Identification of the phase connectivity in distribution systems through constrained least squares and confidence-based sequential assignment

Electrical Power and Energy Systems 143 (2022) 108445 Available online 5 July 2022 0142-0615/© 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect International Journal of Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes Identification of the phase connectivity in distribution systems through constrained least squares and confidence-based sequential assignment M.Á. González-Cagigala, J.A. Rosendo-Macíasa,∗, A. Gómez-Expósitoa,b aDepartment of Electrical Engineering, University of Seville, Spain bENGREEN Laboratory of Engineering for Energy and Environmental Sustainability, Spain ARTICLE INFO Keywords: Constrained least squares Gaussian distribution Phase identification Smart metering Distribution grid ABSTRACT This paper addresses the customer-phase identification problem in three-phase distribution grids including three-phase customers characterized by aggregated energy measurements. The proposed technique first solves a relaxed problem, in which the binary nature of the variables is ignored, which leads to a constrained, leastsquares estimation, using as inputs the active and reactive energy readings provided by the smart meters, along with the energy delivered by each phase at the head of the feeder. With the estimated values of the decision variables, and their corresponding variances, a confidence-based selection technique is then applied for the sequential assignment of the customer with the highest joint probability of being connected to one of the three phases but not to the other two. The performance of the proposed procedure is assessed with five different scenarios in terms of accuracy for increasing number of loads and measurement errors. The robustness of the algorithm is additionally tested in the presence of model errors, and its performance is compared to that of existing methods. 1. Introduction The quality of the service provided to the customers is one of the most important drivers related to the operation and control of distribution networks. In this regard, it is essential to duly characterize the phase which single-phase clients are connected to. In case this information is missing (or inaccurate), feeder unbalance problems may arise, with the associated difficulties, such as excessive voltage variations, which can even violate the grid codes, or a premature deterioration of the grid assets due to the overheating caused by the increased power losses, etc. Moreover, having an accurate topology information benefits the penetration of renewable energy sources, [1,2], in terms of a better per-phase energy balance. 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 tracking 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 constitutes the so-called Customer-Phase Identification (CPI) problem. The relatively recent literature related to the CPI problem is composed of a remarkable number of works, which can be divided into ∗Corresponding author. E-mail address: [email protected] (J.A. Rosendo-Macías). two main groups, [3]: hardware-based and software-based methods. Regarding the first group, the proposed techniques are based on the use of measuring devices designed for this particular purpose, usually deployed at strategic places for a given period of time [4–7]. To reduce the economical impact derived from the deployment of those devices, several software-based methods have arisen. In this category, voltage measurements are considered by some identification methods, in combination with a correlation-based technique [8], a spectral clustering approach [9], or signal processing [10]. Based on voltage observations, [11] presents a procedure to estimate the topology of underground distribution cables. Regarding parameter identification of signal models, new hierarchical (separable) multi-innovation algorithms can be found, in [12] for multi-frequency signals based on the sliding measurement window, and in [13] for signal modeling by using the measurement information. Also on signal modeling, even an optimal adaptive filtering algorithm is proposed in [14] by using the fractional-order derivative technique. The information provided by the smart meters has also been widely used to address the CPI problem, [15,16]. A method considering Least Absolute Shrinkage and Selection Operator (LASSO) is proposed in [17]. In [18], a novel approach for phase identification using graph theory and Principal Component Analysis (PCA) is tested. The https://doi.org/10.1016/j.ijepes.2022.108445 Received 14 June 2021; Received in revised form 11 June 2022; Accepted 21 June 2022 International Journal of Electrical Power and Energy Systems 143 (2022) 108445 2 M.Á. González-Cagigal et al. possible missing information in smart meter data is dealt with in [19] through a correlation analysis. Dynamic State Estimators (DSEs) based on Kalman Filters (KFs) have been recently used in [20], where three KF formulations are compared. Since the variables to be determined in the CPI problem are not time varying, some of the techniques traditionally used in parameter identification can be applied in this context. These include the leastsquares parameter estimation proposed in [21], for multi-input and multi-output systems, the recursive algorithm presented in [22] for signal modeling, and the dynamical window data approach presented in [23] to characterize the frequency response. In this work, an equality-Constrained Least Squares (CLS) problem is proposed to deal with the CPI problem, assuming that the available information comprises exclusively the hourly active and reactive energy readings provided by individual smart meters, along with the aggregated energy delivered by each phase at the head of the feeder (secondary substation). The complexities arising from the involved variables being binary are circumvented by resorting to a confidence-based sequential selection procedure, which uses the first and second-order statistics of the estimated parameters to determine the single-phase client with the highest probability of being connected to a certain phase but not to the other two. The selected candidate is accordingly assigned, and the same procedure is repeated as long as unassigned single-phase loads remain. The proposed method conservatively assumes that other electrical magnitudes which might be provided by smart meters, such as voltage readings for each load, are not available. This calls for the adoption of a simplified loss model, approximately relating the impact of each load on the total losses. Moreover, such model can be easily adapted for different informative scenarios (e.g. when the reactive energy measurements are not provided, or the network topology is not available). As the information considered in this paper for the CPI problem is similar to that assumed in several previous works, such as [17,18,20], the performance of all those techniques will be compared for different noise scenarios and number of customers. The main contribution of the proposed methodology, compared to the identification techniques proposed elsewhere, is that single-phase customers are sequentially assigned to one electrical phase on the basis of key statistical information associated with the variables involved in the problem. Through this approach, the explicit enforcement of binary constraints, inherent to the CPI problem, is avoided. Compared to the previous authors’ work [20], the main differences lie in the capability of directly handling three-phase clients, for which only a single aggregate energy measurement is assumed (rather than three separate measurements), as well as the use of CLS rather than a nonlinear KF. The paper is organized as follows: Section 2presents the proposed technique to approach the CPI problem; Section 3provides a brief description of the case study used for testing; in Section 4, the results obtained in five different scenarios are presented and discussed, while the proposed assignment procedure is compared in Section 5with other published works dealing with this problem; the conclusions are presented in Section 6. 2. Proposed phase-identification technique In this section, the methodology proposed for the phase identification problem is described (see the flowchart of the whole process in Fig. 1). 2.1. Input data The information gathered and processed throughout the phase identification procedure is as follows: •Active and reactive energy measurements at the secondary substation during the sampling interval 𝑘, for each phase 𝑝,𝐸𝑃 𝑝 𝑆𝑘 and 𝐸𝑄𝑝 𝑆𝑘, with 𝑝=𝑎, 𝑏, 𝑐. •Active and reactive energy readings from the smart meter 𝑖during the sampling interval 𝑘,𝐸𝑃𝑖𝑘 and 𝐸𝑄𝑖𝑘. •Identification of the connection type for each customer (singlephase or three-phase), 𝑁𝑠being the number of single-phase consumers and 𝑁𝑡the amount of three-phase loads, yielding a total number of consumption readings 𝑁𝑐=𝑁𝑠+𝑁𝑡, given that only aggregate readings are considered for the three-phase clients. Note the difference in this regard with respect to the previous work [20]. •Topological information, if available, of the electrical distance between each customer and the secondary substation. Although, in this work, hourly intervals are assumed (according to the current Spanish regulation for smart meters), the proposed procedure can work indistinctly with any scanning rate. In fact, the lower the measurement latency, the more accurate the results obtained. 2.2. Equality-constrained least-squares formulation The following optimization problem with equality constraints is considered in this paper to address the CPI problem: min 𝑥(𝑏−𝐴𝑥)𝑇𝑊(𝑏−𝐴𝑥) s.t. 𝐶𝑥 =𝑑 (1) where the objective function is the weighted sum of the squared components of the error vector 𝜖=𝑏−𝐴𝑥, with 𝜖∼𝑁(0, 𝑅)and 𝑊=𝑅−1. As the elements of 𝜖can be assumed to be independent and identically distributed random variables, the covariance matrix will be of the form 𝑅=𝜎2⋅𝐼, where 𝐼is the identity matrix. The elements involved in (1) are described in the sequel: •The matrix 𝐴, related to the consumption of the clients in the distribution grid, is composed in the general case of two submatrices, as follows: 𝐴=⎡⎢⎢⎣ 𝐴𝑃 𝐴𝑄⎤⎥⎥⎦ where 𝐴𝑃has the following structure: 𝐴𝑃=⎡⎢⎢⎣ 𝐸𝑃 0 0 0𝐸𝑃 0 0 0 𝐸𝑃 ⎤⎥⎥⎦ The 𝐻×𝑁𝑐matrix 𝐸𝑃 above comprises the active energy readings 𝐸𝑃𝑖𝑘, as described in Section 2.1. The structure of 𝐴𝑄is similar to that described for 𝐴𝑃, 𝐴𝑄=⎡⎢⎢⎣ 𝐸𝑄 0 0 0𝐸𝑄 0 0 0 𝐸𝑄⎤⎥⎥⎦ but the entries of 𝐸𝑄 are the reactive energy readings. Therefore, the dimension of 𝐴is 6𝐻× 3𝑁𝑐,𝐻being the number of samples (hours with information). In case only active energy measurements are available, the matrix 𝐴𝑄is not considered, so that 𝐴=𝐴𝑃(dimension 3𝐻× 3𝑁𝑐). •The vector 𝑏is composed of the energy measurements at the secondary substation, including the three phases, as follows: 𝑏= ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ 𝐸𝑃 𝑎 𝑆 𝐸𝑃 𝑏 𝑆 𝐸𝑃 𝑐 𝑆 𝐸𝑄𝑎 𝑆 𝐸𝑄𝑏 𝑆 𝐸𝑄𝑐 𝑆 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ International Journal of Electrical Power and Energy Systems 143 (2022) 108445 3 M.Á. González-Cagigal et al. Fig. 1. Flowchart of the proposed methodology. when active and reactive measurements are considered, or 𝑏=⎡⎢⎢⎢⎣ 𝐸𝑃 𝑎 𝑆 𝐸𝑃 𝑏 𝑆 𝐸𝑃 𝑐 𝑆 ⎤⎥⎥⎥⎦ if reactive energy measurements are not provided. It can be noticed that 𝑏is a column vector, of size 6𝐻(or 3𝐻). •The vector 𝑥(dimension 3𝑁𝑐), to be estimated, also embraces the information of the three phases: 𝑥=⎡⎢⎢⎣ 𝑥𝑎 𝑥𝑏 𝑥𝑐⎤⎥⎥⎦ The interpretation of each element 𝑥𝑝 𝑖(with 𝑝=𝑎, 𝑏 or 𝑐) depends on the type of customer. For single-phase clients, this value should be 1if the corresponding customer 𝑖is connected to the phase 𝑝, and 0otherwise. However, in the proposed method, the binary constraint is initially relaxed so that the conventional CLS technique can be applied, and subsequently enforced through a set of confidence coefficients, as discussed later in Section 2.4. Regarding the three-phase loads, 𝑥𝑝 𝑖represents the fraction of energy consumption associated to phase 𝑝throughout the 𝐻 hourly intervals, with 𝑥𝑝 𝑖∈ [0,1]. •Finally, for both single-phase and three-phase clients, the equality constraint 𝑥𝑎 𝑖+𝑥𝑏 𝑖+𝑥𝑐 𝑖= 1 should be satisfied, which can be expressed in compact form as 𝐶𝑥 =𝑑, with 𝐶=[𝐼𝑁𝑐𝐼𝑁𝑐𝐼𝑁𝑐] where 𝐼𝑁𝑐is the identity matrix of size 𝑁𝑐and 𝑑is a column vector of size 𝑁𝑐composed of ones. The above matrix 𝐴fully ignores the network losses. In order to enhance the performance of the identification technique, an estimation of the energy losses is added to the corresponding consumption of each client. For the simplified loss model considered, an average hourly current 𝐼𝑖,𝑘 is calculated for each consumer 𝑖over the interval 𝑘: 𝐼𝑖,𝑘 =√𝐸𝑃 2 𝑖,𝑘 +𝐸𝑄2 𝑖,𝑘 𝑇⋅𝑉𝑛 (2) where 𝑉𝑛is the network rated voltage, and 𝑇is the integration or sampling period. Based on the average current, the active and reactive energy losses attributable to each consumer are accordingly obtained as: 𝐸𝑃 𝑙𝑜𝑠𝑠 𝑖,𝑘 =𝐼2 𝑖,𝑘 ⋅𝑇⋅𝑟𝑐⋅𝑙𝑖(3) 𝐸𝑄𝑙𝑜𝑠𝑠 𝑖,𝑘 =𝐼2 𝑖,𝑘 ⋅𝑇⋅𝑥𝑐⋅𝑙𝑖(4) where 𝑟𝑐and 𝑥𝑐are respectively the conductor resistance and inductance per unit length, and 𝑙𝑖is the estimated electrical distance from load 𝑖to the secondary substation. Those loss terms lead to additional loss matrices 𝐴𝑙𝑜𝑠𝑠 𝑃and 𝐴𝑙𝑜𝑠𝑠 𝑄, featuring the same structure as that presented previously for 𝐴𝑃and 𝐴𝑄. Therefore, in the approximate lossy model, the coefficient matrix 𝐴becomes, 𝐴=⎡⎢⎢⎣ 𝐴𝑃+𝐴𝑙𝑜𝑠𝑠 𝑃 𝐴𝑄+𝐴𝑙𝑜𝑠𝑠 𝑄⎤⎥⎥⎦ 2.3. Solution of the constrained least-squares problem The solution to the optimization problem posed above is provided by the method of Lagrange multipliers. In case all elements of matrix International Journal of Electrical Power and Energy Systems 143 (2022) 108445 4 M.Á. González-Cagigal et al. 𝑅are identical to 𝜎2, as in our problem, the equality constrained formulation reduces to the following system, [24]: [𝐴𝑇𝐴 𝐶𝑇 𝐶0][ 𝑥 𝜎2𝜆]=𝐺[𝑥 𝜎2𝜆]=[𝐴𝑇𝑏 𝑑](5) where 𝜆is the vector of Lagrange multipliers and 𝑥 is the estimate of vector 𝑥. Given that 𝐺is a nonsingular square matrix, the value of 𝑥 can be obtained from: [𝑥 𝜎2𝜆]=𝐺−1 [𝐴𝑇𝑏 𝑑]=[𝐸1𝐸2𝑇 𝐸2𝐸3][𝐴𝑇𝑏 𝑑](6) It can be noticed that, owing to the fact that 𝑅=𝜎2⋅𝐼, the estimate 𝑥 is not actually affected by the value of 𝜎, unlike its covariance, which is given by: cov ( 𝑥) = 𝜎2𝐸1(7) 2.4. Confidence-based selection and assignment Under the CLS customary assumptions, each component of vector 𝑥 is considered as a random variable with a normal distribution, 𝑥𝑖∼ 𝑁(𝑥𝑖, 𝑞𝑖), where 𝑞2 𝑖is the respective diagonal of cov ( 𝑥), [24]. In turn, the scalar 𝜎2, representing the covariance of 𝜖, which is required to obtain 𝑞𝑖, can be estimated from the following expression: 𝜎2=(𝐴 𝑥 −𝑏)𝑇(𝐴 𝑥 −𝑏) 𝑓(8) where 𝑓is the number of degrees of freedom, given by: 𝑓= 6𝐻− 3𝑁𝑐+𝑁𝑐 if active and reactive energy readings are provided, or 𝑓= 3𝐻− 3𝑁𝑐+𝑁𝑐 if only active energy measurements are available. It is worth noting that the real values of the parameters 𝑥𝑖related to three-phase clients might be time-variant, because of the different load distribution of these customers over time for the three phases. However, given that the purpose of the proposed methodology is the identification of the phase connectivity for single-phase loads, only the parameters linked to those consumers will be considered in the sequel for the assignment process. For the selection method proposed in this paper, the elements of 𝑥 are divided into three 𝑁𝑐-dimension vectors, 𝑥𝑎,𝑥𝑏and 𝑥𝑐, with the estimated parameters related to each of the three phases. In the same way, the covariance vector 𝑞is split into 𝑞𝑎,𝑞𝑏and 𝑞𝑐. Based on the above considerations and assumptions, the normal distribution of 𝑥𝑝can be used to quantify the confidence level for a certain single-phase load to be associated with the corresponding electrical phase 𝑝and not to the others. Following [20], the cumulative density function over 0.5, denoted as phase discretization factor, 𝑀𝑝 𝑖, provides information about how close a variable 𝑥𝑝 𝑖is to the discrete value 1. For this particular case, 𝑀𝑝 𝑖is calculated as follows: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑀𝑝 𝑖=1 √2⋅𝜋⋅𝑞𝑝 𝑖 ∫∞ 0.5𝑒 −(𝑥−𝑥𝑝 𝑖)2 2⋅𝑞𝑝 𝑖𝑑𝑥 𝑝=𝑎, 𝑏, 𝑐 𝑖= 1,…, 𝑁𝑠 (9) Fig. 2 illustrates the meaning of this coefficient (shaded area in the density function), for a certain parameter 𝑥𝑝 𝑖. For each single-phase client 𝑖, the information provided by the three coefficients 𝑀𝑎 𝑖,𝑀𝑏 𝑖and 𝑀𝑐 𝑖is combined to calculate the so-called phase confidence factors, 𝐶𝑝 𝑖(confidence in being connected to phase 𝑝and not to the others), using the expressions below: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 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,…, 𝑁𝑠 (10) Fig. 2. Graphic representation of 𝑀𝑝 𝑖. The customer 𝑖with the highest likelihood of being connected to the phase 𝑝is given by the maximum value of the factors 𝐶𝑝 𝑖. Finally, for the selected client 𝑖, the corresponding parameters 𝑥𝑎 𝑖,𝑥𝑏 𝑖,𝑥𝑐 𝑖are assigned integer values (0 or 1). 2.5. Update results In case all the single-phase clients have been assigned, the process ends and the results are shown. Otherwise, the assigned consumption and its corresponding energy losses are removed from matrix 𝐴and introduced as an additional term in vector 𝑏, reducing accordingly the number of single-phase loads to be assigned, 𝑁𝑠. Then, the process is repeated with the remaining loads until 𝑁𝑠= 0. 3. Case study For the results that will be presented in Section 4, a set of synthetic distribution grids with a typical European configuration are considered, including 𝑁𝑠single-phase and 𝑁𝑡three-phase customers, which are not necessarily balanced. A particular example of a distribution network is depicted in Fig. 3, with 𝑁𝑠= 40 single-phase loads (in blue) and 𝑁𝑡= 10 three-phase customers (in red). As illustrated with the small red circles, the information used for the phase assignment is just obtained from the secondary substation and the smart meters installed for all clients in the grid under study. For our experiments, the active and reactive individual energy consumptions are obtained from [25], 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 customer. As customers with null consumption provide no information, the corresponding curves are removed from the raw data. For the single-phase customers, the resulting hourly curves are randomly associated to a certain phase (a, b or c), while a random time-varying load distribution is considered for the three-phase clients. Regarding the resistance per unit length, the value 𝑟𝑐= 0.223 Ω/km has been taken in this work, with a ratio 𝑟𝑐∕𝑥𝑐= 1 in all cases. Finally, a load flow is computed at each hour 𝑘in order to obtain the energy delivered by each phase of the MV/LV secondary substation. This information fully characterizes the distribution grid model involved in the estimation process. International Journal of Electrical Power and Energy Systems 143 (2022) 108445 5 M.Á. González-Cagigal et al. Fig. 3. Single-line diagram of one of the test networks. Table 1 Estimation results. Scenario I. Total number Single-phase Correct Success of loads clients assignments rate 50 40 40 100% 100 80 80 100% 200 160 160 100% 300 240 231 96.25% 400 320 292 91.25% 500 400 343 85.75% 600 480 386 80.42% 4. Numerical results In this section, the proposed sequential assignment technique is tested with several scenarios, which can be grouped into five different categories. 4.1. Scenario I. Original measurements In the first scenario, the actual active and reactive energy consumptions are considered for the proposed sequential assignment methodology, with no additional errors in the measured values. The performance of the identification technique is summarized in Table 1 for increasing number of clients in the distribution grid. In all cases, 20% of customers are three-phase. As expected, the fraction of correct assignments slightly deteriorates as the number of client increases. However, acceptable results have been obtained even for large distribution grids (with 600 loads), with a success rate exceeding 80%. The good performance of the proposed technique is not only determined by the number of correct assignments, but also by the values of the confidence factors, 𝐶𝑝 𝑖, after each iteration. For a case with 80 single-phase loads (a total of 240 𝐶𝑝 𝑖factors), Fig. 4 represents the maximum value of those coefficients at each stage of the process, corresponding with the selected single-phase client in each case. Virtually in all iterations, the maximum value of 𝐶𝑝 𝑖is higher than 0.95, reflecting high confidence in the corresponding assignment. As an illustrative example, Fig. 5 shows the PDFs obtained for the three estimated parameters, 𝑥𝑎 𝑖,𝑥𝑏 𝑖and 𝑥𝑐 𝑖, of a certain single-phase client 𝑖. The numeric values of the phase discretization factors, 𝑀𝑝 𝑖, corresponding to the shaded areas, are also provided in the legends. In this particular case, the resulting confidence factors are 𝐶𝑎 𝑖= 0.024, 𝐶𝑏 𝑖= 0.674 and 𝐶𝑐 𝑖= 0.003, implying a higher probability for the customer 𝑖to be associated with phase 𝑏. Fig. 4. Evolution of the maximum value of 𝐶𝑝 𝑖. Table 2 Success rates, in percentage. Scenario II. Total number Percentage of three-phase clients of loads 10% 20% 30% 40% 50% 50 100 100 100 100 92 100 100 100 100 100 88 150 100 100 100 95.55 85.33 200 100 100 94.28 82.5 74 300 100 96.25 91.90 79.44 68 4.2. Scenario II. Varying number of three-phase clients The objective of the proposed technique is to provide an accurate estimation of the electrical phase connectivity of the single-phase loads in a distribution network. However, the presence of three-phase loads in the grid, for which only aggregate energy readings are available, might difficult the identification process, given the temporal variability of the load balance in three-phase customers, causing changes in the corresponding regression parameters (note that, in [20], it was assumed that individual phase readings were available for three-phase customers). In this scenario, the performance of the proposed technique is evaluated for increasing shares of three-phase consumers. The success rates for the different cases are summarized in Table 2. Also as expected, for a given number of loads, the success rate deteriorates with the share of three-phase clients, as their presence is somehow equivalent to increasing ‘‘noise’’ in the available information. Such deterioration might lead to poor results in large grids with high shares of three-phase clients, as in the rightmost bottom case: 68% success rate for 300 loads with an unrealistically share (50%) of three-phase consumers. 4.3. Scenario III. Lack of data In this scenario, the rate of correct assignments is assessed assuming two less-informative situations, which will be presented separately. 4.3.1. Unavailability of reactive energy readings In the previous sections, the information used for the identification technique corresponds to 480 hourly readings (20 days), both for the active and reactive energy consumption of each client in the grid. In this case, the performance of the proposed method is assessed when only active energy measurements are available. Accordingly, for the simplified loss model presented in (2)–(4), only the active energy losses International Journal of Electrical Power and Energy Systems 143 (2022) 108445 6 M.Á. González-Cagigal et al. Fig. 5. PDFs for a sample single-phase consumer. Table 3 Estimation results with only active energy measurements. Total number Single-phase Correct Success of loads clients assignments rate 50 40 40 100% 100 80 80 100% 200 160 160 100% 300 240 226 94.16% 400 320 279 87.19% 500 400 324 81% 600 480 355 73.96% are calculated assuming unity power factor (𝑐𝑜𝑠(𝜑)=1), yielding the following expression: ⎧ ⎪ ⎨ ⎪ ⎩ 𝐼𝑖,𝑘 =𝐸𝑃𝑖,𝑘 𝑇⋅𝑈𝑛 𝐸𝑃 𝑙𝑜𝑠𝑠 𝑖,𝑘 =𝐼2 𝑖,𝑘 ⋅𝑇⋅𝑟𝑐⋅𝑙𝑖 (11) The remaining assumptions for this case study are similar to those in the first scenario, the corresponding results being summarized in Table 3. Clearly, the relevance of using both active and reactive energy measurements increases with the number of clients. Nevertheless, the success rates obtained with the proposed technique remain within acceptable values, especially for 𝑁𝑐<300. 4.3.2. Amount of hourly snapshots The quality of the phase identification is also analyzed when the number of available measurements decreases. For a fixed number of loads, 𝑁𝑐= 200, with 20% of three-phase clients, Table 4 presents the estimation results obtained when the number of active and reactive energy snapshots decreases. The results in Table 4 show the expected deterioration of the success rate for a reduced number of energy readings. The observed impact is less pronounced when both active and reactive energy readings are gathered. Table 4 Estimation results with decreasing amount of data. Available Success rates with Success rates with data EP and EQ only EP (h) measurements measurements 480 100% 100% 400 100% 100% 300 100% 97.5% 200 100% 90.63% Table 5 Estimation results for increasing noise levels. Scenario IV. Noise Success rate level (%) 2% 100 3% 100 5% 95 7% 91.25 10% 85 A similar study was repeated for different number of clients in the grid. Fig. 6 represents the required number of snapshots to obtain the maximum success rate in each case, with a maximum number of available measurements equal to 480. A roughly linear trend can be noticed for both cases (P&Q or only Q readings), suggesting that more and more snapshots would be needed to assure acceptable results when the number of customers increases, particularly when only 𝑃readings are available. 4.4. Scenario IV. Noisy measurements In this case study, the performance of the proposed technique is tested in the presence of a wide range of measurement errors. In this experiment, a relatively low number of loads is considered: 𝑁𝑐= 100, with 20% of three-phase clients. Increasing levels of Gaussian noise are artificially added to the energy measurements after the load flow is computed. Table 5 gathers the success rates for each noise level. International Journal of Electrical Power and Energy Systems 143 (2022) 108445 7 M.Á. González-Cagigal et al. Fig. 6. Required amount of data for the best performance. Table 6 Estimation results for increasing errors in 𝑟𝑐. Scenario V. Relative error Success in 𝑟𝑐rate (%) 5% 100 10% 100 15% 100 20% 97.5 30% 96.25 From Table 5, it can be concluded that the proposed methodology still features acceptable success rates even with measurement errors of up to 10%. 4.5. Scenario V. Model errors In the simplified loss model considered in the proposed technique, described by Eqs. (2)–(4), there are two main sources of uncertainty: the values of the conductor electrical parameters per unit length, 𝑟𝑐and 𝑥𝑐, and the electrical distance 𝑙𝑖of each client in the distribution grid with respect to the secondary substation. In this scenario, the robustness of the identification method is evaluated against errors in these parameters. Table 6 summarizes, for the same number of customers as in Scenario IV, the success rates obtained for errors in 𝑟ranging from 5 to 30% (assuming 𝑥𝑐=𝑟𝑐). As can be seen, the proposed technique is not significantly affected by this parameter, yielding success rates over 96% even with 30% errors in 𝑟. When the electrical distance of the customers is not available, an alternative loss model is considered, in which the energy losses associated to each client are proportional to the corresponding consumption [18]: 𝐸𝑃 𝑙𝑜𝑠𝑠 𝑖,𝑘 =𝐸𝑃𝑖,𝑘 ∑𝑁𝑐 𝑖=1 𝐸𝑃𝑖,𝑘 ⋅(∑ 𝑝 𝐸𝑃 𝑝 𝑆,𝑘 − 𝑁𝑐 ∑ 𝑖=1 𝐸𝑃𝑖,𝑘)(12) 𝐸𝑄𝑙𝑜𝑠𝑠 𝑖,𝑘 =𝐸𝑄𝑖,𝑘 ∑𝑁𝑐 𝑖=1 𝐸𝑄𝑖,𝑘 ⋅(∑ 𝑝 𝐸𝑄𝑝 𝑆,𝑘 − 𝑁𝑐 ∑ 𝑖=1 𝐸𝑄𝑖,𝑘)(13) where the term (∑𝑝𝐸𝑃 𝑝 𝑆,𝑘 −∑𝑁𝑐 𝑖=1 𝐸𝑃𝑖,𝑘)refers to the difference, at instant 𝑘, between the energy delivered by the three phases of the secondary substation (∑𝑝𝐸𝑃 𝑝 𝑆,𝑘), and the total consumption from all customers (∑𝑁𝑐 𝑖=1 𝐸𝑃𝑖,𝑘), and the same meaning for the counterpart terms in (13). In case reactive energy readings are not available, only Eq. (12) Table 7 Results with no information on the electrical distances. Total number Single-phase Correct Success of loads clients assignments rate 50 40 40 100% 100 80 80 100% 200 160 160 100% 300 240 224 93.33% 400 320 278 86.87% 500 400 321 80.25% 600 480 350 72.92% Table 8 Success rates (%) for the different methods considered. Total number PCA LASSO EnKF Proposed of loads technique 50 100 100 100 100 100 100 100 100 100 200 85 92.5 95 100 300 79.17 80.83 82.5 93.33 400 67.18 70 65.62 86.87 500 50.25 60.50 53.25 80.25 600 47.29 52.92 48.54 72.92 is used. Table 7 shows the results obtained for increasing number of clients, using the simplified loss model provided by (12)–(13). In all cases, 20% of three-phase loads is considered. Compared to the more accurate loss model adopted for the first scenario (Table 1), the performance deterioration is not significant. In all cases, the success rates remain over 85%, giving evidence of the robustness of the algorithm when the exact network electrical model is not available. 5. Comparison with existing methods Finally, a comparison is made in this section among the proposed technique and three different methods, all of them making use of similar input data for the phase-assignment process: •A LASSO-based technique [17]. •The method presented in [18], where PCA is considered with exclusively energy consumption curves from smart meters. •The Ensemble Kalman Filter (EnKF) formulation proposed in [20], which outperformed other KF schemes. Different scenarios are used for the comparison, with increasing number of clients. In all cases, 2% measurement errors are assumed, with no information available about the electrical distances, which means that Eqs. (12) and (13) are used to estimate the energy losses. One every five customers is assumed to be three-phase, each with a single energy reading. Given that all the techniques included in the comparison are formulated exclusively with per-phase measurements, the overall consumption for three-phase customers has been equally divided into the three phases. The rates of correct assignments are summarized in Table 8 for all the techniques. For a reduced number of clients (𝑁𝑐 ≤100), all of the compared methods achieve 100% success rate. As the grid size increases, the PCA, the EnKF and the LASSO-based methods suffer a sharper deterioration of their performance, when compared to that of the proposed technique, which exhibits over 16% better hit rate than the best competing method for 𝑁𝑐≥400. Those results can be in part explained by the weaker treatment of the three-phase clients with aggregate readings, which is the main advantage of the proposed CLS-based technique. International Journal of Electrical Power and Energy Systems 143 (2022) 108445 8 M.Á. González-Cagigal et al. 6. Conclusions In this paper, a methodology is proposed for the sequential phase assignment of single-phase loads in distribution grids, using smart meter information exclusively. The technique is based on a CLS model and the corresponding statistical distributions of the estimated parameters. At each iteration of the procedure, the load with the highest confidence in being connected to a certain phase is selected, so that the binary equality restrictions are avoided as well as the associated computational problems. Five scenarios have been presented to assess the performance of the proposed technique, in terms of accuracy and robustness against model errors. The following conclusions can be highlighted from the obtained results: •The number of correct assignments decreases for increasing number of clients. Nevertheless, a success rate over 80% is still obtained for distribution networks with 600 clients, of which 20% are three-phase. •For larger shares of three-phase consumers, the performance of the identification process deteriorates, yielding poor results (around 70% hit rate) for feeders with 300 clients and 50% of three-phase loads. •The sensitivity of the identification technique to the number of customer is higher, as expected, if only active energy measurements are available. The more loads in the same feeder, the more measurement snapshots are needed to perform an acceptable estimation of the phase connectivity. •Even for measurement errors of up to 10%, the success rates remain over 85% for distribution grids with 200 clients. •Finally, the robustness of the method has been tested against errors in the conductor resistance and reactance used for the calculation of the energy losses. Additionally, a simplified loss model is considered, with very good results, for the cases where the electrical distances are not available. The proposed technique has been compared to other published methods, based on PCA, LASSO and EnKF. While the results obtained for a reduced number of clients are similar for all tested techniques, as the network size increases, the proposed technique has shown remarkably higher success rates than the others. This is mainly a consequence of the proposed CLS-based method being the only one to correctly deal with three-phase clients, when only aggregate energy readings are provided. Further research efforts will be devoted to redesigning the proposed procedure, so that additional electrical quantities can be considered, such as voltage magnitudes, which might be available in future distribution systems. CRediT authorship contribution statement M.Á. 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