A Line Loss Separation Method Based on Topology-aware Correlations and Nonlinear Impedance

Due to the varying order and hierarchical relationship of different meters on the same line, the variation pattern of line loss and electricity consumption is not obvious, making it difficult to separate them accurately. To address this issue, an innovative method for line loss separation based on topological relationship perception and nonlinear impedance is proposed. Firstly, based on the topology recognition algorithm, we identify the order of the electricity meters and the relationship between the father and child-level electricity meters. Then, using the principle of topology relationship, we accurately measure the impact of the father and child level electricity meters on the overall line loss electricity quantity. Combined with data difference and matrix difference methods, the fixed loss value is offset. On this basis, a sliding window method is used to draw the curve change of nonlinear impedance and loss of electricity quantity. By analyzing the approximate continuous relationship between electricity consumption and line loss rate through curves, the line loss electricity consumption can be effectively separated. The results indicate that the proposed method has a better performance compared to other methods.


Introduction
"Announcement No. 42 of 2020" issued by the State Administration for Market Regulation (SAMR) mandates adjustments to the implementation of mandatory management for measuring instruments.According to the announcement, residential electricity meters are required to undergo initial mandatory verification and have a limited period of usage.After the designated period, they need to be replaced or have their usage extended based on the condition of the meters.The "SAMR Metering Letter [2021] No. 352" outlines the key focus areas for national metrology work in 2021, emphasizing the exploration and reform of the mandatory verification system.The letter proposes the establishment of a mechanism that involves "enterprise self-control, user supervision, and government regulation."It also highlights the continued advancement of reforms related to electricity meter status evaluation and replacement.The collaboration between local authorities and power supply departments is encouraged, along with expanding pilot projects and establishing new regulatory mechanisms.These initiatives aim to accumulate experience and ensure the successful implementation of electricity meter status evaluation and replacement.
Various domestic and international scholars have conducted research in different areas.Yuan et al. [1] mentioned that for low-voltage distribution networks with incomplete metering information, line loss calculations are typically performed using methods such as voltage loss, substation loss rate, and equivalent resistance.However, the voltage loss and substation loss rate methods fall under statistical calculations, while the equivalent resistance method takes into account load fluctuations.Nevertheless, these methods have not adequately considered the impact of three-phase imbalances and user load characteristics on the calculations.In practical applications, adjustments are made using three-phase current imbalance coefficients and load shape coefficients to refine the results.However, the improvement in accuracy resulting from these adjustments is limited.Xu et al. [2] proposed a dynamic line loss estimation model that integrates metering errors and line loss estimation.However, the model's precision is not sufficiently high, leading to significant influences on the estimation of error results.Ye et al. [3] filtered measurement data based on similar operating states corresponding to different users' electricity consumption levels at various time periods.The study then utilizes a constrained memorybased recursive least squares algorithm to remotely estimate the operating errors of electricity meters.However, this method assumes that line losses remain constant during calculations, which reduces the accuracy of meter error estimation.Guo [4] put forward a measurement system based on advanced measurement techniques that utilize data analysis methods to calculate and solve meter operating errors.In theory, this approach can achieve "full coverage" of electricity meter status monitoring.However, challenges persist in practical power grid operations, including difficulties in obtaining parameters for loss calculation and severe model ill-conditioning.As a result, significant fluctuations in meter error calculations occur, leading to insufficient reference value provision.Other researchers [5][6][7][8][9][10] have also conducted studies on meter error estimation, but either their applicability is limited, or their accuracy needs improvement.
In conclusion, despite the extensive research conducted by numerous scholars in the field of line loss separation, several challenges remain.Achieving precise and effective separation of line losses is still a key technical obstacle.Based on this, our project proposes a line loss separation method based on topology awareness and nonlinear impedance.The objective is to achieve effective line loss separation, enabling accurate estimation of meter errors.This approach aims to transform the management of measuring equipment from the traditional model of "on-site maintenance + periodic replacement" to a more advanced model of "online maintenance + precise replacement," ensuring the overall stability of measuring devices.

Overall research approach
Considering the current state of the data and in conjunction with the business scenario, we have devised a comprehensive solution approach, as depicted in Figure 1.

Figure 1. Overall solution idea
To begin with, we collect data from the marketing electric power business application system and the electricity consumption information collection system, including archives and energy consumption data.We then address missing values and outliers through diagnostic analysis and appropriate data processing techniques.Next, leveraging the principle of energy conservation, we construct a data matrix.By applying a topology recognition algorithm, we achieve the fitting of loss of electricity data.Additionally, we employ a nonlinear impedance model to analyze the functions of electricity consumption and line loss, allowing us to solve for the relationship between the error and loss of individual electric meters, as well as calculate the line loss value.This process enables us to achieve a precise estimation of meter errors.

Analysis and handling of anomalous data using the Isolation Forest algorithm
The Isolation Forest algorithm is an unsupervised learning method used for detecting anomalies in data.It efficiently identifies outliers by isolating them from the rest of the data.With its linear time complexity and high accuracy, it measures the differences between objects and assigns anomaly scores based on distances or density of points.A higher score indicates a higher likelihood of the object being an anomaly.

Figure 2. Schematic diagram of the Isolation Forest algorithm
In our approach, we apply the Isolation Forest algorithm to analyze and detect anomalies in user electricity consumption data.By clustering the data and incorporating the Isolation Forest algorithm, we can identify abnormal patterns, such as light load and abrupt changes in meter readings.The examples of the identified light load data are as follows in Table 1.
Table 1.Light load data sample table The examples of jump data are shown in Table 2.For the collected light load data, as introducing it into the model would greatly affect its performance, we remove it.As for the collected data with abrupt changes in electricity consumption, we employ a smoothing method on the preceding and succeeding data to ensure more reasonable calculation results.The equation for data smoothing is as follows: . (1)

Construction of line loss identification model based on topological relationships 3.2.1.1 Construction of energy conservation equations
In light of the collected data, such as meter readings and currents, an energy conservation model is constructed by applying the principle of energy conservation.This model takes into account subsequent analysis of topological relationships and nonlinear impedance.By forming relevant data matrices, the patterns of meter data can be analyzed effectively.

Principle and necessity analysis of topological relationship identification
Building upon the energy conservation equations, the topological configurations of both DC and AC circuits are abstracted to analyze the overall topology of the power network comprehensively.As illustrated in Figure 3, it is considered a specific power network where there is only one meter for power supply and 12 meters for power consumption.The topological relationship within this network is depicted below.(2) Condition 2: From point B to consumption meters 1, 2, 3, and 4, there is a presence of resistance, although the resistance values are extremely small compared to the resistance between points A and B (assuming they are 1000 times smaller).Hence, these losses can be neglected.Similarly, from point C to consumption meters 5, 6, 7, and 8, and from point D to consumption meters 9, 10, 11, and 12, there is resistance.However, the resistance values are negligibly small relative to the respective line segments (assuming they are 1000 times smaller), and the loss can be ignored.
(3) Condition 3: It is assumed that each supply meter represents a large resistance for power consumption, and the resistance remains relatively constant.For instance, consumption meter 1 has a resistance value of R1, and so on.
In the given tree-like topology diagram, consumption meters 1, 2, 3, and 4 belong to the same hierarchical level C1.Consumption meters 5, 6, 7, and 8 belong to another hierarchical level C2, and consumption meters 9, 10, 11, and 12 belong to a further hierarchical level C3.Therefore, this topology diagram has one root node, and there are three hierarchical levels.The line segments AB, BC, and CD are shared by the corresponding levels, and they have approximately equal distances to the respective supply meters.C2 and C3 are sub-levels of C1, with C3 being a sub-level of C2.
Based on the above information, it can be inferred that the main losses in this topology diagram occur in the line segments AB, BC, and CD.The losses for the AB segment within a day can be calculated as follows.

𝑊
The line loss of the CD section can be deduced by analogy.The above  is the current value of the electricity meter 1, and  is the user resistance of the meter electricity 1, and so on.
Based on the equation, assuming minimal fluctuations in the current values of the electric meters throughout the day, we can infer that within the topological line.
The line loss of a specific line ≈ The power of the meter itself (in the hierarchical and the subhierarchical levels) * constant + the product of the electricity consumption of each pair of electricity meters (in the hierarchical and the sub-hierarchical levels) * constant.
Consequently, by applying a hierarchical and sub-hierarchical identification approach within the topological network, we can recognize the presence of multiple levels and sub-levels.Based on this recognition, the corresponding constant terms can be calculated.This calculation allows for the precise identification of line loss, separating it from measurement errors and facilitating the estimation of electric meter errors.
For instance, if there are five electric meters in a given hierarchical level and its corresponding sublevel, only fifteen (5+4*5/2=15) constant terms need to be determined, enabling an accurate calculation of line loss separation.Moreover, as the loss calculation in the parent level incorporates similar calculations performed in the sub-levels, it significantly reduces the overall computational complexity.Therefore, the development of a topological recognition algorithm is crucial for accurately identifying line loss and effectively estimating electric meter errors.

Calculation method for power consumption based on topological recognition
Drawing upon the aforementioned analysis of topological relationships and considering the assumptions associated with these relationships, the following conclusions can be derived: 1.When the load side exhibits a high resistance, and the resistance of the power line remains constant, it can be inferred that for two different days with a load power ratio of 4:1, the current ratio will be 2:1.Consequently, the ratio of loss power between the two days will also be 4:1.
2. In the scenario where the load side has a high resistance, and the load resistance varies while the resistance of the power line remains constant, it can be deduced that for two different days with a load power ratio of 2:1, the proportion of loss power decreases as the load power increases.This is attributed to the notion that higher load powers correspond to larger load resistances.Thus, as the load power increases, the proportion of loss power decreases.
Day 1: Assuming a constant line resistance of 1 and a load resistance of 99, the proportion of loss power is 1% with a voltage value of 100.
Day 2: Assuming a constant line resistance of 1 and a load resistance of 199, the proportion of loss power is 0.5% with a voltage value of 100.
Based on this analysis, it is possible to naturally discern the fluctuations in load power and loss of power when they share the same line.
Assuming that the power ratio of two days is 10:1, and the resistance is X and Y, it can be calculated from the power ratio: When the line resistance is 1, the supply resistance is 99 on the first day, then: which is: After conducting calculations, it was determined that the load resistance on the second day was approximately 999.When comparing the numerical values of load power for the two days, using the equation (1^2 * 99) / (0.1^2 * 999), the ratio is estimated to be around 10:1.
Assuming a voltage of 100, the current on the second day is 0.1.Consequently, the ratio of loss power between the two days is derived as (1^2 * 1) / (0.1^2 * 1), resulting in a ratio of 100:1.
Based on the aforementioned analysis, it can be inferred that when there is only one load meter on the line and the load power for two days is represented as X1 and X2, respectively, the corresponding loss power values are denoted as Y1 and Y2.Under the assumption that the ratio of X1 to X2 is C, the approximate ratio of Y1 to Y2 can be expressed as  .This relationship indicates that the growth patterns of load power and loss power follow different curves.Building upon this observation, a constant resistance value is assumed for the line.
For an overall load line, it is possible to identify the day with the minimum loss ratio as the reference point.Using the data from that specific day, the separation of loss power can be performed, yielding the loss power values for each load meter.

𝑆
,  , … ,  , And the load power is: Taking any new data for day h, the different power ratios of n load meters can be calculated: which is: Then the power consumption ratio of the electric meter corresponding to the two days is: With this information, it becomes feasible to calculate the fitted loss power for H days and compare it with the loss power calculated based on meter readings.This facilitates the computation and adjustment of constant coefficients during the process of separating error power, ultimately allowing for an accurate determination of the constant coefficient for the loss power.

Identification of line topology based on time-point voltage data
Building upon the principles and necessity analysis of topology identification explained in section 3.2.1, as well as the low power loss estimation method based on topology identification, an unsupervised algorithm is utilized to identify electricity meters with similar voltage values, grouping them into the same hierarchical level.By considering the voltage magnitudes, a classification is established, enabling the recognition of multi-level topology results.This process involves the selection of meter categories and the determination of coefficients, which ultimately leads to the approximate identification of topology levels.
(1) Selection of identified meter categories Through the utilization of electricity usage data to fine-tune the topology identification algorithm, meters are categorized and recognized according to Table 3 The user has two periods with constant coefficients: one period with a coefficient of 0 and another period with a stable coefficient.

D Significant fluctuations in electricity consumption
Require division into multiple time periods for calculation of user-specific coefficients.
(2) Assumption for constant coefficient calculation Based on the previously mentioned logic, each of the mentioned meters is assumed to have stable periods characterized by constant coefficients (e.g., Period 1, Period n).These assumptions are organized into a matrix, and submatrices are created by selecting specific dates for analysis.By employing a multiple regression analysis, a hierarchical calculation method is implemented to determine the constant coefficients for the topology structure.This approach allows for accurate calculation of the coefficients.

Separation of loss power under nonlinear impedance and its effects
Based on the daily electricity consumption data from the meters, the data is partitioned into n intervals.Within each interval, it is assumed that the resistance values fluctuate within a range of ±5%, and there is no intersection in the resistance variations between adjacent intervals, with a difference in amplitude of over 10%.
By combining the previously discussed relationship between loss power and daily electricity consumption data, it is possible to calculate the non-linear relationships of current-time periods, impedance fluctuation ratios-time periods, current variation patterns-daily electricity consumption values, and impedance fluctuation ratios-daily electricity consumption values using proportional methods.This allows for the separation of loss of power even in the absence of specific resistance values.Based on the prior identification of topology relationships and the results of non-linear impedance analysis, the separation of line losses is achieved.The results are presented in Table 4 It can be observed that the separated line loss closely matches the actual power, indicating that the line loss separation technology based on this method yields better results.
To validate the results, we selected 10 meters and compared the calculated error rates with the actual error rates.The comparison is presented in Table 5 The calculated error rates closely align with the actual error rates, indicating the effectiveness of the method in accurately determining the error power and error rates of the meters.

Conclusion
Validation was conducted using 300 meters with a data time window spanning 330 days.The error rate results are presented in Table 6, and it was observed that 8 meters exhibited higher error rates, identifying them as outliers.Table 6.Validation results

Proposed method Traditional model
T FNR 10% 40% R FPR 0% 20% L TPR 90% 60% Among them, T FNR refers to the false negative rate, R FPR refers to the false positive rate, and L TPR refers to the true positive rate.
Based on the results, this project proposes a line loss separation method based on topological relationship perception and nonlinear impedance, which focuses on the problem of difficulty in effectively and accurately separating line loss electricity.Compared with traditional methods, the proposed method has better performance and provides support for error estimation and stable operation of measuring equipment.Due to limited time and energy, the proposed method is based on time point voltage data for topology relationship recognition, which can be integrated into other data for comprehensive identification and analysis in the future.

Figure 3 .
Figure 3. Schematic diagram of line topological relationship Some of the given conditions are as follows: (1) Condition 1: The distances between consumption meters 1, 2, 3, and 4 and the main supply meter M1 are approximately the same.Additionally, the line segments connect these meters from points A to B. There is no intersection between point B and consumption meters 1 and 2, and so on.(2)Condition 2: From point B to consumption meters 1, 2, 3, and 4, there is a presence of resistance, although the resistance values are extremely small compared to the resistance between points A and B (assuming they are 1000 times smaller).Hence, these losses can be neglected.Similarly, from point C

Table 2 .
Jump data example table

Table 3 .
. Meter categories in topology identification algorithm

Table 4 .
: Results of line loss separation

Table 5 .
. Comparison of error rates