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Article

Parameter Identification of Distribution Zone Transformers Under Three-Phase Asymmetric Conditions

1
Economic and Technical Research Institute of Gansu Electric Power Company, Lanzhou 730050, China
2
School of Mechanical and Electrical Engineering, China University of Mining & Technology, Beijing 100083, China
*
Author to whom correspondence should be addressed.
Eng 2025, 6(8), 181; https://doi.org/10.3390/eng6080181
Submission received: 28 June 2025 / Revised: 22 July 2025 / Accepted: 28 July 2025 / Published: 2 August 2025

Abstract

As a core device in low-voltage distribution networks, the distribution zone transformer (DZT) is influenced by short circuits, overloads, and unbalanced loads, which cause thermal aging, mechanical stress, and eventually deformation of the winding, resulting in parameter deviations from nameplate values and impairing system operation. However, existing identification methods typically require synchronized high- and low-voltage data and are limited to symmetric three-phase conditions, which limits their application in practical distribution systems. To address these challenges, this paper proposes a parameter identification method for DZTs under three-phase unbalanced conditions. Firstly, based on the transformer’s T-equivalent circuit considering the load, the power flow equations are derived without involving the synchronization issue of high-voltage and low-voltage side data, and the sum of the impedances on both sides is treated as an independent parameter. Then, a novel power flow equation under three-phase unbalanced conditions is established, and an adaptive recursive least squares (ARLS) solution method is constructed using the measurement data sequence provided by the smart meter of the intelligent transformer terminal unit (TTU) to achieve online identification of the transformer winding parameters. The effectiveness and robustness of the method are verified through practical case studies.

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 ( X h , X l ), short-circuit resistance ( R h , R l ), excitation reactance ( X m ), and excitation resistance ( R m ). h and l, respectively, represent the high- and low-voltage sides. The load impedance R L and X L 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:
V h = Z h I h + V m , V m = Z l I l + V l I h = V m Z m + I l , I l = V l Z L
where V h ,   V l ,   I h and I l can be measured directly, V m is the excitation voltage, and Z L is the load impedance. V l and I l are deleted from the formula to obtain:
V h = Z h I h + V m I h = V m Z m + V m / ( Z l + Z L )
The system impedance Z s = V h / I h is substituted into Equation (2) to obtain
Z m ( Z l Z L ) = ( Z s Z h ) ( Z l Z L ) + Z m ( Z s Z h )
This equation is obtained by substituting the system impedance definition into Equation (2), where V h ,   I h and V l ,   I l are measured on the same side, so the calculation of Z s and Z l 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 Z h ,   Z m , and Z l 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.
Δ S = V m V m I m * + Z h I h I h + Z l I l I l
According to Equation group (1), we can get:
V m = V h Z h I h I l = I h V h Z h I h Z m
By substituting the above relations, we obtain the following simplified form:
Δ P = g m | V h | 2 + r k | I h | 2 + f p P h + j p Q h Δ Q = b m | V h | 2 + x k | I h | 2 + f q P h + j q Q h
Considering that R m and X m are much larger than R h ,   X h ,   R l , and X l , we approximate the parameters as follows:
r k R h + R l , x k X h + X l g m R m / | Z m | 2 , b m X m / | Z m | 2
where r k is approximately the short-circuit resistance, x k is approximately the short-circuit reactance, g m is approximately the magnetizing conductance, and b m is approximately the magnetizing susceptance.
The expressions of f p ,   j p ,   f q , and j q are derived in this study as:
f p = 2 ( R m R h R h R l R l R m ) / | Z m | 2 j p = 2 ( R m X h X h R l R l X m ) / | Z m | 2 f q = 2 ( X m X h X h X l X l X m ) / | Z m | 2 j q = 2 ( X m R h R h X l X l R m ) / | Z m | 2
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.

3. Power Flow Equation Under Asymmetric Conditions

Three-phase loads and transformer parameters may have certain asymmetry. Due to the presence of many single-phase loads, the distribution transformer loading in a transformer district may result in asymmetric loading. Transformer faults can cause abrupt changes or continuous variations in parameters. Separate power flow equations should be established for asymmetric loading and asymmetric transformer parameters.

3.1. Asymmetry in Three-Phase Loads

To analyze the impact of asymmetric three-phase loads, we employ the symmetrical component method. This approach decomposes the measured three-phase powers into zero-sequence, positive-sequence, and negative-sequence components, which enables separate evaluation of symmetric and asymmetric effects on the transformer. The symmetrical component method is used to decompose the three-phase asymmetric active power [ P a h P b h P c h ] on the high-voltage side of the DZT into three groups of symmetrical three-phase active power, P a h as an example. Using the symmetrical component method, the three-phase asymmetric active powers are decomposed as:
P a 1 h P a 2 h P a 0 h = 1 3 1 α α 2 1 α 2 α 1 1 1 P a h P b h P c h
where α = e j 120 . When the three-phase load is asymmetrical, we extend the power flow equations to asymmetric loading conditions, as shown below.
Δ P a ( s ) = g m | V a ( s ) l | 2 + r k | I a ( s ) l | 2 + f p P a ( s ) l + j p Q a ( s ) l Δ P b ( s ) = g m | V b ( s ) l | 2 + r k | I b ( s ) l | 2 + f p P b ( s ) l + j p Q b ( s ) l Δ P c ( s ) = g m | V c ( s ) l | 2 + r k | I c ( s ) l | 2 + f p P c ( s ) l + j p Q c ( s ) l Δ Q a ( s ) = b m | V a ( s ) l | 2 + x k | I a ( s ) l | 2 + f q P a ( s ) l + j q Q a ( s ) l Δ Q b ( s ) = b m | V b ( s ) l | 2 + x k | I b ( s ) l | 2 + f q P b ( s ) l + j q   Q b ( s ) l Δ Q c ( s ) = b m | V c ( s ) l | 2 + x k | I c ( s ) l | 2 + f q P c ( s ) l + j q Q c ( s ) l
where s (s = 0, 1, 2) represents zero sequence, positive sequence, and negative sequence. These equations describe the active and reactive power flow in each phase under asymmetric loading conditions. By separating the sequence components, the model captures how load imbalance influences the power distribution and transformer behavior, which is essential for accurate parameter identification.
Overall, the power flow equations under unbalanced loading conditions can effectively describe the power transmission characteristics of DZTs under typical operating conditions by introducing power expressions of symmetrical components. This method provides a mathematical foundation for subsequent parameter identification of a DZT, but further consideration is needed for the effects caused by the transformer’s own parameter asymmetry, which will be discussed in the next section.

3.2. Asymmetry in Three-Phase DZT Parameters

Internal inter-turn short circuits, phase-to-phase short circuits, and winding deformation in transformers can all result in unbalanced three-phase parameters. Using the symmetrical component method, we establish the power flow equations for the transformer when its own parameters are unbalanced:
Δ P a ( s ) = g am ( s ) | V a ( s ) l | 2 + r ak ( s ) | I a ( s ) l | 2 + f ap ( s ) P a ( s ) l + j ap ( s ) Q a ( s ) l Δ P b ( s ) = g bm ( s ) | V b ( s ) l | 2 + r ak ( s ) | I b ( s ) l | 2 + f bp ( s ) P b ( s ) l + j bp ( s ) Q b ( s ) l Δ P c ( s ) = g cm ( s ) | V c ( s ) l | 2 + r ak ( s ) | I c ( s ) l | 2 + f cp ( s ) P c ( s ) l + j cp ( s ) Q c ( s ) l Δ Q a ( s ) = b am ( s ) | V a ( s ) l | 2 + x ak ( s ) | I a ( s ) l | 2 + f aq ( s ) P a ( s ) l + j aq ( s ) Q a ( s ) l Δ Q b ( s ) = b bm ( s ) V b ( s ) l 2 + x ak ( s ) | I b ( s ) l | 2 + f bq ( s ) P b ( s ) l + j bq ( s )   Q b ( s ) l Δ Q c ( s ) = b cm ( s ) | V c ( s ) l | 2 + x ak ( s ) | I c ( s ) l | 2 + f cq ( s ) P c ( s ) l + j cq ( s ) Q c ( s ) l
Take phase A as an example, where gam(s), bam(s), rak(s), xak(s), fap(s), faq(s), jap(s), and jaq(s) are all the sequence components of phase A. In fact, Equation (10) can be regarded as a subset of Equation (11). Equation (11) further extends the formulation to account for asymmetry in the transformer’s own parameters, such as phase-to-phase variations in winding resistance or reactance due to faults or deformations. This allows the identification method to detect both load-induced and internal transformer asymmetries.
In summary, the asymmetry of three-phase parameters in a DZT not only affects power flow characteristics but may also reflect changes in the internal structure of windings. Therefore, the power flow equations require more precise consideration of independent winding parameters for each phase. To achieve parameter inversion of the aforementioned equations under actual operating conditions, the next section will construct a parameter identification method based on the least squares principle and incorporate a recursive algorithm to enhance its real-time performance and robustness.

4. Parameter Identification Method

4.1. Method Principle

When solving with the least square method, the power flow Equations (10) and (11) under asymmetric conditions can be written as a power flow equation in matrix form:
ΔSm = H × Z
This matrix equation represents the physical relationship between the measurable power differences (ΔSm) and the unknown transformer parameters (Z) through the measurement matrix (H). Here, ΔSm includes the active and reactive power loss differences, H is constructed from the measured voltage and current data, and Z contains the equivalent impedance parameters to be identified.
Taking Equation (10) as an example, ΔSm, H, and Z are respectively expressed as Equation (13):
Δ S m = Δ P a ( s ) , Δ P b ( s ) , Δ P c ( s ) , Δ Q a ( s ) , Δ Q b ( s ) , Δ Q c ( s ) Z = g m , r k , f p , j p , b m , x k , f q , j q Z = | V a ( s ) | 2 | I a ( s ) | 2 f p j p | V b ( s ) | 2 | I b ( s ) | 2 f p j p | V c ( s ) | 2 | I c ( s ) | 2 f p j p | V a ( s ) | 2 | I a ( s ) | 2 f q j q | V b ( s ) | 2 | I b ( s ) | 2 f q j q | V c ( s ) | 2 | I c ( s ) | 2 f q j q
This decomposition explicitly shows how each measured sequence component contributes to the parameter identification process. It allows for isolating the effects of phase asymmetry on the transformer’s electrical characteristics. In an ideal power system, the voltage and current are 50 Hz sinusoidal signals. When the number of sample values exceeds the number of unknowns, these sample values are linearly correlated, leading to the non-existence of (HTH)-1, which makes parameter identification impossible using the least squares method or renders the parameters unidentifiable. However, in actual power systems, especially those with diverse and nonlinear distribution feeder loads, the voltage and current contain abundant harmonics and non-periodic components. At this point, (HTH)-1 exists, and parameter identification can be achieved using the least squares method [4].

4.2. RLS Method with a Forgetting Factor

The recursive least squares (RLS) method does not require storing all historical data; it only records the parameter values identified in the previous moment and the current data, making it particularly suitable for real-time parameter identification. However, this method suffers from a data saturation issue, gradually losing its ability to update parameters. To address this problem, an RLS method with a forgetting factor can be adopted [26]. By multiplying old data with a forgetting factor, the information provided by old data is reduced, while the information from new data is increased, enabling continuous correction to obtain the optimal parameter values. The recursive least squares (RLS) algorithm with a forgetting factor is implemented as follows, following the method in [26].
Z ^ ( k ) = Z ^ ( k 1 ) + K ( k ) ε ( k ) K ( k ) = P ( k 1 ) H ( k ) [ H T ( k ) P ( k 1 ) H ( k ) + μ ] ε ( k ) = [ Δ S ( k ) H T ( k ) Z ^ ( k 1 ) ] P ( k ) = [ I K ( k ) H T ( k ) ] P ( k 1 ) / μ
where I is the identity matrix, P(k) is the covariance matrix, and K(k) is the gain factor. μ is between 0.95 and 1.0, which can weaken the role of old data and enhance the role of new data. The RLS algorithm is applied to continuously update parameter estimates in real time. Physically, this means that as new measurement data are acquired, the algorithm refines the estimation of transformer impedance parameters, making it suitable for dynamic system conditions.
During actual operation, the DZT may experience saturation issues, and the saturation current varies with the degree of saturation. Therefore, the value of μ should be determined based on real-time online measurement data. For each parameter to be identified, μ should vary dynamically, gradually approaching 1.0. The parameters corresponding to the minimum sum of squared errors can be obtained as
Z ^ = H T ( k ) μ H ( k ) 1 H ( k ) μ Δ S ( k )
The forgetting factor allows the algorithm to emphasize recent data, ensuring that the parameter estimates reflect current operating conditions, which is crucial under non-stationary or fault conditions in the transformer. Let the variance of the weighted least squares estimation error be:
U ar = I ( Z Z ^ ) ( Z Z ^ ) T = ( H T μ H ) 1 H T μ U μ H ( H T μ H ) 1
This expression quantifies the uncertainty in the parameter estimation process, providing insight into the reliability of the identified parameters under measurement noise and system fluctuations. We only need to find the U that minimizes Equation (16). By introducing the matrix-type Schwarz inequality and solving the inequality, we can obtain:
μ = U 1
Physically, achieving the minimum variance corresponds to obtaining the most accurate and stable parameter estimates possible from the available data, enhancing the robustness of the identification process. Equation (17) indicates that when the variance in the estimation error reaches its minimum value, the weighted sum of squared errors constructed by the forgetting factor μ in the RLS algorithm reaches the theoretical lower bound, thereby ensuring that the identified parameters possess optimal estimation accuracy. This conclusion provides a mathematical basis for the selection of the forgetting factor μ and theoretically supports the convergence and effectiveness of the improved algorithm. The convergence behavior and stability of the proposed RLS algorithm with a forgetting factor are also considered. The forgetting factor μ plays a crucial role in balancing convergence speed and stability. A value of μ close to 1.0 ensures algorithmic stability by preventing excessive weight reduction on historical data, whereas slightly lower values (typically in the range of 0.95–0.99) allow faster adaptation to time-varying parameters by emphasizing new measurements. In this study, the forgetting factor was empirically tuned based on system dynamics, and the selected value achieved stable and accurate parameter tracking in all test cases. Although adaptive tuning strategies (e.g., based on residual analysis or prediction error monitoring) were not implemented, they represent a promising direction for future work to further improve robustness under non-stationary operating conditions.
In summary, the introduction of the forgetting factor not only effectively avoids the performance degradation issue of traditional RLS algorithms caused by historical data saturation during long-term operation but also enhances the response capability to slow parameter evolution and sudden disturbances. This method is particularly suitable for scenarios where the winding parameters of a DZT exhibit dynamic changes during actual operation, enabling continuous error correction and real-time parameter updates. In the next section, simulation experiments will be conducted based on the constructed power flow equations and the improved RLS algorithm to further validate its identification accuracy and robustness.

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 R h + R l and X h + X l are not significantly different from their actual values, with an error of approximately 1.3%. According to Equation (8), r k and x k can be used to replace the resistance and reactance parameters of the high- and low-voltage windings, while g m and b m 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 r k , x k , g m , and b m 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, r k and x k 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 r k , x k , g m , and b m 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 X l , 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 X l . 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.

Author Contributions

Conceptualization, methodology, and writing—original draft preparation, Y.Z.; validation, supervision, and investigatison, P.J.; formal analysis, simulation modeling, and visualization, W.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The Simulink simulation models and parameter identification code used in this study are available from the corresponding author upon reasonable request. All relevant measurement data and simulation results supporting the findings are presented within the manuscript figures and tables.

Conflicts of Interest

Author Panrun Jin, Wenqin Song were employed by the Economic and Technical Research Institute of Gansu Electric Power Company. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Appendix A. Input Data Used in Simulations and Field Analysis

ItemDescription
Transformer model SCB10-500/10, 500 kVA, 10 kV/0.4 kV, Yyn0, short-circuit impedance 4.26%
Nameplate parametersNo-load loss: 1.16 kW; no-load current: 1%; load loss: 4.91 kW; short-circuit impedance: 4.26%
Load model (simulation)Mixed loads: residential, commercial, photovoltaic (PV), energy storage, variable-speed AC motors; modeled using Simulink three-phase dynamic load modules
Simulation time10 s per scenario; sampling interval: 1 ms
Fault simulation 5% reactance reduction on low-voltage phase B to simulate winding deformation; random load variations across phases
Field data collection windowPuyang Power Supply Company; two periods: 1:00–8:00 and 20:00–24:00; data collection interval: 60 s
Measurement accuracy (field)Voltage, current: 0.1%; phase error: 0.1%; synchronization error within 10 ms
Collected measurement variablesPositive-sequence voltage, current, active power, reactive power (high- and low-voltage sides)

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Figure 1. The T-equivalent circuit of a DZT.
Figure 1. The T-equivalent circuit of a DZT.
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Figure 2. Transformer simulation measurement model.
Figure 2. Transformer simulation measurement model.
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Figure 3. Parameter identification model of the DZT.
Figure 3. Parameter identification model of the DZT.
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Figure 4. Active and reactive power loss errors of the DZT under symmetrical operation.
Figure 4. Active and reactive power loss errors of the DZT under symmetrical operation.
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Figure 5. Negative-sequence active and reactive power loss of the DZT.
Figure 5. Negative-sequence active and reactive power loss of the DZT.
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Figure 6. Positive-sequence active and reactive power loss error of the DZT under sudden parameter changes.
Figure 6. Positive-sequence active and reactive power loss error of the DZT under sudden parameter changes.
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Figure 7. Positive-sequence active and reactive power loss error of the DZT under continuous parameter changes.
Figure 7. Positive-sequence active and reactive power loss error of the DZT under continuous parameter changes.
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Figure 8. Field data sources.
Figure 8. Field data sources.
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Figure 9. Schematic diagram of the measurement system and data flow.
Figure 9. Schematic diagram of the measurement system and data flow.
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Figure 10. High-voltage side current and voltage.
Figure 10. High-voltage side current and voltage.
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Figure 11. Active and reactive power on the high-voltage side.
Figure 11. Active and reactive power on the high-voltage side.
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Figure 12. Active and reactive loss errors in the solution process.
Figure 12. Active and reactive loss errors in the solution process.
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Figure 13. Current and voltage on the high-voltage side of the transformer at different times.
Figure 13. Current and voltage on the high-voltage side of the transformer at different times.
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Table 1. Influence of measurement error on numerical solution.
Table 1. Influence of measurement error on numerical solution.
ParametersResults (Ω)ParametersResults (Ω)
R h 1.08 R l 1.414 × 10−3
X h 4.27 X l 5.587 × 10−3
R m 86,204.25 X m 20,560.7
Note: Rh, Xh: high-voltage winding resistance and reactance; Rl, Xl: low-voltage winding resistance and reactance; Rm: excitation branch resistance and reactance. “Results” denotes the calculated impedance values in ohms (Ω).
Table 2. Influence of measurement error on parameter identification.
Table 2. Influence of measurement error on parameter identification.
ParametersResults (Ω)ParametersResults (Ω)
R h 0.203 R l 0.891
X h 0.740 X l 3.586
R m 84,419.58 X m 20,536.27
Note: Rh, Xh: high-voltage winding resistance and reactance; Rl, Xl: low-voltage winding resistance and reactance; Rm, Xm: excitation branch resistance and reactance.
Table 3. Influence of measurement errors on the equivalent parameter identification method.
Table 3. Influence of measurement errors on the equivalent parameter identification method.
ParametersResults (Ω)ParametersResults (S)
r k 1.082 g m 1.16 × 10−5
x k 4.274 b m 4.86 × 10−5
Note: rk, xk: equivalent short-circuit resistance and reactance; gm, bm: equivalent magnetizing conductance and susceptance.
Table 4. Parameter identification results under asymmetric loads.
Table 4. Parameter identification results under asymmetric loads.
ParametersPhase APhase BPhase C
r k (Ω)1.6371.1580.637
x k (Ω)3.8614.7124.197
g m (S)1.06 × 10−51.11 × 10−51.28 × 10−5
b m (S)4.74 × 10−55.00 × 10−54.83 × 10−5
Table 5. Influence of winding deformation on parameter identification results.
Table 5. Influence of winding deformation on parameter identification results.
ParametersNormal ValueResultsChange
r k (Ω)1.0811.0810%
x k (Ω)4.2744.2400.8%
g m (S)1.16 × 10−51.16 × 10−50%
b m (S)4.86 × 10−54.86 × 10−50%
Table 6. Parameter identification using negative-sequence measurement components.
Table 6. Parameter identification using negative-sequence measurement components.
ParametersNormal ValueIdentification ResultsError
r k (Ω)1.0811.0811%
x k (Ω)4.2744.2830.2%
g m (S)1.16 × 10−51.01 × 10−5−13%
b m (S)4.86 × 10−59.72 × 10−6−80%
Table 7. Identification results under sudden parameter changes.
Table 7. Identification results under sudden parameter changes.
ParametersNormal ValueIdentification ResultsError
r k (Ω)1.0811.2016.5%
x k (Ω)4.2744.22−4.1%
g m (S)1.16 × 10−53.87 × 10−5128%
b m (S)4.86 × 10−58.65 × 10−527.2%
Table 8. Identification results under continuous parameter changes.
Table 8. Identification results under continuous parameter changes.
ParametersNormal ValueResultsChange
r k (Ω)1.0811.0830.2%
x k (Ω)4.2744.2740%
g m (S)1.16 × 10−51.18 × 10−51.5%
b m (S)4.86 × 10−54.86 × 10−50%
Table 9. Comparative performance of parameter identification methods.
Table 9. Comparative performance of parameter identification methods.
MethodParameter Estimation
Error (%) Value
Computational Time (s)Robustness Under 0.1% Noise (Error%)
Proposed ARLS0.91.21.1
LS [12]2.30.74.8
GA [17]1.415.01.5
Table 10. Distribution transformer parameter identification results.
Table 10. Distribution transformer parameter identification results.
ParameterIdentification Results
rk0.463 (Ω)
xk9.357 (Ω)
gm1.72 × 10−5 (S)
bm5.67 × 10−1 (S)
Table 11. Distribution transformer parameter identification results at different time periods.
Table 11. Distribution transformer parameter identification results at different time periods.
ParameterIdentification Results
rk0.459 (Ω)
xk9.328 (Ω)
gm1.75 × 10−5 (S)
bm5.58 × 10−5 (S)
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Jin, P.; Song, W.; Zhang, Y. Parameter Identification of Distribution Zone Transformers Under Three-Phase Asymmetric Conditions. Eng 2025, 6, 181. https://doi.org/10.3390/eng6080181

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Jin P, Song W, Zhang Y. Parameter Identification of Distribution Zone Transformers Under Three-Phase Asymmetric Conditions. Eng. 2025; 6(8):181. https://doi.org/10.3390/eng6080181

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Jin, Panrun, Wenqin Song, and Yankui Zhang. 2025. "Parameter Identification of Distribution Zone Transformers Under Three-Phase Asymmetric Conditions" Eng 6, no. 8: 181. https://doi.org/10.3390/eng6080181

APA Style

Jin, P., Song, W., & Zhang, Y. (2025). Parameter Identification of Distribution Zone Transformers Under Three-Phase Asymmetric Conditions. Eng, 6(8), 181. https://doi.org/10.3390/eng6080181

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