Algorithmic foundations of automated monitoring of commercial and technical power losses in distribution networks

. It is known that significant power losses in power distribution networks (PDNs) caused by unauthorized power withdrawals (UPWs) lead to a decrease in their technical and economic performance. There are no digital technologies designed for separate assessment and monitoring of technical and commercial power losses in modern automatic system for commercial metering of power consumption (ASCMPC), which are currently widely used for automation and computerization of processes in distribution networks. The article proposes a new algorithm for solving the above problem on the basis of data obtained from electricity meters included in the ASCMPC structure. In order to ensure the solvability of the identification problem, the concept of a virtual network model characterizing the desired state of the PDN in the absence of UPW is introduced. On its basis, algebraic equations are obtained, the solution of which allows to identify technical and commercial power losses in three-phase networks. The obtained results are oriented on further improvement of modern ASCMPC and increase of their efficiency and reliability.


Introduction
Automation of information processes in power distribution networks (PDNs) is currently carried out on the basis of implementation of automatic system for commercial metering of power consumption (ASCMPC) [1], which are elements of Smart Grid [2] technology.As it is known, ASCMPC hardware-software complexes belong to the class of information-measuring systems, the main function of which is commercial accounting of electric power in PDNs.The analysis of the functional structure of existing ASCMPCs shows that they do not solve the tasks of separate identification and operational monitoring of technical and commercial power losses in the network [3], as well as do not perform optimization tasks [4][5][6][7], which reduces their efficiency and technical and economic performance of distribution companies.In particular, this is due to the fact that to date, adequate and constructive methods for solving the above problems have not been sufficiently developed, taking into account such factors as the presence of unauthorized power withdrawals (UPW) in the network [8 -10], and the asymmetry of PDN operation modes [11][12][13][14].The problem also lies in the fact that most PDNs belong to the class of large dynamic systems, which have a complex structure and function under conditions of incomplete information about their states and parameters.In [15,16] some approaches to identification of power losses in 0.4 kV distribution networks are proposed.The article considers algorithmic problems related to the development of these approaches.

Task definition
We consider a four-conductor 0.4 kV PDN, the conditional design scheme of which is shown in Fig. 1, where ,  -index variables, representing phase numbers, respectively А, В, С  1,3 and the network's electrical circuits  1,  ;  -instantaneous electromotive force of power supply source for  phase;  ,  ,  -sinusoidal instantaneous current, voltage and complex load resistance of the customer with coordinate ,  correspondingly; ̃ ,  -sinusoidal instantaneous current and complex resistance of  -intercustomer site of  phase;  ,  -instantaneous current and complex resistance of  -site of neutral conductor;  ,  -instantaneous currents and voltages at the inputs of the corresponding phases.Let us assume that the following conditions are fulfilled: 1.The distribution network operates in asymmetrical mode; 2. There are unauthorized power withdrawals (UPW) in the network; 3. Cross-sections of phase and neutral conductors are assumed to be the same, i.e.   ; 4. At discrete time moment   synchronous measurements of the effective values of currents  ,  and voltages  ,  at the inputs of phases and loads of the network, respectively, as well as power factors  cos  and  cos  between them are performed using the ASCMPC electricity meters.The received information is recorded in the ASCMPC database and vectors of input phase currents ′  ,  ,  are formed on their basis.According to the above data, the active  ,   and reactive  ,  powers consumed by the phases and customers of the network, respectively, are determined using known formulas.At the moment   of synchronous measurements of electricity meter readings included in the ASCMPC, the distribution network may be in operative (normal, desired) ( ) or disturbed ( ) state.There are no uncontrolled power losses in the network in the first case (state  ).There are uncontrolled power losses in at least one of its phases in the disturbed state of the network С .In this case, the following balance ratios for capacities are valid: where   -total capacity, consumed by  phase at the time moment   ;   -total complex power consumed by customers of  phase;   -technical power losses in  phase; ∆ -uncontrolled power losses in  phase of the network, caused by the presence of UPWs in the PDN.
Thus powers   and   are known values, as in the existing ASCMPC it is possible to calculate their values according to the data of active and reactive powers received from the balance meter and customer meters using known formulas: The solution of the generated identification problem includes the following main stages: 1. Building a virtual network (VN) model.

Identification of the current condition of the real
network. 3. Formation of the general scheme of identification of power losses in PDN. 4. Estimation of input desired phase currents of the virtual network. 5. Estimation of currents of unauthorized customers.6. Construction of the algorithm for identification of power losses in the network.

Building a virtual network (VN) model
It can be noted that the identification problem under consideration belongs to the class of those problems in which the initial data and conditions for its solution are not fully specified, i.e. the problem should be solved under conditions of insufficient information about the state of the real network.This uncertainty is due to the lack of data on unauthorized consumers, such as their coordinates and load parameters.Under these conditions, in order to solve the identification problem formulated above, we further introduce the concept of virtual network (VN) model, which describes the desired state of the real PDN under certain conditions.The analysis shows that for this purpose it is reasonable to introduce two types of virtual networks (Fig. 2): 1.
Virtual network of the first type (ВС 1 ), which is characterized by the fact that there are no uncontrolled power losses ∆ , and the inter-customer resistances  have zero values (Fig. 2a): Second virtual network (VN 2 ) -there are UPWs, i.e., the following conditions are fulfilled (Fig. 2b): ∆ 0,  1,3 , (5) As well as ratios (4).Here  -complex electromotive force of network supply source;  , complex current and resistance of the virtual customer load, having the coordinate ,  ;  ,  -complex current at the inputs of  phase ВС and ВС respectively; ∆ , ∆ -complex current and resistance of load of unauthorized customer connected to the corresponding phase of the PDN.Next, without loss of generality of the task and for the sake of brevity, let us consider the  phase of the virtual network to which n customers are connected.Input phase currents  ВС 1 characterize the normal operation mode of the considered real distribution network in the absence of uncontrolled currents ∆ .Therefore, hereafter we will call them the desired phase currents.A comparative analysis of the structures of the original PDN (Fig. 1) and virtual networks (Fig. 2) under conditions (4) and (5) shows that the following ratios for the complex currents  ,  and ∆ are valid: Complex resistances  of loads of network customers are known values which are calculated on the basis of readings (data) of customer electricity meters according to the formulas: Relevant conductivities of loads of network customers: where  = .At the same time, the powers  , consumed by the network customers and included in expressions (3) are determined by the formulas:

Identification of the current state of the real network
Under the conditions of the task under consideration, the current values of phase currents    at the input of the initial real distribution network (Fig. 1) at time   are measured by the balance three-phase meter, which are contained in the ASCMPC database in the form of vector ′  ,  ,  .The complex currents of unauthorized consumers ∆ can be represented as: where ∆ , ∆ -modules and phases of complex currents ∆ respectively.Estimates of the effective values of ∆ will be identified later.
The analysis shows that it is reasonable to use the following conditions to identify the current state of the PDN: where ∆ -maximum permissible error of current measurement in ASCMPC.
Obviously, if at least one of the ratios (13) is not fulfilled, there are UPW in the network, and their fulfillment means that the PDN operates in the normal mode.Thus, ratios (13) can be used as a criterion condition for identifying the current state of the PDN.

Formation of a general scheme for identification of capacity losses
We will consider the virtual network diagrams shown in Fig. 2. Let us denote by ′ the powers consumed by the loads of the k-phase of BC2.The analysis shows that there are following balance ratios between the powers ′ ,  and △  : +△ ′ ,  1,3 , (14) Where the total power  consumed by the loads of the k-phase is determined by formula (3).Each of these components, when conditions (4) and ( 5 After simple transformations the last equations will be written in the form:

,3 . 18
As a result, the required load resistances △ of unauthorized consumers will be determined: Then the required values of power losses Δ caused by unauthorized withdrawals in the phases of the real network, taking into account (8), are calculated by the formulas: where the effective values of currents of unauthorized consumers Δ  will be defined further.Now, based on the balance ratios (1), we determine the values of technical losses  of capacities in the phases of the real network: Where   complex capacity, consumed by  phase of the real network at the time moment   , which is calculated from the data of the balance three-phase electricity meter according to formulas (2).
Let us assume that data collection from the ASCMPC meters is performed at discrete moments of time   ,   ,   , …,   .Then the use of the above computational procedure on the basis of these data makes it possible to perform automated operational control and monitoring of power losses in PDN in real time.

Estimation of input desired phase currents of the virtual network
To calculate the effective values of the input desired currents  ВС 1 , we write the first ratios of the system (15) in the form:

Estimation of currents of unauthorized consumers
For this purpose, based on ratios (7)

Fig. 1 -
Fig.1 -Conditional design scheme of a three-phase network.
The task is to identify commercial ∆  and technical   power losses in real time based on measurement data obtained from ASCMPC electricity meters.