Analysis of reliability of power supply of consumers in non-full-phase modes using adaptive models of their calculation

Reviewed and analyzed issues related to the reliability of electricity supply in non-full-phase modes of electrical networks, to calculate which use adaptive methods, is built on one Central principle of modeling the modes of electric power systems (EPS). Special attention is paid to the calculation of damages from no full provided of electricity in non-full-phase modes. Introduction At damages (failures, incidents) of the main power supply equipment (power lines, groups of single-phase transformers, shunt reactors) in networks of 110 kV and above which are working with effectively grounded neutral, the main part of shutdowns is connected with steady single-phase short circuits which share reaches on the average 65%. This creates an alternative: 1. For the period of repair of damaged equipment if you want to disconnect the damaged item at all, thus reducing network throughput with feasible energyeconomic consequences in the form of limits to some consumers, the undersupply of electricity to them and thereby causing economic damage or 2. Using single phase control switches or highlighting with disconnectors’ help of damaged phase, and translate the work of a damaged element in incomplete phase mode "two wires – ground", leading to the emergence of a network components of currents and voltages of inverse and zero sequences, with the corresponding consequences in the form of possible overloads in the remaining phases and overheating of transformers, generators and motors currents reverse sequence. To prevent these overloads, it will be necessary limits the loads of consumers with the appearance no full provided of electricity and economic damage from power supply disruptions. The duration of incomplete phase modes can range from a few hours (for repair of the overhead line phase or replacement of the transformer’s phase) to several days and months (for factory repair of the damaged transformer’s phase in the absence of the reserve phase) [1]. Operation in such modes is one, and often the only way to improve the reliability of the operation of the EPS and power supply of consumers in post-accident and repair modes. For estimation the admissibility of non-full-phase modes of operation of the electric network with a grounded neutral, it is necessary: the first step is to calculate the parameters of the normal mode and the parameters of the non-full-phase steady-state mode of the EPS, to determine the parameters of asymmetry in case of damage and subsequent shutdown of one of the phases; the second step is to identify the level of overcurrent of "healthy" phases, the degree of asymmetry of currents and the magnitude of the reverse sequence currents as the most dangerous for the occurrence of overheating of generators, transformers and motor load of consumers working in the network; the third step is the assessment of measures for unloading generating capacities and limiting the loads of consumers to remove re-loads from the network elements (generators, transformers, "healthy" phases of power lines). To identify the economic efficiency of the use of non-full-phase modes of the EPS and to improve the reliability of the network, it is necessary: the first is to estimate the no full provided of electricity to consumers with a limited load and the resulting economic damage when operating in a nonfull-phase mode; the second is to determine the probability and frequency arising of non-full phase regimes; the third is to determine the likely damage to consumers when the electric network operates in a nonfull-phase mode: НПФР ущ З . For comparison, you should do similar calculations for the alternative variant of the reliability of the network with disconnection of damaging element (transmission line, transformer) by, if it is necessary, measures to be introduced in allowable of mode settings area and entering the required limits of consumer loads. Next, it is necessary to determine the under-supply of electricity and damage to consumers from violations of their power supply in the shutdown mode of the damaged network element, to assess the probability and , 0 Web of Conferences https://doi.org/10.1051/e3sconf/20191390 E3S 139 (2019) 1


Introduction
At damages (failures, incidents) of the main power supply equipment (power lines, groups of single-phase transformers, shunt reactors) in networks of 110 kV and above which are working with effectively grounded neutral, the main part of shutdowns is connected with steady single-phase short circuits which share reaches on the average 65%.
This creates an alternative: 1. For the period of repair of damaged equipment if you want to disconnect the damaged item at all, thus reducing network throughput with feasible energyeconomic consequences in the form of limits to some consumers, the undersupply of electricity to them and thereby causing economic damage or 2. Using single phase control switches or highlighting with disconnectors' help of damaged phase, and translate the work of a damaged element in incomplete phase mode "two wires -ground", leading to the emergence of a network components of currents and voltages of inverse and zero sequences, with the corresponding consequences in the form of possible overloads in the remaining phases and overheating of transformers, generators and motors currents reverse sequence. To prevent these overloads, it will be necessary limits the loads of consumers with the appearance no full provided of electricity and economic damage from power supply disruptions.
The duration of incomplete phase modes can range from a few hours (for repair of the overhead line phase or replacement of the transformer's phase) to several days and months (for factory repair of the damaged transformer's phase in the absence of the reserve phase) [1]. Operation in such modes is one, and often the only way to improve the reliability of the operation of the EPS and power supply of consumers in post-accident and repair modes.
For estimation the admissibility of non-full-phase modes of operation of the electric network with a grounded neutral, it is necessary: the first step -is to calculate the parameters of the normal mode and the parameters of the non-full-phase steady-state mode of the EPS, to determine the parameters of asymmetry in case of damage and subsequent shutdown of one of the phases; the second step -is to identify the level of overcurrent of "healthy" phases, the degree of asymmetry of currents and the magnitude of the reverse sequence currents as the most dangerous for the occurrence of overheating of generators, transformers and motor load of consumers working in the network; the third step -is the assessment of measures for unloading generating capacities and limiting the loads of consumers to remove re-loads from the network elements (generators, transformers, "healthy" phases of power lines).
To identify the economic efficiency of the use of non-full-phase modes of the EPS and to improve the reliability of the network, it is necessary: the first -is to estimate the no full provided of electricity to consumers with a limited load and the resulting economic damage when operating in a nonfull-phase mode; the second -is to determine the probability and frequency arising of non-full phase regimes; the third -is to determine the likely damage to consumers when the electric network operates in a nonfull-phase mode: For comparison, you should do similar calculations for the alternative variant of the reliability of the network with disconnection of damaging element (transmission line, transformer) by, if it is necessary, measures to be introduced in allowable of mode settings area and entering the required limits of consumer loads.
Next, it is necessary to determine the under-supply of electricity and damage to consumers from violations of their power supply in the shutdown mode of the damaged network element, to assess the probability and  To select the option of doing network mode: unbalanced or disconnection of the damaged element, it is necessary to match the likely impact of the limitations of loads in unbalanced modes with the damages caused by outages of corrupted items: (1) and choose an option with minimal damage to consumers [2,3]: (2)

Models of calculation of non-full-phase modes of EPS
Mathematical models and algorithms for solving the problems of calculation and analysis of the modes of EPS, in addition to the general requirements of reliability, speed, convenience of representation of the source and output information, are subject to increasingly high requirements for the adequacy of the display of real modes, adaptability to changes in the network structure, modes and types of disturbances.
On the basis of the conducted analysis of existent methods of calculation of unbalance EPS [4][5][6][7][8][9][10] the conclusion about expediency of application for calculation and analysis of substitute complex schemes (SCS) based on symmetrical components.
Linear mathematical models of calculation of nonfull-phase modes of EPS are used for the purposes of relay protection and automation (RPA). Emergency modes of EPS caused both by longitudinal asymmetry, and modes of short circuits (SC) in the electric networks working in the incomplete phase mode are calculated.
The developed method of calculation of non-fullphase modes is based on the use of complex substitution schemes without direct electrical connections between the sequences.
For the formation of SCS, the substitution schemes of individual sequences of symmetric components in the places of asymmetries are connected to each other on the basis of boundary conditions. Modeling of the boundary conditions of asymmetry in the preparation of SCS is carried out with the help of multi poles communication (MPC) [6,7] or sets of ideal transformers (IT) [11,12] with transformation coefficients Kit = 1, a, a 2 , where a = exp(j2π/3), corresponding to the coefficients of the equations of boundary conditions. These two different methods of modeling boundary conditions are based on fundamentally different approaches to the representation of asymmetry locations. In the first case the matrix of self and mutual conductancies of the place asymmetry; in the second case, with set of IT compiled boundary conditions of unbalance.
The use IT for communication between schemes of direct (DS), reverse (RS) and zero (ZS) sequences allows to provide the uniform principle of modeling of elements of a network within SCS. In addition, if the numbering of the SCS nodes is carried out so that the IT nodes receive the last numbers, then the matrix of the node conductivities of the SCS will have a block-diagonal structure with a bordering part in which the conductivities of the IT connections with the nodes of asymmetry are located. The block-diagonal structure of the SCS matrix allows avoiding the procedure of end-toend optimal renumbering of the complex circuit nodes in the analysis of non-full-phase modes and making maximum use of the results of calculations of the previous steady-state mode. At the same time, the advantages of calculating such matrices by methods of optimal elimination of variables are used to the maximum. The use of MPC for modeling of longitudinal type asymmetries violates the block-diagonal structure of the matrix of node conductivities of SCS due to the appearance of direct connections between the nodes of asymmetry of schemes DS, RS and ZS.
Based on the advantages of modeling the boundary conditions of IT over MPC, a universal algorithm for calculating incomplete-phase and complex-symmetric modes of EPS by complex substitution schemes is developed [12].
SCS, compiled for the calculation of non-full-phase and complex-symmetric modes of EPS, is described by a system of linear nodal equations: where Y КСЗis the matrix of nodal conductivities of SCS; where Y (1) , Y (2) , Y (0) are the node conductance matrices of the DS, RS and ZS circuits, respectively; The use of linear models of electrical systems used for the purposes of RPA for the calculation of non-fullphase steady-state regimes (NPSR) leads to large errors. NFRS are essentially load modes, so their calculation, as well as the calculation of the normal steady-state modes of power systems, should be carried out when setting the power values of load and generator units, that is, by nonlinear models.
Substitution schemes of individual sequences of DS, RS and ZS are formed on the basis of the same linear models, except for the account of loads and generators. As you know, in the coordinates of symmetric components synchronous generators are power sources only DS. Sources of currents and voltages of RS and ZS are places of asymmetries. Therefore, loads and generators in the calculation of NPSR are taken into account as follows: in the scheme of DS-the power of the normal (pre-emergency) mode, power losses in loads and generators caused by the flow of currents in them RS and ZS-the corresponding resistances.
The system of nonlinear nodal equations (4) is described. The equations of nodal stresses for DS nodes are nonlinear, since the driving currents in DS nodes are determined through the power of the nodes: where i Ŝ and i Û are the conjugate values of the power and voltage of the i-th node.
A method for solving a system of nodal equations describing the SCS is developed. Previously, the direct Gauss move excludes passive nodes of the RS and ZS circuits and additional it nodes. The result is a transformed system of nonlinear equations DS: The method of calculation of NPSR on complex schemes of substitution possesses sufficient simplicity of algorithm as is based on nodal equations, and universality for electric systems of any complexity. In addition, the obtained matrix ) 1 ( * Y is a matrix of nodal conductivities of the power system taking into account non-full-phase inclusions and can be used for the study of static aperiodic stability of NPSR.

Probabilistic characteristics of non-fullphase modes and damage to consumers from violations of their power supply
In unbalanced mode, de-energizing phase "A" and conservation in the "healthy" phases "b" and "C" determination of the coefficient of over-current in the remaining phases "b" and "C" is carried out according to the following formula: where п р ед ВЛ I is the maximum permissible heating current in the "healthy" phase. The maximum asymmetry of the currents will be equal to the ratio of the total current in phase "B"(or in phase "C") to the direct sequence current: and the current asymmetry of the reverse sequence will be equal to: These three factors will characterize the state of the network operating in the incomplete phase mode.
In [1] it is specified that at work of a network in the incomplete phase mode the asymmetry of currents in phases of generators shall not exceed the following values: a) for turbogenerators -12%; b) for hydrogenerators with air cooling of stator windings with a capacity of 125 MVA and below -20%, with a capacity of more than 125 MVA -15%, and for hydrogenerators with water cooling of stator windings -10%; c) for synchronous compensators -20%.
In this case, the reverse sequence current during long-term operation should not exceed 5% for turbogenerators and 10% for hydrogenerators.
If switching measures to reduce the asymmetry of currents and currents of the reverse sequence do not give the desired results, it is necessary to limit the loads of consumers and reduce the overall load of overloaded power lines and generating equipment.
Unloading of the network should be carried out purposefully with the choice to limit the loads of those nodes that lead to current limitation in the most overloaded transmission line operating in incomplete phase mode. For this purpose it is necessary to use wellknown system of coefficients of distribution of currents The current in the overloaded branch j is defined by the expression: where (i) I у is the resulting current in the i-th node.
To limit loads , nodes with coefficients that affect the unloading of the most overloaded j-th branch before the condition is met are selected: If the current asymmetry coefficients modulo (8) and the reverse sequence (9) are greater than their limit (normative) values, it will require additional limitation of consumer loads in the network nodes until the condition is reached: With this in mind, the limited load nodes will be: and the system load limit will be equal to the sum of the node constraints: Slightly more complicated is the determination of the limits of the consumers in case of unbalance in open distribution equipment (ODE) nodes due to system failures or tire Assembly of switches, which leads to a transition in unbalanced mode all connected to this switchyard, power transmission lines and transformers. The difficulty lies in the fact that the restrictions of the loads of substations are fed from these power lines have an impact not just one but several working in unbalanced mode transmission lines. To do this, it is necessary to rank all substations connected with the damaged node according to the values of their influence on overload and current asymmetry in this node (ODE) and sequentially introduce load restrictions until the condition (13) is met.
Probabilistic characteristics of non-full-phase modes are: 1. The average annual frequency of single-phase steadystate power lines on the power lines entering the circuit and communication transformers (the latter, in principle, can be neglected, given their extremely rare damage relative to the power lines):   of power supply of consumers of the i-th node, rub/ kWh, taken according to [3].
As shown by preliminary calculations for several circuits of electrical networks economically expedient to use in stable single-phase SC incomplete phase mode than to work with the disconnection of damaged elements.

Conclusion
1. The use of adaptive nonlinear models for calculation of non-full-phase steady-state modes of EPS allows to significantly increase the accuracy of calculations.
2. Proposed model of electric networks of 110 kV and above, taking into account the reliability of power supply allows to estimate the parameters of unbalance when single-phase short circuits in networks with grounded neutral and the resulting asymmetry of the currents in the healthy phases on the module and the currents reverse order and carry out the input modes allowable under asymmetry region by selectively targeted restrictions loads of customers in compliance with terms of at least economic damages.
3. Comparison of results of work of electric networks of 110 kV and above in incomplete phase modes and their work with shutdown of the damaged elements allows to justify economically the most effective mode of their functioning in the emergency situations arising at steady single-phase short circuits.