Identification of Transformer Parameters Using Dandelion Algorithm

Abstract

Researchers tackled the challenge of finding the right parameters for a transformer-equivalent circuit. They achieved this by minimizing the difference between actual measurements (currents, powers, secondary voltage) during a transformer load test and the values predicted by the model using different parameter settings. This process considers limitations on what values the parameters can have. This research introduces the application of a new and effective optimization algorithm called the dandelion algorithm (DA) to determine these transformer parameters. Information from real-time tests (single- and three-phase transformers) is fed into a computer program that uses the DA to find the best parameters by minimizing the aforementioned difference. Tests confirm that the DA is a reliable and accurate tool for estimating the transformer parameters. It achieves excellent performance and stability in finding the optimal values that precisely reflect how a transformer behaves. The DA achieved a significantly lower best fitness function value of 0.0136101 for the three-phase transformer case, while for the single-phase case it reached 0.601764. This indicates a substantially improved match between estimated and measured electrical parameters for the three-phase transformer model. By comparing DA with six competitive algorithms to prove how well each method minimized the difference between measurements and predictions, it could be shown that the DA outperforms these other techniques.

1. Introduction

In recent years, access to reliable electrical power has become a fundamental right for humanity, underpinning both technological and social development. To meet the ever-growing demand at load centers, a complex infrastructure has been established for generating, transforming, transporting, distributing, and selling electricity. Among the most crucial devices in this chain, and in the electrical sector as a whole, are transformers. These versatile devices play a critical role in interconnecting power generation plants with transmission networks. They also serve as the link between transmission and sub-transmission networks, ultimately delivering power to residential, industrial, and commercial end users. This is achieved by adjusting the voltage level (high, medium, or low) at each stage of the transmission and distribution process. Transformers can highly efficiently transmit energy via transmission lines depending on parameters and losses [1].

Several studies have been conducted to determine transformer-equivalent circuit parameters in order to reduce losses, increase performance, and lower operational costs. The frequency dependence of unknown transformer characteristics complicates the accuracy of transformer modeling [2]. Also, the accuracy of transformer parameter estimation can be significantly impacted by factors like harmonics in the voltage waveform, saturation of the transformer core, and transient events on the power grid. Optimal transformer design requires accurate parameter estimates to meet standards and specifications. Hence, precise estimation of transformer parameters was achieved by employing either frequency response [3] or time domain analysis through real-time measurements [4]. To account for the saturation of the transformer core, inrush current measurements [5] were utilized during the parameter estimation process. Moreover, short-circuit and open-circuit tests have been the cornerstone methods for determining transformer parameters. However, these are limited to laboratory settings and require disconnecting transformers from the power grid. This disconnection disrupts service to multiple users in distribution systems, negatively impacting reliability indices [6]. Additionally, the sheer number of transformers in a distribution system makes this approach economically impractical. Traditionally, analytical methods based on finite-element analysis have been the primary tools for rapid transformer physical sizing. However, these methods can struggle with increasingly complex transformer geometries and non-linear material properties. To address these limitations, researchers have begun exploring the application of non-conventional techniques, including evolutionary computation algorithms [7,8]. Optimization algorithms have become a cornerstone of tackling various electrical engineering challenges. From optimizing the design of electric machines, transformers, and transmission lines to enhancing the performance of fuel cells, photovoltaic modules, power system stabilizers, etc., these techniques offer powerful solutions [9,10,11,12]. In the context of parameter estimation, optimization techniques are employed to establish a correspondence between observed data and their predicted counterparts. This correspondence is quantified by a cost function, which the optimization algorithm aims to minimize. The minimization process effectively reduces the discrepancy between the observed and estimated values. As reported in the literature, there are many optimization algorithms employed in the parameter estimation of transformers, such as evolutionary programming [13], the coyote optimization algorithm [14], the jellyfish optimizer algorithm [15], and the chaotic optimization approach [16]. The aforementioned optimization algorithms possess the advantageous characteristic of being implementable using either load data or nameplate data acquired from the transformer while it remains operational. This eliminates the need for service disruptions associated with traditional parameter estimation methods.

Table 1. Recent algorithms in parameter estimation of transformers.

Ref.	Optimization Algorithm	Transformer Type	Main Outcomes/Shortcomings
[7]	Artificial hummingbird optimizer	Single-phase Three-phase	Emphasizing accuracy under various loading conditions
[14]	Coyote optimization algorithm	Single-phase Three-phase	The greatest concordance with experimentally measured values
[15]	Jellyfish search optimizer	Single-phase	Fast convergence
[16]	Chaotic optimization approach	Single-phase	No statistical analysis
[17]	Artificial bee colony	Single-phase	Using various scenarios for validation There is no comparison with competitive algorithms
[18]	Slime mold optimizer	Single-phase Three-phase	Fast convergence and accuracy
[19]	Improved Tasmanian devil	Single-phase	Improved algorithm Robustness and accuracy
[20]	Generalized normal distribution optimizer	Single-phase	Least computational efforts and fast convergence
[21]	Hurricane optimization algorithm	Single-phase	Validation through three different sizes of transformers

introduces recent algorithms in parameter estimation of transformers and their main outcomes.

While metaheuristics offer promising results for transformer modeling, their effectiveness hinges on two key factors: a thorough exploration of the model’s complex search landscape and fine-tuning of algorithm parameters like population size and iteration count. Furthermore, specific control settings in some algorithms significantly impact performance. Inappropriate settings can lead to increased computation time, reduced accuracy, and a higher risk of getting stuck in suboptimal solutions (local optima). This highlights the need for improvement in existing methods or the development of new variants, acknowledging the “no free lunch” theorem [22], which states that there is no single best approach for all optimization problems. This has motivated researchers to improve existing methods or create entirely new ones, aiming for strong performance across diverse and challenging optimization problems. Therefore the dandelion algorithm (DA) [23] is introduced in this study.

Encouraged by its success in solving engineering problems like maximum power point tracking of photovoltaic systems [24,25], allocation of energy storage devices [26], parameters extraction of energy storage devices [27], controller tuning in various applications [28,29,30,31], and economic dispatch [32], even in optimal design of steel frames [33], the authors explore the application of the DA in identifying optimal parameters for electrical transformers, ultimately aiming to ensure reliable power system operation. This approach has proven effective in finding near-optimal solutions for various real-world engineering challenges. The key to its success lies in its ability to balance exploration (searching for new possibilities) and exploitation (refining promising solutions), leading to competitive performance compared to other algorithms.

The key contributions of this research can be enumerated in the following points:

i.

This work presents the application of the DA for estimating the parameters of transformers.

ii.

To validate the effectiveness of the proposed method, experiments are conducted on both single-phase and three-phase transformers.

iii.

The performance of the DA is evaluated by comparing its results with those obtained from other optimization techniques.

The remaining sections of this research are organized as follows: Section 2 focuses on the transformer mathematical modeling and the employed fitness function while Section 3 introduces the DA formulation. In Section 4, the authors delve into a specific case study, showcasing the results obtained through simulation. The subsequent Section 5 then provides concluding remarks based on these findings.

2. Problem Mathematical Formulation

The per-phase transformer’s steady-state equivalent circuit referred to its primary side is presented in Figure 1 for subsequent analysis. By applying Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL) to the equivalent circuit of the transformer depicted in Figure 1, the following relationships are derived [18]:

$$\underset{\_}{E_1}=\underset{\_}{V_1}-\underset{\_}{Z_1} \underset{\_}{I_1}$$

(1)

$$\underset{\_}{E_1}=\underset{\_}{V_2^{\prime }}+\underset{\_}{Z_2^{\prime }} \underset{\_}{I_2^{\prime }}$$

(2)

$$\underset{\_}{V_2^{\prime }}=\underset{\_}{Z_{Load}^{\prime }} \underset{\_}{I_2^{\prime }}$$

(3)

$$\underset{\_}{E_1}=\underset{\_}{I_o} \underset{\_}{Z_m}$$

(4)

$$\underset{\_}{I_1}={\displaystyle \frac{{\underset{\_}{V}}_1}{{\underset{\_}{Z}}_{eq}}}=\underset{\_}{I_o}+\underset{\_}{I_2^{\prime }}$$

(5)

$$\underset{\_}{I_o}={\displaystyle \frac{{\underset{\_}{E}}_1}{{\underset{\_}{Z}}_m}}={\displaystyle \frac{{\underset{\_}{E}}_1}{R_c}}+{\displaystyle \frac{{\underset{\_}{E}}_1}{j{X}_m}}={ I}_{e+h}-j{I}_m$$

(6)

$$\underset{\_}{I_2^{\prime }}=\underset{\_}{I_1}{\displaystyle \frac{{\underset{\_}{Z}}_m}{{\underset{\_}{Z}}_m+{\underset{\_}{Z}}_2^{\prime }+{\underset{\_}{Z}}_{Load}^{\prime }}}$$

(7)

where $\underset{\_}{Z_1}$, $\underset{\_}{I_1}$, and $\underset{\_}{E_1}$ represent the primary side impedance, current, and induced voltage, respectively. Also, $\underset{\_}{V_2^{\prime }}, \underset{\_}{Z_2^{\prime }}, \underset{\_}{I_2^{\prime }}, \mathrm{a}\mathrm{n}\mathrm{d} \underset{\_}{Z_{Load}^{\prime }}$ represent the secondary side voltage, impedance, current, and load impedance referred to the transformer’s primary side correspondingly. The equivalent impedance is symbolized by ${\underset{\_}{Z}}_{eq}$. The total excitation current of the transformer $\underset{\_}{I_o}$ can be broken down into two components: $I_{e+h}$, the current representing core losses, and $I_m$, the magnetizing current, while the core impedance is defined by $\underset{\_}{Z_m}$. The input power $P_{in}$, output power $P_{out}$, and efficiency $\eta$ can be calculated using the following equations:

$$P_{in}=\mathfrak{R}\left(V_1{I_1}^{*}\right)$$

(8)

$$P_{out}=\mathfrak{R}\left(V_2^{\prime }{I_2^{\prime }}^{*}\right)$$

(9)

$$\eta ={\displaystyle \frac{P_{out}}{P_{in}}}$$

(10)

Traditionally, transformer parameter estimation relies on experimental methods like short-circuit and open-circuit tests. This research, however, proposes an alternative approach that estimates parameters directly from terminal current and voltage readings. To achieve this, the sum of squared relative differences (SSRD) between the measured powers, terminal voltages, and currents and the corresponding values calculated from the single-phase equivalent circuit model shown in Figure 1 is minimized. This minimization process defines the objective function ($O.F$) presented in (11) complemented by (12–16).

$$O.F=\mathit{min}\left(SSRD\right)=\underset{ }{\mathit{min}}\left\{f_1+f_2+f_3+f_4+f_5\right\}$$

(11)

$${f_1=\sum _{i=1}^N\left({\displaystyle \frac{I_{1-e}\left(i\right)-I_{1-m}\left(i\right)}{I_{1-m}\left(i\right)}}\right)}^2$$

(12)

$$f_2=\sum _{i=1}^N{\left({\displaystyle \frac{I_{2-e}\left(i\right)-I_{2-m}\left(i\right)}{I_{2-m}\left(i\right)}}\right)}^2$$

(13)

$${f_3=\sum _{i=1}^N\left({\displaystyle \frac{P_{1-e}\left(i\right)-P_{1-m}\left(i\right)}{P_{1-m}\left(i\right)}}\right)}^2$$

(14)

$$f_4=\sum _{i=1}^N{\left({\displaystyle \frac{P_{2-e}\left(i\right)-P_{2-m}\left(i\right)}{P_{2-m}\left(i\right)}}\right)}^2$$

(15)

$${f_5=\sum _{i=1}^N\left({\displaystyle \frac{V_{2-e}\left(i\right)-V_{2-m}\left(i\right)}{V_{2-m}\left(i\right)}}\right)}^2$$

(16)

Equation (11) must satisfy the boundary constraints imposed on the control variables of the problem.

3. Dandelion Algorithm

In 2022, Shijie Zhao introduced the dandelion algorithm (DA) [23], a nature-inspired algorithm that mimics how dandelions spread their seeds. Dandelions rely on wind to carry their seeds to new locations. Here is how the DA algorithm reflects this process:

Rising Phase: Similar to wind currents lifting dandelion seeds, the algorithm explores the search space for potential solutions. Favorable conditions (like sunshine) might lead to a broader search.

Descending Phase: Once a certain level of exploration is achieved, the algorithm focuses on refining promising areas, mimicking seeds descending towards the ground.

Landing Phase: Finally, just as seeds settle in various locations, the algorithm allows solutions to converge towards optimal values based on wind and weather (representing the optimization problem).

Dandelions’ population evolves through three rounds of seed dispersal to the potential solutions population as follows:

During the initialization phase, within the DA, each dandelion seed represents a candidate solution in the search space. The population of solutions within a DO instance can be mathematically expressed as:

$$Pop.=\left[\begin{array}{ccc}{x}_1^1& \cdots & x_1^{dim}\\ ⋮& \ddots & ⋮\\ x_{N_{Pop}}^1& \cdots & x_{N_{Pop}}^{dim}\end{array}\right]$$

(17)

In the context provided, $N_{Pop}$ represents the population size while dim represents the dimension of the variable. Within the designated range of the given problem, encompassing the upper limit (${Var}_{Max}$) and lower limit (${Var}_{Min}$), every conceivable solution is generated in a random manner. It is worth noting that the individual $X_i$, denoting the ith individual, can be mathematically represented by (18). The variable $i$ is an integer ranging from 1 to $N_{Pop}$, while $rand$ denotes a random number within the interval of 0 to 1.

$$X_i=rand\times \left({Var}_{Max}-{Var}_{Min}\right)+{Var}_{Min}$$

(18)

$${Var}_{Max}=\left[{{Var}_{Max}}_1 , {{Var}_{Max}}_2,\cdots ,{{Var}_{Max}}_{dim}\right]$$

(19)

$${Var}_{Min}=\left[{{Var}_{Min}}_1 , {{Var}_{Min}}_2,\cdots ,{{Var}_{Min}}_{dim}\right]$$

(20)

In the DA, the primary elite is the individual possessing the highest fitness value, denoted as the optimal position for the dandelion seed to thrive. The mathematical representation of the primary elite, utilizing the minimum value as an example, is:

$$f_{Opt.}=\underset{ }{\min }\left(f\left(X_i\right)\right)$$

(21)

$$X_{elite}=X\left(find\left(f_{Opt.}==f\left(X_i\right)\right)\right)$$

(22)

The function $find\left(\right)$ denotes two indices that possess identical values.

Round of Raising: To achieve dispersal from their parent plants, dandelion seeds must attain a precise elevation during the ascending phase. The ascent of dandelion seeds to diverse altitudes is contingent upon factors such as air humidity, wind speed, and other variables. In this particular case, the two prevailing weather conditions are as delineated below.

• Case (1)

Wind velocities during a cloudless day may be conceptualized as conforming to a lognormal distribution, with $\mathit{ln}Y~N\left(\mu ,{\sigma }^2\right)$ as given in (23), where $\mu$ and $\sigma$ represent the mean value and standard deviation, respectively. The variable $y$ is of a random nature drawn from the standard normal distribution with a mean of zero and a standard deviation of one. The velocity of the wind impacts the altitude to which a dandelion seed will ascend. In the event of stronger winds, the dandelion is propelled to greater heights, resulting in the seeds dispersing over a wider distance.

$$\mathit{ln}Y=\left\{\begin{array}{c}\begin{array}{cc}{\displaystyle \frac{1}{y\sqrt{2\pi }}}{e}^{\left[-{\displaystyle \frac{1}{2{\sigma }^2}}{\left(\mathit{ln}y\right)}^2\right]}& 0\le y\\ 0& 0>y\end{array}\\ \end{array}\right.$$

(23)

$$X_{t+1}=X_t+\alpha \times {\upsilon }_x\times {\upsilon }_y\times \mathit{ln}Y\times \left(X_s-X_t\right)$$

(24)

Within each iteration (denoted by $t$), the dandelion seed position at that iteration is denoted by $X_t$. This position is influenced by a randomly selected position $X_s$, which is generated during the same iteration and can be expressed by (25). Also, the variable $\alpha$ represents a random perturbation uniformly distributed within the closed interval zero and can be calculated using (26).

$$X_s=rand(1,dim)\times \left({Var}_{Max}-{Var}_{Min}\right)+{Var}_{Min}$$

(25)

$$\alpha =rand\left( \right)\times \left({\displaystyle \frac{1}{T^2}}{t}^2-{\displaystyle \frac{2}{T}}t+1\right)$$

(26)

The coefficients for the lift component of the dandelion ${\upsilon }_x$ and ${\upsilon }_y$ can be formulated using (27), while the variable $\theta$ is subject to random variation within the range of $-\pi$ to $\pi$.

$$\begin{gathered} r={\displaystyle \frac{1}{e^{\theta }}} \\ {\upsilon }_x=r\cos \theta \\ {\upsilon }_y=r\sin \theta \end{gathered}$$

(27)

• Case (2)

The presence of moisture and air resistance hinders the ability of dandelion seeds to effectively disperse with the wind during humid conditions. Dandelion seeds primarily explore their immediate surroundings during dispersal. This localized search behavior can be mathematically represented by (28). The parameter $k$ is employed to control the size of the local search area for a dandelion, while (29) is utilized to compute the extent of the domain.

$$\begin{gathered} X_{t+1}=X_t\times k \\ k=1-rand\left( \right)\ast q \end{gathered}$$

(28)

$$q={\displaystyle \frac{1}{T^2-2T+1}}{t}^2-{\displaystyle \frac{1}{T^2-2T+1}}t+1+{\displaystyle \frac{1}{T^2-2T+1}}$$

(29)

Ultimately, the mathematical representation of dandelion seeds during the round of raising is:

$$X_{t+1}=\left\{\begin{array}{c}\begin{array}{cc}{X}_t+\alpha \times {\upsilon }_x\times {\upsilon }_y\times \ln Y\times \left(X_s-X_t\right)& randn<1.5\\ X_t\times k& else\end{array}\end{array}\right.$$

(30)

Round of Descending: During this round, the dandelion seeds ascend to a specific altitude before gradually descending (exploration stage). Brownian motion is utilized in dispersal optimization to simulate the path of a dandelion seed as it travels. This is mathematically formulated by (31), with the Brownian motion specified by ${\beta }_t$.

$$X_{t+1}=X_t-\alpha \times {\beta }_t\times \left(X_{mean\_t}-\alpha \times {\beta }_t\times X_t\right)$$

(31)

The average position of the entire population of dandelion seeds at iteration $t$ is denoted by $X_{mean\_t}$. This population mean can be mathematically expressed as:

$$X_{mean\_t}={\displaystyle \frac{1}{N_{Pop}}}\sum _{i=1}^{N_{Pop}}{X}_i$$

(32)

Round of Landing: This is the final stage of the DA, in which the focus shifts towards exploitation. Leveraging the information gleaned from the preceding stages (rising and local search), each dandelion seed randomly selects its landing location. As the number of iterations increases, the population is expected to converge towards the optimal solution within the search space. This convergence process ultimately leads to the identification of the following global optimal solution. This behavior is expressed by (33), where each dandelion seed aspires to achieve its own optimal position, denoted by $X_{elite}$.

$$X_{t+1}=X_{elite}+levy\left(\lambda \right)\times \alpha \times \left(X_{elite}-X_t\times \delta \right)$$

(33)

The function denoted by $levy\left(\lambda \right)$ represents the Lévy flight utilized within the algorithm. The specific mathematical formulation for this function is provided in (34).

$$levy\left(\lambda \right)=s\times {\displaystyle \frac{w\times \sigma }{{\left|t\right|}^{\frac{1}{\beta }}}}$$

(34)

The variable $\beta$ is a stochastic variable sampled from a uniform distribution between 0 and 2. Also, the variable $s$ is a constant value set at 0.01. Both variables $w$ and $t$ are also stochastic variables drawn from uniform distributions ranging from 0 to 1. The mathematical representation of the standard deviation $\sigma$ can be expressed as follows.

$$\sigma =\left({\displaystyle \frac{\Gamma \left(1+\beta \right)\times \mathit{sin}\left(\frac{\pi \beta }{2}\right)}{\Gamma \left(\frac{1+\beta }{2}\right)\times \mathit{sin}\left(\frac{\beta -1}{2}\right)}}\right)$$

(35)

where $\delta$ is a function that exhibits linear growth within the interval [0, 2] and can be determined using the following equation:

$$\delta ={\displaystyle \frac{2t}{T}}$$

(36)

The flowchart presented in Figure 2 offers a visual representation of the DA algorithm’s steps.

The DA algorithm’s effectiveness is assessed by comparing estimated values to measured data. This assessment procedure uses the fitness function specified in Equation (11) and runs until the maximum number of iterations is achieved. Figure 3 represents an overview of the methods involved in parameters.

4. Case Studies: Results and Discussion

To confirm the effectiveness and capability of the DA to identify the transformer-equivalent circuit parameters, two test cases were utilized in the experiments: a single-phase transformer with a rating of 300 VA and a voltage ratio of ${\displaystyle \frac{230}{2\times 115}}$ V, as well as a three-phase transformer with a rating of 300 VA and a voltage ratio of ${\displaystyle \frac{400}{2\times 200}}$ V. To determine the actual parameters of the equivalent circuit, a load test was conducted and reported in [18], and will be utilized in this research. The obtained measurements were used to calculate the SSRD as the objective function to be minimized by the DA for estimating the parameters of the transformers. These tests provide valuable insights into the performance and characteristics of the transformers. The effectiveness of the DA is evaluated by comparing its results to those obtained from previously established algorithms: the slime mold optimizer (SMOA) [18], atom search optimization (ASO) [34], interior search algorithm (ISA) [35], sunflower optimization (SFO) [36], sinh cosh optimizer (SCHO) [37], and hippopotamus optimizer (HO) [38]. In order to ensure a fair comparison, all optimization algorithms were run with a uniform population size of 30 and a maximum of 50 iterations. The following subsections illustrate the application of the DA to extract the equivalent circuit parameters of both single-phase and three-phase transformers.

4.1. Single-Phase Transformer: Parameters Identification

The DA utilized the recorded load test measurements to identify the equivalent circuit parameters of a single-phase transformer with a rating of 300 VA and a voltage ratio of ${\displaystyle \frac{230}{2\times 115}}$ V. The optimization process involved minimizing a pre-defined fitness function given in (11) that quantified the difference between the predicted and measured electrical quantities (e.g., currents, powers, secondary voltage) during load testing. After multiple iterations of the DA, the convergence curve depicted in Figure 4 illustrates the achievement of a best-fitness function value of 0.601764. This value indicates a close match between the estimated and measured electrical characteristics of the transformer. Furthermore, as shown in Figure 5, the DA algorithm exhibited notable improved performance compared to other optimization algorithms considered for this task, demonstrating its effectiveness in this specific application.

The estimated equivalent circuit parameters of the transformer obtained using the DA are presented in

Table 2. The successfully identified optimal parameters for the single-phase transformer using DA.

${\mathbf{R}}_1$ (Ω)	${\mathbf{R}}_2^{\mathbf{\prime }}$ (Ω)	${\mathbf{X}}_1$ (Ω)	${\mathbf{X}}_2^{\mathbf{\prime }}$ (Ω)	${\mathbf{R}}_{\mathbf{c}}$ (Ω)	${\mathbf{X}}_{\mathbf{m}}$ (Ω)
3.0000	0.750	0.0375	0.0086	4000	1453

. These parameters (e.g., resistances, reactance) represent the electrical characteristics of the transformer’s windings and core losses. To assess the accuracy of the identified parameters, a validation step was conducted. The identified parameters using the DA were used to predict the primary and secondary voltages, currents, and power values under various load conditions. These predicted values were then compared to the actual measurements obtained during load testing, as shown in Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10. Although slight discrepancies were observed between the estimated and measured quantities, this comparison serves as a crucial validation of the effectiveness of the DA in accurately identifying the equivalent circuit parameters of the transformer.

4.2. Three-Phase Transformer: Parameters Identification

Building upon its success with the single-phase transformer, the DA was also employed to estimate the equivalent circuit parameters of a three-phase transformer. This three-phase transformer has a rated power of 300 VA and a voltage ratio of ${\displaystyle \frac{400}{2\times 200}}$ V. As with the single-phase case, the DA minimizes a pre-defined fitness function that quantifies the discrepancy between the predicted and measured electrical quantities during load testing.

After multiple iterations, the convergence curve in Figure 11 depicts the achievement of a best-fitness function value of 0.0136101. This significantly lower value compared to the single-phase case (0.601764) suggests a much closer match between the estimated and measured electrical characteristics for the three-phase transformer. Furthermore, as shown in Figure 12, the DA again demonstrated superior performance compared to other optimization algorithms like the SMOA and SFO. Notably, the relative percentage difference in fitness function values between the DA and SMOA is 84.5%, while it reaches 89.5% compared to SFO, highlighting the clear advantage of the DA in this application. The estimated equivalent circuit parameters obtained using the DA for the three-phase transformer are presented in

Table 3. The successfully identified optimal parameters for the three-phase transformer using DA.

${\mathbf{R}}_1$ (Ω)	${\mathbf{R}}_2^{\mathbf{\prime }}$ (Ω)	${\mathbf{X}}_1$ (Ω)	${\mathbf{X}}_2^{\mathbf{\prime }}$ (Ω)	${\mathbf{R}}_{\mathbf{c}}$ (Ω)	${\mathbf{X}}_{\mathbf{m}}$ (Ω)
17.8601	19.4189	4.1000	4.0017	13054	3246.2

. These parameters will again represent the electrical characteristics of the transformer’s windings and core losses but in a three-phase configuration.

Upon examination of the predicted primary and secondary voltages, currents, and powers against the actual measurements depicted in Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17, the accuracy of the estimated transformer parameters was confirmed. There is a relative coincidence between the estimated and measured values, and this comparison serves to validate the efficiency of the parameter identification process.

5. Conclusions

The estimation of unknown transformer-equivalent circuit parameters through load testing offers an attractive approach due to its relatively low data requirement compared to alternative methods. Furthermore, the application of optimization algorithms facilitates the minimization of discrepancies between the estimated and measured values extracted from the load test data. This paper proposes the dandelion algorithm (DA) as a means to address this challenge. The DA is presented as a precise, efficient, and reliable method for identifying optimal values of unknown transformer parameters. The core objective function employed minimizes the sum of squared relative differences (SSRDs) between computed and measured currents, powers, and secondary voltages during a transformer load test. The application of the DA for transformer parameter estimation has been thoroughly investigated in this study, revealing its remarkable speed and accuracy compared to well-established optimizers. The obtained results provide solid evidence supporting the efficiency and reliability of the proposed DA approach, emphasizing its benefits in terms of quicker convergence and improved precision. The optimization process yielded a superior fitness function value of 0.0136101 for the three-phase transformer model, demonstrating a considerably closer alignment between simulated and experimental electrical characteristics, while the single-phase counterpart was 0.601764. These results underscore the efficacy of the proposed approach in accurately modeling complex three-phase transformer systems. These significant findings strongly indicate that the DA holds great value as a tool for electrical engineers aiming to optimize various parameters.