to increase the speed of calculations several times for complex circuits with high calculation accuracy compared to the Seidel method.
To confirm the conclusion obtained, the authors performed additional calculations of various electrical circuits, including more complex ones in composition and number of nodes.
Next, we present the results of applying the developed topological model to various test circuits.As we can see, the maximum deviation of the voltage modulus is 1.4%, and it is good results.When considering networks with a lot of nodes with possible random dynamic changes, associated, for example, with various types of emergency failures of nodes, it is necessary to consider random graphs.They can be described and studied using pseudo-finite models of standard graph theory.General laws for well-defined systems can be investigated using statistical and modeltheoretic methods.From a model-theoretic point of view, one can approach the approximation [11].The family of such pseudo-finite theories, as well as their topological properties, were studied in [12].

Conclusion
The considered iterative process has reliable and faster convergence than the Gauss-Seidel method using a matrix of nodal conductivities.The proposed method, unlike the method using Zу, does not require determining and storing a matrix of nodal resistances.A feature of the new topological method for calculating the mode of an electrical network is that the current distribution coefficients in a complex circuit can be determined at the stage of generating initial data on the network topology, which significantly reduces the computational costs of real calculations of a complex electrical network.

Fig. 2 .
Fig. 2. IEEE test 5-node circuit rucase_5_4.txt.The initial and calculated data are presented in the following tables.

Fig. 3 .
Fig. 3. Dependence of the number of iterations on the calculation accuracy in semi-logarithmic coordinates.

Fig. 4 .
Fig. 4. Dependence of program execution time on the number of iterations.

Table 3 .
Calculated and initial data for nodes of the test circuit rucase_5_4.txt.

Table 4 .
Initial data for the branches of the test circuit rucase_5_4.txt.

Table 5 .
Calculated data for a 5-node circuit rucase_5_4.txt

Table 6 .
Calculated data for a 5-node circuit rucase_5_4.txt with a generation capacity of 4400 MW.

Table 7 .
Initial data for the branches of the test circuit

Table 7 .
Initial data for the branches of the test circuit

Table 8 .
Comparative results of calculations.

Table 9 .
Calculated data for the circuit