# Optimal parameter extraction of equivalent circuits for single- and three- phase Power transformers based on arctic puffin algorithm accomplished with experimental verification

Amlak Abaza, Ragab A El-Sehiemy, Mohamed Assaad Hamida, Raphael Chenouard, Ahmed Bayomyi

► To cite this version:

Amlak Abaza, Ragab A El-Sehiemy, Mohamed Assaad Hamida, Raphael Chenouard, Ahmed Bayomyi. Optimal parameter extraction of equivalent circuits for single- and three- phase Power transformers based on arctic puffin algorithm accomplished with experimental verification. Results in engineering, 2025, 26, pp.104888. 10.1016/j.rineng.2025.104888. hal-05230329

HAL Id: hal-05230329

https://hal.science/hal-05230329v1

Submitted on 29 Aug 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Research paper

# Optimal parameter extraction of equivalent circuits for single- and three-phase Power transformers based on arctic puffin algorithm accomplished with experimental verification

Amlak Abaza<sup>a</sup>, Ragab A. El-Sehiemy<sup>a,b,c</sup>, Mohamed Assaad Hamida<sup>d,e,f</sup>, Raphael Chenouard<sup>d</sup>, Ahmed Bayomyi<sup>e,f</sup>

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

**Keywords:**

Parameter estimation

Equivalent circuit

Single and three phase transformers

Arctic puffin optimization algorithm

Robustness

## ABSTRACT

The power transformer is a critical device in power systems. This paper addresses one of the major problems which hopes to enhance the accuracy of estimation of parameters, which is critical in power transformer modeling, maintenance, and operating efficiency. In that context, this work estimates the parameters of single- and three-phase power transformers by a new optimizer called Arctic Puffin Optimization Algorithm (APO). The algorithm is intended to improve estimation of transformer parameters with the goal of reducing the error incurred between the estimated and actual values of the parameters. To verify the accuracy of the APO, experimental measurements were conducted on single- and three-phase transformers. The assessment of the algorithm's effectiveness was performed against the effectiveness of other commonly used estimating methods. The results have shown that APO increases the accuracy of estimation of the parameters of both single- and three-phase transformers to considerable levels. Dependability of the APO was established by experimental verification, which disclosed an ultimate connection between the resultant quantities and actual measurements. The study also confirmed APO can be useful for transformer parameter estimation because APO converges more rapidly and more precisely compared with traditional methods of the literature.

## 1. Introduction

The optimization of transformer parameters has been a major research topic in the electrical engineering fields today, with the main goal being the reliability and performance of the power systems [1–3]. The transformer is a key part of the electrical network, and it provides the means of distribution and transfer of electrical energy [4].

The measurement of transformer parameters with accuracy is a very critical task, and the overall performance of the power systems depends directly on their working performance [5]. Single-phase transformers and three-phase transformers are the two main categories of power transformers. Three-phase transformers are necessary for industrial and utility levels, and single-phase transformers are widely used in

residential and small business settings [6,7]. Different methods of estimation for each transformer's parameters are necessary due to their different working specifications and requirements [8–10]. Transformer performance and its efficiency depend mostly on its specifications, including resistance, reactance, and ratio of turns [11–13]. Transformer design, working, and maintenance depend on accurate calculation of these values.

The traditional methods of transformer parameter estimation, such as the least square method and Newton-Raphson technique [14–16], rely mostly on empirical measurement assumptions and therefore can be inaccurate and take a long time because they may not be able to deal with the inherent nonlinearity. The inherent measurement errors of measuring processes can also make transformer parameter estimation more complex.

<footnote><sup>a</sup> Electrical Engineering Department - Faculty of Engineering, Kafrelsheikh University, 33516, Egypt</footnote>

<footnote><sup>b</sup> The Academy of Scientific Research and Technology (ASRT), Cairo, Egypt</footnote>

<footnote><sup>c</sup> Széchenyi István University, Egyetem tér 1., 9026 Györ, Hungary</footnote>

<footnote><sup>d</sup> Nantes Université, Ecole Centrale de Nantes, LS2N, 44321 Nantes, France</footnote>

<footnote><sup>e</sup> Electrical Department, Faculty of Engineering and Technology, Badr University in Cairo (BUC), 11899 Badr, Egypt</footnote>

<footnote><sup>f</sup> Physics and Engineering Mathematics Department, Faculty of Engineering, Kafrelsheikh University, 33516, Egypt</footnote>

<footnote>* Corresponding author.
E-mail address: mohamed.hamida@ec-nantes.fr (M.A. Hamida).</footnote>
Nomenclature

<table><tr><td>R<sub>P</sub></td><td>Per phase primary resistance, Ω.</td></tr><tr><td>R<sub>S</sub></td><td>Per phase secondary resistance, Ω.</td></tr><tr><td>X<sub>P</sub></td><td>Per phase primary reactance, Ω.</td></tr><tr><td>X<sub>S</sub></td><td>Per phase secondary reactance, Ω.</td></tr><tr><td>X<sub>m</sub></td><td>Per phase magnetizing reactance, Ω.</td></tr><tr><td>R<sub>C</sub></td><td>Per phase core loss resistance, Ω.</td></tr><tr><td>V<sub>P</sub></td><td>Per phase primary voltage, V.</td></tr><tr><td>V<sub>S</sub></td><td>Per phase secondary voltage, V.</td></tr><tr><td>I<sub>S</sub></td><td>Per phase secondary current, A.</td></tr><tr><td>I<sub>P</sub></td><td>Per phase primary current, A.</td></tr><tr><td>I<sub>o</sub></td><td>No load current, A</td></tr><tr><td>I<sub>c</sub></td><td>No load current, active element, A</td></tr><tr><td>I<sub>m</sub></td><td>No load current, reactive element, A</td></tr><tr><td>P<sub>in</sub></td><td>Input power, W</td></tr><tr><td>P<sub>o</sub></td><td>Output power, W</td></tr><tr><td>PF</td><td>Rated power factor</td></tr></table>

<table><tr><td>η</td><td>Transformer efficiency</td></tr><tr><td>r<sub>1</sub>, r<sub>2</sub> and r<sub>3</sub> and β</td><td>Random variables between 0 and 1</td></tr><tr><td>V'<sub>s</sub></td><td>Per phase secondary voltage, V. All secondary voltage, current, resistance and reactance are referred to primary side</td></tr><tr><td>I'<sub>s</sub></td><td>Per phase secondary current, A. Per phase secondary current, A. All secondary voltage, current, resistance and reactance are referred to primary side</td></tr><tr><td>R'<sub>s</sub></td><td>Per phase secondary resistance, Ω. All secondary voltage, current, resistance and reactance are referred to primary side</td></tr><tr><td>X'<sub>s</sub></td><td>Per phase secondary reactance, Ω. All secondary voltage, current, resistance and reactance are referred to primary side</td></tr><tr><td>u<sub>b</sub> and l<sub>b</sub></td><td>Refer to the maximum and minimum limits of decision variables</td></tr><tr><td>min and max</td><td>Refer to the minimum and maximum operators</td></tr></table>

<figure><img src="image_3.png" /><figcaption>Fig. 1. Equivalent circuit of single-phase transformer; (a) with two actual sides; (b) with secondary side referred to primary side.</figcaption></figure>

The application of optimization algorithms offers a workable option that can increase the accuracy of the results and speed up the parameter estimation procedure in light of these challenges and the advancement of advanced computer techniques [17]. There are basis to build up metaheuristic optimization algorithms such as evolutionary-based [18], physics-based [19], swarm intelligence-based [20,21] and human behavior-based [22] which have been generally applied in different engineering fields, including power systems.

For instance, the quadratic interpolation optimizer (QIO) was employed to estimate PV parameters' models considering the electric field impacts [23], A new improved hybrid method, that combines grey wolf optimizer (GWO)- sine cosine algorithm (SCA), is used to evaluate the solar photovoltaic module equivalent circuit's optimal parameters including thin film [24] after diode elements' contributions to the equivalent circuits of thin-film solar cells have been determined using the current separation method [25], Newton-Raphson method (NRM) hybrid with metaheuristic algorithms is used to extract the parameters of multijunction solar cell [26], chaotic variant of Gorilla Troops Optimizer (GTO) was introduced to estimate the parameters of PV models [27], artificial rabbits' optimization algorithm ARO has been used to identify the proton exchange membrane fuel parameters [28], research has indicated that PSO can efficiently approximate transformer parameters by reducing the discrepancy between the computed and observed values [29], the Evolutionary Algorithms (EA) is another metaheuristic

algorithm that has gained traction in the field of power transformer parameter estimation [30]. In addition to the development and application of these algorithms, experimental verification plays a crucial role in validating the effectiveness of the proposed methods [31]. In [7], the Generalized Normal Distribution Optimizer (GNDO) was introduced to extract the parameters of single phase transformer from voltage and current measurements.

Additionally, the experimental confirmation not only strengthens the validity of the approach being proposed, but also enlightens the limitation and difficulty of every algorithm [32]. However, because of the "no free lunch" (NFL) theorem, where different algorithms perform differently when solving different kinds of optimization problems with different characteristics and where there exists no algorithm capable of performing best under every situation [33].

The Arctic puffin's extraordinary adaptability and efficiency in their food quest serves as the inspiration for the Arctic Puffin Optimization (APO) algorithm. This algorithm mimics nature and has a population approach for searching the solution space, much like methods utilized by puffins. Because of its capacity for achieving a trade-off between exploration and exploitation, APO finds its optimal use in multidimensional and nonlinear objective function optimization problems. The APO algorithm offers a way of achieving more precise parameter estimation with lower process expense for transformer parameter estimation. A growing tendency toward biomimicry—the use of computer intelligence to solve complicated engineering problems—is reflected in the incorporation of nature-inspired algorithms, such as the APO algorithm, into engineering processes [34].

The main contribution of this paper is to employ both the manufacture name-plate data and the experimental data to estimate an accurate model for transformers which has a great importance in power system analysis. The single phase and three phase transformers with different sizes are modeled accurately in this paper using the proposed APO algorithm. The proposed APO algorithm is used for estimating the parameters of single-phase and three-phase power transformers, followed by experimental verification of the estimated parameters. While there are currently many optimization algorithms available, the ongoing development of new metaheuristic algorithms is required. The paper major novelty issues are highlighted as follows:

* The proposed method estimates the transformer parameters based on experimental and name plates for single and three phase transformers.

* The proposed method is assessed considering four famous optimizers
<figure><img src="image_4.png" /><figcaption>Fig. 2. Flow chart of APO for transformer estimation problem.</figcaption></figure>

* The robustness of the proposed method is validated using statistical indices

The structure of the rest of this work involves the followings: Section 2 presents the mathematical formulation and the equivalent circuits of the considered transformers. Section 3 presents the proposed APO and its implementation for extracting the transformers’ parameters. In Section 4, the simulation results for 5 cases are taken into account. Section 5 concludes the paper findings and reports the future trends.

## 2. Mathematical formulation and equivalent circuits of transformers

The equivalent circuit of single-phase transformer at the steady state operation is shown in Fig. 1(a) and (b). Fig. 1(a) explains the actual equivalent circuit of a single-phase transformer with their actual primary and secondary parameters. Fig. 2(b) shows the equivalent circuit of the transformer referred to the primary side.

The primary side is represented by its series resistive and reactive elements, $R_p$ and $X_p$. These two elements represent the copper losses and the leakage reactance in the primary side. In addition, $R_s$ and $X_s$ represent the same behaviour in secondary side. The behaviour of the core of the transformer is represented by a magnetizing branch, $X_m$, and core

losses, $R_c$. The turns ratio, a, affects the values of voltages, currents, and impedances of both sides of the transformer. The relations of voltages and currents in each side, ($V_p$, $I_p$ at primary side and $V_s$, $I_s$ at secondary side) can be explained using Kirchhoff voltage and current laws. The primary current and voltages are expressed, as follows:

$$V_p = E_p + I_p Z_p \quad (1)$$

$$E_p = V'_s + I'_s Z'_s \quad (2)$$

$$E_p = I_o Z_m \quad (3)$$

$$I_p = I_o + I'_s \quad (4)$$

$$I_o = I_c - J I_m \quad (5)$$

where, $a = (V_p/V_s)$, $I'_s = (I_s/a)$, $V'_s = (V_s \times a)$, $Z'_s = (Z_s \times a^2)Z_p = R_p + JX_p$, $Z_m = (R_c \times JX_m / R_c + JX_m)$, $Z_s = R'_s + JX'_s$ where, $I_o$ is the no load current, which divides into active element, $I_c$ and reactive element, $I_m$. The referred elements of the secondary sides are $V'_s, I'_s$, and $Z'_s$.

The performance of the transformer, voltage regulation, VR, and efficiency, $\eta$ is calculated as follows:
<table><caption>**Table 1**
Five different cases of studies of single and three transformers.</caption><thead><tr><th>Transformer</th><th>Phases</th><th>Rating (kVA)</th><th>Voltages (V)</th><th>Frequency</th><th>Parameters based on</th></tr></thead><tbody><tr><td>Case 1</td><td>1-phase</td><td>&lt; 1</td><td>230/230</td><td>50</td><td rowspan="5">Open and Short circuit tests</td></tr><tr><td>Case 2</td><td>1-phase</td><td>2</td><td>230/115</td><td>50</td></tr><tr><td>Case 3</td><td>3-phase</td><td>4</td><td>380/380</td><td>50</td></tr><tr><td>Case 4</td><td>1-phase</td><td>10</td><td>500/125</td><td>50</td></tr><tr><td>Case 5</td><td>1-phase</td><td>15</td><td>2400/240</td><td>50</td></tr></tbody></table>

<table><caption>**Table 2**
Transformers boundaries.</caption><thead><tr><th rowspan="3">Parameters</th><th rowspan="3">Bounds</th><th colspan="6">Estimated parameters of different transformers</th></tr><tr><th colspan="3">Based on measurements (O.C.&amp; S.C.)</th><th colspan="3">Based on nameplate data</th></tr><tr><th>Case 1</th><th>Case 2</th><th>Case 3</th><th>Case 4</th><th>Case 5</th></tr></thead><tbody><tr><td>R<sub>p</sub> (Ω)</td><td>min</td><td>1.6</td><td>0.35</td><td>1.6</td><td>0.8</td><td>2.3</td></tr><tr><td></td><td>max</td><td>1.9</td><td>0.5</td><td>2</td><td>1</td><td>2.7</td></tr><tr><td>X<sub>p</sub> (Ω)</td><td>min</td><td>10.6</td><td>0.1</td><td>10.6</td><td>0.8</td><td>2.5</td></tr><tr><td></td><td>max</td><td>11</td><td>0.3</td><td>11</td><td>1</td><td>3.5</td></tr><tr><td>R<sub>s</sub> (Ω)</td><td>min</td><td>1.6</td><td>0.4</td><td>1.6</td><td>1.4</td><td>1.6</td></tr><tr><td></td><td>max</td><td>2</td><td>0.6</td><td>2</td><td>1.7</td><td>2.5</td></tr><tr><td>X<sub>s</sub> (Ω)</td><td>min</td><td>10.6</td><td>0.01</td><td>10.6</td><td>0.3</td><td>2</td></tr><tr><td></td><td>max</td><td>11</td><td>0.05</td><td>11</td><td>0.6</td><td>2.5</td></tr><tr><td>R<sub>c</sub> (Ω)</td><td>min</td><td>3411</td><td>1400</td><td>59,100</td><td>640</td><td>104,200</td></tr><tr><td></td><td>max</td><td>3420</td><td>1500</td><td>59,400</td><td>750</td><td>105,500</td></tr><tr><td>X<sub>m</sub> (Ω)</td><td>min</td><td>1940</td><td>290</td><td>1750</td><td>220</td><td>9000</td></tr><tr><td></td><td>max</td><td>1960</td><td>300</td><td>2300</td><td>270</td><td>9300</td></tr></tbody></table>

$$VR = \frac{E_p - V_s}{V_s} \qquad (6)$$

$$\eta = \frac{P_o}{P_{in}} \qquad (7)$$

where $P_o$ and $P_{in}$ represent the output and input active power in Watts, and are calculated as follows:

$$P_{in} = real(V_p \times I_p^*) \qquad (8)$$

$$P_o = real(V_s \times I_s^*) \qquad (9)$$

The primary power factor $PF_p$, can be calculated as follows:

$$PF_p = cos(\angle(I_p)) \qquad (10)$$

From Eqs. (1)-(10), it is clear the importance of using the accurate values of the transformer parameters ($R_p, X_p, R'_s, X'_s, R_c$, and $X_m$) in evaluating the performance of single-phase as well as three-phase transformer. These parameters can be calculated using the open circuit and short circuit tests or by using the nameplate data of transformer. On the other hand, many optimization techniques tried to extract accurate values for these parameters to realize the accuracy required in determining the performance of transformers.

### 3. Arctic puffin optimization (APO)

In this research, the APO is used to extract the optimal parameters of a single-phase transformer as well as three-phase transformer. The transformer has six parameters namely $R_p, X_p, R_s, X_s, R_c$, and $X_m$. The behavior of Arctic puffin is described as follows:

A tiny, uncommon bird that is native to the Arctic is the Arctic puffin. These birds, who mostly live in oceans, typically hunt in flocks or

groups. They have exceptional flying and fishing capabilities, and they cooperate to find food in the water [35].

When at sea or during aerial activity, the Arctic puffin favors remaining together in groups. They are better able to live in the Arctic thanks to their large-scale mixed lifestyle. During their journeys in the Arctic, Arctic puffins will surely confront danger. The puffins will immediately launch their flock into the air and sound a warning as soon as they identify an opponent [36].

#### 3.1. Fundamental ideas and the mathematical model

This section presents the APO algorithm, which is modeled after the survival strategies of the Arctic Puffin. Three basic stages comprise the mathematical model of APO: initialization of the population, the stage of aerial flight (exploration), and the stage of submerged foraging (exploitation). It also includes the switch between these two tactics, which is brought about by the puffin's behavioral transformation factor B.

##### 3.1.1. Initial population

Every Arctic Puffin in the optimization process is a possible solution. The following equation characterizes the initialization of the population generation process:

$$x_i^t = rand \ast (u_b - l_b), \quad i = 1, 2, 3...N \qquad (11)$$

where N is the total number of individuals in the population, $u_b$ and $l_b$ denote the higher and lower limits, respectively, $x_i^t$ is the position of the i-th Arctic Puffin and rand generates a random number between 0 and 1.

##### 3.1.2. Stage of aerial flight (exploration)

When navigating in the air, puffins have two key tactics to deal with different scenarios. Plunging for prey while in the air is the second method, and searching from the air is the first. These two stages' different behavioral tactics show how adaptable Arctic puffins are to a variety of situations, which helps them survive and procreate successfully.

The first strategy is aerial search: Arctic puffins usually fly in groups or formations, working together to improve flying efficiency and create chances for collective hunting. They keep their flight altitudes somewhat low in order to make it easier to harvest possible undersea food supplies. They focus on surveying possible prey during this phase and keep an eye out for any potential predators nearby. When conditions are right, especially when there are lots of fish and few predators, fish quickly move to the surface of the water in order to decently catch their prey. The equations position update of this approach are as follows.

$$Y_i^{t+1} = X_i^t + (X_i^t - X_r^t) \ast L(D) + R \qquad (12)$$

$$R = round(0.5 \ast (0.05 + rand)) \ast \alpha \qquad (13)$$

$$\alpha = Normal(0, 1) \qquad (14)$$

where $X_i^t$ denotes the population's current i-th candidate solution, $X_r^t$ is a candidate solution randomly selected from the current population, with $X_i^t \neq X_r^t$; L(D) denotes a random number generated through Levy flight; D is the dimensionality; $\alpha$ is a random number following a standard normal distribution and r is a random integer between 1 and N - 1, excluding i.

Finding good points for fishery is a key step of the aerial searching strategy for puffins. The flight plan for this step strengthens the global searching ability of the algorithm, enabling it to explore the problem space more thoroughly and boost the likelihood of finding the universal optimal solution.

The second strategy is swooping predation: Swooping refers to a major strategy of Arctic puffin predators, quickly changing their flight path for more speedy catching of the target. To survive, when there is
<figure><img src="image_5.png" /><figcaption>(a) Case 1 (single-phase transformer)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(b) Case 3 (three-phase transformer);</figcaption></figure>

<caption>Fig. 3. Experiment's photograph of transformers.</caption>

<table><caption>**Table 3**

Parameters calculated based on short- and open-circuit tests and nameplate data of five cases applying APO.</caption><thead><tr><th>Transformer</th><th>R<sub>p</sub>(Ω)</th><th>X<sub>p</sub>(Ω)</th><th>R<sub>s</sub>(Ω)</th><th>X<sub>s</sub>(Ω)</th><th>R<sub>c</sub>(Ω)</th><th>X<sub>m</sub>(Ω)</th><th>I<sub>p</sub>(A)</th><th>Efficiency (%)</th><th>PF<sub>p</sub></th></tr></thead><tbody><tr><td>Case 1</td><td>1.7</td><td>10.843</td><td>1.7</td><td>10.843</td><td>3413.25</td><td>1947.7</td><td>1.08</td><td>92.2</td><td>0.994</td></tr><tr><td>Case 2</td><td>0.428</td><td>0.21</td><td>0.508</td><td>0.03</td><td>1437.5</td><td>294.8</td><td>8.896</td><td>94.9</td><td>0.996</td></tr><tr><td>Case 3</td><td>1.7</td><td>10.84</td><td>1.7</td><td>10.84</td><td>59,252</td><td>2021</td><td>7.0424</td><td>90.1314</td><td>0.9999</td></tr><tr><td>Case 4</td><td>0.9</td><td>0.94</td><td>1.6</td><td>0.44</td><td>700</td><td>250</td><td>20.9025</td><td>87.43</td><td>0.994865</td></tr><tr><td>Case 5</td><td>2.45</td><td>3.14</td><td>2</td><td>2.2294</td><td>105,000</td><td>9106</td><td>6.4135</td><td>98.19</td><td>0.832</td></tr></tbody></table>

<table><caption>**Table 4**

Simulation results of optimal parameters of five different transformers using APO.</caption><thead><tr><th rowspan="2">Parameters</th><th colspan="5">Estimated parameters of different transformers</th></tr><tr><th colspan="3">Based on measurements (O.C&amp; S.C)</th><th colspan="2">Based on nameplate data</th></tr></thead><tbody><tr><td>Value</td><td>Case 1</td><td>Case 2</td><td>Case 3</td><td>Case 4</td><td>Case 5</td></tr><tr><td>R<sub>p</sub> (Ω)</td><td>1.690</td><td>0.400</td><td>1.71</td><td>0.903</td><td>2.411</td></tr><tr><td>X<sub>p</sub> (Ω)</td><td>10.769</td><td>0.224</td><td>10.79</td><td>0.904</td><td>3.086</td></tr><tr><td>R<sub>s</sub> (Ω)</td><td>1.716</td><td>0.500</td><td>1.69</td><td>1.596</td><td>2.056</td></tr><tr><td>X<sub>s</sub> (Ω)</td><td>10.850</td><td>0.032</td><td>10.85</td><td>0.431</td><td>2.244</td></tr><tr><td>R<sub>c</sub>(Ω)</td><td>3411.000</td><td>1439.344</td><td>59,200.00</td><td>699.249</td><td>105,298.822</td></tr><tr><td>X<sub>m</sub>(Ω)</td><td>1945.000</td><td>293.000</td><td>2022.01</td><td>249.978</td><td>9101.288</td></tr><tr><td>SSAE</td><td>9.97E-08</td><td>1.04E-08</td><td>8.77E-10</td><td>8.09E-27</td><td>3.82E-30</td></tr></tbody></table>

<table><caption>**Table 5**

Actual and estimated data at full load condition for different studied cases.</caption><thead><tr><th>Case</th><th>Data</th><th>I<sub>p</sub>(A)</th><th>η<sub>FL</sub></th><th>PF<sub>p</sub></th><th>SSAE</th></tr></thead><tbody><tr><td rowspan="3">1</td><td>Actual</td><td>1.08</td><td>0.922</td><td>0.994</td><td>0</td></tr><tr><td>Estimated</td><td>1.0798</td><td>0.92198</td><td>0.9942</td><td>9.97E-08</td></tr><tr><td>Error</td><td>-2.27E-04</td><td>-1.83E-05</td><td>2.18E-04</td><td></td></tr><tr><td rowspan="3">2</td><td>Actual</td><td>8.896</td><td>0.949</td><td>0.996</td><td>0</td></tr><tr><td>Estimated</td><td>8.895431</td><td>0.94894</td><td>0.995952</td><td>1.04E-08</td></tr><tr><td>Error</td><td>-6.29E-05</td><td>-6.40E-05</td><td>-4.87E-05</td><td></td></tr><tr><td rowspan="3">3</td><td>Actual</td><td>7.0424</td><td>0.901</td><td>0.9999</td><td>0</td></tr><tr><td>Estimated</td><td>7.04239</td><td>0.901314</td><td>0.99987</td><td>8.77E-10</td></tr><tr><td>Error</td><td>-7.26E-12</td><td>-1.38E-06</td><td>-2.96E-05</td><td></td></tr><tr><td rowspan="3">4</td><td>Actual</td><td>20.9025</td><td>0.8743</td><td>0.994865</td><td>0</td></tr><tr><td>Estimated</td><td>20.9025</td><td>0.874</td><td>0.9948</td><td>8.09E-27</td></tr><tr><td>Error</td><td>-1.83E-05</td><td>-2.27E-04</td><td>2.18E-04</td><td></td></tr><tr><td rowspan="3">5</td><td>Actual</td><td>6.4135</td><td>0.9819</td><td>0.832</td><td>0</td></tr><tr><td>Estimated</td><td>6.4135</td><td>0.9819</td><td>0.832</td><td>3.82E-30</td></tr><tr><td>Error</td><td>-3.39E-16</td><td>-1.25E-15</td><td>-1.47E-15</td><td></td></tr></tbody></table>
<figure><img src="image_7.png" /><figcaption>(a) Case 1</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(c) Case 3</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(b) Case 2</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(d) Case 4.</figcaption></figure>

<caption>Fig. 4. Convergence behaviour of APO for estimating the transformer parameters.</caption>

competition, they need to use this strategy of swooping for more speedy and effective catching of the target. We implemented a velocity coefficient S in the puffin's movement for the purpose of mimicking this behavior of swooping. The equation for updating the location is.

$$Z_{i}^{t+1} = Y_{i}^{t+1} * S \qquad (15)$$

$$S = \tan((rand - 0.5) * \pi) \qquad (16)$$

The velocity coefficient S, established by the Arctic puffin in this flight strategy, has a significant function of governing the flight's speed and direction. S, being a mathematical velocity coefficient, with its changing direction and amplitude of its flight speed, can enable the puffin to dynamically satisfy changing requirements for foraging. With the employment of parameter S, the algorithm becomes more similar to the flight behavior of the puffin, and its flexibility when faced with uncertainty and competition, and its adaptation ability for a more complex air environment, are enhanced. Besides, the employment of parameter S's distributional properties strengthens the algorithm's diversification and randomness and the Arctic puffin's exploration ability. Thus, the algorithm's searching performance and searching efficiency in the solution space are enhanced even more with the design of

the swooping predation phase. The algorithm's flexibility and searching performance under varying scenarios are improved. The algorithm decides on integrating the candidate positions created in both rounds into a new solution for generating the optimal solutions under varying scenarios. The top N individuals are then chosen after fitness ordering of these solutions for generating the new generation. The description of equations is as follows:

$$P_{i}^{t+1} = Y_{i}^{t+1} \cup Z_{i}^{t+1} \qquad (17)$$

$$new = sort(P_{i}^{t+1}) \qquad (18)$$

$$X_{i}^{t+1} = new (1:N) \qquad (19)$$

where sort is used to choose the new population $X_{i}^{t+1}$ by sorting the new population based on fitness values, which range from small to large.

### 3.1.3. Underwater foraging stage (Exploitation)

The three primary underwater foraging techniques are acquiring forage, intensification of searching effort, and avoidance of predators. The sections that follow will detail the steps and discuss their critical
<figure><img src="image_11.png" /><figcaption>Fig. 5. performance of transformer: (a) efficiency of Case 1; (b) voltage regulation of Case 1; (c) efficiency of Case 3; (d) voltage regulation of Case 3.</figcaption></figure>

<table><caption>**Table 6**
Simulation results of optimal parameters of Case 1 applying APO and other optimizers.</caption><thead><tr><th>Parameters</th><th>APO</th><th>GWO</th><th>PSO</th><th>SCA</th><th>SSA</th></tr></thead><tbody><tr><td>R<sub>p</sub> (Ω)</td><td>1.690</td><td>1.690</td><td>1.690</td><td>1.690</td><td>1.690</td></tr><tr><td>X<sub>p</sub> (Ω)</td><td>10.769</td><td>10.850</td><td>10.718</td><td>10.717</td><td>10.850</td></tr><tr><td>R<sub>s</sub> (Ω)</td><td>1.716</td><td>1.716</td><td>1.716</td><td>1.716</td><td>1.716</td></tr><tr><td>X<sub>s</sub> (Ω)</td><td>10.850</td><td>10.850</td><td>10.850</td><td>10.850</td><td>10.850</td></tr><tr><td>R<sub>c</sub> (Ω)</td><td>3411.000</td><td>3411.000</td><td>3411.000</td><td>3411.000</td><td>3411.000</td></tr><tr><td>X<sub>m</sub> (Ω)</td><td>1945.000</td><td>1945.000</td><td>1945.000</td><td>1945.000</td><td>1945.000</td></tr><tr><td>SSAE</td><td>9.9657E-08</td><td>9.9657E-08</td><td>9.9657E-08</td><td>9.9657E-08</td><td>9.9657E-08</td></tr></tbody></table>

roles in the Arctic Puffin's existence.

Gathering foraging is the primary underwater strategy: Arctic puffins also forage in group fashion quite regularly, where they congregate around fish aggregations near the water's surface. The rate and effectiveness of hunting are improved with this group predation strategy. The probability of predation being successful is improved with their ability for working in unison and encircling and catching aggregations of fish. This becomes possible with collective foraging. Apart from this, puffins on the water's surface also learn from their peers and apply this knowledge for identifying where to forage for, and where there could be

<figure><img src="image_12.png" /><figcaption>**Table 7**
Simulation results of optimal parameters of Case 2 applying APO and other optimizers.</figcaption></figure>

<table><thead><tr><th>Parameters</th><th>APO</th><th>GWO</th><th>PSO</th><th>SCA</th><th>SSA</th></tr></thead><tbody><tr><td>R<sub>p</sub> (Ω)</td><td>0.400</td><td>0.400</td><td>0.400</td><td>0.400</td><td>0.400</td></tr><tr><td>X<sub>p</sub> (Ω)</td><td>0.224</td><td>0.237</td><td>0.250</td><td>0.210</td><td>0.232</td></tr><tr><td>R<sub>s</sub> (Ω)</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td></tr><tr><td>X<sub>s</sub> (Ω)</td><td>0.032</td><td>0.032</td><td>0.032</td><td>0.032</td><td>0.032</td></tr><tr><td>R<sub>c</sub> (Ω)</td><td>1439.344</td><td>1439.344</td><td>1439.344</td><td>1439.344</td><td>1439.343</td></tr><tr><td>X<sub>m</sub> (Ω)</td><td>293.000</td><td>293.000</td><td>293.000</td><td>293.000</td><td>293.000</td></tr><tr><td>SSAE</td><td>1.04283E-08</td><td>1.04283E-08</td><td>1.04283E-08</td><td>1.04283E-08</td><td>1.04283E-08</td></tr></tbody></table>

sources of food.

The position update is represented by the following equation.

$$W_i^{t+1} = \begin{cases} X_{r1}^t + F * L(D) * (X_{r2}^t - X_{r3}^t), & \text{rand} \ge 0.5 \\ X_{r1}^t + F * (X_{r2}^t - X_{r3}^t), & \text{rand} < 0.5 \end{cases} \quad (20)$$

where F is the cooperation factor that modifies the Arctic puffins' pre- dation behavior. The variables r1, r2, and r3 represent random integers within the range of 1 and N - 1 (excluding i), whereas the candidate solutions $X_{r1}^t$, $X_{r2}^t$ and $X_{r3}^t$ are chosen at random from the existing
<caption>**Table 8**</caption>

Simulation results of optimal parameters of Case 3 applying APO and other optimizers.

<table><caption>**Table 9**</caption><thead><tr><th>Parameters</th><th>APO</th><th>GWO</th><th>PSO</th><th>SCA</th><th>SSA</th></tr></thead><tbody><tr><td>R<sub>p</sub> (Ω)</td><td>1.7099</td><td>1.7099</td><td>1.7099</td><td>1.7099</td><td>1.7098</td></tr><tr><td>X<sub>p</sub> (Ω)</td><td>10.7940</td><td>10.7000</td><td>10.7000</td><td>10.8464</td><td>10.8500</td></tr><tr><td>R<sub>s</sub> (Ω)</td><td>1.6900</td><td>1.6900</td><td>1.6900</td><td>1.6900</td><td>1.6900</td></tr><tr><td>X<sub>s</sub> (Ω)</td><td>10.850</td><td>10.850</td><td>10.850</td><td>10.850</td><td>10.850</td></tr><tr><td>R<sub>c</sub>(Ω)</td><td>59,200.00</td><td>59,200.00</td><td>59,200.00</td><td>59,200.00</td><td>59,200.00</td></tr><tr><td>X<sub>m</sub>(Ω)</td><td>2022.01</td><td>2022.09</td><td>2022.01</td><td>2022.01</td><td>2022.34</td></tr><tr><td>SSAE</td><td>8.77387E-10</td><td>8.77463E-10</td><td>8.77387E-10</td><td>8.77387E-10</td><td>8.79263E-10</td></tr></tbody></table>

Simulation results of optimal parameters of Case 4 applying APO and other optimizers.

<table><thead><tr><th>Parameters</th><th>APO</th><th>GWO</th><th>PSO</th><th>SCA</th><th>SSA</th></tr></thead><tbody><tr><td>R<sub>p</sub> (Ω)</td><td>0.903</td><td>0.892</td><td>0.890</td><td>0.902</td><td>0.896</td></tr><tr><td>X<sub>p</sub> (Ω)</td><td>0.904</td><td>0.919</td><td>0.950</td><td>0.927</td><td>0.950</td></tr><tr><td>R<sub>s</sub> (Ω)</td><td>1.596</td><td>1.608</td><td>1.612</td><td>1.597</td><td>1.606</td></tr><tr><td>X<sub>s</sub> (Ω)</td><td>0.431</td><td>0.433</td><td>0.450</td><td>0.422</td><td>0.447</td></tr><tr><td>R<sub>c</sub>(Ω)</td><td>699.249</td><td>699.789</td><td>701.070</td><td>698.601</td><td>700.911</td></tr><tr><td>X<sub>m</sub>(Ω)</td><td>249.978</td><td>250.048</td><td>250.064</td><td>250.011</td><td>250.510</td></tr><tr><td>SSAE</td><td>8.0904E-27</td><td>1.0849E-11</td><td>0.0</td><td>5.6781E-20</td><td>2.2902E-09</td></tr></tbody></table>

population and r1 ≠ r2 ≠ r3 and $X_{r2}^t \neq X_{r3}^t$.

To illustrate foraging behavior, two position equations explain the approach of obtaining forage. In these equations, when rand<0.5, Arctic puffins participate in cooperative foraging with other puffins, utilizing the cooperative factor F to explore the surrounding environment through random movements, hunting for food or other resources. When rand≥0.5, the alternative equation shows a more intricate method of finding food. When a puffin notices a school of fish during the gathering foraging phase, it may quickly swim after its fellows and change its position to join a more advantageous predation group for better cooperative foraging. Therefore, the Levy flight factor is utilized once more to augment the puffin's capacity for swift predation, guaranteeing successful prey acquisition. By balancing exploration and exploitation during optimization, this dual-strategy approach helps conduct a more thorough search of the solution space.

Intensifying research is the second underwater tactic: After some amount of time, Arctic puffins can experience their current foraging spot's sources of food being depleted or depleted due to predators. The puffins must change their underwater positions in order for them to be able to find substitute fish or other underwater foods for them to be able to continue meeting their nutritional needs. The formula for this action of intensifying research is:

$$Y_i^{t+1} = W_i^{t+1} * (1 + f) \quad (21)$$

$$f = 0.1 * (rand - 1) * \frac{(T - t)}{T} \quad (22)$$

<caption>**Table 10**</caption>

Simulation results of optimal parameters of Case 5 applying APO and other optimizers.

<table><thead><tr><th>Parameters</th><th>APO</th><th>GWO</th><th>PSO</th><th>SCA</th><th>SSA</th></tr></thead><tbody><tr><td>R<sub>p</sub> (Ω)</td><td>2.4112</td><td>2.4717</td><td>2.4000</td><td>2.4602</td><td>2.4000</td></tr><tr><td>X<sub>p</sub> (Ω)</td><td>3.0861</td><td>3.0926</td><td>3.0000</td><td>3.1282</td><td>3.1872</td></tr><tr><td>R<sub>s</sub> (Ω)</td><td>2.0559</td><td>1.9927</td><td>2.0660</td><td>2.0046</td><td>2.0655</td></tr><tr><td>X<sub>s</sub> (Ω)</td><td>2.2436</td><td>2.2082</td><td>2.2000</td><td>2.2153</td><td>2.2000</td></tr><tr><td>R<sub>c</sub>(Ω)</td><td>105,298.82</td><td>105,280.99</td><td>105,161.92</td><td>105,292.70</td><td>105,175.42</td></tr><tr><td>X<sub>m</sub>(Ω)</td><td>9101.29</td><td>9099.81</td><td>9101.03</td><td>9099.91</td><td>9100.60</td></tr><tr><td>SSAE</td><td>3.823E-30</td><td>8.658E-13</td><td>0.0</td><td>6.688E-19</td><td>5.719E-12</td></tr></tbody></table>

where T represents the iterations being performed and T represents the overall iterations. A random value referred to as Rand adds some randomness into f. The step is modeled with variations on the initial point in Eq. (21). The parameter (1 + f) plays a key role in the enhanced searching process, where f represents an adaptive value modifying the Arctic puffin's location in water in the second step.

The ability of puffins to accommodate their environment when searching for sustenance has been the source of inspiration for this adaptive aspect's design. To be more precise, the parameter is gradually adjusted with the rise in iterations, such that the puffin can select on the basis of the improvement and randomness of its quest for more plentiful sources of sustenance. With this approach, Arctic puffins can fulfill their survival requirements when under predation and be resilient when adapting towards the sustenance available in their environment. The adaptation and improvement of the algorithm when faced with changing difficulties and new solutions are made possible with its exploration of the space of searching.

Avoiding predators represents the third underwater strategy.

When Arctic puffins realize predators are present near them, they behave in this technique's description. The puffins signal other puffins of incoming threats with a precise call or noise. The call informs other puffins of predators and commands them away from the spot.

It functions as a warning signal. At the same time, when they sense that a predator is approaching, Arctic puffins quickly change course, swimming quickly down a longer route in the direction of safety to avoid the threat. The position update equation for this method is as follows:

$$Z_i^{t+1} = \begin{cases} X_i^t + F * L(D) * (X_{r1}^t - X_{r2}^t), & rand \ge 0.5 \\ X_i^t + \beta * (X_{r1}^t - X_{r2}^t), & rand < 0.5 \end{cases} \quad (23)$$

where β is a random number between 0 and 1 that is evenly distributed.

This method involves two distinct behavioral patterns: one involves averting danger quickly, and the other involves avoiding danger gradually. This dual-behavior approach mimics how to break out of local optima in various scenarios within the algorithm. When rand ≥ 0.5 suggests the presence of a predator, Arctic puffins opt to duck right away, significantly altering their posture. By using known knowledge to improve leaping ability, this technique helps the computer locate better solutions in potential local optima. On the other hand, Arctic puffin behavior is generally more cautious when rand < 0.5 shows that the birds seek to avoid predators proactively. They shift locations at random to avoid possible threat. This is an exploratory strategy that enables the algorithm to more carefully scan the surroundings in the hopes of finding better solutions.

As a result, in many circumstances, the algorithm is assisted in maintaining balance by this dual-behavior approach. It improves the algorithm's ability to search globally and helps avoid local optima by enabling the methodical study of new regions while swiftly adjusting to known information. To guarantee resilience and flexibility, the algorithm simulates this balancing mechanism as:

$$P_i^{t+1} = W_i^{t+1} \cup Y_i^{t+1} \cup Z_i^{t+1} \quad (24)$$

$$new = sort(P_i^{t+1}) \quad (25)$$
<table><caption>**Table 11**
Statistical results of APO and other optimizers (30 runs).</caption><thead><tr><th>Case</th><th>Parameters</th><th>APO</th><th>GWO</th><th>PSO</th><th>SCA</th><th>SSA</th></tr></thead><tbody><tr><td rowspan="6">Case 1</td><td>Min.</td><td>9.966E-08</td><td>9.966E-08</td><td>9.966E-08</td><td>9.966E-08</td><td>9.966E-08</td></tr><tr><td>Max.</td><td>9.966E-08</td><td>9.966E-08</td><td>1.038E-07</td><td>9.966E-08</td><td>9.969E-08</td></tr><tr><td>Average</td><td>9.966E-08</td><td>9.966E-08</td><td>9.970E-08</td><td>9.966E-08</td><td>9.966E-08</td></tr><tr><td>Var.</td><td>4.086E-34</td><td>2.832E-26</td><td>1.723E-19</td><td>8.674E-38</td><td>2.290E-23</td></tr><tr><td>Med.</td><td>9.966E-08</td><td>9.966E-08</td><td>9.966E-08</td><td>9.966E-08</td><td>9.966E-08</td></tr><tr><td>STD</td><td>2.021E-17</td><td>1.683E-13</td><td>4.150E-10</td><td>2.945E-19</td><td>4.785E-12</td></tr><tr><td rowspan="6">Case 2</td><td>Min.</td><td>1.0428E-08</td><td>1.0428E-08</td><td>1.0428E-08</td><td>1.0428E-08</td><td>1.0428E-08</td></tr><tr><td>Max.</td><td>1.0428E-08</td><td>1.0454E-08</td><td>1.0569E-08</td><td>1.0446E-08</td><td>1.0569E-08</td></tr><tr><td>Average</td><td>1.0428E-08</td><td>1.0429E-08</td><td>1.0430E-08</td><td>1.0429E-08</td><td>1.0441E-08</td></tr><tr><td>Var.</td><td>2.3867E-35</td><td>1.0063E-23</td><td>1.9907E-22</td><td>4.8775E-24</td><td>5.7335E-22</td></tr><tr><td>Med.</td><td>1.0428E-08</td><td>1.0429E-08</td><td>1.0428E-08</td><td>1.0428E-08</td><td>1.0433E-08</td></tr><tr><td>STD</td><td>4.8854E-18</td><td>3.1722E-12</td><td>1.4109E-11</td><td>2.2085E-12</td><td>2.3945E-11</td></tr><tr><td rowspan="6">Case 3</td><td>Min.</td><td>8.77387E-10</td><td>8.77463E-10</td><td>8.77387E-10</td><td>8.7739E-10</td><td>8.7926E-10</td></tr><tr><td>Max.</td><td>8.77387E-10</td><td>9.77397E-10</td><td>1.11344E-09</td><td>8.8412E-10</td><td>7.7183E-09</td></tr><tr><td>Average</td><td>8.77387E-10</td><td>9.05759E-10</td><td>9.01898E-10</td><td>8.7895E-10</td><td>2.1545E-09</td></tr><tr><td>Var.</td><td>2.04203E-40</td><td>5.12195E-22</td><td>4.23910E-21</td><td>4.1399E-24</td><td>1.3977E-18</td></tr><tr><td>Med.</td><td>8.77387E-10</td><td>8.99486E-10</td><td>8.80269E-10</td><td>8.7764E-10</td><td>1.8435E-09</td></tr><tr><td>STD</td><td>1.4290E-20</td><td>2.2637E-11</td><td>6.5108E-11</td><td>2.0347E-12</td><td>1.1823E-09</td></tr><tr><td rowspan="6">Case 4</td><td>Min.</td><td>8.0904E-27</td><td>1.0849E-11</td><td>0.0000E+00</td><td>5.6781E-20</td><td>2.2902E-09</td></tr><tr><td>Max.</td><td>7.50E-22</td><td>7.85E-10</td><td>3.80E-23</td><td>1.09E-16</td><td>7.65E-08</td></tr><tr><td>Average</td><td>2.59E-23</td><td>1.86E-10</td><td>3.91E-25</td><td>1.39E-17</td><td>2.37E-08</td></tr><tr><td>Var.</td><td>9.02E-45</td><td>1.85E-20</td><td>1.45E-47</td><td>2.22E-34</td><td>2.64E-16</td></tr><tr><td>Med.</td><td>1.31E-24</td><td>1.57E-10</td><td>1.25E-32</td><td>9.82E-18</td><td>2.07E-08</td></tr><tr><td>STD</td><td>9.50E-23</td><td>1.36E-10</td><td>3.80E-24</td><td>1.49E-17</td><td>1.62E-08</td></tr><tr><td rowspan="6">Case 5</td><td>Min.</td><td>3.823E-30</td><td>8.658E-13</td><td>0.000E+00</td><td>6.688E-19</td><td>5.719E-12</td></tr><tr><td>Max.</td><td>2.22E-25</td><td>6.43E-11</td><td>2.36E-12</td><td>3.46E-12</td><td>8.82E-10</td></tr><tr><td>Average</td><td>1.35E-26</td><td>1.97E-11</td><td>4.72E-14</td><td>3.00E-13</td><td>2.42E-10</td></tr><tr><td>Var.</td><td>1.41E-51</td><td>2.02E-22</td><td>1.10E-25</td><td>4.09E-25</td><td>3.22E-20</td></tr><tr><td>Med.</td><td>2.46E-27</td><td>1.60E-11</td><td>0.00E+00</td><td>3.54E-14</td><td>1.96E-10</td></tr><tr><td>STD</td><td>3.75E-26</td><td>1.42E-11</td><td>3.32E-13</td><td>6.39E-13</td><td>1.80E-10</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>(a) Case 3</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(b) Case 4.</figcaption></figure>

<caption>Fig. 6. Convergence behaviour of APO compared to GWO, PSO, SCA and SSA.</caption>

<table><caption>**Table 12**
Friedman ANOVA indices for APO.</caption><thead><tr><th></th><th>Source</th><th>'SS'</th><th>'df'</th><th>'MS'</th><th>'Chi-sq'</th><th>ρ >Chi-sq</th></tr></thead><tbody><tr><td rowspan="3">Case 1</td><td>Columns</td><td>978.7</td><td>4</td><td>244.675</td><td>391.48</td><td>1.93E-83</td></tr><tr><td>Error</td><td>21.3</td><td>396</td><td>0.053788</td><td></td><td></td></tr><tr><td>Total</td><td>1000</td><td>499</td><td></td><td></td><td></td></tr><tr><td rowspan="3">Case 2</td><td>Columns</td><td>856.76</td><td>4</td><td>214.19</td><td>342.704</td><td>6.59E-73</td></tr><tr><td>Error</td><td>143.24</td><td>396</td><td>0.361717</td><td></td><td></td></tr><tr><td>Total</td><td>1000</td><td>499</td><td></td><td></td><td></td></tr><tr><td rowspan="3">Case 3</td><td>Columns</td><td>811</td><td>4</td><td>202.75</td><td>324.4</td><td>5.89E-69</td></tr><tr><td>Error</td><td>189</td><td>396</td><td>0.477273</td><td></td><td></td></tr><tr><td>Total</td><td>1000</td><td>499</td><td></td><td></td><td></td></tr></tbody></table>

$$X_i^{t+1} = new \ (1:N) \qquad (26)$$

This all-encompassing approach enables the model to consider several foraging scenarios at the same time, selecting the best position as the ideal outcome in the end.

### 3.1.4. *Behavior conversion factor B*

The Arctic puffin in the APO algorithm always likes to dive for food regularly to carry out local exploitation in the later stage of the iteration, while it always tends to make frequent aerial flights in the initial stage of the iteration to realize global search. The lifestyle of the Arctic puffin served as the model for this search technique. The Arctic puffin is more likely to look for appropriate feeding waters in its early stages than it is to concentrate on diving for food in its later stages. To accomplish a seamless transition from global search to local exploitation, the APO algorithm creates a behavioral transition coefficient B based on this behavioral pattern. Here is how it is defined.
<figure><img src="image_15.png" /><figcaption>Fig. 7. Box plot for APO and the other four competitive optimizers.</figcaption></figure>

$$B = 2 * \log\left(\frac{1}{\text{rand}}\right) * (1 - t/T) \qquad (27)$$

where t and T represent the current iteration number and the maximum iteration number, respectively, and the rand is a random number in (0, 1). In the APO algorithm, B is an adaptively changing parameter that simultaneously incorporates stochasticity to simulate the dynamic randomness of the behavior of Arctic puffins. B is not arbitrarily selectable; rather, it is dynamically derived based on the ratio between the current and maximum iteration numbers. Because of this architecture, the value of B can be dynamically changed as the iterations continue, allowing it to react to the needs of the search at various points.

In the meantime, we add a new parameter C to the APO algorithm, which determines the search strategy to be used at the current iteration step by comparing B with C. The Arctic puffin, whose life patterns allow them to flexibly transition between aerial flight (exploration) and diving for foraging (exploitation) depending on their needs, served as an inspiration for the creation of this comparison mechanism. The algorithm will go into the air flying phase when B is greater than C. This implies that in order to identify viable, high-quality solutions, the algorithm will thoroughly search for a sizable portion of the existing solution space. By employing this tactic, the algorithm is prevented from entering a local optimum too soon and is guaranteed to search the whole solution space. To illustrate foraging behavior Conversely, the algorithm switches into a local exploitation strategy and proceeds into the foraging and diving process when B's value becomes equal to, or even lower, than C. To also increase the accuracy of the solution, the algorithm will undertake a fine-grained exploration of the vicinity of the identified high-quality solution. The approach raises the accuracy and rate of convergence of the algorithm and also enables the algorithm to explore more feasible, higher quality solutions.

As a result, the behavioral switching coefficient B becomes critical for maintaining a balance between exploration and exploitation of the algorithm and represents the Arctic puffin's transition from its two ways of being—under water foraging and flight in the air. The algorithm may adjust the search strategy flexibly to different iteration stages by comparing it with parameter C. This approach enhances the algorithm's robustness and performance by enabling it to strike a suitable balance between local exploitation and global search. The flow chart of the algorithm is shown in Fig. 2.

## 3.2. Extraction of transformer parameters applying APO

The parameters of a single-phase transformer as well as three-phase transformer are vital elements in performance evaluation. These parameters

are six elements and namely R<sub>p</sub>, X<sub>p</sub>, R<sub>s</sub>, X<sub>s</sub>, R<sub>c</sub>, and X<sub>m</sub>. The optimal values of these parameters can be defined by applying an optimization algorithm. The optimization process depends on minimizing the error between the actual values, obtained from experiment results or using nameplate data, and the estimated ones applying the APO optimization method. Here, the objective function of the optimization problem based on minimizing the summation of square absolute error (SSAE) of primary current, rated efficiency and the power factor at full load, as follows:

$$f_1 = \frac{\eta_e - \eta_m}{\eta_m} \qquad (28)$$

$$f_2 = \frac{I_{pe} - I_{pm}}{I_{pm}} \qquad (29)$$

$$f_3 = \frac{PF_e - PF_m}{PF_m} \qquad (30)$$

$$SSAE = f_1^2 + f_2^2 + f_3^2 \qquad (31)$$

where η<sub>e</sub> and η<sub>m</sub> refer to estimated and the measured efficiency, respectively. The estimated and measured values of the primary current are I<sub>pe</sub> and I<sub>pm</sub>. In addition, PF<sub>e</sub> and PF<sub>m</sub> are the estimated and the measured power factors, respectively.

The objective function aims at achieving the minimum of SSAE. The flow chart of the proposed APO is explained in Fig. 2. APO is applied with 100 population and 500 iterations, which is considered as the stopping criterion of the optimization process. The objective function (OF) aims to minimize SSAE as given in Eq. (32).

$$\text{OF} = \min(\text{SSAE}) \qquad (32)$$

subject to:

$$\begin{align}
R_p^{\min} &\le R_p \le R_p^{\max} \quad \text{and} \quad R_s^{\min} &\le R_s \le R_s^{\max} \nonumber \\
X_p^{\min} &\le X_p \le X_p^{\max} \quad \text{and} \quad X_s^{\min} &\le X_s \le X_s^{\min} \nonumber \\
R_c^{\min} &\le R_c \le R_c^{\max} \quad \text{and} \quad X_m^{\min} &\le X_m \le X_m^{\max} \tag{33}
\end{align}$$

## 4. Results and discussions

To approve the proposed APO method efficiency, the six optimized parameters of single- and three-phase transformers are extracted based on the measurements such as open-circuit and short circuit tests or based on the nameplate data. The transformers used to study the problem are described in Table 1. The study in the first three cases, Cases 1–3,
depends on open- and short-circuit tests, while the study in Cases 4–5 depends on the nameplate data. In Table 1, the number of phases, KVA rating, voltage and operating frequency are defined for each transformer. Table 2 shows the transformers boundaries in Cases 1–5. Fig. 3(a) and (b) illustrate the photograph of the testes of Cases 1 and 2. The tests results of Case 3 are taken from [30].

The reported results of open-circuit test are employed to calculate the elements of magnetizing circuit ($R_c$ and $X_m$). The short-circuit measurements are used to calculate the resistive and reactive elements of the primary and secondary windings, $R_p$, $X_p$, $R_s$, and $X_s$, respectively. Additionally, the full load nameplate data are the base to calculate these parameters in Cases 4 and 5. The calculated parameters from measurements and nameplate are considered as the actual data of a transformer. Table 3 records the actual parameters of the five transformers studied. The parameters of each transformer are the input to the proposed APO algorithm to calculate the SSAE for estimating the optimal parameters of each transformer.

Table 4 shows the simulation results of five transformers. The results in Table 4 are the six estimated parameters ($R_p, X_p, R_s, X_s, R_c$, and $X_m$). The extracted parameters are employed to calculate the estimated primary current, primary power factor and efficiency at full load condition. The actual primary current and power factor and efficiency calculated using the actual transformer parameters are introduced in Table 5. It is noticed that the error between the actual and estimated data does not exceed 2.27E-04. This emphasizes on the accuracy of the results.

Additionally, SSAE values are much lower as they reach 9.97E-08, 1.04E-08, 8.77E-10, 8.09E-27, and 3.82E-30 for Case 1- Case 5, respectively. The lower values of SSAE confirm the good convergence between the estimated and the actual parameters. Moreover, the convergence behavior of the proposed APO algorithm is explained in Fig. 4(a)-(d) for Cases 1-4.

It is clear that APO has a fast and a good convergence behavior in optimizing the transformer parameters. The effectiveness of the APO algorithm in estimating the transformer parameters are validated through calculating the performance of the transformer using the estimated parameters and compare the reported results with that using the actual parameters. Fig. 5(a)-(d) illustrates a good accuracy of estimating the efficiency and voltage regulation in Cases 1 and 3.

To show the capability, effectiveness, and efficacy of the APO algorithm, a comparative study is applied using four well-known optimizers such as grey wolf optimizer GWO [37], particle swarm optimization (PSO) [38], sine cosine algorithm (SCA) [39], and salp swarm optimization algorithm (SSA) [40].

Tables 6, 7, 8, 9, 10 present the simulation results of the transformers studied through Cases 1–5. The results concerning the transformer parameters show the optimal estimated values of each parameter and the value of SSAE. In addition, statistical indices are recorded in Table 11 showing the best, worst, average, variance, median, and standard deviation of each optimizer over 100 runs. In Case 3, the minimum (best) value is 8.77387E-10 via APO compared to 8.77463E-10 using GWO. Additionally, the variance is 2.04203E-40 via APO where it is 5.12195E-22 via GWO. These results emphasize the superiority of APO over GWO in estimating the parameters of the optimization problem. Also, in Tables 6, 7, 8, 9,10,11, the proposed APO algorithm competes the other well-known optimizers in realizing the lower value of SSAE. The variance (var.) and the standard deviation (STD) also prove the efficacy of the optimizer. The results indicate that APO has the lower variance in Case 1, the lower best, average, variance and STD in Case 2, the lower best, max., average, median, variance and STD in Case 3. The proposed APO outperforms the others in 3 out of 5 Cases and it ranked as the second-best SSAE in Cases 4 and 5 while it ranked as the first at max, average, variance, median and STD in Case 5. All results indicate that the APO is effective, accurate in extracting the optimal parameters of transformers assessed to GWO, PSO, SCA and SSA. The convergence behaviour of the APO optimizer compared to others confirms its ability

and efficacy in optimizing different optimization problem. Fig. 6(a) and (b) shows a good convergence behaviour of APO compared to others at Cases 3 and 4, respectively.

For more validation of the quality of APO, ANOVA test is performed to statistical results of Cases 1–3 alongside the four competitive optimizers and APO. The test checks the realization of the existence of a significant deviation between the mean values of groups. The existence of this difference confirms the effectiveness of using the different algorithms in estimating the optimal parameters of the optimization problem. The results of the test among five optimizer's statistical results are reported in Table 12. The lower value of $\rho$ approves the existence of a significant difference between means of optimizers results (null hypothesis ($p < 0.05$)). Additionally, the best mean (lower value) with APO optimizer is explained using the box of plot, as shown in Fig. 7(a), (b) and (c) for Cases 1–3. It is clear that different statistical indices and tests confirm the efficacy and supremacy of the proposed APO compared to the GWO, PSO, SCA and SSA.

## 5. Conclusions

This paper concludes that the proposed the Arctic Puffin Optimization (APO) for finding the optimal parameter extraction of equivalent circuits for single- and three- phase Power transformers. The APO demonstrates exceptional efficiency in proved for both single- and three-phase power transformers. Applications are conducted to involve five transformers with different size. The APO outperforms competitive algorithms, GWO, PSO, SCA and SSA, by converging faster to an optimal solution with reduced computational time and good convergence behaviour. In the paper, the obtained simulation results are accomplished with experimental verification. The experimental verification shows that the APO maintains high accuracy and reliability across various load and operating conditions, reflecting its adaptability to real-world complexities such as fluctuating loads or non-ideal operating environments. It is particularly beneficial in scenarios requiring quick and accurate modeling for real-time transformer analysis. The experimental results associate closely with the simulated outcomes, confirming that the APO provides accurate parameter estimation under real operating conditions. This ensures its direct applicability in transformer diagnostics, maintenance, and design optimization. For extra validation of the solution quality of APO, ANOVA test is performed to statistical results of three cases alongside the competitive optimizers and APO. In the future studies, the algorithm is not only effective for power transformers, but it can also be extended to other systems requiring equivalent circuit modeling such as polyphase induction motors, transmission lines and the modeling of renewable energy sources.

### CRediT authorship contribution statement

**Amlak Abaza:** Software, Funding acquisition, Conceptualization.
**Ragab A. El-Sehiemy:** Writing – review & editing, Visualization, Supervision, Conceptualization.
**Mohamed Assaad Hamida:** Writing – review & editing, Validation, Supervision, Resources.
**Raphael Chehouard:** Writing – review & editing, Supervision, Methodology, Formal analysis.
**Ahmed Bayomyi:** Writing – original draft, Validation, Software, Methodology.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgement

Ragab A. El-Sehiemy would like to thank the Academy of Scientific Research and Technology for Funding the project # 25158 with STAR
program – First Cycle 2023.

## Data availability

No data was used for the research described in the article.

## References

[1] S.P. Agrawal, P. Jangir, L. Abualigah, S.B. Pandya, A. Parmar, A.E. Ezugwu, A. Smerat Arpita, The quick crisscross sine cosine algorithm for optimal FACTS placement in uncertain wind integrated scenario based power systems, Results. Eng. 25 (2025) 103703, https://doi.org/10.1016/j.rineng.2024.103703.

[2] H. Yang, L. Wang, Y. Zhang, H.M. Tai, Y. Ma, M. Zhou, Reliability evaluation of power system considering time of use electricity pricing, IEEe Trans. Power. Syst. (2019) 34, https://doi.org/10.1109/TPWRS.2018.2879953.

[3] E.S. Ali, R.A. El-Sehiemy, A.A. Abou El-Ela, Optimal partitioning of unbalanced active distribution systems for supply-sufficient micro-grids considering uncertainty, Int. Trans. Electr. Energy Syst. (2021) 31, https://doi.org/10.1002/ 2050-7038.13210.

[4] K. Dawood, S. Tursun, Optimized gap positions for improved leakage impedance in dry-type transformer design, results eng. 23 (2024) 102632. doi:10.1016/j. ri neng.2024.102632.

[5] H. Yu, W. Xiong, K. Xu, Y. Yu, X. Yuan, X. Zou, L. Xiao, A risk assessment method of power transformer based on three-parameter interval grey number decision- making, Appl. Sci. 12 (2022), https://doi.org/10.3390/app12073480.

[6] J. Xue, F. Wang, D. Boroyevich, Z. Shen, Single-phase vs. three-phase high density power transformers, in: 2010 IEEE Energy Convers. Congr. Expo. ECCE 2010 - Proc, 2010, https://doi.org/10.1109/ECCE.2010.5618452.

[7] J.D. Camelo-Daza, D.N. Betancourt-Alonso, O.D. Montoya, E. Gómez-Vargas, Parameter estimation in single-phase transformers via the generalized normal distribution optimizer while considering voltage and current measurements, Results. Eng. 21 (2024), https://doi.org/10.1016/j.rineng.2024.101760.

[8] M.M. Islam, G. Lee, S.N. Hettiwatte, A review of condition monitoring techniques and diagnostic tests for lifetime estimation of power transformers, Electr. Eng. 100 (2018), https://doi.org/10.1007/s00202-017-0532-4.

[9] M.P. Calasan, A. Jovanovic, V. Rubezic, D. Mujicic, A. Deriszadeh, Notes on parameter estimation for single-phase transformer, IEEe Trans. Ind. Appl. 56 (2020), https://doi.org/10.1109/TIA.2020.2992667.

[10] M.I. Abdelwanis, A. Abaza, R.A. El-Sehiemy, M.N. Ibrahim, H. Rezk, Parameter estimation of electric power transformers using coyote optimization algorithm with experimental verification, IEEe Access. 8 (2020), https://doi.org/10.1109/ ACCESS.2020.2978398.

[11] D. van Niekerk, B. Schoombie, P. Bokoro, Design of an experimental approach for characterization and performance analysis of high-frequency transformer core materials, Energies 16 (2023), https://doi.org/10.3390/en16093950.

[12] M. Hazrat Ali, U. Chakma, D. Howlader, M. Tawhidul Islam, K.A. Khan, Studies on performance parameters of a practical transformer for various utilizations, Microsyst. Technol. 26 (2020), https://doi.org/10.1007/s00542-019-04711-w.

[13] S.V. Kulkarni, S.A. Khaparde, Transformer Engineering: Design, 2nd Edition, Technology, and Diagnostics, 2017, https://doi.org/10.1201/b13011.

[14] P.B. Petrović, D. Rozgić, Computational effective modified Newton-Raphson algorithm for power harmonics parameters estimation, IET Signal Process 12 (2018), https://doi.org/10.1049/iet-spr.2017.0573.

[15] H. Ramsin, P. Å. Wedin, A comparison of some algorithms for the nonlinear least squares problem, BIT 17 (1977), https://doi.org/10.1007/BF01932400.

[16] M.S. AbouOmar, A. Eltayeb, M.S. Al-Quraishi, S. El Ferik, Parameter estimation of proton exchange membrane fuel cells using chaotic Newton-Raphson-based optimizer, Results. Eng. 24 (2024) 103369, https://doi.org/10.1016/j. rineng.2024.103369.

[17] K. Deb, Optimization For Engineering Design Algorithms and Examples, ISBN-978- 81-203-4678-9, 2nd Ed., PHI Learn. Priv. Ltd, 2012.

[18] N.E. Karkalos, A.P. Markopoulos, J.P. Davim, Evolutionary-based methods, SpringerBriefs Appl. Sci. Technol. (2019), https://doi.org/10.1007/978-3-319- 92393-2_2.

[19] D. Demirol, F. Oztemiz, A. Karci, Performance comparison of physics based meta- heuristic optimization algorithms, in: 2018 Int. Conf. Artif. Intell. Data Process. IDAP 2018, 2019, https://doi.org/10.1109/IDAP.2018.8620742.

[20] A. Banerjee, D. Singh, S. Sahana, I. Nath, Impacts of metaheuristic and swarm intelligence approach in optimization, Cogn. Big Data Intell. Metaheuristic App. (2021), https://doi.org/10.1016/B978-0-323-85117-6.00008-X.

[21] X. Liu, Mathematical scheduling model of complex industrial process combining swarm intelligence algorithm and swarm dimension reduction technology, Results. Eng. 21 (2024), https://doi.org/10.1016/j.rineng.2024.101796.

[22] S.A. Ahmadi, Human behavior-based optimization: a novel metaheuristic approach to solve complex optimization problems, Neural Comput. Appl. 28 (2017), https:// doi.org/10.1007/s00521-016-2334-4.

[23] A.S.A. Bayoumi, R.A. El Sehiemy, M. El-Kemary, A. Abaza, Advanced extraction of PV parameters' models based on electric field impacts on semiconductor conductivity using QIO algorithm, Sci. Rep. 14 (2024) 1-13, https://doi.org/ 10.1038/s41598-024-65091-4.

[24] R. Devarapalli, B.V. Rao, A. Al-Durra, Optimal parameter assessment of Solar Photovoltaic module equivalent circuit using a novel enhanced hybrid GWO-SCA algorithm, Energy Reports 8 (2022), https://doi.org/10.1016/j.egyr.2022.09.069.

[25] N.E. Gorji, Modified equivalent circuits for CuGaSe2 thin films including space charge limited current, Opt. Quantum. Electron. (2015) 47, https://doi.org/ 10.1007/s11082-014-0072-3.

[26] M.M. Elzalabani, D.M. Atia, A.Y. Eliwa, B.A.A. Zalam, M.S. AbouOmar, Multijunction solar cell parameter estimation based on metaheuristic algorithms, results eng. (2025) 104287. doi:10.1016/j.rineng.2025.104287.

[27] M. Calasan, S. Vujosvić, M. Micev, S.H.E. Abdel Aleem, H.M. Hasanien, Accurate modeling and parameters estimation of photovoltaic models: Analytical and artificial intelligence solutions, Results. Eng. 24 (2024) 103433, https://doi.org/ 10.1016/j.rineng.2024.103433.

[28] F.B. Baz, R.A. El Sehiemy, A.S.A. Bayoumi, A. Abaza, Parameter extraction of proton exchange membrane fuel cell based on artificial rabbits' optimization algorithm and conducting laboratory tests, Sci. Rep. 14 (2024) 1-19, https://doi. org/10.1038/s41598-024-70886-6.

[29] F. Mobeen, S. Ahmad, A. Shoukat, S. Batool, A generalized modeling and analysis for transformer parameterization, in: ICET 2021 - 16th Int. Conf. Emerg. Technol. 2021, Proc, 2021, https://doi.org/10.1109/ICET54505.2021.9689869.

[30] H. Ascencion-Mestiza, S. Maximov, E. Mezura-Montes, J.C. Olivares-Galvan, R. Ocon-Valdez, R. Escarela-Perez, Estimation of the equivalent circuit parameters in transformers using evolutionary algorithms, Math. Comput. Appl. 28 (2023), https://doi.org/10.3390/mca28020036.

[31] J. Yin, Y. Zhu, G. Yu, Power transformer fault diagnosis based on support vector machine with cross validation and genetic algorithm, in: APAP 2011 - Proc. 2011 Int. Conf. Adv. Power Syst. Autom. Prot., 2011, https://doi.org/10.1109/ APAP.2011.6180419.

[32] R.M. Rizk-Allah, M.I. Abdelwanis, R.A. El-Sehiemy, A.S. Abd-Elrazek, An interior search algorithm based on chaotic and crossover strategies for parameter extraction of polyphase induction machines, Neural Comput. Appl. 35 (2023), https://doi.org/10.1007/s00521-022-08055-x.

[33] D.H. Wolpert, W.G. Macready, No free lunch theorems for optimization, IEEe Trans. Evol. Comput. 1 (1997), https://doi.org/10.1109/4235.585893.

[34] W. Wang, W. Tian, D. Xu, H. Zang, Arctic puffin optimization: a bio-inspired metaheuristic algorithm for solving engineering design optimization, Adv. Eng. Softw. 195 (2024) 103694, https://doi.org/10.1016/j.advensoft.2024.103694.

[35] K.K. Burnham, J.L. Burnham, J.A. Johnson, A. Huffman, Migratory movements of Atlantic puffins Fratercula arctica naumanni from high Arctic Greenland, PLoS. One 16 (2021), https://doi.org/10.1371/journal.pone.0252055.

[36] A.L. Fayet, G.V. Clucas, T. Anker-Nilssen, M. Syposz, E.S. Hansen, Local prey shortages drive foraging costs and breeding success in a declining seabird, the Atlantic puffin, J. Anim. Ecol. 90 (2021), https://doi.org/10.1111/1365- 2656.13442.

[37] Y. Li, B. Feng, G. Li, J. Qi, D. Zhao, Y. Mu, Optimal distributed generation planning in active distribution networks considering integration of energy storage, Appl. Energy 210 (2018) 1073-1081, https://doi.org/10.1016/j.apenergy.2017.08.008.

[38] A.A. Fathy, M.S. Elbages, R.A. El-Sehiemy, F.M. Bendary, Static transmission expansion planning for realistic networks in Egypt, Electr. Power Syst. Res. 151 (2017) 404-418, https://doi.org/10.1016/j.epsr.2017.06.007.

[39] E.S. Ali, R.A. El-Sehiemy, A.A.A. El-Ela, S. Kamel, B. Khan, Optimal planning of uncertain renewable energy sources in unbalanced distribution systems by a multi- objective hybrid PSO-SCO algorithm, IET Renew. Power Gener. 16 (2022) 2111-2124, https://doi.org/10.1049/rpg2.12499.

[40] R.A. El Sehiemy, F. Selim, B. Bentouati, M.A. Abido, A novel multi-objective hybrid particle swarm and salp optimization algorithm for technical-economical- environmental operation in power systems, Energy 193 (2020), https://doi.org/ 10.1016/j.energy.2019.116817.