The problem of parameters identification for transformer equivalent circuit can be solved by optimizing a nonlinear formula. The objective function attempts to minimize the sum of squared relative errors amongst the accompanying calculated and actual points of currents, powers, and secondary voltage during the load test of the transformer subject to a set of parameters constraints. The authors of this paper propose applying a new and efficient stochastic optimizer called the slime mold optimization algorithm (SMOA) to identify parameters of the transformer equivalent circuit. The experimental measurements of load test of single- and three-phase transformers are entered to MATLAB code for extracting the transformer parameters through minimizing the objective function. Experimental verification of SMOA for parameter estimation of single- and three-phase transformers shows the capability and accuracy of SMOA in estimating these parameters. SMOA offers high performance and stability in determining optimal parameters to yield precise transformer performance. The results of parameters identification of transformer using SMOA are compared with the results using three optimization algorithms namely atom search optimizer, interior search algorithm, and sunflower optimizer. The comparisons are fairly performed in terms of the smallness of objective function. Comparisons shows that SMOA outperforms other contemporary algorithms at this task.

Transformers are major components in power systems that are important in both transmission and distribution networks. They are also used in industrial and household devices. Transformer maloperation significantly affects system performance, reliability, and stability [

A transformer’s equivalent circuit has parameters for resistance and reactance that should be identified for accurate analysis of electric power grids. Transformer parameters also have a major effect on the transformer’s performance in different operating conditions. Accurately estimating transformer parameters arises from the need to improve transformer performance in both steady-state and transient operating conditions. Realistic studies in system behavior require a reasonable transformer model, which plays an important role in the integration between the other components [

A transformer is modeled by considering its nonlinearities [

Generally, transformer parameters can be computed in various ways: standard test executions (no-load and short circuit tests) [

Recent optimization techniques have made major strides in solving power problems such as optimal power flow [

Other researchers have implemented numerous optimization techniques for identifying transformer parameters such as a chaotic optimization algorithm [

Despite this brief survey, the no-free-lunch theorem demonstrates that the possibility of further improvement in estimating transformer parameters remains. To this end, the authors of this paper consider using slime mold optimization algorithm (SMOA), which was created in 2020 to estimate unknown transformer parameters. SMOA was inspired as a novel meta-heuristic algorithm by slime mold oscillation modes in nature and applied effectively to the designs of pressure containers and the welded beams [

The main contribution of this paper can be summarized as follows:

• Application of SMOA to estimate transformer parameters.

• Experimentation of single- and three-phase transformers for validation of the proposed method.

• Comparison of SMOA with other optimization techniques based on their results.

The paper is organized as follows. In Section 2, introduce the equivalent circuit of a transformer. Section 3 gives a short overview of the SMOA method. Section 4 presents our experiment, the numerical results of the application of SMOA, and discussion. Finally, Section 5 introduces our conclusions.

The per-phase equivalent circuit of a three-phase power transformer with respect to its primary side is shown in

Using the fundamental laws of electric circuits, the following equations can be formulated:

Minimizing the sum of squared relative errors (SSRE) amongst the estimated and measured points is the objective function (F_{obje}) for the transformer parameters. It is extracted as,

where F_{obje} is subject to constraints, which are defined by the lower and upper limits of the transformer parameters.

SMOA is inspired by slime mold oscillation modes in nature. The slime mold that inspired this algorithm is

The SMOA method has several features. It is characterized by a unique mathematical model. The adaptive weights permit the SMOA to maintain a specific perturbation rate and guarantee fast convergence that prevents the falling into local optima. It also displays exceptional exploratory and exploitative abilities. To establish the optimal route for obtaining food, SMOA uses adaptive weights to simulate a slime mold’s generation of positive and negative feedback as it spreads, a bio-oscillator created by its search for food. SMAO can also make correct decisions based on historical data due to its excellent utilization of individual fitness values. The description of SMOA model in mathematical terms is in the following paragraphs.

The slime mold uses odors in the air to find food.

and

where

In these equations, the condition referring to

The location of the slime mold

The value of vb changes randomly between [−a, a], and it steadily approaches zero as the iterations increase, while vc oscillates between [1,0] and approaches zero.

The experiments were conducted with two test cases: a single-phase transformer rated at 300 VA, 230/2·115 V, and a three-phase transformer rated at 300 VA, 400/2·200 V. One primary winding and two secondary windings were in the single-phase transformer and in each phase of the three-phase transformer. The load test was carried out on the two transformers and used the measurements to calculate SSRE as F_{obje} to be minimized by SMOA for estimating transformers parameters. SMOA’s results are compared to those from ASO [

Algorithm 1 | ||
---|---|---|

Initialize the parameters, population (popu), |
||

Initialize the slime mold positions |
||

While ( |
||

Determine the fitness of all slime mold; | ||

Update the bestFitness, X_{b}; |
||

Determine W using |
||

For each search | ||

Update p, vb, vc; | ||

Update the positions using |
||

End For | ||

End While | ||

Return bestFitness, X_{b}; |

_{obje} as these optimizers are stochastic.

R_{L} (Ω) |
V_{1} (V) |
V_{2} (V) |
I_{1} (A) |
I_{2} (A) |
P_{1} (W) |
P_{2} (W) |
η (%) |
---|---|---|---|---|---|---|---|

492.0 | 230 | 114.5 | 0.24 | 0.19 | 36.40 | 23.2 | 63.7 |

445.0 | 230 | 114.5 | 0.24 | 0.20 | 37.00 | 24.4 | 65.9 |

398.0 | 230 | 114.4 | 0.25 | 0.23 | 40.30 | 28.1 | 69.7 |

351.0 | 230 | 114.4 | 0.26 | 0.27 | 43.10 | 32.7 | 75.9 |

304.0 | 230 | 114.4 | 0.28 | 0.34 | 50.90 | 41.1 | 80.7 |

257.0 | 230 | 114.3 | 0.34 | 0.47 | 65.79 | 56.9 | 86.5 |

210.0 | 230 | 114.1 | 0.41 | 0.63 | 83.00 | 76.2 | 91.8 |

163.0 | 230 | 113.9 | 0.44 | 0.72 | 95.50 | 90.4 | 94.7 |

116.0 | 230 | 113.4 | 0.58 | 1.00 | 125.40 | 120.0 | 95.7 |

92.5 | 230 | 113.0 | 0.70 | 1.22 | 153.00 | 146.4 | 95.7 |

69.0 | 230 | 112.5 | 0.94 | 1.66 | 209.70 | 197.5 | 94.2 |

R_{L} (Ω) |
I_{1} (A) |
I_{2} (A) |
η (%) | ||||
---|---|---|---|---|---|---|---|

4920 | 242 | 241 | 0.10 | 0.05 | 16.0 | 11.5 | 71.9 |

3980 | 242 | 240 | 0.11 | 0.06 | 18.5 | 14.5 | 78.4 |

3040 | 242 | 239 | 0.12 | 0.08 | 22.6 | 19.0 | 84.1 |

2100 | 242 | 238 | 0.15 | 0.11 | 30.0 | 26.5 | 88.3 |

1160 | 242 | 235 | 0.23 | 0.20 | 52.0 | 47.0 | 90.4 |

690 | 242 | 230 | 0.36 | 0.34 | 86.0 | 79.0 | 91.9 |

502 | 242 | 226 | 0.47 | 0.45 | 110.0 | 100.0 | 90.9 |

361 | 242 | 220 | 0.63 | 0.60 | 150.0 | 132.0 | 88.0 |

SMOA | popu = 30, k_{max} = 50 |

ASO | popu = 30, k_{max} = 50, α = 50, β = 0.2 |

ISA | popu = 30, k_{max} = 50 |

SFO | popu = 30, k_{max} = 50, p = 0.05, m = 0.05 |

After applying SMOA, the estimated transformer parameters are utilized to calculate the currents, powers, and secondary voltages via the fundamental laws of electric circuits. The smallest values of the resultant SSRE were 0.601768 and 1.16306 for the single- and three-phase transformers, respectively.

The parameters of single- and three-phase transformers are extracted using SMOA and calculated the percentage error as shown in

SMOA | 3.0024 | 0.750 | 0.0375 | 0.0070 | 4000 | 1453 |

Datasheet | 3.1000 | 0.775 | 0.0382 | 0.0067 | 3933 | 1437 |

Error | 3.1% | −3.2% | −1.8% | 4.5% | 1.7% | 1.1% |

SMOA | 21.4 | 20.6 | 4.10 | 3.90 | 13054 | 4081 |

Datasheet | 20.9 | 20.9 | 3.97 | 3.97 | 12778 | 4169 |

Error | 2.3% | 1.4% | 3.3% | −1.8% | 2.2% | −2.1% |

Algorithm | SMOA | ASO | ISA | SFO |
---|---|---|---|---|

SSRE | 0.601768 | 0.605171 | 0.601805 | 0.616262 |

Average processing time per run (s) | 4.498597 | 5.963200 | 4.920652 | 5.983119 |

Algorithm | SMOA | ASO | ISA | SFO |
---|---|---|---|---|

SSRE | 1.16306 | 1.19276 | 1.16538 | 1.23168 |

Average processing time per run (s) | 3.460343 | 4.551267 | 3.176482 | 4.338592 |

The plots of I_{1}-R_{L}, I_{2}-R_{L}, V_{2}-R_{L}, P_{1}-R_{L}, P_{2}-R_{L}, and η-R_{L} of the transformers extracted by SMOA and their measured values are displayed in

_{obje} are written. Smaller SD values emphasize the effectiveness of SMOA in identifying the unknown parameters of the two transformers.

Indicator | Single-phase transformer | Three-phase transformer |
---|---|---|

Best | 0.601768 | 1.16306 |

Worst | 0.638345 | 1.22699 |

SD | 0.010234 | 0.01718 |

Obtaining unknown parameters of the transformer equivalent circuit by load tests, is preferred because it requires less data than other methods. The use of optimization algorithms minimizes the deviations between the estimated and measured values of load test data. The authors of this paper propose using SMOA as a precise, quick, and reliable means for generating the best values of the unknown transformer parameters. Our proposed objective function seeks to minimize the sum of squared relative errors (SSREs) between the computed and measured currents, powers, and secondary voltages in a load test of the transformer. Our investigation into a test implementation of SMOA for transformer parameter estimation reveals its improved speed and accuracy compared to existing optimizers. The results show that our proposed SMOA is efficient and dependable, outperforms other approaches in terms of quicker convergence, and has superior accuracy. The authors of this paper conclude that SMOA is a precise algorithm that can be used to optimize a broad variety of parameters in the field of electrical engineering.

slime mould optimization algorithm

the resistance of the primary winding

the leakage reactance of primary winding

the refereed resistance of the secondary winding

the refereed leakage reactance of secondary winding

the core loss resistance

the magnetizing reactance

the primary voltage (V)

the refereed secondary voltage (V)

the impedance of the primary winding

the refereed impedance of the secondary winding

the magnetizing impedance

the total transformer impedance

the input current (A)

the refereed secondary current

the no-load current (A)

the input power (W)

the output power (W)

the efficiency (%)

the sum of squared relative errors

the number of measurements

the measured values

the estimated values

the objective function

the location of the slime mould

the candidate with the highest order concentration in the iteration k

iteration number

the maximum number of iterations

two randomly selected candidates from the slime mould swarm

variable lies in the range [-a, a]

variable that linearly decreased from one to zero

the slime mould weight

the best fitness in the current iteration

the worst fitness in the current iteration

the fitness of candidate X

the best-obtained fitness throughout all iterations

and r random vector in the range of [0,1]

the border limits of search space

population

atomic search optimizer

interior search algorithm

sunflower optimizer

The authors gratefully acknowledge the approval and the support of this research study by Taif University Researchers Supporting Project number (TURSP-2020/86), Taif University, Taif, Saudi Arabia.