Application of nonlinear Kalman filters to the identification of customer phase connection in distribution grids

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Elsevier in International Journal of Electrical Power and Energy Systems, Vol. 125, on February 2021, available at: https://doi.org/10.1016/j.ijepes.2020.106410 Copyright 2000 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND Application of nonlinear Kalman filters to the identification of customer phase connection in distribution grids M. A. Gonz´alez-Cagigal, J. A. Rosendo-Mac´ıas and A. G´omez-Exp´osito Department of Electrical Engineering, University of Seville Abstract This paper presents a state estimation approach to address the problem of identifying the phase to which single-phase customers are connected in three-phase distribution grids. The proposed method performs Kalman filtering on the information provided simultaneously by the smart meter of every customer and the aggregated energy consumption measured at each phase of the secondary substation feeding the set of customers. Different nonlinear formulations of the Kalman filter are tested and their performance compared, showing that the ensemble Kalman filter provides better estimation results when the system size increases. The accuracy, robustness and limitations of the estimator are also tested when measurement errors are considered. Keywords: Ensemble Kalman filter, Cubature Kalman Filter, Unscented Kalman Filter, phase identification, state estimation. 1. Introduction Correct operation monitoring and control in distribution systems are essential in order to assure a good quality of the service provided to the customers. In this context, it is most important for grid operators to unambiguously know which loads are connected to each of the three phases of the system (the European feeder topology is assumed in this work). An accurate connectivity information is a prerequisite to promote the correct phase balance of LV consumers, alleviating in this way the problems derived from feeder unbalances, such as sharper voltage drops, which can even violate the grid codes, and increasing power losses which also affect the lifespan of the equipment due to temperature rise. Moreover, the penetration of renewable energy resources at the distribution level also benefits from the phase identification, since it helps to establish a better production-consumption balance for each phase of the grid. In this regard, despite the efforts undertaken by distribution companies, they frequently lack enough information about the phase connection of their single-phase customers, owing for instance to network reconfiguration after faults, phase switching derived from improper maintenance, or inaccurate recording of the true load-to-phase connectivity. In these circumstances, a method must be developed to estimate as accurately as possible the actual phase to which a customer is connected in LV feeders, which is known as the customer-phase identification (CPI) problem. The CPI problem has been approached in several ways by previous works. In [1], a signal processing perspective is applied to voltage observations, which are also used both in [2], for a correlation-based methodology, and in [3], where a spectral clustering technique is proposed. The connecting phase of underground distribution transformers is determined in [4] through phase voltage measurements. Smart meters have improved the communication between the loads and the substations and can be also used for the CPI problem, [5], [6]. A method based on Least Absolute Shrinkage and Selection Operator (LASSO) is proposed in [7], also using smart meter data from a LV distribution network. In [8], a novel approach for phase identification using graph theory and principal component analysis (PCA) is tested. The possible missing information in smart meter data is dealt with in [9] through a correlation analysis. In this work, the CPI is addressed by applying a Kalman filtering (KF) state estimation technique to a set of hourly consumption curves obtained from real loads, along with hourly energy measurements taken at each phase of the secondary substation. The proposed method conservatively assumes that other electrical magnitudes potentially provided by smart meters, such as voltage readings or reactive power consumed by each load, are not available. Moreover, a simplified loss model is adopted, allowing the impact of each load on the energy delivered by each phase of the transformer to be estimated for a given topology. The KF is a dynamic state estimator (DSE) widely used in electric power systems in any of its diverse forms, [10]-[12]. Particularly, the application of KF to parameter estimation in power systems has been successfully tested, e.g. in [13]-[15], providing evidence of the DSE potential for the CPI problem. In this work, three nonlinear KF schemes are tested Preprint submitted to Electrical Power and Energy Systems and compared, namely the so-called unscented Kalman filter (UKF), the cubature Kalman filter (CKF), and the ensemble Kalman filter (EnKF). The proposed technique takes advantage of the hourly information provided by smart meters to sequentially assign customers to the most likely phase. The proposed KF implementation does not explicitly enforce, beforehand, binary constraints for the state variables, but rather adopts a novel, statisticallybased inference logic successively rounding state variables to their nearest binary value. As the input information considered in this work is similar to that assumed in [7] and [8], the KF-based estimation is also compared to those competing techniques for different scenarios. The paper is organized as follows: Section 2 provides a brief background on the different KF schemes considered. Next, the feeder used for testing and the simplified loss model adopted are presented in section 3. The implementation details of the KF based estimation techniques, as applied to the CPI problem, are given in section 4. In section 5, the results obtained in two different scenarios are presented and discussed, while the proposed KF technique is compared in section 6 with other published works dealing with CPI. The conclusions derived from the case studies considered are drawn in section 7. 2. Kalman filter background In this section, three different Kalman filter formulations are succinctly reviewed, as applied to continuoustime, discrete-measurement nonlinear systems. In the discretetime framework, the associated equations may be expressed as follows, xk=f(xk−1, uk−1) + wk(1) zk=g(xk, uk) + vk(2) where xkis the state vector at instant k,ukthe system input, and zkthe vector of available measurements. Gaussian processes are considered for the model and measurement noises, wkand vk, with covariance matrices Qand R, respectively. The iterative processes of the Kalman filter schemes considered in this work, all of them involving prediction and correction stages, are described below. 2.1. Unscented Kalman Filter At instant k, a cloud of 2L+ 1 vectors, the so-called σ-points, is obtained from the previously estimated state vector, ˆxk−1(dimension L), and the covariance matrix of the state estimation error, Pk−1, as follows, [16]:        x0 k−1= ˆxk−1 xi k−1= ˆxk−1+ [p(L+λ)Pk−1]i xi+L k−1= ˆxk−1−[p(L+λ)Pk−1]i+L i= 1, ..., L (3) [p(L+λ)Pk−1]ibeing the ith column of the corresponding matrix, and λa scaling factor calculated as follows: λ=α2(L+κ)−L(4) where αand κare two filter parameters to be tuned. These σ-points are evaluated through equation (1), yielding 2L+1 vectors, xi− k, from which the a priori estimations ˆx− kand P− kare obtained: ˆx− k= 2L X i=0 Wmixi− k(5) P− k= 2L X i=0 Wci(xi− k−ˆx− k)(xi− k−ˆx− k)T+Qk(6) where the weighting vectors Wmand Wcare calculated from:    Wm0=λ L+λ Wc0=λ L+λ+ 1 −α2+β Wmi =Wci =1 2(L+λ)i= 1, ..., 2L (7) βbeing another tunable parameter. On the basis of the a priori information, the correction stage starts with the calculation of a new cloud of vectors, x− k, which are evaluated with the measurement function g(·) in equation (2), and weighted with the vectors Wm, yielding γi− k=g(xi− k, uk)i= 0, ..., 2L(8) ˆz− k= 2L X i=0 Wmiγi− k(9) Then, the covariance matrix of the measurement estimation error, P− zk, and the cross-covariance matrix of state and measurements, P− xzk, are obtained using the vector Wc as follows: P− zk = 2L X i=0 Wci(γi− k−ˆz− k)(γi− k−ˆz− k)T+Rk(10) P− xzk = 2L X i=0 Wci(xi− k−ˆx− k)(γi− k−ˆz− k)T(11) The correction stage concludes with the a posteriori predictions, ˆxk= ˆx− k+Kk(zk−ˆz− k)T(12) Pk=P− k−KkP− zkKT k(13) which are based on the a priori predictions at instant k and the so-called Kalman gain, Kk, calculated from Kk=P− xzk(P− zk)−1(14) 2 2.2. Cubature Kalman Filter This KF formulation uses a set of 2Lcubature points calculated from ˆxk−1and Pk−1through the following expressions, [17]: Sk−1ST k−1=Pk−1(15) xi k−1=Sk−1ξi√L+ ˆxk−1i= 1, ..., 2L(16) where Sis a positive-definite square root of matrix P(the Cholesky factorization of matrix Pis customarily used), and ξiis the ith cubature node, obtained as the intersection of the unit sphere and the RLaxis. The state function f(·) in (1) is evaluated for the set of cubature points, yielding a set of 2Lvectors xi− k, from which the a priori estimation is computed, ˆx− k=1 2L 2L X i=1 xi− k(17) P− k=1 2L 2L X i=1 xi− kxi−T k−ˆx− kˆx−T k+Qk(18) For the correction stage, the covariance matrix P− kis factorized in order to calculate both the matrix S− k, S− kS−T k=P− k(19) and a new set of 2Lcubature points, x− k, at which function g(·) in (2) is evaluated to obtain γ− k. Then the measurement estimation, ˆz− k, its covariance matrix, P− zk, and the cross-covariance matrix of state and measurements, P− xzk, are calculated as follows: ˆz− k=1 2L 2L X i=1 γi− k(20) P− zk =1 2L 2L X i=1 γi− kγi−T k−ˆz− kˆz−T k+Rk(21) P− xzk =1 2L 2L X i=1 xi− kγi−T k−ˆx− kˆz−T k(22) The a posteriori predictions of the state vector, ˆxk, and the covariance Pkare calculated with the Kalman gain using the same equations (14)-(13) as in the UKF algorithm. 2.3. Ensemble Kalman Filter The EnKF, [18], is a Monte Carlo approximation of the original KF which has proven accurate enough in highdimensional state-space problems. The ensemble is represented by an LxNmatrix, Nbeing the number of samples considered. The ensemble is first propagated through the state and measurement functions,    xi− k=f(xi k−1, uk−1) zi− k=g(xi k, uk) i= 1, ..., N (23) and then the mean values are calculated: x− k=1 N N X i=1 xi− k(24) zk=1 N N X i=1 zi− k(25) The EnKF correction stage is based on the calculation of the intermediate matrices PkHT k=1 N N X i=1 (xi− k−x− k)(zi− k−zk)T(26) HkPkHT k=1 N N X i=1 (zi− k−zk)(zi− k−zk)T(27) which allow the Kalman gain and the updated values of each sample in the ensemble to be obtained: Kk=PkHT k(HkPkHT k+R)−1(28) xi k=xi− k+Kk(zk−zi− k)i= 1, ..., N (29) Finally, the corrected covariance matrix, Pk, is calculated as follows: xk=1 N N X i=1 xi k(30) Pk=1 N N X i=1 (xi k−xk)(xi k−xk)T(31) 3. Problem statement and modeling In order to assess the ability of the KF-based methodology to address the CPI problem, a typical distribution grid is considered comprising Nssingle-phase and Ntthreephase customers, not necessarily balanced, resulting in Nc=Ns+ 3Nttotal consumption curves in the network. In this work, it is assumed that the number of three-phase customers is 20% of the total. The proposed methodology, as presented in section 4, is easily extensible to other load connection arrangements, such as two-phase loads, still found in some areas. Figure 1 shows an example of a distribution network with 100 loads, used in the sequel to test the estimation techniques. The energy consumption of each load iat a certain hour k, denoted by Ei,k, is obtained from [19], where real hourly data from a European distribution company, comprising smart meters readings for 20 days, are provided, leading to a total of 480 energy measurements for each 3 customer. In case of three-phase loads, a single-phase consumption is assigned to each phase. As customers with null consumption cannot be identified (they provide no information), the corresponding curves are removed from the raw data. Figure 1: Single-line diagram of one of the test networks To fully characterize the energy consumption of the customers, the reactive energy is also obtained from the raw data in [19]. Finally, the resulting hourly curves are randomly associated to a certain phase (a, b or c). Along with the previous customer information, a typical distribution feeder topology is considered, each customer being associated to one of the grid nodes. Then, a load flow can be solved at each hour kin order to obtain the energy delivered by each phase of the MV/LV secondary substation, Ea SS,k,Eb SS,k and Ec SS,k, which is used by the proposed KF-based estimation technique as a measurement. This fully defines the distribution grid model involved in the estimation process. 4. Kalman Filter Implementation The application of the different KF schemes described in Section 2 to the CPI problem is illustrated in the flowchart represented in Figure 2, as summarized in the sequel. 4.1. Parallel filtering For every iteration of each tested technique, three independent KF-based estimators run in parallel, one for each phase pof the distribution grid, being the state vector composed of Ncvariables, xp i, with p=a, b, c, so that L=Nc. Any three-phase customer is characterized by a set of three consecutive state variables, {xp i, xp i+1, xp i+2}, Read Data Update assigned consumers KF tuning Ini  aliza  on for Phase b  lter Ini  aliza  on for Phase a  lter Ini  aliza  on for Phase c  lter Phase b es  ma  on Phase a es  ma  on Phase c es  ma  on Calcula  on of the Phase a discre  za  on factor   Calcula  on of the Phase b discre  za  on factor Calcula  on of the Phase c discre  za  on factor     Calcula  on of the Phase con  dence factors   Select maximum   Phase assignment All consumers assigned? Show results Figure 2: Flowchart of the CPI methodology for each phase. Those variables indicate if the consumption iis associated to the corresponding phase p. For these static variables, the state function, f(·), in equation (1), is taken as a random walk, independent of the system input, uk, yielding the following expression: xp k=xp k−1+wp k(32) where the model noise, wp k, is considered to have the same covariance matrix, Q, for each phase. The system input, uk, used in the measurement function g(·) in equation (2), is determined by the hourly energy consumption of each customer, uT k=E1, E2, ..., ENck. Regarding the network losses, as no measurements about the voltage and power factor of each load are available, a simplified loss model is adopted by the KF estimators, for which a given rated voltage and power factor cos ϕi,k = 1 are assumed for all loads points. With these assumptions, an average hourly current, Ii,k is calculated for each consumer ifrom the corresponding hourly energy consumption, Ei,k: Ii,k =Ei,k T·U(33) where Uis the voltage magnitude, and Tis the energy integration period. In this work, as the energy measure4 ments are supposed to be obtained hourly, T= 1h. Accordingly, the energy loss attributable to customer i is given by: Eloss i,k =I2 i,k ·T·r·li(34) where ris the conductor resistance per unit length and liis the electrical distance from each consumer to the secondary substation, both assumed to be known (r= 0.223Ω/km, has been adopted in this paper). The vector ukalso includes the variables xp icorresponding to customers which have been previously assigned to a phase, as described below. The measurement of each filter, zp k, is taken as the energy delivered at each hour by the corresponding phase of the secondary substation, zp k=Ep SS,k, so the measurement function g(·) in equation (2) reduces to: zp k= Nc X i=1 xpi,k ·(Ei,k +Eloss i,k ) + vp k(35) where the measurement noise, vp k, is considered to have the same covariance matrix, R, for each phase. 4.2. Initialization and tuning As explained formerly, before the KFs are applied, a set of input data are gathered, including the hourly consumption of each customer and the energy delivered by the MV/LV transformer for the time window available. A separation is made between single-phase and three-phase loads. For the last ones, it is assumed that three energy measurements are available, one per phase, but the phase labels (a, b or c) are unknown. At the end of the estimation processes, the actual distribution of customers among the three phases is used to evaluate the performance of each estimation procedure. The KF formulations require that an initial estimation is adopted, which is determined by the state vector ˆx0 and the covariance matrix of the initial estimation error, P0. For the UKF and CKF implementations, all state variables in the three phases are initialized as 0, assuming a complete lack of knowledge about their real values. In the case of EnKF, the samples are given random binary values as initial estimation, which enhances the convergence of the estimation algorithm. The covariance matrix, P0is considered as a diagonal matrix with Pii = 10 in the three KF schemes. The covariance matrices Qand Rare defined as diagonal matrices, considering typical values of Qii = 10−4and Rii = 9 ·10−4respectively, the last one being equivalent to assuming a s.d. of 3% for the measurement errors. For the estimation based on UKF, it is necessary to define the tunable parameters of the filters introduced in equations (4) and (7). A study is made in [20] over the influence of these parameters in the estimation process, concluding that α= 10−4,β= 2 and κ= 3 −Nc, are reasonable values for good estimation results. Regarding the EnKF, the number of samples in the ensemble is taken as N= 10 ·Ncso that it can suit the different sizes of the state vector. 4.3. Candidate selection and assignment Once the three parallel estimators have provided a customer distribution for their corresponding phase, a single consumption is assigned and removed from the subsequent estimation processes. The candidate selection at each iteration of the CPI procedure is based on the estimated values of the state variables, ˆxp i, and the covariance of their estimation error, Pii. Forcing the state variables to be only 0 or 1, as required by the nature of the problem in hand, would involve equality constraints of the form ˆx·(ˆx−1) = 0. However, the application of the KF estimation algorithms to such nonconvex model is prone to convergence problems. For this reason, the approach proposed in this work to enforce the binary character of the state variables relies on the confidence level of a given consumption being associated to a certain phase and not to the others. This is quantified through the so-called phase confidence factors, Cp i, calculated with the probability density function (PDF) of the state variables, as follows. First, the cumulative density function for ˆxp i>0.5, denoted as Mp i, is calculated as follows:      Mp i=1 √2·π·Pii R∞ 0.5e− (x−ˆxp i)2 2·Pii dx p=a, b, c i= 1, ..., Nc Since Mp iprovides information on the discrete value associated to the corresponding state variable, this coefficient is called phase discretization factor. Figure 3 includes a graphic representation of this factor, corresponding with the shaded area in the Gaussian density function, considering 0.5 as lower limit. Then, for each energy consumption curve i, the phase confidence factor combine the information derived from the three phases of the corresponding load through the following expression:        Ca i=Ma i·(1 −Mb i)·(1 −Mc i) Cb i=Mb i·(1 −Mc i)·(1 −Ma i) Cc i=Mc i·(1 −Ma i)·(1 −Mb i) i= 1, ..., Nc The maximum value of the phase confidence factors Cp i is selected, which determines the consumption iwith the highest probability of being associated to the phase pand not associated to the other phases. Finally, once a consumption is selected using the previously defined phase confidence factors, the three variables associated with that customer are assigned integer values as follows: 5 -3 -2 -1 0 1 2 3 Estimated value 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Probability density Mi f Figure 3: Representation of Mp i •If the consumption is related to a single-phase client, ˆxp i= 1 for the phase the client is connected to, and ˆxp i= 0 for the other two phases. •For three-phase clients, assume without loss of generality that the consumption iis related to the phase a. Then, ˆxa i= 1, ˆxb i= 0 and ˆxc i= 0, as in the case of single-phase customers. Additionally, as two energy consumptions from a certain three-phase load cannot be associated to the same phase, the variables ˆxa i+1 and ˆxa i+2 are set to 0. 4.4. Update results If all the loads have been assigned, then the estimation process ends. Otherwise, the state variables and the system input are updated. For each filter, the state variables of the assigned consumptions are extracted from the state vector xand introduced as part of the system input, so that the number of unassigned consumers Ncis reduced accordingly. The estimation process is then repeated with the remaining consumers until Nc= 0, meaning that the final estimation result has been achieved. 5. Case studies In this section, the proposed KF formulations are compared on several case studies, which can be grouped into four different scenarios. By performing a series of preliminary tests in which the percentages of three-phase loads range from 10 to 30%, no significant differences have been observed in the performance of the proposed methodology. For this reason, as stated above, 20% of three-phase customers has been assumed in all scenarios. 5.1. Scenario I: Original measurements In this scenario, the actual consumption is used for the KF-based estimation, assuming that all the measurements are correctly obtained. The performance of UKF and CKF with increasing number of loads is shown in Tables 1 and 2, respectively. Note that, while both estimators correctly assign 100% of customers for Nc= 50, the behavior of the UKF deteriorates faster as the number of loads increases. Consumption Correct Wrong Percentage curves assignments assignments 50 50 0 100% 100 95 5 95% 200 181 19 90.5% 300 216 84 72% 400 233 167 58.25% Table 1: Estimation results for the UKF. Scenario I Consumption Correct Wrong Percentage curves assignments assignments 50 50 0 100% 100 100 0 100% 200 195 5 97.5% 300 262 38 87.33% 400 313 87 78.25% Table 2: Estimation results for the CKF. Scenario I Consumption Correct Wrong Percentage curves assignments assignments 50 50 0 100% 100 100 0 100% 200 200 0 100% 300 283 17 94.33% 400 346 54 86.5% Table 3: Estimation results for the EnKF. Scenario I As shown in Table 3, the EnKF response to increasing system sizes is better than that of the other formulations, which confirms the expected behavior of this KF scheme for high-dimensional problems. The convergence of the estimation processes is not only determined by the value of ˆxp i, but also by the covariance of the estimation error for each state variable, Pii. An illustrative example is shown in Figure 4, representing the PDFs obtained for the three state variables of a certain single-phase consumer, given the values of ˆxp iand Pii from the KF-based estimation process. The phase discretization factors, Mp i, corresponding to the shaded areas in 6 the graphics, and their numeric values are included in the respective legends. In this particular case the computed values of the phase confidence factors are Ca i= 0.014, Cb i= 0.711 and Cc i= 0.001, meaning a confidence of 0.711 that the consumer is associated to phase b. This value would be compared with those of the rest of the clients in order to obtain the selected candidate in the corresponding iteration of the proposed methodology. Finally, in order to illustrate the evolution of the phase confidence factor, Cp i, Figure 5 represents this coefficient in descending order at different stages of the estimation process for 100 loads (i.e., 300 Cp ifactors), using the EnKF formulation. It can be noticed that the maximum of the confidence factors, used to select the next candidate at the corresponding iteration, is close to 1 in all cases. In the top graph of Figure 5, when the whole set of 100 loads is still unassigned, a large number of phase confidence factors remain with a small value (Cp i= 0.125), in accordance to the common initial value Mp i= 0.5 adopted for the three-phase discretization factors, which means a complete lack of information, at this early stage of the iterative process, on the phase to which those consumption curves should be associated. Then, as more loads are assigned, the coefficients of the remaining loads more clearly show a trend towards 1 or 0 (about one third tends to 1 whereas the remaining two thirds tend to 0). Phase a -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Estimated value 0 0.5 1 Probability density Mi a=0.15 Phase b -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Estimated value 0 0.5 1 1.5 Probability density Mi b=0.9 Phase c -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Estimated value 0 0.5 1 1.5 Probability density Mi c=0.07 Figure 4: Probability density functions for a sample consumer 5.2. Scenario II: Analysis of the required amount of data For the results shown above, hourly smart meter readings for 20 days are used, leading to 480 total energy measurements for each consumer. This section analyzes the quality of the estimation as the number of available measurements decreases. Table 4 summarizes the performance 50 100 150 200 250 300 100 loads remaining 0 0.5 1 Ci p 20 40 60 80 100 120 140 50 loads remaining 0 0.5 1 Ci p 5 10 15 20 25 30 10 loads remaining 0 0.5 1 Ci p Figure 5: Evolution of the phase confidence factor at different stages of the three estimators for the case in which Nc= 200 consumption curves (Ns= 116 single-phase and Nt= 28 three-phase customers). Available UKF CKF EnKF data Hit Rate (%) Hit Rate (%) Hit Rate (%) 480 90.5 97.5 100 400 90.5 97.5 100 300 90 96 100 200 85 90 95 100 67 70 72 Table 4: Estimation results for different KF formulations. Scenario II Those results suggest a deterioration of the performance of the KF-based estimation techniques when the number of available measurements is lower than 200 (around 8 days) in this particular scenario. Further tests with feeders comprising different numbers of customers show that the required number of measurement snapshots increases with the number of loads, as intuitively expected. For instance, for Nc= 100 (one half of that shown in Table 4), the success rate of the three estimators does not deteriorate substantially, even when only 100 energy measurements (around 4 days) are considered for each load. 5.3. Scenario III: Noisy measurements The performance of the different KF formulations is evaluated in a realistic scenario where errors in the measurements are considered. As the objective of these case studies is to determine the robustness of the KF schemes against measurement errors, a relatively low number of loads is considered, namely Ns= 58 single-phase and Nt= 14 three-phase clients, which leads to a total of Nc= 100 consumption curves. 7 Gaussian noise is artificially added to each measurement after the load flow is computed. The rates of correct phase-to-customer assignments for increasing measurement errors are summarized in Table 5 for the three estimators. Noise UKF CKF EnKF level Hit Rate (%) Hit Rate (%) Hit Rate (%) 1% 95 100 100 2% 92 96 100 3% 89 93 98 5% 83 89 92 Table 5: Estimation results for different KF formulations. Scenario III In light of those results, it can be concluded, also as expected, that the number of correct assignments decreases with increasing measurement noise, for every KF formulation, being the robustness of the CKF and EnKF similar, superior in any case to that of the UKF formulation. Nevertheless, all formulations show acceptable results when typical noise levels are considered in the measurements. 5.4. Scenario IV: Model errors In the proposed implementation of the KF for the CPI problem, a simplified loss model is considered for which the value of the conductor resistance per unit length, r, is required as per equation (34). In this scenario, the performance of the different KF formulations is evaluated when errors in rare considered. For the same number of customers as in Scenario III, Table 6 summarizes the rates of correct phase-to-customer assignments for errors in rranging from 5 to 20%. Error UKF CKF EnKF in rHit Rate (%) Hit Rate (%) Hit Rate (%) 5% 95 100 100 10% 91 95 98 15% 86 91 95 20% 80 84 89 Table 6: Estimation results for different KF formulations. Scenario IV It can be concluded that the results remain acceptable, at least for the EnKF, when the assumed resistance error does not exceed 10%. 6. Comparison with existing methods Considering the nature of the information on which the proposed KF methods are based, a comparison can be easily made with the performance of the methods proposed in [7], where a LASSO-based technique is applied to the CPI problem using exclusively energy consumption curves obtained from smart meters, and in [8], where PCA is considered for the same purpose. Different scenarios are considered for the comparison, with increasing number of consumers and 1% s.d. in the measurement errors for all cases. The rate of correct assignments for each method in the different cases studied are summarized in Table 7, where it is observed that: •The performance of the PCA method is similar to that of the UKF in all the studied scenarios. •The LASSO-based method has proven to be more sensitive to the number of loads than the CKF and the EnKF formulations, being the success rates of the three techniques very close for Nc≤200. •The results obtained by the CKF and the EnKF schemes are substantially better when the number of loads increases, as can be noticed particularly for Nc= 400. Number of clients PCA LASSO UKF CKF EnKF 50 100 100 100 100 100 100 100 100 95 100 100 200 92 100 90.5 97 100 300 72.67 87.67 71 86.33 94 400 60.25 71.75 57.75 78.25 86.5 Table 7: Rate of correct assignments for the PCA, the LASSO-based and the proposed KF-based methods 7. Conclusions In this paper, a technique based on Kalman filtering and smart meters information is presented to identify the electrical phase which individual loads are connected to in distribution grids. For this purpose, the performance of three KF schemes, UKF, CKF and EnKF, is tested and compared. The proposed estimation algorithm iteratively selects the customer with the highest probability to be connected to a certain phase, based on the estimated value of the corresponding state variable and the covariance of the estimation error. This way of handling binary variables prevents the computational problems potentially arising by the enforcement of the customary equality constraints, and might find application in other binary-constrained problems. Four separate scenarios have been considered to test the accuracy and robustness of the KF formulation for feeders with increasing number of loads and increasing measurement and model errors. The amount of measurements required for a good performance of the algorithm is also analyzed. From the results obtained, the following conclusions are drawn: 8