# Identification of Transformer Parameters Using Dandelion Algorithm
## 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]:  
\[\underline{E_{1}}=\underline{V_{1}}-\underline{Z_{1}}\ \underline{I_{1}} \tag{1}\] \[\underline{E_{1}}=\underline{V_{2}^{\prime}}+\underline{Z_{2}^{ \prime}}\ \underline{I_{2}^{\prime}}\] (2) \[\underline{V_{2}^{\prime}}=\underline{Z_{Load}^{\prime}}\ \underline{I_{2}^{ \prime}}\] (3) \[\underline{E_{1}}=\underline{I_{0}}\ \underline{Z_{m}}\] (4) \[\underline{I_{1}}=\frac{V_{1}}{\underline{Z_{eq}}}=\underline{I_{ 0}}+\underline{I_{2}^{\prime}}\] (5) \[\underline{I_{0}}=\frac{E_{1}}{\underline{Z_{m}}}=\frac{\underline {E_{1}}}{R_{c}}+\frac{\underline{E_{1}}}{jX_{m}}=\ I_{e+h}-jl_{m}\] (6) \[\underline{I_{2}^{\prime}}=\underline{I_{1}}\frac{\underline{Z_{m} }}{\underline{Z_{m}}+\underline{Z_{2}^{\prime}}+\underline{Z_{Load}^{\prime}}} \tag{7}\]where \(Z_{1}\), \(I_{1}\), and \(E_{1}\) represent the primary side impedance, current, and induced voltage, respectively. Also, \(\underline{V_{2}^{\prime}}\), \(\underline{Z_{2}^{\prime}}\), \(\underline{I_{2}^{\prime}}\), and \(\underline{Z_{Load}^{\prime}}\) represent the secondary side voltage, impedance, current, and load impedance referred to the transformer's primary side correspondingly. The equivalent impedance is symbolized by \(\underline{Z_{eq}}\). The total excitation current of the transformer \(\underline{I_{0}}\) 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 \(\underline{Z_{m}}\). The input power \(P_{in}\), output power \(P_{out}\), and efficiency \(\eta\) can be calculated using the following equations:  
\[P_{in}=\Re(V_{1}{I_{1}}^{*}) \tag{8}\]  
\[P_{out}=\Re\Big{(}V_{2}^{\prime}{I_{2}}^{*}\Big{)} \tag{9}\]  
\[\eta=\frac{P_{out}}{P_{in}} \tag{10}\]  
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).  
\[O.F=min(SSRD)=min\{f_{1}+f_{2}+f_{3}+f_{4}+f_{5}\} \tag{11}\]  
\[f_{1}=\sum_{i=1}^{N}\bigg{(}\frac{I_{1-e}(i)-I_{1-m}(i)}{I_{1-m}(i)}\bigg{)}^ {2} \tag{12}\]  
\[f_{2}=\sum_{i=1}^{N}\bigg{(}\frac{I_{2-e}(i)-I_{2-m}(i)}{I_{2-m}(i)}\bigg{)}^ {2} \tag{13}\]  
\[f_{3}=\sum_{i=1}^{N}\bigg{(}\frac{P_{1-e}(i)-P_{1-m}(i)}{P_{1-m}(i)}\bigg{)}^ {2} \tag{14}\]  
Figure 1: Per-phase equivalent circuit of transformer.  
\[f_{4} =\sum_{i=1}^{N}\biggl{(}\frac{P_{2-e}(i)-P_{2-m}(i)}{P_{2-m}(i)} \biggr{)}^{2} \tag{15}\] \[f_{5} =\sum_{i=1}^{N}\biggl{(}\frac{V_{2-e}(i)-V_{2-m}(i)}{V_{2-m}(i)} \biggr{)}^{2} \tag{16}\]  
Equation (11) must satisfy the boundary constraints imposed on the control variables of the problem.