1. Introduction
The loads in the low-voltage distribution network are diverse. In particular, more and more distributed photovoltaic, electric vehicle (EV) charging stations, energy storage devices, and power electronics loads increase the pressure and risk of distribution zone transformers (DZTs) [
1]. Under long-term operation, affected by various factors, such as short circuits, overloads, and load imbalance, the winding’s health status will gradually deteriorate or even deform, resulting in transformer parameters that deviate from their factory values [
2]. The accuracy of transformer parameters is crucial for system analysis, control, and protection, and parameter changes can be utilized for online monitoring of winding deformation, fault diagnosis, and health condition evaluation [
3,
4,
5,
6,
7]. Therefore, the real-time and accurate identification of transformer parameters is of great significance.
Therefore, this study aims to propose a parameter identification method for distribution zone transformers under three-phase unbalanced conditions. The specific objectives are to (1) derive the power flow equations without requiring high- and low-voltage data synchronization, (2) develop an adaptive recursive least squares (ARLS) method for parameter identification, and (3) validate the proposed method through practical case studies.
In order to improve the accuracy of winding deformation and fault diagnosis, an equivalent circuit model comprising the iron core, winding, and insulation (including resistance, inductance, and capacitance) is proposed to represent the transformer. By leveraging the fact that mechanical deformation of the transformer winding leads to changes in the equivalent circuit parameters, comparing these parameters with measured values enables diagnosing the location and severity of winding deformation. The online parameter identification methods based on equivalent circuits mainly include solving transformer differential equations or vector equations [
8,
9,
10,
11], least squares and its improved methods [
2,
3,
12,
13,
14,
15,
16], and intelligent optimization methods [
6,
17,
18,
19,
20,
21,
22,
23,
24,
25], etc.
Using differential or vector equations is conceptually simple, but the results depend on the accuracy of the equation coefficients. The authors of one study developed a method to establish a dynamic vector model equation for transformer windings based on the winding current and voltage under no-load conditions. They used the least squares method to solve for the model parameters, but this method could only identify the winding resistance and leakage inductance [
4]. A differential equation method was proposed for parameter identification using real-time data of transformers, which was simple and straightforward [
8]. However, the results depended on the accuracy of the equation coefficients, and due to interference introduced by differentiation, it may lead to either no solution or incorrect solutions. The authors of [
9] utilized a second-order generalized integrator frequency-locked loop to extract the fundamental instantaneous phasors of voltage and current signals from both the high- and low-voltage sides of a transformer. Subsequently, based on the transformer’s vector equations, the short-circuit parameters were calculated. This method imposed high requirements for the synchronization of instantaneous data from both sides. The authors of [
10] represented the equivalent circuit of a two-winding transformer in the form of a quadripole, established the vector equations of the quadripole, and then determined the winding parameters by calculating the quadripole coefficients based on phasor measurement unit (PMU) measurement data. The authors developed an iterative method to calculate circuit parameters based on the transformer’s T-equivalent circuit using open-circuit and short-circuit measurement data; however, this method did not consider the influence of load [
11].
The essence of the least square method is to solve the overdetermined equation composed of transformer differential equations at the adjacent sampling time. Document [
12] established the magnetic flux balance equation based on the transformer model and the electromagnetic relationships between the primary and secondary sides. It then applied a least squares method based on orthogonal decomposition to identify the winding parameters; this method required derivatives of the voltage and current as input data. To reduce the influence of rounding errors during the recursive process, a least squares algorithm based on UD decomposition was used to identify the transformer’s equivalent circuit parameters [
13]. The difficulty of this method was obtaining the winding current on the delta side. An improved model was established by the authors of another study for transformer parameter identification based on Wide Area Measurement Systems (WAMSs) and Supervisory Control And Data Acquisition (SCADA) data, incorporating relative error into the objective function and employing the least squares method for solving [
2]. However, this method imposed excessively high requirements on data acquisition facilities. A least squares method was proposed to identify the equivalent resistance and leakage inductance based on the transformer circuit balance equation; however, only these two parameters could be identified simultaneously [
3]. Reference [
5] established a transformer equivalent circuit model, including winding resistance, leakage inductance, and capacitance, and applied the recursive least squares method for online identification of two parameters: the winding leakage inductance and the capacitance. A two-step identification technique was proposed for transformer positive-sequence parameters based on median estimation and least squares methods, where the transformer equivalent circuit model relied on PMU measurement data [
14]. Reference [
15] used sampled values of the primary- and secondary-side transformer voltages and currents, along with the relationships between the transformer’s steady-state equations and reactance parameters. They employed the recursive least squares method to solve these equations but could only identify the equivalent leakage reactance. Study [
16] simultaneously estimated transmission line and transformer parameters using a nonlinear weighted least squares method for solving, which required PMU devices at both ends of the transmission system to measure synchronized voltage and current phasors, without considering the transformer’s inherent three-phase unbalance.
Intelligent optimization methods set a feasible region, select an appropriate objective function, and iteratively approach the optimal solution, ultimately obtaining parameters under the optimal objective function. In one study, a genetic algorithm was employed to identify the parameters of the transformer’s R-L-C-M model without requiring any analytical formulas, although it required measured frequency response data to construct the objective function [
17]. The authors of [
18] modeled a transformer as a two-winding equivalent network and used a genetic algorithm and a Gauss-Newton iterative algorithm to identify the parameters of the equivalent network. The objective function was limited to amplitude-frequency information and could not fully encompass the mechanical information of the transformer windings. The authors of [
19] established a dual-winding high-frequency trapezoidal network, obtained actual parameters through finite element simulation, and implemented parameter identification using a genetic algorithm and a particle swarm algorithm. The authors developed a methodology to establish a transformer network model without winding design data by utilizing end-to-end open-circuit frequency responses and other terminal test results. They then applied a genetic algorithm to approximate the unknown parameters of the model [
20]. The authors of [
21] established a transformer dual-winding trapezoidal equivalent network, utilizing amplitude–frequency and phase-frequency information from the frequency response curve to construct the objective function and employing the improved Whale Optimization Algorithm (WOA) to identify the equivalent network parameters. Study [
22] injected a pulse signal into the transformer bushing, measured the response at the neutral point terminal to acquire data, and then used the Zoom Genetic Algorithm (ZGA) to identify the winding parameters. The authors of [
23] utilized transformer port information and a particle swarm algorithm to identify the leakage impedance parameters of a transformer’s T-equivalent circuit through two iterative processes. Reference [
6] considered the transformer’s voltage circuit equation, active loss equation, input impedance equation, no-load current equation, and no-load loss equation when establishing a parameter estimation model and then solved using particle swarm optimization. A Slime Mold Optimization Algorithm (SMOA) was employed to solve nonlinear optimization problems for identifying transformer equivalent circuit parameters, which required measured data of the transformer load [
24]. Finally, the authors of [
25] employed the Tasmanian Devil Optimization (TDO) algorithm to identify transformer parameters.
Existing transformer parameter identification methods often assume symmetric three-phase conditions, require synchronized measurements between the high- and low-voltage sides, or rely on computationally intensive AI-based optimization techniques. These limitations hinder their practical application in low-voltage distribution networks, where data acquisition is constrained by the capabilities of transformer terminal units and asymmetric operating conditions are common. To address these research gaps, this study proposes the following main contributions.
For low-voltage DZTs, a parameter identification method under three-phase unbalanced conditions is proposed, with the following main contributions: First, based on the transformer T-equivalent circuit under load, the power flow equations are derived without involving synchronization issues between high- and low-voltage side data, and the sum of high- and low-voltage side impedances is treated as an independent parameter. Then, a novel power flow equation under three-phase unbalanced conditions is established, and an ARLS solution method is constructed using the measurement data sequence provided by the smart meter of the intelligent TTU, achieving online identification of transformer winding parameters.
The remaining content of this paper is organized as follows:
Section 2 establishes the DZT power flow equations based on the transformer T-equivalent circuit;
Section 3 establishes the DZT power flow equations under three-phase asymmetry conditions, including load asymmetry and transformer parameter asymmetry;
Section 4 describes the principles and implementation of the parameter identification method; and, finally, practical case studies are used to verify the effectiveness and robustness of the proposed method.
2. Materials and Methods
This section outlines the research methods and main steps used in this study for transformer parameter identification under three-phase unbalanced conditions. First, a T-equivalent circuit model is established, taking into account the transformer’s load characteristics. Based on this model, power flow equations are derived without requiring synchronized high- and low-voltage side measurements. Then, a novel power flow formulation for asymmetric three-phase conditions is proposed. To estimate the transformer’s equivalent parameters, an ARLS algorithm is developed using real-time measurement data from the transformer terminal unit (TTU). Finally, the proposed method is validated through simulation and case studies to demonstrate its effectiveness and robustness.
2.1. T-Equivalent Circuit of DZTs
Transformer equivalent models mainly include Г-type and T-type circuits, in which the T-type circuit can more accurately reflect the internal electromagnetic relationship of a transformer [
11]. The parameters include four types: short-circuit reactance (
), short-circuit resistance (
), excitation reactance (
), and excitation resistance (
). h and l, respectively, represent the high- and low-voltage sides. The load impedance
and
are added to the T-equivalent circuit, as shown in
Figure 1.
Based on the transformer’s T-equivalent circuit, as shown in
Figure 1, we derive the following equations:
where
and
can be measured directly,
is the excitation voltage, and
is the load impedance.
and
are deleted from the formula to obtain:
The system impedance
is substituted into Equation (2) to obtain
This equation is obtained by substituting the system impedance definition into Equation (2), where and are measured on the same side, so the calculation of and does not involve the synchronization problem of high- and low-voltage measurements, and TTU can effectively control the synchronization time of data on both sides to ensure the accuracy and real-time performance of the measured data. The measurement time of system impedance and load impedance can be regarded as the same time, so they are known quantities, and the unknown quantities are equivalent parameters , and of the DZT.
However, Equation (3) is nonlinear and requires numerical methods for its solution; it is inevitably affected by measurement errors from the instrument transformers. Measurement errors of system impedance and load impedance are transmitted in series to the parameter identification error. For example, in the T-equivalent circuit, the winding impedance is connected in series with the load impedance. Since the winding impedance is much smaller than the load impedance, even a small error in the load impedance measurement leads to a large error in the calculated winding impedance. Therefore, Equation (3) cannot be directly used for the parameter solution of distribution transformers.
2.2. Power Flow Equation of DZTs
To address these issues, we consider a T-equivalent circuit including the load and treat the sum of the high- and low-voltage winding impedances as an independent parameter. Power flow equations are established using measurement data sequences provided by the TTU’s smart meter. Solving these equations with the least squares method minimizes the influence of measurement errors.
The transformer’s complex power loss ∆
S consists of three components: the excitation loss and the winding losses on the high- and low-voltage sides. We derive the following expression for the complex power loss of the transformer.
According to Equation group (1), we can get:
By substituting the above relations, we obtain the following simplified form:
Considering that
and
are much larger than
, and
, we approximate the parameters as follows:
where
is approximately the short-circuit resistance,
is approximately the short-circuit reactance,
is approximately the magnetizing conductance, and
is approximately the magnetizing susceptance.
The expressions of
, and
are derived in this study as:
These parameters are related to the phase and can be identified but have no practical significance. In fact, if the low-voltage side data VL, IL, PL, and QL are used, a similar T-type circuit power flow equation can also be established. That is, the excitation and short-circuit impedance with the same accuracy can be obtained by using the high- and low-voltage side data.
Equation (6) can be solved in two ways. The first is the static method, which uses four different load data to form four independent equations and solve four unknown parameters. The second is the continuous method, which uses the least squares method to solve the measurement data sequence of the distribution transformer, which can minimize the error.
5. Simulation Analysis of Parameter Identification
The validation is conducted through practical case studies. The nameplate parameters of the dry-type DZT are as follows: rated capacity of 500 kVA, rated voltage of 10 kV/0.4 kV, no-load loss of 1.16 kW, no-load current of 1.0%, load loss of 4.91 kW, and short-circuit impedance of 4.0%. The connection method is Yyn0 with neutral points grounded on both sides. The loads of the DZT, including residential loads, commercial loads, grid connected photovoltaic, variable-frequency speed-regulating AC motors, and energy storage devices, are simulated using Simulink’s three-phase dynamic load module and a custom load controller (version MATLAB R2022a). The measurement model and parameter identification model of the transformer are shown in
Figure 2 and
Figure 3.
The detailed simulation parameters and configurations are provided in
Appendix A.
5.1. Numerical Solution of Parameters
To compare, first, the impedance parameters of both sides and the excitation parameter
Xm are solved using numerical methods based on Equation (3), and the results are shown in
Table 1.
From
Table 1, the equivalent parameters obtained through numerical methods are consistent with those calculated based on the transformer nameplate data. To validate the impact of measurement errors on the numerical solution method of Equation (3), we assume a 0.1% measurement error in the load impedance, using the positive-sequence components of the measurement data for parameter calculation, as shown in
Table 2.
As shown in
Table 2, except for the relatively small error in excitation impedance, the errors of the other parameters are quite large. The fundamental reason lies in the fact that in the T-equivalent circuit, the winding impedance is series-connected with the load impedance, but its value is much smaller than the load impedance. Consequently, even a small error in the load impedance corresponds to a significant error in the winding impedance. Therefore, the system of equations formed by (3) is ill conditioned, making the solution process highly sensitive to measurement errors and resulting in larger parameter identification errors. Hence, Equation (3) is not suitable for directly solving the equivalent parameters of distribution transformers.
5.2. Parameter Identification Under Symmetrical Operation
From
Table 2, it can also be observed that the numerical calculation results of
and
are not significantly different from their actual values, with an error of approximately 1.3%. According to Equation (8),
and
can be used to replace the resistance and reactance parameters of the high- and low-voltage windings, while
and
can be used to replace the excitation resistance and reactance parameters. Then, the proposed identification method is applied, and the results are shown in
Table 3, with the errors in active and reactive power losses illustrated in
Figure 4.
From
Table 3, it can be seen that
,
,
, and
are basically consistent with the sum of the high- and low-voltage side winding resistances, the sum of the leakage reactances, and the excitation conductance and susceptance in the transformer equivalent circuit, respectively. Therefore,
and
can be identified as independent parameters with very small errors.
Figure 5 also shows the changes in active and reactive power loss errors during the parameter identification process, as well as a comparison between them.
Figure 5 demonstrates that the identification algorithm exhibits good robustness against measurement errors, with reactive power loss errors significantly smaller than active power loss errors, indicating higher stability in the identification of excitation conductance and susceptance parameters.
To validate that the parameter identification method is independent of DZT load variations, simulations were conducted using different load intervals. The results showed that , , , and remained unchanged, thus no data tables are provided. Unlike the numerical solution in Equation (3), the proposed parameter identification method is unaffected by load variations.
5.3. Parameter Identification Under Unbalanced Three-Phase Loading Conditions
Solving the system of Equation (7) requires four sets of different load data, which are obtained by setting four different three-phase unbalanced loads for parameter identification, as shown in
Table 4.
From
Table 4, it can be seen that when there is an unbalanced three-phase load, the errors in the identification results of DZT parameters are relatively large, making it impossible to determine whether a fault or deformation has occurred. The structure of a DZT is typically three-phase and three-column. For Yyn0 and Dyn11 connection groups, the zero-sequence impedance differs significantly from the positive- and negative-sequence impedances. For Yyn connection groups, there is no zero-sequence path, resulting in infinite zero-sequence impedance. Due to the inconsistency between zero-sequence impedance and positive/negative-sequence impedances, the identification results have significant errors. Therefore, we filter out the zero- and negative-sequence components of the measurement data, retaining only the positive-sequence components for DZT parameter identification. The identification results then closely match the actual DZT parameters, indicating that load asymmetry does not affect the positive-sequence component method. Therefore, the data table is omitted.
5.4. Identification of the DZT with Parameter Asymmetry
We assume that the positive, negative, and zero sequences of the DZT have the same magnetic flux path, but the parameters of each phase are inconsistent. If the winding deformation occurs in one phase of the low-voltage side, resulting in a 5% decrease in the reactance value
, a random load is applied, and parameter identification is performed using the positive-sequence component of the measurement data. The results are shown in
Table 5.
As shown in
Table 5, the short-circuit reactance
xk also changes, but the change of 0.8% is smaller than the reduction of 5% in
X1. Although the identified parameters are close to the actual values, the sensitivity is reduced, which is not conducive to early detection of single-phase faults. Since the measurement errors of load impedance directly affect the calculation errors of short-circuit impedance, and the positive-sequence load impedance is significantly larger than the transformer’s short-circuit impedance, the use of the positive-sequence component method cannot avoid introducing significant errors in the short-circuit impedance calculation.
To address this issue, parameter identification was performed using the negative-sequence components of the measurement data, as shown in
Table 6. The corresponding active and reactive power loss errors are illustrated in
Figure 5.
As shown in
Table 6, even with significant measurement errors, the identification results for short-circuit impedance still maintain a high level of precision, with much smaller errors compared to the positive-sequence component method. This is because the unbalanced degree of the power grid’s voltage is much smaller than that of the load, resulting in much smaller errors caused by impedance measurements. However, the identification results for magnetizing impedance are relatively poor, also due to the very small negative-sequence unbalance degree of the power source, meaning that the negative-sequence measurement components can only be used for short-circuit impedance identification.
Figure 6 shows the variations in active and reactive power loss errors when using negative-sequence measurement components for parameter identification. It is observed that the reactive power loss error fluctuates significantly, indicating lower precision in identifying excitation parameters. However, it is still smaller than the active power loss error, as excitation parameters are primarily influenced by the source voltage.
5.5. Identification of DZT Parameters Under Fault Conditions
Some sudden faults can cause step changes in transformer parameters, such as the occurrence of a large short-circuit current on the load side, leading to sudden deformation of the windings. There are also continuously deteriorating faults where parameter deviations gradually increase. Analysis of parameter identification under both scenarios is conducted.
In the first scenario, assuming a short circuit occurs on the low-voltage load side during parameter identification, resulting in deformation of a phase winding and a sudden 5% reduction in reactance
. Parameter identification is performed using the positive-sequence component method, with results shown in
Table 7. The errors in active and reactive power losses are shown in
Figure 6.
As shown in
Table 7, both the impedance and excitation parameter identifications have relatively large errors. In the second scenario, during parameter identification, it is assumed that the low-voltage side winding reactance
Xl linearly decreases from its normal value to simulate continuous parameter changes. The identification results are shown in
Table 8, and the errors in active and reactive power losses are illustrated in
Figure 7.
The active and reactive power loss errors in
Figure 7 also exhibit significant changes. Although these errors are considerable, the parameter identification results and the loss error curves support one another, serving as a basis for assessing winding deformation.
As shown in
Table 8, the identification errors under continuously varying parameters are not significant. Specifically, resistance parameters are more susceptible to influence, while the deviation in short-circuit reactance parameters is minimal. As observed in
Figure 7, the comparison of active and reactive power loss errors shows that the former is larger, while the latter exhibits a divergent trend. This is related to uniformly distributed random loads, and actual loads will be more complex.
5.6. Comparative Validation
To further validate the effectiveness and robustness of the proposed ARLS method, comparative studies were conducted against two classical parameter identification approaches: the least squares (LS) method [
12] and the genetic algorithm (GA) approach [
17]. The comparison was performed under identical simulation conditions, including asymmetric three-phase loading and parameter variation scenarios, using the Simulink platform.
The comparison focused on three key performance indicators:
- (1)
Parameter estimation error;
- (2)
Computational time;
- (3)
Robustness to measurement noise (evaluated under 0.1% Gaussian noise added to measurement data).
The results are summarized in
Table 9.
The results show that the proposed ARLS method achieves the lowest parameter estimation error among the three methods while maintaining fast computational speed suitable for online identification. Although the LS method is computationally faster, it is significantly less robust to measurement noise. The GA approach achieves acceptable estimation accuracy but requires considerably higher computational time due to its iterative nature.
These comparative results highlight the superior performance of the proposed ARLS method in terms of accuracy, speed, and robustness, making it highly applicable to real-time transformer monitoring under practical operating conditions.
5.7. Field Data Analysis
The field data come from the intelligent fusion terminal of a transformer in a certain area of Puyang Power Supply Company in Henan Province, China, as shown in
Figure 8. To further clarify the measurement system and data flow,
Figure 9 presents a schematic overview of the key components involved, including the transformer, TTU, smart meter, and data export, which provide input for the ARLS-based parameter identification.
The distribution transformer model is SCB10-500/10, and the only nameplate parameter is the short-circuit impedance of 4.26%. There are five load branches on the low-voltage side of the transformer, including one lighting load and four induction motor loads. The current and voltage measurement accuracy of SIT (Smart Integrated Terminal) is 0.1%, and the phase error is 0.1%. Its data freezing function can ensure that the synchronization error of the high- and low-voltage side data acquisition time is within 10 ms. Within this time error, the load change can be ignored, so the high- and low-voltage side measurement data can be regarded as the measurement data under the same load point.
SIT can collect the following data: positive-sequence voltage and current RMS values on the high- and low-voltage sides, as well as positive-sequence active and reactive power, with a collection interval of 60 s. The data from 1:00 to 8:00 on a certain day are shown in
Figure 10 and
Figure 11.
Based on the field-measured data, the proposed method is used to identify the parameters of the distribution transformer. The calculation results are shown in
Table 10. The active and reactive loss errors of the distribution transformer during the identification process are shown in
Figure 12.
The power loss error curve is similar to the MATLAB (R2022a) simulation result of the aforementioned random error. Moreover, the short-circuit impedance parameter of the distribution transformer estimated by using the identified rk and xk is 4.25%, which is very close to the nameplate parameter, proving that the algorithm is applicable to the actual site and has practical value.
The data from 20:00 to 24:00 on the same day were collected again, and the power supply voltage and load current are shown in
Figure 13. Comparing the two sets of current and voltage curves, it can be seen that the transformers in the two periods work in different load ranges, but the transformer parameters obtained are similar, as shown in
Table 11. In other words, the proposed method can obtain consistent parameter identification results in different load ranges, verifying the applicability of the method to load fluctuations.
5.8. Comparison with Recent Novel Power Flow Methods
Recent research has seen the emergence of various novel power flow and parameter identification methods, particularly those leveraging artificial intelligence (AI), optimization algorithms, and data-driven techniques. For example, genetic algorithms (GAs) [
17], particle swarm optimization (PSO) [
23], and slime mold optimization (SMOA) [
24] have been employed to solve transformer parameter estimation problems, offering global search capabilities but often at the cost of high computational complexity and longer computation times. In contrast, machine learning approaches and neural networks provide predictive models trained on historical data, yet they require large datasets and lack transparency in physical interpretation.
Compared to these recent methods, the proposed ARLS approach offers several advantages: (1) it achieves real-time online identification using only local measurements without the need for synchronized high- and low-voltage data; (2) it maintains low computational burden, making it suitable for deployment in practical distribution networks; and (3) it demonstrates superior robustness to measurement noise, as shown in the comparative results (
Table 9). While heuristic and AI-based methods are powerful for offline analysis and design, the ARLS method fills a critical gap by enabling efficient, interpretable, and real-time parameter tracking under three-phase asymmetric conditions.
6. Conclusions
For parameter identification of the DZT under three-phase asymmetry conditions, we propose a recursive least squares method based on the transformer’s T-equivalent circuit power flow equations. This method effectively overcomes the issue of unsynchronized data between the high- and low-voltage sides. The proposed method treats the sum of high- and low-voltage impedances of the transformer’s equivalent circuit as an independent parameter, establishes power flow equations under load asymmetry and transformer parameter asymmetry conditions, and solves them using a data sequence collected by the TTU with a recursive least squares method incorporating a forgetting factor. The proposed method was analyzed and validated through a practical case study on the Simulink simulation platform. The results indicate that the impedance parameters on both the high-voltage and low-voltage sides are significantly affected by load variations and must be identified collectively. Under three-phase symmetry conditions, the parameter identification achieves high precision. In cases of load asymmetry, using only the positive-sequence component yields accurate identification results. For a DZT with three-phase parameter asymmetry, combining positive and negative-sequence component methods can promptly indicate the occurrence of asymmetry faults. In scenarios where transformer faults cause parameter abrupt changes or continuous variations, although the identification accuracy is relatively low, it still satisfies the requirement for transformer fault judgment. Additionally, the impact of power voltage variations on magnetizing impedance and the influence of ambient temperature on winding resistance require further investigation. Furthermore, the proposed method is well-suited for practical utility deployment. It features low computational complexity, as it relies only on real-time local measurements without requiring large datasets or complex iterative calculations. By utilizing data from existing TTUs and avoiding the need for synchronized high- and low-voltage measurements, the approach minimizes data acquisition burden and deployment complexity, facilitating integration into modern low-voltage distribution networks.
In the future, further studies could focus on extending the proposed method to more complex transformer models, such as those considering nonlinear magnetic characteristics and temperature effects. In addition, exploring the integration of this approach into large-scale smart grid systems and improving the real-time performance and robustness of the algorithm would provide valuable directions for continued research. It should be noted that while the proposed method accounts for local three-phase unbalance at the transformer level, it does not consider large-scale upstream disturbances, such as voltage sags or frequency variations. Addressing these broader dynamics is an important direction for future work to improve the method’s robustness in real-world applications.