# Estimation of Equivalent Circuit Parameters of Single-Phase Transformer by Using Chaotic Optimization Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Transformer Equivalent Circuit

_{load}, is presented in Figure 1 [9,10]. In this figure, R

_{1}, R

_{2}′, R

_{c}, X

_{1}, X

_{2}′ and X

_{m}represent primary coil resistance, secondary coil resistance referred to the primary side, core loss resistance, primary coil reactance, secondary coil reactance referred to the primary side, and magnetizing reactance, respectively. In the entire text, apostrophe ‘denotes the secondary value referred to the primary side.

## 3. Chaotic Optimization Approach

_{1},x

_{2},…,xn] contains the variables ${x}_{i}\in \left[{L}_{i},{U}_{i}\right]$, limited to the lower (L

_{i}) and upper (U

_{i}) permitted value. In this paper, for estimation transformer equivalent circuit parameters, we adopt $n=6$ and $X=\left[{x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6}\right]=\left[{R}_{1},{X}_{1},{{R}^{\prime}}_{2},{{X}^{\prime}}_{2},{R}_{c},{X}_{m}\right]$.

## 4. Application of COA for Transformer Equivalent Circuit Parameters Estimation

_{g}= 600, λ = 0.1 and M

_{L}= 300 in all simulations.

#### 4.1. COA Application for Transformer Parameter Estimation Based on the Nameplate data

_{1}= 6.2A, I

_{2}’ = 6.2A, V

_{2}’ = 2383.8V [9,10].

_{1}), secondary current referred to the primary side (I

_{2}′) and secondary voltage referred to the primary side (V

_{2}′), i.e. [9,10]:

_{1}= 2.0924 Ω, X

_{1}= 1.0358 Ω, R

_{2}′ = 0.5234 Ω, X

_{2}′ = 0.081 Ω, R

_{c}= 1.06∙107 Ω, X

_{m}= 4.77∙106 Ω) we get I

_{1}= 6.2002 A, I

_{2}′ = 6.1999 A, V

_{2}′ = 2383.8 V, i.e. we get better matching between the measured end the estimated full load data.

#### 4.2. COA Application for Transformer Parameters Estimation Based on the Load Data Obtained from Experiments

## 5. Experimental Results and Application of COA

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Transformer equivalent circuit referred to the primary side with connected active load R

_{load}.

Global Search | |

Step 1 | Choose the parameter range. |

Step 2 | Set the initial conditions y_{i}(0) for i = 1,…, n. |

Step 3 | Determine the maximum number of iterations M_{g} for the chaotic global search. |

Step 4 | Form variables x_{i}(k) = L_{i}+y_{i}(k)∙(U_{i}-L_{i}), i = 1, …, n. |

Step 5 | In the k-th iteration for X(k), calculate the fitness function F(X(k)). |

Step 6 | Minimize the fitness function F yielding the vector X* which is input to the local search procedure. |

Local Search | |

Step 1 | Determine the number of iterations for the local search M_{L}. |

Step 2 | In the k-th iteration, form variables x_{i}(k) = x_{i}* ± λ∙y_{i}(k), i = 1, …, n. The sign + or—is selected randomly with equal probability. |

Step 3 | In the k-th iteration for X(k), calculate the fitness function F(X(k)). |

Step 4 | Minimize the fitness function F yielding the final vector X, whose coordinates are R_{1}, X_{1}, R_{2}′, X_{2}′, R_{c} and X_{m} are parameter estimates of the transformer equivalent circuit. |

R_{1} [Ω] | X_{1} [Ω] | R_{2}′ [Ω] | X_{2}′ [Ω] | R_{C} [Ω] | X_{m} [Ω] | |
---|---|---|---|---|---|---|

Actual * | 2.45 | 3.14 | 2 | 2.2294 | 105000 | 9106 |

PSO [9] | 2.25 | 4.082 | 2.2 | 1.8526 | 99517 | 9009 |

GA [9] | 2.76 | 3.414 | 1.68 | 1.846 | 97001 | 8951 |

ICA [10] | 2 | 3 | 1.80 | 2 | 120000 | 9200 |

GSA [10] | 2 | 3.11 | 1.81 | 2.26 | 104281 | 9094.87 |

COA | 1.9854 | 2.6117 | 1.4851 | 1.5203 | 131010 | 10074 |

I_{1} | I_{2}′ | V_{2}′ | |
---|---|---|---|

Measurement | 6.2 | 6.2 | 2383.8 |

PSO * | 6.1979 | 6.1671 | 2371.1 |

Error ** PSO | 0.0021 | 0.0329 | 12.7 |

GA * | 6.1993 | 6.1678 | 2371.4 |

Error GA | 0.0007 | 0.0322 | 12.4 |

ICA * | 6.2051 | 6.1784 | 2375.5 |

Error ICA | 0.0051 | 0.0216 | 8.3 |

GSA * | 6.2081 | 6.1781 | 2375.3 |

Error GSA | 0.0081 | 0.0219 | 8.5 |

COA | 6.2079 | 6.1843 | 2377.7 |

Error COA | 0.0079 | 0.0157 | 6.1 |

**Table 4.**Load data of a single-phase transformer [11].

Load [%] | V_{1} [V] | V_{2} [V] | I_{1} [A] | I_{2} [A] | P_{1} [W] | P_{2} [W] |
---|---|---|---|---|---|---|

50 | 226 | 109 | 4.6 | 8.7 | 1000 | 948.3 |

60 | 225 | 108 | 5.4 | 10.4 | 1180 | 1123.2 |

70 | 225 | 108 | 6.3 | 12.2 | 1400 | 1317.6 |

80 | 223 | 107 | 7.1 | 13.9 | 1568 | 1487.3 |

90 | 223 | 106 | 8 | 15.7 | 1768 | 1664.2 |

100 | 223 | 105 | 8.7 | 17.39 | 1940 | 1826 |

R_{1} | X_{1} | R_{2}′ | X_{2}′ | R_{C} | X_{m} | |
---|---|---|---|---|---|---|

Actual * [11] | 0.428 | 0.21 | 0.508 | 0.03 | 1437.5 | 294.8 |

BFA [11] | 0.428 | 0.43 | 0.493 | 0.024 | 1437.5 | 294.226 |

COA ** | 0.5048 | 0.6048 | 0.7070 | 0.0105 | 2012.6 | 287.179 |

_{1}= 1.002 Ω, X

_{1}= 0.8637 Ω, R

_{2}′ = 0.3308 Ω, X

_{2}′ = 0.0394 Ω, R

_{c}= 5038 Ω and X

_{m}= 170.9988 Ω.

**Table 6.**Measured and estimated values for a single-phase transformer [11].

Load [%] | V_{1} [V] | V_{2} [V] | I_{1} [A] | I_{2} [A] | P_{1} [W] | P_{2} [W] | |
---|---|---|---|---|---|---|---|

50 | Measured | 226 | 109 | 4.6 | 8.7 | 1000 | 948.3 |

BFA [11] | 226 | 110.8 | 4.64 | 8.84 | 1032.9 | 979.1 | |

COA | 226 | 110.06 | 4.57 | 8.78 | 1015.9 | 966.8 | |

Error * BFA | 1.8 | 0.04 | 0.14 | 32.9 | 30.8 | ||

Error COA ** | 1.06 | 0.03 | 0.08 | 15.9 | 18.5 | ||

60 | Measured | 225 | 108 | 5.4 | 10.4 | 1180 | 1123.2 |

BFA [11] | 225 | 109.9 | 5.49 | 10.58 | 1223.4 | 1162.1 | |

COA | 225 | 109.04 | 5.4162 | 10.50 | 1203.8 | 1144.7 | |

Error BFA | 1.9 | 0.09 | 0.15 | 42.4 | 38.9 | ||

Error COA ** | 1.04 | 0.0162 | 0.1 | 23.8 | 21.5 | ||

70 | Measured | 225 | 108 | 6.3 | 12.2 | 1400 | 1317.6 |

BFA [11] | 225 | 109.4 | 6.38 | 12.36 | 1423.8 | 1352.9 | |

COA | 225 | 108.51 | 6.2866 | 12.25 | 1401.0 | 1329.5 | |

Error BFA | 1.4 | 0.08 | 0.16 | 23.8 | 35.3 | ||

Error COA ** | 0.51 | 0.0144 | 0.05 | 1.0 | 11.9 | ||

80 | Measured | 223 | 107 | 7.1 | 13.9 | 1568 | 1487.3 |

BFA [11] | 223 | 108.1 | 7.21 | 14.04 | 1597.1 | 1516.5 | |

COA | 223 | 107.01 | 7.1009 | 13.90 | 1571.1 | 1486.7 | |

Error BFA | 1.1 | 0.11 | 1.04 | 29.1 | 29.2 | ||

Error COA ** | 0.01 | 0.0009 | 0 | 3.1 | 0.6 | ||

90 | Measured | 223 | 106 | 8 | 15.7 | 1768 | 1664.2 |

BFA [11] | 223 | 107.6 | 8.15 | 15.94 | 1808.8 | 1714.8 | |

COA | 223 | 106.44 | 8.0278 | 15.76 | 1778.3 | 1676.8 | |

Error BFA | 1.6 | 0.15 | 0.24 | 40.8 | 50.6 | ||

Error COA ** | 0.44 | 0.0278 | 0.06 | 10.3 | 12.6 | ||

100 | Measured | 223 | 105 | 8.7 | 17.39 | 1940 | 1826 |

BFA [11] | 223 | 107.2 | 9.06 | 17.75 | 2010.7 | 1902.3 | |

COA | 223 | 105.90 | 8.9106 | 17.53 | 1975.5 | 1855.5 | |

Error BFA | 2.2 | 0.36 | 0.36 | 70.7 | 76.3 | ||

Error COA ** | 0.9 | 0.2106 | 0.14 | 35.5 | 29.5 |

_{1}= 1.002 Ω, X

_{1}= 0.8637 Ω, R

_{2}′ = 0.3308 Ω, X

_{2}′ = 0.0394 Ω, R

_{c}= 5038 Ω and X

_{m}= 170.9988 Ω the matching between the measured and estimated values of the observed variables will be much better.

R_{1} | X_{1} | R_{2}′ | X_{2}′ | R_{C} | X_{m} | |
---|---|---|---|---|---|---|

Standard test | 0.5965 | 0.3772 | 0.5965 | 0.3772 | 2494.8 | 235.84 |

Method [9,10] + COA | 0.4519 | 0.1137 | 0.5528 | 0.2001 | 2129.6 | 200.5981 |

Load data + COA + OF_{1} | 0.4414 | 0.1984 | 0.4459 | 0.5039 | 2132.3 | 238.8488 |

Load data + COA + OF_{2} | 0.4275 | 0.2010 | 0.4264 | 0.5039 | 2011.3 | 241.0986 |

Load data + COA + OF_{3} | 0.4321 | 0.2204 | 0.4169 | 0.4928 | 2149.8 | 238.6202 |

R_{Load} [Ω] | V_{1} [V] | V_{2} [V] | I_{1} [A] | I_{2} [A] | P_{1} [W] | P_{2} [W] |
---|---|---|---|---|---|---|

69.4 | 220.23 | 110.25 | 1.28 | 1.59 | 195.7 | 175.2 |

49.5 | 220.09 | 109.99 | 1.51 | 2.22 | 265.9 | 244.6 |

38.7 | 220.30 | 109.91 | 1.78 | 2.84 | 334.2 | 311.9 |

25.2 | 220.25 | 109.44 | 2.45 | 4.34 | 5009 | 474.9 |

19.4 | 219.95 | 108.90 | 3.05 | 5.62 | 640.9 | 611.5 |

12.6 | 220.19 | 108.12 | 4.51 | 8.59 | 972.1 | 928.7 |

9.97 | 219.95 | 107.33 | 5.58 | 10.76 | 1210.4 | 1155.0 |

8.32 | 220.53 | 106.96 | 6.65 | 12.89 | 1450.1 | 1378.6 |

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**MDPI and ACS Style**

Ćalasan, M.; Mujičić, D.; Rubežić, V.; Radulović, M.
Estimation of Equivalent Circuit Parameters of Single-Phase Transformer by Using Chaotic Optimization Approach. *Energies* **2019**, *12*, 1697.
https://doi.org/10.3390/en12091697

**AMA Style**

Ćalasan M, Mujičić D, Rubežić V, Radulović M.
Estimation of Equivalent Circuit Parameters of Single-Phase Transformer by Using Chaotic Optimization Approach. *Energies*. 2019; 12(9):1697.
https://doi.org/10.3390/en12091697

**Chicago/Turabian Style**

Ćalasan, Martin, Danilo Mujičić, Vesna Rubežić, and Milovan Radulović.
2019. "Estimation of Equivalent Circuit Parameters of Single-Phase Transformer by Using Chaotic Optimization Approach" *Energies* 12, no. 9: 1697.
https://doi.org/10.3390/en12091697