Abstract
Accurate information for consumer phase connectivity in a low-voltage distribution network (LVDN) is critical for the management of line losses and the quality of customer service. The wide application of smart meters provides the data basis for the phase identification of LVDN. However, the measurement errors, poor communication, and data distortion have significant impacts on the accuracy of phase identification. In order to solve this problem, this paper proposes a phase identification method of LVDN based on stepwise regression (SR) method. First, a multiple linear regression model based on the principle of energy conservation is established for phase identification of LVDN. Second, the SR algorithm is used to identify the consumer phase connectivity. Third, by defining a significance correction factor, the results from the SR algorithm are updated to improve the accuracy of phase identification. Finally, an LVDN test system with 63 consumers is constructed based on the real load. The simulation results prove that the identification accuracy achieved by the proposed method is higher than other phase identification methods under the influence of various errors.
IN recent years, with the application of information and communication technology (ICT) in power systems, “digitization” has become an important feature of the modern power system. Among them, advanced metering infrastructure (AMI) provides the foundation to transform the planning, operation, and management of distribution networks [
The low-voltage distribution network (LVDN) is located at the edge of the power system and directly connects consumers. It is critical to ensuring the quality of the power supply and improving the consumer experience. However, the consumer phase connectivity information in LVDN is generally missing or inaccurate, which has become a bottleneck that restricts the planning and operation management of LVDN [
The principle of voltage correlation means that the correlation factors between the voltage profiles of consumers reflect the electrical distance between them, and consumers with a close electrical distance will have a greater probability of being in the same phase connectivity or the same branch [
The principle of energy conservation means that the current injected from the upstream nodes (busbars of each phase) of the LVDN at any point in time is equal to the sum of the currents flowing to the downstream nodes (consumers). Reference [
Compared with integer programming methods, regression analysis and machine learning methods have higher accuracy in phase recognition, which has recently attracted researchers’ attention. Reference [
Category | Reference | Method | Advantages and disadvantages |
---|---|---|---|
Voltage correlation |
[ | Correlation analysis |
(+) Only require voltage data, with high computational efficiency (-) Have poor identification performance on account of short electrical distances, light load, or balance three-phase load |
[ | Linear regression |
(+) Identify the connections of branches, consumer phase, and line impedance simultaneously (-) Be unavailable for LVDNs with more complex structures or with a large number of consumers | |
[ | Supervised learning |
(+) Achieve high accuracies based on sufficient training data samples (-) Be difficult to obtain training data labels in practice | |
[ | Clustering |
(+) Be easy to implement and tune (-) Be sensitive to algorithm parameter | |
Energy conservation |
[ | Integer programming |
(+) Only require current data (-) Be sensitive to bad and incomplete data |
[ | MILP |
(+) Have high computational efficiency (-) Need to collect phase angle information | |
[ | Lasso regression |
(+) Achieve higher accuracies based on strict parameter (-) Be sensitive to bad and incomplete data | |
[ | Clustering |
(+) Adapt to incomplete data (-) Require high data synchronization | |
[ | PCA |
(+) Only require load data and have high computational efficiency (-) Be sensitive to bad and incomplete data |
Note: the symbols (+) and (-) represent advantages and disadvantages, respectively.
In general, the identification methods of LVDN topology based on energy conservation usually require high-quality measurement data. However, in fact, due to the impacts of meter measurement errors, clock synchronization errors, communication interruptions, and other negative factors, the measurements may be seriously distorted, and the identification accuracy cannot be guaranteed [
Therefore, this paper proposes to apply a stepwise regression (SR) algorithm to effectively identify phase connection for consumers in LVDN. SR is a systematic algorithm for adding and removing terms from a multiple linear model based on their statistical significance in a regression [
The main contributions of this paper are as follows.
1) This paper applies the SR algorithm to identify phase connection for consumers in LVDN for the first time. The algorithm identifies the phase connectivity according to the significance test.
2) The significance correction factors are defined in this paper to correct the results of the SR algorithm, which can improve the identification accuracy with a variety of errors.
3) The size of errors when selecting current and active power as regression variables is analyzed, and the effects of selecting different regression variables on the accuracy of the phase identification are compared.
4) The influence of different algorithm parameters on the identification accuracy of the proposed method is analyzed, and on the premise of considering various errors, it is compared with the least square (LS) method [
Different from the LVDN in North America, LVDN in China generally has a three-phase four-wire structure [

Fig. 1 Illustration of a simple LVDN.
The phase topology of LVDN can be considered to be the connectivity relationship between consumers and each phase feeder. As shown in

Fig. 2 Phase topology of LVDN.
Then, the vector of phase current phasor , the matrix of consumer current phasor , and the vector of regression coefficient can be expressed as:
(1) |
(2) |
(3) |
where represents the phase connectivity information for the
(4) |
The phase identification essentially solves (4) to obtain the regression coefficient vector, which reflects the corresponding phase connection. Considering that the phase angle data cannot be measured by sensors or smart meters in LVDN [
(5) |
The phase current or active power is taken as the dependent variable Y, and the current or active power obtained by the consumer meter is taken as the independent variable X.
In (5), the errors include the measurement errors , model errors , and hidden errors , i.e.,
(6) |
The measurement errors are from the meter reading and the clock synchronization. can also be modelled to be Gaussian distribution with an expected value of 0 [
(7) |
(8) |
(9) |
(10) |
(11) |
where is the accuracy level of the meter, which is generally 0.2, 1, 2, or 5 [
The model errors refer to the errors caused by ignoring the phase angle and technical losses. These errors are mainly related to the grid structure and the network load level.
The hidden errors refer to the missing or serious distortion of measurement data due to the problems such as electricity theft [
With the access to distributed energy resources (DERs), there are also integrated prosumers. Since the DERs are generally installed behind consumer meters, the meter outputs are the net imbalance of local demand and supply. Thus, (5) inherently considers the integrated prosumers in LVDN [
To solve (5), the traditional methods used to convert it into an optimization problem [
This paper proposes to apply an SR algorithm to identify the consumer phase of LVDN according to their significances, which can be checked through F-test. Instead of focusing on the specific value of the regression coefficient obtained through optimization, the SR method is used to solve (5) based on the P-value, which can reflect the significant contribution of the corresponding independent variable to the observations. Therefore, the SR method provides a systematic way of identifying the consumer phase connectivity in a statistical framework, which considers errors and can achieve a higher identification accuracy. The key steps of SR algorithm are described as follows.
1) Based on the observations of current or active power, the multi-linear model as shown in (5) can be established.
2) The SR algorithm is used to add and remove the independent variables from (5) based on their statistical significances. The consumer phase is then determined according to the significant independent variable subsets for each phase.
3) Based on the correction factor, the calculation results of the SR algorithm are corrected to obtain the final identification of the consumer phase connectivity.
4) For cases where consumer phase connectivity cannot be determined due to light load or error influence, the methods such as voltage correlation analysis or field testing can be adopted.
The independent variable significantly influences the observations of the dependent variable when there is a linear correlation independent variable and a dependent variable. This means that the corresponding regression coefficients should be significantly different from 0. From the perspective of hypothesis testing, it is equivalent to testing whether hypothesis (12) is accepted.
(12) |
In this paper, the F-test is used to test the significance of the regression coefficient of a single variable, and the F-test value of the variable is constructed as:
(13) |
(14) |
(15) |
where SSE is the residual sum of squares obtained by linear regression (5) of the dependent variable on the N independent variables; SSEj is the residual sum of squares obtained by linear regression (5) of the dependent variable on the remaining independent variables after removing the independent variable; and is the partial residual sum of squares, and its value is equal to the difference between and SSE.
Assuming regression error satisfies the normal distribution, Fj will obey the F-distribution with degrees of freedom when , i.e.,
(16) |
Then, the probability that the hypothesis holds is:
(17) |
where is called the P-value for the F-test when regression coefficient . A smaller P-value indicates a higher likelihood that the corresponding independent variable has significant contributions to the observation of dependent variable, and vice versa.
SR is an iterative procedure to find the subset of the independent variable and corresponding regression coefficients that “best” explain the observations of the dependent variable. The main idea of the SR algorithm is to introduce the variables one by one, and if the independent variable meets the introduction criteria based on its significance, i.e., , this new variable is introduced. Each time a new variable is introduced, the old variables of the selected equations are tested one by one. If the non-significant exclusion condition is met, i.e., , the old variable is removed to ensure that the variables in the independent variable subset are all significant. This process is repeated by several times until no new variables can be introduced.
In order to avoid falling into the infinite loop of introducing-removing-introducing the same variable, it is generally required that is smaller than , i.e., . The detailed calculation procedure of the SR algorithm is explained below.
Algorithm 1 : SR algorithm |
---|
Inputs: observation vector Y, design matrix X, significance thresholds and |
Outputs: significant independent variable subset Step 1: start with initial regression model only with the DC component Step 2: select and add one independent variable to the regression model. The significance is checked using the F-test to obtain the P-value Step 3: if , shall be added to the regression model, and . If , go directly to Step 6 Step 4: the significance of all independent variables in regression model shall be checked using the F-test to obtain a set of significant P-value Step 5: let . If , shall be removed to the regression model, and . If , go directly to Step 6 Step 6: Steps 2-5 are repeated until no independent variable needs to be added or removed from the regression model according to F-test Step 7: End |
As explained above, using the SR algorithm, the significant independent variable subset for each phase can be obtained as:
(21) |
Considering the influence of errors and the settings of significance threshold, there could be intersections among , , and , as shown in

Fig. 3 Intersections among subsets of variables for each phase.
According to the linear correlation principle, for the consumers with smaller errors and heavier load, the expected value of the corresponding regression coefficient should be closer to 1 and the variance should be smaller. It means that the likelihood of is larger and the likelihood of is smaller. Based on this principle, the significance correction factor can be defined as:
(18) |
where P1 represents the P-value for the F-test of . The smaller the value, the smaller the likelihood of . Its calculation method is similar to that for P0.
The values are in the range of . If , . In extreme cases, if and , this means that the likelihood for will be highest. Conversely, if , . In extreme cases, if and , it means that the likelihood for will be highest. If , . This means that the likelihood for or will be the same at the highest uncertainty.
For , if , the reliability of the phase identification result on the corresponding consumer shall be unacceptable. For the intersections of subsets , , and , taking the intersection set XinAB as an example , if , the phase of consumer is more likely to be phase A rather than phase B. In this way, the identification results from the SR algorithm can be corrected based on the values of . The detailed steps are described as follows.
Algorithm 2 : correction of results from SR algorithm |
---|
Inputs: subset of significant independent variables for phase Φ: |
Outputs: corrected subset of significant independent variables for phase Φ X'inΦ Step 1: calculate the significant correction factor sets corresponding to XinΦ Step 2: find the elements that are less than 0 in ζinΦ, and remove the corresponding independent variables from XinΦ Step 3: for , if , remove xi from XinA; if , remove from ; otherwise, remove xi from both and Step 4: for , if , remove xi from XinA; if , remove xi from ; otherwise, remove xi from both and Step 5: for , if , remove xi from XinB; if , remove xi from ; otherwise, remove xi from both and Step 6: repeat Steps 3-5 to get form Step 7: end |
The final identification result of the consumer phase connectivity is obtained according to . In order to evaluate the performance of the algorithm, two indicators, i.e., precision rate and recall rate , are proposed as:
(19) |
(20) |
where Noutput is the number of consumers with identifiable phase connectivity information from
The output results of the algorithm under different significance thresholds are calculated since it is difficult to obtain the optimal threshold in advance in practical applications. To facilitate the comparison, define the credible precision rate , which represents the average precision rate under different threshold values when the recall rate is larger than g:
(21) |
where is the precision rate when ; and is the lower limit of the allowable recall rate.
The real LVDN of Guangdong Province in China is used to test the performance of the proposed method. Only the single line diagram of phase A of the test network is shown in

Fig. 4 LVDN test system with 63 consumers.
The consumer load data are collected with a sampling interval of 15 min. The sampling period is 2 days with a total of 192 time instants. The power consumption summary of each phase consumed in 2 days is presented in

Fig. 5 Power consumption summary of each phase.
Subset | Consumer ID |
---|---|
XinA | 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 19, 23, 31, 38, 44, 48, 53 |
XinB | 3, 10, 14, 15, 21, 22, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45 |
XinC | 1, 4, 39, 40, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63 |
There will be a number of misidentifications (, ) with a low precision rate of 72.4%. Also, there are a lot of intersections between the independent variable subsets for each phase in
To improve the accuracy of identification, the significance correction factor of each variable in is calculated, as shown in

Fig. 6 Significance correction factors for variables of each phase. (a) Phase A. (b) Phase B. (c) Phase C.

Fig. 7 Significance correction factors for variables in intersection.
According to the significance correction factors in
Subset | Consumer ID |
---|---|
1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 19 | |
3, 21, 22, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42 | |
43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63 |
It can be observed from
In order to analyze the influence of significance thresholds setting on the performance of the proposed method, the value of is varied in the range of (0, 0.5]. The corresponding and under different significance thresholds are shown in

Fig. 8 and under different significance thresholds.
When the significance threshold is raised, the conditions for adding the independent variables to the regression model will be more relaxed, so the recall rate will continue to increase, and the corresponding precision rate will continue to decrease in
(%) | (%) | (%) | ||
---|---|---|---|---|
0.001 | 0.0005 | 100 | 63.5 | 95.1 |
0.500 | 0.2500 | 91.9 | 98.4 | 95.1 |
To evaluate the computational burden, the SR algorithm implemented by MATLAB statistics toolbox (version R2019b) through the function “stepwisefit” is applied 100 times when varies in the range of (0, 0.5]. The computational time of the SR process is counted and averaged.
On average, the computation time used by the SR algorithm is approximately 32 s using a computer with an Intel Core i5-8265 CPU of 3.4 GHz and a RAM of 8 GB. Then, the computational time will increase insignificantly when the number of consumers in the network increases. Therefore, the proposed method is not applied to real-time applications.
When the influence of hidden error is not considered, the relative error e/y of the regression model (5) is mainly affected by the model error and measurement error, wherein the model error is related to the type of the selected regression variable. Taking phase A of the LVDN shown in

Fig. 9 Cumulative distribution function curves of e/y under different measurement errors. (a) . (b) . (c) .
As can be observed from
In order to further compare the identification results when the current and active power are used as regression variables, the test system shown in
εs (%) | (%) | |
---|---|---|
Current | Active power | |
1 | 100.0 | 100.0 |
4 | 99.9 | 98.4 |
8 | 95.1 | 93.7 |
This subsection compares the proposed method (M4) with LS method (M1) [
Under different measurement errors, the precision rate and recall rate of M1, M2, M4, and M5 are calculated as shown in
(%) | M1 | M2 | M4 | M5 | ||||
---|---|---|---|---|---|---|---|---|
(%) | (%) | (%) | (%) | (%) | (%) | (%) | (%) | |
1 | 98.41 | 100 | 100.0 | 100 | 83.2 | 100 | 100.0 | 100.0 |
4 | 96.80 | 98.4 | 77.5 | 100.0 | 93.7 | |||
8 | 85.70 | 95.2 | 70.2 | 96.5 | 90.5 |
Comparing the identification results of different methods, it can be observed that under the same measurement error, the proposed method has the highest accuracy. But when the measurement error is large (), the recall rate of the proposed method cannot reach 100% due to the large measurement error. The accuracy corresponding to M5 is nonideal since the voltage profiles of customers within short electrical distances are similar.
One case is that the current observations of consumers 9-39 are either 0 or missing due to electricity theft or interruption of communication at partial time instants. Then, under different measurement errors, the precision rate of M1 and M2 are calculated, as shown in
(%) | (%) | |
---|---|---|
M1 | M2 | |
1 | 80.9 | 73.0 |
4 | 79.3 | 69.8 |
8 | 66.7 | 63.5 |
(%) | (%) | |
---|---|---|
M3 | M4 | |
1 | 82.7 | 98.2 |
4 | 81.7 | 95.2 |
8 | 71.5 | 84.4 |
It can be observed from
It can be observed from
Another case is that the phase current observations measured by the sensors are missing at partial time instants. To avoid making mistakes in algorithm operation, these time instants with data missing should be abandoned. Considering the reduction of valid data samples, it could lead to the deterioration of results.
When the measurement error ratio is 4%, the precision rate and recall rate of M3 and M4 can be presented, as shown in

Fig. 10 and curves of M3 and M4. (a) M3. (b) M4.
It can be observed from
This paper has proposed an SR-based phase identification method of LVDN. The SR algorithm is used to identify the consumer phase connectivity based on their significances, and the significance correction factor is proposed for result correction. Through case studies based on a test system, the following conclusions can be drawn.
1) Compared with the LS and IQP methods, the proposed method has higher identification accuracy, especially when there is a hidden error. But the recall rate of the proposed method cannot reach 100% when the errors are large.
2) Compared with using active power, when the current is used as the regression variable, the multi-linear model error is smaller, which is beneficial for improving the accuracy of the proposed method.
3) Compared with the Lasso regression method, when there is a hidden error and the recall rate is higher than 80%, the accuracy of the proposed method is increased by an average of 14%.
For the practical application of the proposed method, future research will focus on improving the recall rate without reducing the precision rate. Also, the impact of distributed generation on the identification results of the proposed method will also be studied.
Nomenclature
Symbol | —— | Definition |
---|---|---|
A. | —— | Sets |
—— | Subset of significant correction factors corresponding to XinΦ | |
H | —— | Set of indices of measurements |
J | —— | Set of indices of phases |
L | —— | Set of indices of consumers |
—— | Subset of significant independent variables for phase Φ | |
—— | Corrected subset of significant independent variable for phase Φ | |
B. | —— | Vectors and Matrices |
—— | Vector of regression coefficients | |
—— | Vector of regression coefficients for phase Φ | |
e | —— | Vector of errors |
—— | Vector of measurement errors | |
—— | Vector of model errors | |
eh | —— | Vector of hidden errors |
Φ | —— | Vector of current phasor for phase Φ |
M | —— | Matrix of consumer current phasor |
X | —— | Design matrix of current magnitudes or active power measurements of consumers in low-voltage distribution network (LVDN) |
Y | —— | Vector of current magnitudes or active power measurements of phase |
C. | —— | Variables |
βΦj | —— | Regression coefficient of the |
βj | —— | Regression coefficient of the |
σsi | —— | Standard deviation of measurement error at the |
ζ | —— | Significance correction factor of independent variable regression coefficient |
—— | Precision rate | |
—— | Recall rate | |
Φ | —— | Index of phases |
Fj | —— | F-test value of the independent variable |
Φi | —— | Injection current for phase Φ at the |
Mij | —— | Load current of the |
Ncorrect | —— | Number of consumers with identifiable phase connectivity information from algorithms |
Noutput | —— | Number of consumers with correct phase identification from the outputs of algorithms |
xΦ(i) | —— | Consumer ID corresponding to the |
D. | —— | Parameters |
α | —— | Accuracy level of meter |
η | —— | Deviation of meter clock relative to reference clock |
εs | —— | Measurement error ratio |
λentry | —— | Significance introduced threshold |
λremove | —— | Significance remove threshold |
λlasso | —— | Regularization parameter of Lasso regression |
g | —— | Lower limit of allowable recall rate |
nΦ | —— | Total number of elements for subset XinΦ |
P0 | —— | P-value for the F-test that regression coefficient is equal to 0 |
P1 | —— | P-value for the F-test that regression coefficient is equal to 1 |
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