GB2616276A - Network identification - Google Patents

Abstract

A network comprises a first set of nodes, each having a smart meter, and a second set of nodes, each having a µ-phasor measurement unit, µ-PMU. A partial admittance matrix is determined using power and reactive power values measured by the smart meters. The bus admittance matrix is determined using the partial admittance matrix, voltage phasor and current phasor values measured by the µ-PMUs, voltage and current magnitude values measured by the smart meters and estimated voltage phase angles. Determining the partial admittance matrix may involve determining approximate conductance and susceptance values, calculating changes in active and reactive power, and determining more accurate conductances, susceptances and phase angles iteratively. Determining the bus admittance matrix may involve determining complete voltage and current phasors, performing orthogonal triangular decomposition and obtaining linearly independent and dependent rows of the phasors, and formulating and solving an objective function using alternating direct method of multipliers.

Description

Network Identification

Field

The present invention relates to a method of determining topology information, in particular admittance matrices, branch parameters, connectivity between buses, and other details of a power distribution network.

Background

For power distribution networks (for example Western power distribution, SP energy ro network, UK power network, Scottish Sr southern electricity network, Northern powergrid, and Electricity northwest) it is useful to know information about the topology of the network and the value of branch parameters. Determination of these network details is known as 'Network Identification'. Examples of topology information include admittance and impedance matrices; examples of branch parameters include conductance and susceptance, variation in these branch parameters, variation in network configuration.

The network structure varies over time and network details are often not available to a network operator. Thus, Network Identification plays a vital role in performing tasks that enable the operator to achieve optimal operation of the power distribution network. These tasks include: * Fault detection and localization, for example in renewable energy resources (RESs) penetrated low-voltage feeder; * Voltage, current, and internal states of the distribution network; * Estimation of critical lines and real-time contingency analysis; * Distributed state-estimation; * Distribution load flow analysis; and * Planning and commissioning of the distribution network.

Network Identification often involves determining network details such as active and reactive power and voltage and current at certain nodes within the network-in order to calculate network details such as admittance matrices. Typically, phasor measurement units (PMUs) are used to measure the phase and magnitude of current and voltage at nodes of the network. This data is measured by synchronized global positioning system mechanism. PMUs have good estimation accuracy for long transmission lines because power distribution networks have significant phase differences between the buses (or "nodes"). However, this is not case for low-voltage power distribution networks (or "feeders"). Instead, micro-phasor measurement units (p-PMUs) can be used to improve the estimation accuracy for these low-voltage distribution networks.

Smart meters are an emerging device for determining network details in low-voltage power distribution networks. In particular, smart meters are used to measure active power, reactive power and voltage magnitude. This data is unsynchronized (or "asynchronous"). Thus, SMs and p-PMUs are both known in the art for their use in jo Network Identification and the performance of tasks for optimizing the operation of a network. For example, the article by A. Prostejovsky, 0. Gehrke, A. Kosek, et al., "Distribution line parameter estimation under consideration of measurement tolerances," IEEE Trans. Ind. Informat., vol. 12, 110. 2, pp. 726-735, (April 2016) discloses a method of estimating distribution line parameters using only root mean /5 square voltage and power measurements under consideration of measurement tolerances, noise, and asynchronous timestamps. This method fails to accomplish the network structure/configuration, event detection and branch parameters having an for unbalanced structure of the feeder.

It is known in the art that tasks, for example state-estimation, are successful only when the network topology and branch parameters are accurately known. Therefore, there exists in the art various algorithms to identify network topology and branch parameters accurately using synchronized voltage and current measurements. For example, the article by Z. S. Hosseini, A. Khodaei and A. Paaso, "Machine learning-enabled distribution network phase identification," IEEE Trans. Power Syst., vol. 36, no. 2, pp. 842-850, (March 2021) describes a machine learning based data mining method for an accurate and efficient phase identification of residential customers in a distribution network by leveraging power consumption data collected through the advanced metering infrastructure (AMI). However, these methods unable to identify the branch parameters and it has no capability to detect the change in the system configuration.

The machine-learning algorithm described by Z. S. Hosseini et a/. needs enough time to let the algorithm learn and develop enough to comply their purpose with a considerable amount of accuracy and relevancy. This algorithm is autonomous but highly susceptible to errors, which may not be immediately discoverable. Furthermore, this graphical algorithm is not able to identify the line parameters, change in branch parameters, etc.. -3 -

F. Si, Y. Han, J. Wang, et al., "Connectivity verification in distribution systems using smart meter voltage analytics: A cloud-edge collaboration approach," IEEE Trans. Ind. Informat., vol. 17, no. 6, pp. 3929-3939, (June 2021) Another algorithm known in the art is described in F. Si, Y. Han,.!. Wang, et al., "Connectivity verification in distribution systems using smart meter voltage analytics: A cloud-edge collaboration approach," IEEE Trans. Ind. Informat., vol. 17, no. 6, pp. 3929-3939, (June 2021). This affinity propagation clustering-based approach is for to connectivity identification in a low-voltage distribution feeder using smart meter data.

Although the algorithm identifies the network configuration, it is not capable of identifying the network branch parameters with the presence of Gaussian noise in the measurements. -4 -

Summary

According to a first aspect of the present invention, there is provided a method of determining a bus admittance matrix for a power distribution network, the network comprising a first set of nodes, each node having a smart meter arranged to measure parameters of the node, and a second set of nodes, each node having a p-phasor measurement unit, R-PMU, arranged to measure parameters of the node. The method comprises determining a partial admittance matrix using power and reactive power values measured by the smart meters, determining the bus admittance matrix using the partial admittance matrix, voltage phasor and current phasor values measured by the io p-PMUs, and voltage and current magnitude values measured by the smart meters and estimated voltage phase angles.

Thus, a method of determining a bus admittance matrix is provided which uses a combination of p.-PMU data (synchronised) and SM data (unsynchronised). The /5 method presents a cheaper alternative to approaches involving p-PMU measurements only, but a more accurate method compared to approaches involving SM measurements only. Furthermore, the method provides accurate network details (admittance matrices, branch parameters, connectivity between buses, change in system configuration, variation in branch parameters etc.) irrespective of whether the network is balanced or unbalanced.

The power distribution network may be a low-voltage power distribution network.

The power distribution network may be national power network or a local network 25 which forms part of a national power network.

Determining the partial admittance matrix may involve determining approximate values of conductance and susceptance for each node in the first set of nodes, calculate change in active power and reactive power over time using the approximate values, determine more accurate values of conductance, susceptance, and voltage phase angle using an iterative method, constructing the partial admittance matrix using the more accurate values.

The iterative method may be the Newton-Raphson method. -5 -

After determining the partial admittance matrix, determining the bus admittance matrix may involve determining complete voltage and current phasors for the first and second sets of nodes, performing orthogonal triangular decomposition of the complete voltage and current phasors, obtaining linearly independent and dependent rows of the complete voltage and current phasors, formulating and solving an objective function using alternating direction method of multipliers, and determining the bus admittance matrix using the solved objective function.

According to a second aspect of the present invention, there is provided a computer io program which, when executed by at least one or more processors, causes the processors to perform a method according to the first aspect.

According to a third aspect of the present invention, there is provided a computer readable medium, which stores or carries the computer program according to the second aspect.

According to a fourth aspect of the present invention, there is provided a hardware processor configured to perform a method according to the first aspect. -6 -

Brief Description of Drawings

Certain embodiments of the present invention will now be description, by way of example, with reference to the accompanying drawings, in which: Figure 1 shows a simplified example of a power distribution network; Figure 2 shows a first method of determining a bus admittance; Figure 3 shows a second method of determining a bus admittance; Figure 4 shows an Iterative framework between OpenDSSO and MATLABO; Figure 5 shows schematics of a benchmarked IEEE 13-bus system; Figure 6(a) shows a representation of individual phase admittance matrix estimation /0 errors for a phase a'; Figure 6(b) shows a representation of individual phase admittance matrix estimation errors for a phase b'; Figure 6(c) shows a representation of individual phase admittance matrix estimation errors for a phase "c'; /5 Figure 6(d) shows a plot of relative error of estimated conductance and susceptance for phase a'; Figure 6(e) shows a plot of relative error of estimated conductance and susceptance for phase b'; Figure 6(f) shows a plot of relative error of estimated conductance and susceptance for phase 'c'; Figure 7(a) shows a plot of estimation error in line conductance and line susceptance with respect to smart meter power measurements; Figure 7(b) shows a plot of standard deviation in smart meter power measurements against MAPE variation of conductance and susceptance; Figure 8(a) shows a plot of estimation error in line conductance and line susceptance with respect to micro-phasor measurement unit, p-PMU, measurements; Figure 8(b) shows a plot of standard deviation in micro-phasor measurement unit measurements against MAPE variation of conductance and susceptance; Figure 9 shows a plot of the number of iterations for convergence for the method shown 30 in Figure 3 for three different p-PMU standard deviations; Figure 10 shows a representation of the effect of change in standard deviation in gPMU measurements on the phase-a' admittance matrix error; Figure 11(a) shows a plot of variation of the estimation error of conductance and susceptance in various phase-'a' branches with optimal p-PMU locations with 35 PMUs; -7 -Figure 11(b) shows a plot of variation of the estimation error of conductance and susceptance in various phase-a' branches with optimal p-PMU locations with 4 pPMUs; Figure 12(a) shows a representation of individual phase admittance matrix estimation errors, following the switching event for phase-a'; Figure 12(b) shows a representation of individual phase admittance matrix estimation errors, following the switching event for phase-'b'; Figure 12(c) shows a representation of individual phase admittance matrix estimation errors, following the switching event for phase-"c' io Figure 12 (d) shows a plot of relative error of the estimated conductances and susceptances of each of the branches in three phases, following the switching event for phase-'a'; Figure 12(e) shows a plot of relative error of the estimated conductances and susceptances of each of the branches in three phases, following the switching event for /5 phase-'b'; Figure 12(f) shows a plot of relative error of the estimated conductances and susceptances of each of the branches in three phases, following the switching event for phase-'c'; Figure 13(a) shows a representation of the performance of the system with the presence of p-PMU devices for phase-'a'; Figure 13(b) shows a representation of the performance of the system with the presence of R-PMU devices for phase-'13'; Figure 13(c) shows a representation of the performance of the system with the presence of R-PMU devices for phase-'c'; Figure 14(a) shows a detailed schematic of the benchmarked IEEE 123-bus system; Figure 14(b) shows a representation of phase admittance matrix estimation errors for phase-'a'; Figure 15 shows network identification of the IEEE 13-bus feeder with the presence of renewable energy sources; Figure 16(a) shows a heat map of identification error using a prior art method; Figure 16 (b) shows a heat map of identification error using the method shown in Figure 3; Figure 17(a) shows a plot tracking performance of conductance and susceptance of branch parameters using prior art framework; Figure 17(b) shows a plot tracking performance of conductance and susceptance of branch parameters using the method shown in Figure 3; -8 -Figure 18(a) shows a plot showing estimation error in conductance and susceptance using a prior art algorithm; and Figure 18(b) shows a plot showing estimation error in conductance and susceptance using a the method shown in Figure 3;

Detailed Description

The present application is concerned with a method of determining information about the topology of a power distribution network, in particular a bus admittance matrix. This bus admittance matrix includes admittance values between nodes of the network.

The method utilises measurements made by both smart meters (SMs) and microphasor measurement units (u-PMUs) coupled to nodes of the network such that voltage, current values etc. can be measured at each node. An admittance matrix is determined for two sets or groups of nodes; each set of nodes is coupled to either SMs or p-PMUs. As will be hereinafter explained, the way in which the matrices are calculated for each set differs.

Power distribution network 1 Referring to Figure 1, a simplified power distribution network 1 (herein also referred to as "network") is shown. Preferably, this network 1 is a low-voltage power distribution network.

This network 1 may be a network on a national scale, for example Western power distribution, SP energy network, UK power network, Scottish SE southern electricity 25 network, Northern powergrid, Electricity northwest. The network 1 may be a local network which forms part of a wider network.

This simplified power distribution network 1 will be used hereinafter to show how details of a power distribution network can be determined using a method according to 30 the present application, as shown in Figure 2.

The network 1 includes a plurality of buses 2 (or "nodes"), each bus 2 is coupled to a load 3. The network 1 further includes a power generator 4 arranged to supply a power signal to each load 3. The loads 3 may be domestic or industrial buildings, for example.

A measurement device 5 is connected between each bus 2 and their respective load 3 such that certain measurements can be taken of the power signal supplied to the load 3 -9 - (the bus 2 and measurement device 5 are schematically depicted in a single block in Figure 1).

The measurement devices 5 in the network 1 consist of either smart meters or p-PMUs. 5 In other words, measurements at each bus 2 are taken either by a smart meter or a tiPMUs.

Each smart meter is configured to measure active power, reactive power, and voltage and current magnitude at its respective bus 2. The measurements taken by the smart jo meters are unsynchronised. Each p-PMU is configured to measure voltage phasor and current phasor values. The measurements taken by the p-PMUs are synchronized.

As will be hereinafter explained, the measurements taken by both the SMs and p-PMUs will be used to determine or calculate details of a power distribution network, for is example the network 1 shown in Figure 1. Details of the power network include admittance matrices, and line conductance and line susceptance (herein also simply referred to as "conductance" and "susceptance") at nodes.

Method of determining network details Referring also to Figure 2, an overview of the method of determining details of the network 1, including determination of a bus admittance matrix, will now be described.

As will be hereinafter explained, the method involves: * determining a partial admittance matrix corresponding to the nodes 2 at which smart meters 4 record measurements ( herein also referred to as "SM nodes"); then * determining a bus admittance matrix for the network 1.

Firstly, the measurement devices 5 perform measurements of the supply signal and 3o data from both the SMs and p-PMUs are collected (step Si.o). The data collected by the SMs consist of active power, reactive power, voltage and current magnitude for each measured node. The data collected by the p-PMUs consists of voltage and current phase and voltage and current magnitude.

The data sets may have been collected prior to the start of the method.

-10 -Approximate values of the conductance and susceptance (branch parameters) between the buses 2 of the network 1 are determined (step S1.1) using the measurements collected from the smart meters 4. The conductance and susceptance is determined only for those nodes to which smart meters 4 are coupled and record measurements.

Next, more accurate values of conductance and susceptance are determined by first determining the change in active and reactive power for each SM node (step S1.2).

Then, more accurate values of the conductance, susceptance, and voltage phase angle calculated in step S1.1 are determined using the Newton-Raphson (NR) method (step S1.3).

Using the more accurate values obtained in step S1.3, a partial admittance matrix for the SM nodes can be formulated. This admittance matrix constitutes partial topology information of the network 1.

Next, the voltage and current phasors for each u-PMU are determined using the voltage and current magnitude and phase angle measurements from the p-PMUs (step S1.4). These phasors are presented in a voltage phasor matrix and a current phasor matrix, 20 each having rows and columns.

Then, as will be explained in more detail herein below, orthogonal triangular decomposition of the voltage and current phasors is performed (step S1.5) and the linearly independent and dependent rows of the voltage and current phasors are obtained (step St.6).

As will be hereinafter explained, an objective function is then formulated (step S1.7) using the dependent and independent rows obtained in step S1.6. This objective function is then solved by dividing the function into sub-problems (step S1.8). This is 3o done using alternating direction method of multipliers (ADMM).

The bus admittance matrix for the network is then obtained by using the partial admittance matrix and solving the objective function (step S1.9).

Referring to also Figure 3, a more detailed method of determining details of a power distribution network will now be described.

Measurements are obtained from the SMs and p-PMUs installed at various nodes in the network (step S2.o), as hereinbefore described. Then, a method to obtain the partial information of the topology using only the smart meter measurements is described.

Finally, the p-PMU measurements are used to obtain the complete network identification. The topology information, as well as the conductances and susceptances of each of the lines, is obtained through this method.

The method is organized in three subsections, described as follows.

A. Available Measurements The measurements are available from the smart meters and the distributed phasor measurement units. The measurements available from the smart meters are the real and reactive power injections and the voltage magnitudes, while the measurements with distributed micro-phasor measurement units include the time-stamped voltage phasors and the current phasors. it is assumed that the smart meters are installed at certain buses, while the phasor measurement units are installed at the rest of the buses.

Assuming that 'CV and 'IV are the susceptance and conductance of a line between the nodes 1' and T (i, j EN), then the active and reactive power injections at a given bus can be written as: Pt = IKIE7-1117,11(Gti cos O0 B,i sin 0,j) (1) = 'VII Ein=11VAGEJ sin 61 -B,J cos 0 ti) (2) where Pi' and (),' are the nodal real power and reactive power injections, while I Vil represents the voltage magnitude at a node T. The smart meter measurements are initially processed using (1)-(2) as base equations, in order to extract the partial information of the topology, while the p-PMU measurements are then utilized to extract the complete topology information.

B. Extraction of Partial Topology Information using Smart Meter Measurements Assuming that the smart meter measurements are available only at certain nodes, the 35 partial information about the network topology is extracted using the corresponding -12 -measurements. This information is extracted in two main steps. The first step involves a regression process to evaluate the approximate values of the partial topology parameters, i.e. the line susceptances and the conductances, while the second step involves obtaining their exact estimates.

0 Step-1: Parameter Evaluation through Regression From (1)-(2), for a given number of nodes 'n', it is possible to write, = f221 1 (3) T [GK Qtht GBr ' * * Brn 1 where = ti cos 0" By sin OH) = -(G11 sin 0ii -B11 cos B j) (4) Initially, using the available smart meter measurements (i.e. real power, reactive power and the voltage magnitude) at different time stamps, the following matrices are formulated for 'K' instants of measurements, D(i) pr v (1) i 1 [PVC] = 1 Vp;:l (KK)) 12i1VJ1(1) 11/1 'VI 1(K) [(2119 = (5) (i) Q(K)

TI Q,

-1,11-1(1) IV 7-1( 10 The superscripts c(1)' to (K)' represent the time instants of measurements, while the subscripts '1' to 'r' represent the numbers corresponding to a node assuming that the smart meters are installed at t' nodes (r c N). Using (3),(5), the following expression 25 can be written, Gn ** *af 11711(1) * * * 11711(K) =F * (6) *** GiEn iVri(1) A pseudo-inverse technique [23], is then used to compute the approximate values of 'GC' matrix as, [G4] = [Pv9 [vK]avK][v99-1 (7) where, [Gt] corresponds to the first matrix of the right-hand part of (6), while FIN is the corresponding second matrix. Likewise, '13b#' matrix is computed as, [W] = -[Q1/1(][VinaV19[VIT)-1 (8) It may be noted that only the information of conductances and susceptances corresponding to the smart meter nodes is obtained from (7)-(8). The voltage angle 'Oil in the distribution feeder, is usually within at rad, thereby, sin 0 o, cos 0 1. Therefore, the obtained values of Gij#' and 13ij#' from (7)-(8) (step 52.1), form the approximate values of Gui' and 13u;', to be utilized in the succeeding step as depicted in Fig. 1.

2) Step-II: Parameter Evaluation through Newton-Raphson Analysis In this step, the approximate values of conductances and the susceptances obtained in step-I, are used to obtain the corresponding exact values of the partial topology information (corresponding to the SM nodes) as depicted in Figure 3.

The Newton-Raphson (NR) method is employed here, where the approximate conductances and susceptances serve as a fine initial start for this method. Using the available active and reactive power injection measurements, at the buses with smart meters, the change in the active and reactive power matrices are built as (step S2.2 and step 52.3), Pfl) -PIT [MIK] = -pLi) n0) [aQ9 = F tel Lel c (K) -PIP li pnyo _ p(K) r _0,("<) c e1, Q7c u io _ o,Qc) Qic.1) -Qa) (9) -14 -where, the prefix 'c' denotes the calculated values. Initially, these real and reactive powers are calculated using the approximate susceptances and conductances. The Jacobian matrix in the NR method is as, APi aP

_ [AQI ag

Ab (10) op at, 29 at, [Al 86 AO where [gg kit on(i) The problem (10) is solved using the pseudo-inverse as (step S2.4), op ab 29 ab an Fag' rP ag Al, = AO ag Do 29 pLi opi p]AQ (12) The conductances, the susceptances as well as the voltage angles are updated as, [g b 0]?; = [g b OF + [Ag Ab AOF (13) The new values of the line parameters (conductance and susceptance) and the voltage angles (phases) are then used for the next NR iteration, while these iterations are carried out until the convergence point is reached. The threshold for convergence, computation of the pseudo-inverse (t), as well as the Jacobian matrix, are disclosed in J. Zhang, Y. Wang, Y. VVeng, et al., "Topology identification and line parameter estimation for non-PMU distribution network: A numerical method," IEEE Trans. Smart Grid, vol.11, no. 5, PP. 4440-4453. Sept. 2020. A threshold is set for the topology modification, where the small values of conductances and susceptances are identified as wrong branches and thus eliminated. The NR iteration is run again whenever a branch is less than this threshold. The accuracy of the voltage angle estimation is also improved using a pseudo power flow conducted with the known information, before every iteration.

Thus, application of the NR method results in more accurate values of the line parameters and voltage phases; these values are then used to determine a partial admittance matrix i.e. 'Yu' (step S2.6), as will be explained herein below.

-15 -The overall process to obtain the accurate network parameters is illustrated in Figure 3.

C. Complete Topology Information with a-PAIU Measurements The partial information of the line susceptances and conductances are obtained, using the smart meter measurements, by solving the NR method in section B-2. However, to obtain the complete information of the topology, the p.-PM15 measurements are required.

/ 0 As hereinbefore explained, for SMs installed at buses '1' to 'r'', the information obtained from solving equation (12), are the elements of the true (or more accurate) conductance and susceptance matrices, corresponding to the approximate matrices [G# ] and [B#] in equations (6) and (7) respectively. Thus, as the 'GI; and from the rows '1' to 'r' are accurately known by step B-2, which implies that the top 'r' rows of the bus admittance /5 matrix are known accurately, i.e. from the system's bus admittance matrix (14), the submatrix Tr' corresponding to the matrix with first 'r' rows, is entirely known.

Ybus Yr,tt (14) Yr+1,tt The matrix 'Yin' is the matrix with the last 'n-r' rows of 'Ys,,,' in equation (14), associated with the unknown values. Thus the main goal is to estimate the elements of the matrix Yrn' using the available R-PMU voltage phasor and the current phasor measurements.

For a distribution network with 'n' number of nodes, the admittance matrix is associated with the current injections at the 'n' nodes and the voltages at the 'n' nodes in the following way, U1(k) h(k) *** h(k)iT = Yhus* [Y1(k) V1(k) *** Yn(k)F (15) It may be noted that the elements of the current injections vector are.10 t=1,9 n) and the nodal voltages vector are both of which are the complex quantities i.e. phasors at a particular instant of time 'k'. These phasors are obtained from the phasor measurement units installed in the network. The information available from the phasor -16 -measurement units are the time-stamped current and voltage phasors with nodes 'r' to 'n', which are given as (step S2.7), Vr_1(1) VmpPhausor = F LIid (1) Jr-vi (K) P1Matisor = I Ip (16) n(1) In(K) The indices in the brackets in equation (16) represent the voltage and current sample instants from '1' to 'K'. With this available phasor information, the matrix Tit' needs to io be estimated. For this purpose, the problem statement is formulated as, %Pm= argmin SM (17) rphasol, pmu yrn. , p mu -1 phasor v phasor

-

phasor subject to E S(n-r)xn ts where V1(1) * * * VI (K)1 V S ' (18) phasor ' ' r Vr (10 The vector in equation (i8) is the phasor vector of voltages corresponding to the buses with smart meters, i.e. buses 1' to r'. The information of this vector is known from the measured magnitudes '1Vir, while the angle information is also known, from equation (13). Therefore, the feasible solution to the problem statement (17) could be obtained from the available information. By solving (17), the accurate is obtained, while the accurate Yir' is acquired in section B-2, thereby, the complete information of 'Yip.' matrix is obtained i.e. the bus admittance matrix. This information depicts both the 25 topology, along with the line parameters of the network.

However, obtaining the 'Yin' matrix directly from equation (17) is challenging, owing to the sparsity in the matrix and the matrices vichtsum.' and liftcuo,' being rank-deficient matrices. Therefore, the procedure to obtain Yrn' accurately is reported as follows.

-17 -Initially, the tipper triangular elements of the admittance matrix are grouped in the function G(Yrn) as, "Yrn) = [+i,1 +1,2 *** Yr+1,11 Yr +2,n r+2,2 *** Yr/A (19) Yr+2,2 Yr+2,3 *** Yr+3,3 * ** A binary operator 'OX' is formulated, that could convert the G(Y) to vec(Yo.), which implies that vec(Yi..)=0xxG(t..). Thus, equation (17) is rewritten using equation (19) /0 as, 01,1120r)11 G(t.")= argmin 11010,",", 100x.x -vec (20) xec0,2+7,-)/ 2 Xl where, 0 represents Kronecker product.

Owing to the low-rank structure of the matrices (16), a transformation matrix 'TNT' (size: nxn matrix) is formulated, that separate the linearly independent rows from the linearly dependent rows (step 82.10) as, TM. VphasOr = [11/ NT (21) In (21) 'Iv' is a matrix with the p' linearly independent rows of Arpbasoc and 'Dv' with the remaining 'n-p' rows of the Vphasm.' matrix.

To obtain 'TM', firstly, the orthogonal triangular decomposition of %Mast); is done and the diagonal elements of the upper triangular matrix are sorted (step S2.8 and step S2.9). The overall process to obtain 'TM' is illustrated in Figure 3. Then, the first p' elements of the permutation matrix thus obtained, are selected as the indices of the linearly independent rows. The sub-matrices 'Iv' and 'Dv' are thus obtained and in the similar way, sub-matrices 'II and 'DJ' corresponding to the trs,' are also obtained (step 82.11) i.e., TM. IptUsor = [11- (22) The transformation operation for the admittance matrix is evaluated from (21) and (22) as, TA/. ipPrtaUsOr = (TM. Y111. T441) (TM. VPhaSeT) (23) [YT1 YT21 YT3 YT4 As 'IN; is a linearly independent vector, 'Dv' could be written as Dv=17/1.I-V, where, Di Di2 nv(Dv) (24) DI (= [D13 D 141) = Iv X p lo where, pinv represents Moore-Penrose psendoinverse. Thus, from (21)-(24), [1 j 1]'-'T1 [iv Dvp' Yri YT4 YT3 YT4 (25) Finally, the optimization problem (20) is reformulated and estimated admittance submatrices (Itn, YT4) is computed using (23)-(25) (step S2.13) as, [G (V1.1) G(214)] 11[-(D1 0 D1)0D1i ODJ.x xec(n2+n),/zxl -vec ((Iv x pinv(Iv)) -(pinv(Iv) x II). D011 (26) To simplify the problem statement (26), the elements of (26) are assigned to new variables as G IL x -[G (tT1) G(71-4)] (27) ILXL -(Dr 0 DOOni, °Dm] (28) VI"' = vec x pinv(1))-(pinv(lv) X If). D1) (29) -19 -The size L' in (27)-(29) can be identified as ((2(n-02-(2(n-r)+1).p+p2)Y2. The rest of sub-matrices (?12. V13) can be obtained through (25) (step S2.14). Therefore, from (26)(29), the system at hand, is described through following equation, V = Y. G + £ (30) where, 'c' is a finite Gaussian measurement noise. For a single ith row, (3()) can be written as, /0 V1 = Yi. + Et I i=1,2, 1, (31) To obtain the "Y; from the (31) and thus the matrix 'Y.' (step S2.15), the regression problem is formulated using alternating direction method of multipliers (ADMM) approach as follows.

argmin(1/2)11Y1Gi +

YE

subject to Gi -zi = o (32) where, a is a regularization parameter factor; zi are typically weighted vector and 20 intermediate vector, zi E Rm. Based on the formulation of the ADMM method, the Lagrange multiplier (371) is incorporated into the existing formulated problem (32) as follows.

z y i) = (1/2)11Y161 - + allz;Ili + Yr (G; -zi) (P /2)11 (33) where, p is a positive penalty parameter. In ADMM method, Gi and zi are updated in an alternating or sequential fashion, which accounts for the term alternating direction.

The minimization problem is divided into two subproblems to reduce the computational burden and executes the minimization independently, unlike the Lasso framework known in the art. The iterative steps of the ADMM algorithm are expressed as, 6k+1 = argttun F(Gi,zi, yi) (34) -20 -Zk+1 = argmin F(Gi, z1, y) (35) yr = ± Gr1 (36) The iteration (34)-(36) is updated until the stopping criterion is satisfied. The stopping criterion is defined as, 116f, -4112 pabs + prcimax (11012,114,112) (37) zr1112S /labs + c/P ilYn12 where, pabs and pro are absolute tolerance, relative tolerance, respectively.

Application of method in example network To validate the effectiveness of method according to the present application (as illustrated in, for example, Figure 3), the bench marked IEEE 13-bus and IEEE 123-bus feeders are considered herein with a coupled household load.

The Electric Power Research Institute (EPRTC)) has developed the open distribution system simulator (OpenDSS®) software, which facilitates the input/output information to study the dynamics of the benchmarked distribution feeder. The main advantage of OpenDSS® platform is that it is capable to perform multi-phase power flow, unbalanced power flow analysis, fault/event analysis, and stability analysis, unlike MATPower® and PowerWorld® simulator platform.

Referring also to Figure 4, the detailed process of an iterative procedure of MATLAB® software with OpenDSS® platform is shown.

The component object model (COM) interface platform is used to communicate between OpenDSS and MATLAB® software. The detailed script of location and rating of integrated renewable energy sources, variation in branch parameters, type of events, etc., are written and recorded in the MATLAB® environment and this script is processed as input to the OpenDSS® platform. Likewise, the event-logger of the OpenDSS® platform also has an inherent feature to track/record the operation, control action, switching operation, and event for the given network.

-21 -The voltage measurement is directly obtained from the node. The current measurement is obtained by taking the difference between the net injected current and net outgoing current at a certain bus. A command-separated value (.csv) report of the voltage-current dataset is obtained and processed into the MALTAB® software to accomplish the identification objectives (in other words, determination of the bus admittance matrix etc.). The salient points of the iterative procedure are depicted in Figure 4.

Referring also to Figure 5, the detailed configuration of the modified IEEE 13-bus system is shown.

From Figure 5, it is clear that the IEEE 13-bus system is highly unbalanced lateral as each phase is not connected with all buses. For the reliability of the electric power supply, the normally open (NO) and normally closed (NC) breakers are coupled between 692-680 and 692-671 buses as depicted in Figure 5.

The detailed installation of the SMs and p-PMUs are demonstrated in Figure 5. Based on optimal location, the SMs (e.g., at 6n, 645, 646, 652, 671, 68o, 684) and p-PMUs (e.g., at 632, 633, 634, 65o, 675) are considered herein to suffice for Network Identification.

The necessary data at each bus of the IEEE 13-bus system is acquired from the OpenDSSO software, then, processed further in the MATLAB® platform. The real-time measurements of the local load profile for the benchmarked IEEE 13-bus system are acquired from the Autonomous Decentralised Renewable Energy Systems (ADRES) project repository. The performance method according to the present application is tested under several operating scenarios such as basic network identification, network identification with the consideration of noise in the measurements, impact of change in standard deviation in measurements, and identification under a change in the system configuration. The detailed analysis of the method according to present application with various cases is analyzed as follows.

Case-I Basic Identification of Topology Referring also to Figure 6, estimation of branch parameters of for the IEEE 13-bus feeder using the method according to the present invention is shown. As will be hereinafter explained, figures 6 (a) to (c) show a coloured representation of individual -22 -phase admittance matrix estimation errors for (a) Phase-a', (b) Phase-b' and, (c) Phase-'c' respectively (see a, b, and c labels in Figure 5).

Figures 6 (a-c) demonstrate the basic identification of the modified IEEE 13-bus system. As smart meter data and u-PMUs data are obtained from the defined buses, the presented ADMM algorithm segregates the data based on the type of measurements and identifies the network configuration using equations (7)-(8), (13) and (31).

Figure 6(a) shows the identification of phase-a' of the given JEFF 13-bus low-voltage io distribution system. From Fig. 4 (a), it can be observed that the identification error in the admittance matrix is quite low in the range of To-I. Similarly, the identification in the admittance matrix for the phase-'b' and phase-'c' are illustrated in Figs. 4 (b-c). The typical value of error in identification is very low as the presented method provides robust identification and has good accuracy due to having insusceptibility against poor /5 conditioning. The detailed analysis of the identification error in conductance and susceptance of each branch (e.g. for phase-a', phase-1)% and phase-'c') is plotted in Figs. 4 (d-f). Tt can be easily seen that the error in estimated parameters is achieved within 2%.

Case-IIPerformance of Network Identification with Consideration of Noise in SM Measurements Referring also to Figure 7, estimate error of the algorithm for different noise levels in the SM power measurements is shown. In particular, Figure 7 (a) shows g' and error variation with the standard deviation in each of phase-a' branches, and Figure 7 (b) MAPE variation of g' and 'b' in all three phases Figure 7 (a) and Figure 7 (b) demonstrate the robust performance of the network identification algorithm with consideration of noise in the SM measurements. Figure 7 (a) shows the illustrative 3-D heat map of estimation error (%) of each branch versus standard deviation in SM measurements. It is easy to observe that identification error is quite noticeably low for having a low standard deviation of SM measurements.

The Newton-Raphson algorithm (see equations (9)-(13)) has inherent advantages of swiftness and quadratic convergence rate. As standard deviation is increased in the SM measurements, the identification error is considerably increased as depicted in Figure 7 (a).

-23 -Furthermore, the mean absolute percentage error for the estimated branch parameters of each phase is analyzed in Figure 7 (b) with the variation of standard deviation in the SM measurements. The 'blue' ("MAPE of 'CI' line with circle plot points) and 'red' ("MAPE of 13-line with cross plot points) colour graph indicates the mean absolute percentage error' for estimated conductance and susceptance, respectively. The presented ADMM based network identification algorithm (in other words, the method according to the present application described with respect to Figure 3) provides accurate results as it follows the decomposition-coordination procedure and local-Jo subproblems equations (34)-(36) are coordinated to find a solution to a large global problem. In general, the typical variation in the range of io% is observed in the estimated branch parameters when any dynamic reconfiguration/event has occurred in the system. Therefore, it may lead to an indication of the change in the network configuration for having erroneous results in estimated conductance and susceptance parameters. Hence, the limit line is chosen at io% in the maximum absolute percentage error (MAPE) graph in Figure 7 (b). One can be easily observed that the permissible standard deviation is found in range of 0.07 for the satisfactory identification as estimation error breaches the maximum allowable change (io%) in branch parameters afterwards.

Case-III Performance of Network Identification with Consideration of Noise in wliMIT Measurements Referring also to Figure 8, estimation error of the algorithm for different noise levels in the R-PMU measurements is shown. Figure 8 (a) shows g' and 'b' error variation with the standard deviation in each of phase-'a' branches, and Figure 8 (b) shows MAPE variation of g' and 'b' in all three phases Figure 8 and Figure 9 demonstrates the robustness of the presented algorithm to suffice the identification objectives with the consideration of noise in the p-PMU measurements. Figure 8 (a) shows the illustrative 3-D heat map of estimation error (%) of each branch versus standard deviation in p-PMU measurements. It shows that the identification error is quite low for having a small value of the standard deviation.

The method according to the present application has better noise rejection capability 35 than, for example, W. Pan, A. Sootla, and G. Stan, "Distributed reconstruction of nonlinear networks: An ADMM approach," IFAC Proceedings Volumes, vol. 47, no. 3, -24 -PP. 3208-3213, (2014) and S. Boyd, N. Pa rikh, E. Chu, et al., A Distributed Optimization and Statistical Learning via The Alternating Direction Method of Multipliers, 1st Edition, Now Publisher Inc., Hanover, USA, (2011). Thereby, the identification objective is not compromised as depicted in Figure 8 (a). As the standard deviation is increased in the measurements, the identification error is increased as depicted in Figure 8 (a).

The detailed error analysis using 'mean absolute percentage error' is analyzed in Figure 8 (b) for the estimated conductance and susceptance of each branch. It can be easily io observed that the permissible standard deviation is accomplished within the range of 3 X 10-3 for the satisfactory operation of the identification algorithm as estimation error breaches the maximum allowable change (10%) in branch parameters afterward. The salient point of the network identification is described in Figure 8 (b).

Figure 9 shows the required number of iterations to estimate the accurate branch parameters with a variation in the standard deviation. The method according to the present has a better convergence rate with a low number of iterations, which is explained as follows. The Newton-Raphson algorithm has a quadratic converge rate and obtained information is processed further to estimate the branch parameters through the ADMM algorithm. In the ADMM algorithm, it explicitly targets the minimization problem by splitting it into two distinct objectives and provides better optimization than W. Pan, A. Sootla, and G. Stan, "Distributed reconstruction of nonlinear networks: An ADMM approach," IFAC Proceedings Volumes, vol. 47, no. 3, pp. 3208-3213, (2014) and S. Boyd, N. Parikh, E. Chu, et al., A Distributed Optimization and Statistical Learning via The Alternating Direction Method of Multipliers, 1st Edition, Now Publisher Inc., Hanover, USA, (2011).

In addition, the presented ADMM algorithm naturally decouples the non-smooth term from the smooth term, which is computationally advantageous over state-of-art algorithms. Furthermore, the actual effects on the identification of the admittance matrix are illustrated in Figure 10. One can easily notice that there is a significant error in the identified branch parameters in the range of ten. Therefore, this estimation error in the admittance matrix will propagate in the state estimation and will lead to the non-optimal operation of the system.

Case-IV Effect of Change in Standard Deviation with Number of p-PMUs -25 -Referring also to Figure n (a) and (b), variation of the estimation error of 'g' and 'b' in various phase-a' branches with optimal p-PMU locations are shown in (a) With 3 pPMUs and, (b) With 4 p-PMUs.

Figure n (a) shows the identification error for having three u-PMUs in the IEEE 13-bus (e.g., 632, 650, 675) low-voltage distribution system. The identification error for having three p-PMUs is accomplished within range of 40 with consideration of standard deviation of 10-' in the measurements. The identification error can be reduced by having more numbers of p-PMUs in the system as depicted in Figure n (b).

The identification performance is shown in Figure 11 (b) for having four p-PMUs in the system (e.g., 632, 634, 650, 675). The identification error is achieved within the range of 30 with consideration of a standard deviation of to-1 in the measurements. The mean absolute percentage error for parameter identification is analyzed for having three, four, and five ii-PMUs in the IEEE 13-bits feeder. As the number of p-PMUs increases, the identification error is reduced, however, it is a trade-off between accuracy and overall cost of the network monitoring system.

Case-VPerformance of Network Identification with Change in Network Configuration Referring also to Figures 12(a -c), coloured representation of individual phase admittance matrix estimation errors, following the switching event (a) Phase-'a', (b) Phase-'b' and, (c) Phase-'c' is shown. Referring also to Figure 12 (d -f), relative error of the estimated conductances and susceptances of each of the branches in three phases, following the switching event for phaseja', phase-b' and phase-'c' is shown.

Figs. 12 (a-c) show the identification of events in the IEEE 13-bus distribution system with an altered status of the three-phase breaker between the buses 671-692 and 68o692. These results are captured by taking the difference of estimated admittance matrix, which are computed before the event and after the event, in the network.

Figures 12 (a-c) show the amount of change in the admittance matrix with the specific bus number for each phase, respectively. It is worth noticing that the change is symmetrical (e.g., Y671-692, Y671-671, and Y692-671) in the heat map as the three-phase breaker between the buses 671-692 is altered from its nominal status. Likewise, the change in the conductance and susceptance (e.g., Y680-692, Y680-68o, and Y692680) of the corresponding buses are also detected in Figures 10 (a-c) as the three-phase -26 -breaker between the buses 680-692 is altered from its nominal status. The typical change in the estimated admittance matrix is about to be in the range of 15 to 20 for each phase. The detailed change in the conductance and susceptance of the estimated admittance matrix is analyzed in Figures 12 (d-f).

In addition, the typical changes in conductance and susceptance values are captured through its variation in the estimated conductance and susceptance at two different instants (e.g., before and after the event). For ease of understanding, the change in branches 671-692 has been analyzed in Figures 10 (d-f) as the branches 680-692 did io not exist before the event. Hence, the typical variation of the branch parameters (e.g., 671-692) for each phase is illustrated in Figures 12 (d-fl.

Case-VI Performance of Network Identification with Only p-PMLIS Measurement Referring also to Figures 13 (a -b), performance of the system with the presence of ftPMU device (a) Phase-a', (b) Phase-b' and, (c) Phase-'c' is shown.

Figures 13 (a-c) show the performance of the system with only the presence of R-PMU measurements (i.e., placed at all buses) for the given benchmarked IEEE 13-bus feeder.

The error in the estimated nodal admittance matrix is plotted in Figures 13 (a-c). It can be easy to observe that identification error for phase-'a' is quite low and attained within satisfactory limits as depicted in Figure 13 (a). As these measurements include the voltage phase angle and current phase angle, the identification computational time quite low (i.e., phase angle of the distribution buses is not required to estimate through the Newton-Raphson method). Likewise, the network identification for phase-'b' and phase-'c' is illustrated in Figures 13 (b-c). Hence, the method according to the present invention effectively identifies the network structure. Therefore, the identification task can be accomplished quickly as compared to nominal topology identification.

3o Case-VIII Performance of Network Identification for Benchmarked IEEE 123-Bus Feeder Referring also to Figure 14 (a -b), network identification of the IEEE 123-bus feeder using the method according to the present application, specifically (a) detailed schematics of the benchmarked IEEE 123-bus system, and (b) coloured representation 35 of phase admittance matrix estimation errors for phase-a' is shown.

-27 -Figure 14(a) shows the schematics of the benchmarked IEEE 123-bus feeder. The detailed configuration, connection, and location of power system components (e.g., voltage regulator, switch, transformer, etc.) are described in Figure 14 (a). The real-time measurements of the local load profile (e.g., the household electrical load, commercial load, and industrial load) for the benchmarked IEEE 123-bus system are acquired from the Autonomous Decentralised Renewable Energy Systems (ADRES) project repository (ADRES-Concept, "Autonomous decentralised renewable energy systems,". [Online] Available: https://www.ea.tuwien.ac.at/proj ects/ adres concept/EN/. (Accessed on: Feb. 2021) and G. Brauner, D. Tiefgraber, C. io Leitinger, et al., "ADRES: autonomous decentralized regenerative energy-systems," in Proc. European PV-Hybrid and Mini-Grid Conference, Greece, pp. 514-521, (May 2008).

For simplicity, the identified network for phase-a of the IEEE 123-feeder is /5 demonstrated in Figure 14 (b). It shows the error in the estimated nodal admittance matrix of the identified system configuration. One can easily notice that identification error in the admittance matrix (e.g., conductance and susceptance) is quite low and accomplished within permissible limits as depicted in Figure 14 (b). The presented framework provides swift network identification as compared to the state-of-art least absolute shrinkage and selection operator (Lasso) algorithm. The advantage of the method lies in the formulation of optimization problem and an update of search variable can be decomposed, which means the problem can be easily parallelized or scalable even for large network/topology. Hence, the method works satisfactorily for the benchmarked IEEE 123-bus feeder, and it successfully illustrates the scalability and efficacy for the large feeder system.

Case-VII Performance of Network Identification with Integration of Renewable Energy Sources Figure 15 shows the performance of the system with the presence of renewable energy sources, specifically network identification of the IEEE 13-bus feeder with the presence of renewable energy sources. The location of the renewable energy sources at certain buses is given as follows: 650, 632, 671, 633, 680, 684, and 652. Certain points are needed to be clear to understand the impact of renewable energy sources on the network identification objectives. The heat bar in the left-hand side of the heat map manifests the error in the estimated nodal admittance matrix before the connection of the renewable energy sources. Likewise, the bubble size and bubble colour represent an -28 -increment of identification error in the estimated nodal admittance matrix with the presence of renewable energy sources. One can observe that the identified error in an estimated admittance matrix is accomplished within the range of 0.35, which is quite lower than the nominal operating scenario. It is easy to notice that most of the bubble in the heat map is orange, yellow, and sky-blue colour, which denotes a minor increment in identification error with integration of renewable energy sources Nonetheless, the presented framework identifies the network configuration satisfactorily with the presence of the renewable energy sources.

jo Comparative performances Comparative performance is carried out between the method according to the present application and prior art algorithms. The two distinct scenarios are considered to evaluate the effectiveness of the method according to the present application over the state-of-art framework. Firstly, the comparative performance is carried out on the is bench marked IEEE 33-bus balanced feeder with the presence of Gaussian noise. The location of smart meters and p-PMUs are given as follows: p-PMUs are connected at certain buses of the 1EEE-33 bus feeder such as 16, 17, 18, 22,30, 31, 32, and 33; whereas smart meter is connected at rest of the buses. Secondly, the event detection is carried out on the benchmarked IEEE 13-bus unbalanced feeder. The detailed analysis is described as follows.

Case-I Comparative Performance Between Presented Framework and State-of-art Framework for Balanced Feeder Referring also to Figures 16 (a-b) and Figures 17 (a-b), these figures show the response of the state-of-art framework described in J. Zhang, Y. Wang, Y. Weng, et al., "Topology identification and line parameter estimation for non-PMIJ distribution network: A numerical method," IEEE Trans. Smart Grid, vol. 11, no. 5, PP. 4440-4453, (Sept. 2020) to identify the benchmarked IEEE 33-bus feeder structure with the presence of Gaussian noise in the measurements.

Figure 16 (a) shows that identification error is attained within the range of 0.7 using the state-of-art framework "Topology identification and line parameter estimation for nonPMU distribution network: A numerical method", which is quite higher than basic network identification. Nevertheless, the performance of the identification is attained within permissible limits and the tracking of branch parameters is illustrated in Figure 17 (a).

-29 -It shows that the identification of the branch parameters (e.g., conductance and susceptance) fall within the range of 1 % to 4%. The performance of the method according to the present application to identify the topology structure is described in Figure 16 (b) and Figure 17 (b). It is easy to notice that the error in the estimated nodal admittance matrix is quite low (e.g., 0.25) as compared to the state-of-art framework "Topology identification and line parameter estimation for non-PMU distribution network: A numerical method", which is clearly described in Figure 16 (a) and Figure 16 (b). The tracking performance of the branch conductance and susceptance is demonstrated in Fig. 17 (b). In contrast with state-of-art framework "Topology identification and line parameter estimation for non-PMU distribution network: A numerical method", it shows that present method has better estimation accuracy as the estimation error is accomplished within the range of 0.5 % to 2 %. Hence, the presented framework provides an excellent response as compared to the state-of-art framework /5 "Topology identification and fine parameter estimation for non-PMU distribution network: A numerical method".

Case-II Comparative Performance Between Presented Framework and State-of-art Framework for Unbalanced Feeder Referring also to Figure 18 (a-b), this figure shows the performance of the network identification with the steady-state estimation of branch parameters variation of the branch parameters (e.g., io% variation in branch 671-692).

Figure 18 (a) shows the network identification using the known algorithm disclosed in 0. Ardakanian, V. Wong, R. Dobbe, et al., "On identification of distribution grids," IEEE Trans. Control Netw. Syst., vol. 6, no. 3, pp. 950-960, (Sept. 2019) for the revised branch parameters. It shows that identification error using the known algorithm "On identification of distribution grids," in the estimated branch parameter of phase-a' is found in the range of 3% to 6% as depicted in Figure 18 (a).

In contrast with the known "On identification of distribution grids" method, the method according to the present application estimates the updated branch parameters of phase-a' with the estimation error in the range of I% to 2% as illustrated in Figure (b).

-30 -Furthermore, the comparative analysis between several methods is described in Table-H. It shows that the algorithm "On identification of distribution grids" is not robust with respect to noise, unable to identify the change in branch parameters, and not feasible to identify the unbalanced network. Likewise, the graphical and comprehensive model (disclosed for example in both M. Babakmehr, M. Sim5es, M. Wakin, et al., "Compressive sensing-based topology identification for smart grids," IEEE Trans. Ind. Informat., vol. 12, 110. 2, pp. 532-543, (April 2016) and D. Deka, M. Chertkov and S. Backhaus, "Joint estimation of topology and injection statistics in distribution grids with missing nodes," IEEE Trans. Control Netw. Syst., vol. 7, no. 3, pp. 1391-1403, /0 (Sept. 2020)) fails to identify the network parameters and system configuration under unbalanced or noisy measurements.

However, Lasso and Newton-Raphson algorithms ( "On identification of distribution grids" method previously mentioned and also J. Zhang, Y. Wang, Y. Weng, et al., /5 "Topology identification and line parameter estimation for non-PMU distribution network: A numerical method," IEEE Trans. Smart Grid, vol. 11, no. 5, pp. 4440-4453, (Sept. 2020)) are robust to the input noise and capable to identify the change in branch parameters with help of u-PMUs and smart meter measurements, respectively.

However, they both fail to accomplish the network identification objectives when both measurements are available from the network. In contrary to the state-of-art methods hereinbefore mentioned, the method according to the present application copes up with both non-synchronous and synchronous measurements, robust with respect to noise, capable to identify the branch parameters even under an unbalanced network, etc. -31 -The salient points of various algorithms are described in Table T: Description Method 1 Methods 2 & 3 Method 4 Presented Algorithm Type of algorithm Identification with hybrid data set Robustness Accuracy Wein ilicat ion in unbala need network Identification with variation in branch parameters MIQP Lasso NR ADIVIM Not feasible Not feasible Not feasible Feasible No Yes Yes Yes Average Good Relatively good Better Not feasible Feasible Not feasible Feasible Not feasible Feasible Feasible Feasible "Method 1" refers to the method disclosed in Z. Tian, W. Wu, et al., "A mixed integer 5 quadratic programming model for topology identification in distribution network," IEEE Trans. Power Syst., vol. 31, no. 1, pp. 823-824, (Jan. 2016).

"Method 2" refers to the method disclosed in 0. Ardakanian, Y. Yuan, R. Dobbe, et al., "Event detection and localization in distribution grids with phasor measurement units," rn in Proc. IEEE Power & Energy Society General Meeting, Chicago, IL, USA, pp. 1-5, (2o17).

"Method 3" refers to the method disclosed in 0. Ardakanian, V. Wong, R. Dobbe, et al., "On identification of distribution grids," TREE Trans. Control Netw. Syst., vol. 6, no. 3, /5 pp. 950-960, (Sept. 2019).

"Method 4" refers to the method disclosed in.1. Zhang, Y. Wang, Y. Weng, et al., "Topology identification and fine parameter estimation for non-PMU distribution network: A numerical method," IEEE Trans. Smart Grid, vol. 11, no. 5, pp. 4440-4453, (Sept. 2020).

Advantages of method As hereinbefore outlined and explained, the method according to the present application (for example, shown in Figure 2 and Figure 3) has a number of advantages

over the prior art. -32 -

The presented ADMM based algorithm leverages the measurements from both SMs and p-PMUs, for accurate network identification, whereas, the prior art algorithms (for example, methods 1, 2, and 4) deal with uniform measurements (e.g., either p-PMUs or smart meters) to fulfil the identification objectives (e.g. calculation of bus admittance matrix).

The method according to the present application provides robust and efficacious identification performance even with wide variations of standard deviation in the pPMUs and smart meters measurements. In addition, the scalability and efficacy of the jo presented framework are validated through the bench ma rked TEEE 123-bus feeder. The present method accomplishes the network identification objectives even with the presence of the stochastic nature of renewable power generation. To validate this feature, the performance is validated on benchmarked IEEE 13-bus feeders with the presence of renewable energy sources on certain buses-as hereinbefore described.

Furthermore, as the present method does not depend upon the types of loads, it accomplishes the network identification objectives without having particular knowledge of the number of households and its load profile.

In contrast with the prior art methods (for example methods 1 and 4), the presented algorithm efficiently estimates the network configuration, branch parameters, change in branch parameters, and change or event in network structure. It is validated for an unbalanced IEEE 13-bus network with the opening of a three-phase branch (e.g. branch 671-692 in Figure 5) and connection of a three-phase branch (e.g., branch between 680 to 692 in Figure 5).

As hereinbefore explained, the performance of the present method has been compared to traditional alfgorithms. The comparative identification result is demonstrated to estimate the revised branch parameters with the io% variation in branch parameters 30 (e.g. branch between 671 to 68o) on the IEEE 13-bus unbalanced feeder. -33 -

The disadvantages or limitations of some prior art methods compared to the present method are summarised in Table II: References Type of Topology Identification Event Detection Parameter Estimation Network Prostejovsky el al. Balanced Not Feasible Not Feasible Feasible Unbalanced Not Feasible Not Feasible Not Feasible &bal.:mein et al. Balanced Feasible Feasible Feasible unbalanced Not Feasible Not Feasible Not ireasible osseini c/at Balanced Feasible* Not Feasible Not Feasible Unbalanced Feasible Not Feasible Not Feasible Si et al Balanced Feasible Feasible Not Feasible Unbalanced Not Feasible Not Feasible Not Feasible Gandturu et at Balanced Feasible Feasible Not Feasible Unbalanced Feasible Feasible Not Feasible Zhang al Balanced Feasible Feasible Feasible Unbalanced Not Feasible Not Feasible Not Feasible He et at Balanced Feasible Feasible Not Feasible Unbalanced Not Feasible Not Feasible Not Feasible Tian c/at Balanced Feasible Feasible Not Feasible Unbalanced Not Feasible Not Feasible Not Feasible Tian etal. is method 1 and Zhang etal. is method 4.

Prostejovsky et at refers to the method disclosed in A. Prostejovsky, 0. Gehrke, A. Kosek, et al., "Distribution line parameter estimation under consideration of measurement tolerances," IEEE Trans. Ind. Informat., vol. 12, no. 2, pp. 726-735, April ro 2016.

Babakmehr etal. refers to the method disclosed in M. Babakmehr, M. SimOes, M. Wakin, et al., "Compressive sensing-based topology identification for smart grids," IEEE Trans. Ind. Informat., vol. 12, no. 2, PP. 532-543, April 2016.

Hosseini et al. refers to the method disclosed in Z. S. Hosseini, A. Khodaei and A. Paaso, "Machine learning-enabled distribution network phase identification," IEEE Trans. Power Syst., vol. 36, no. 2, pp. 842-850, March 2021.

Si et al refers to the method disclosed in F. Si, Y. Han, J. Wang, et al., "Connectivity verification in distribution systems using smart meter voltage analytics: A cloud-edge collaboration approach," IEEE Trans. Ind. Informat., vol. 17, no. 6, pp. 3929-3939, June 2021.

-34 -Gandluru et al. refers to the method disclosed in A. Gandlurti, S. Poudel and A. Dubey, "Joint estimation of operational topology and outages for unbalanced power distribution systems," IEEE Trans. Power Syst., vol. 35, no. 1, pp. 605-617, Jan. 2020.

Zhang et al. refers to the method disclosed in J. Zhang, Y. Wang, Y. Weng, et al., "Topology identification and line parameter estimation for non-PMU distribution network: A numerical method," IEEE Trans. Smart Grid, vol. 11, no. 5, PP. 4440-4453, Sept. 2020.

io He et al. refers to the method disclosed in X. He, R. C. QM, Q. Ai, et al., "A hybrid framework for topology identification of distribution grid with renewables integration," IEEE Trans. Power Syst., vol. 36, 110. 2, pp. 1493-1503, March 2021.

Claims (9)

1. -35 -Claims 1. A method of determining a bus admittance matrix for a power distribution network, the network comprising a first set of nodes, each node having a smart meter arranged to measure parameters of the node, and a second set of nodes, each node having a n-phasor measurement unit, R-PMU, arranged to measure parameters of the node, the method comprising: determining a partial admittance matrix using power and reactive power values ro measured by the smart meters; determining the bus admittance matrix using the partial admittance matrix, voltage phasor and current phasor values measured by the rr-PMUs, and voltage and current magnitude values measured by the smart meters and estimated voltage phase angles.
2. 2. A method according to claim 1, wherein the power distribution network is a low-voltage power distribution network.
3. 3- A method according to claim 1 or 2, wherein the power distribution network is a national power network or a local network which forms part of a national power network.
4. 4. A method according to any preceding claim, wherein determining the partial admittance matrix involves: or determining approximate values of conductance and susceptance for each node in the first set of nodes; calculate change in active power and reactive power over time using the approximate values; determine more accurate values of conductance, susceptance, and voltage phase angle using an iterative method; constructing the partial admittance matrix using the more accurate values.
5. 5- A method according to claim 4, wherein the iterative method is the Newton-Raphson method.
6. 6. A method according to any preceding claim, wherein after determining the partial admittance matrix, determining the bus admittance matrix involves: determining complete voltage and current phasors for the first and second sets of nodes; performing orthogonal triangular decomposition of the complete voltage and current phasors; obtaining linearly independent and dependent rows of the complete voltage and current phasors; formulating and solving an objective function using alternating direction method /o of multipliers; determining the bus admittance matrix using the solved objective function.
7. 7. A computer program which, when executed by at least one or more processors, causes the processors to perform a method according to any one of claims ito 6.
8. 8. A computer readable medium, which stores or carries the computer program according to claim 7.
9. 9. A hardware processor configured to perform a method according to any one of claims ito 6.