Development of Matrix Based Single Equation to Compute OC-SC Tests Parameters for 1-Phase Transformer

Advantages of single-phase transformers are If two and more transformers are connected in parallel the service continuity ensures and the system converts reliable and economical. If more transformers are works in parallel the chance of overburdening of particular transformer reduces. The power supply in a single-phase transformer is through one conductor, while in three-phase transformers, the power supply is through three conductors. A single-phase transformer needs two wires to complete the circuit, while a three-phase transformer needs four wires to complete the circuit. A current carrying electrical power transformer has to withstand various impacts such as: thermal, mechanical, chemical, electrical and electromagnetic during steady state and transient loading conditions. Otherwise it leads to failure due to single or multiple reasons.  Then, it is necessary to comprehend the various causes of power transformer failures. A failure occurs, when the withstand strength of one of the above key properties is exceeded by operating stresses on insulating material and as well as conducting (passive parameters) material. An insulating material testing method as per (Zhenyu Wu, 2020) is used for the judgment of winding faults based on the resonant frequency of the HV-transformer. This testing method provides a new opening to evaluate parameter identification on axial movement and Short-Circuit (SC) fault under minimum time consideration. Likewise, the transformer winding construction can be considered as complex-nonlinear network, which is composed of distributed parameters, such as resistance, capacitance, and inductance according to (M. Florkowski, 2018) and (K. Preis, 2018). 

In case of students to start up the practical knowledge on indirect tests i.e. Open Circuit(OC)-Short Circuit (OC-SC) tests need to calculate efficiency and regulations of 1-phase transformer. These tests are executed with MATLAB simulation environment under no-load, SC test and load test conditions are achieved by (Hammad, 2020) and however in this method transformer saturation analysis is missing. In case of small capacity of 1-phase transformer the performance parameters such as regulations and efficiencies are evaluated using direct load test as per (Abanihi, 2014), yet this direct load test consumes more power and further more rating leads-to more challenging. In (http://www.ee.uidaho.edu/ee/power/jlaw/COURSES/ECE420/S09/Lab/Lab1.pdf, 2007) an experimental technique is applied to determine the phase equivalent parameters using OC-SC tests for three phase transformer and afterwards computed the regulation and efficiency. However, the method claims that tests gives approximate values.  

As per (Teixeira, 2018) to understand influence of harmonics voltage distortion is evaluated by the OC-SC tests for transformer equivalent circuit and losses. In this case, it is able to validate that the harmonic distortions of voltage, which are not able to affect the parameters and its circuit equivalent and losses for 3-phase, 15 kVA transformer. Analysis reported in (Eslami, 2018) for three phase circuits having different types of faults including short circuits in nodes and lines found that for open circuits with falling conductors and having internal faults of transformer, both separately or simultaneously. 

As per (Masaya Hiraide, 2017) standards such as International-Electro-Technical-Commission (IEC) and (IEC62271-100) requires that the Transformer-Limited-Fault (TLF) interruption is verified based on test duty T10 with an amplitude factor equals to 1.7 for the Transient-Recovery-Voltage (TRV). In (Masaya Hiraide, 2017) authors were measured Short-Circuit(SC) inductances as parameter of model transformer in two parts first considering an iron core and second by not considering the same. The short-circuit inductance with the iron core research to value equal to negligibly small and equals to one without the iron core. For the TRV frequency region in (Masaya Hiraide, 2017) observed that is the reason for the phenomena of low amplitude factor. Further, the same report reveals that the judgment of incipient may be due reactance and resistance may cause power transformers faults in the early stage is plays an important role like being long-lived, soft-operation, avoiding possible power discussion, and ensuring a good an energy system. In this (Ali Kirkbas, 2020) analyzed that the performance of transformer due to physical faults, wherein the fault types will vary based on the electrical test methods, in continuation the method transformer confirms the validity and reliability aspects. In case of distribution transformer operation and construction, it is observed on the state quantities that influence, and fuzzy binary quantification is performed using the state quantities. The association between the state quantities and the fault is explored by the fuzzy-algorithm, and the key state-variables that make the transformer fault are extracted in the paper (Zhichun Yang, 2020). Forecast of life of transformer, plays a role, a simple program is submitted in (Rajesh Reddy D, 2020) to follow the health condition.  

Furthermore, winding parameters the leakage inductance act as a performance parameters of transformer because it determines the voltage drop and the short-circuit currents. There are number of ways given (I. Hernández, 2014) to control the leakage inductance: (a) add spaces between the windings; (b) insert a second core between primary and secondary; (c) leave unwound sectors around the core. 

To link to the proposed work, the OC and SC lab test, experimental students can refer to this basic online available, which provides information in the form of spreadsheet for transformer specification as reported (https://electrical-engineering-portal.com/download-center/electrical-software/transformer-losses-calculation, 2021). In recent year (Edwell. T. Mharakurwa, 2022) author done review on life time issues of expectancy of transformer during the operational and performance status in order to minimize failures and operating costs. 

3.1 Modified equation for open circuit test

The open circuit test with different meters like voltmeter (V), wattmeter (W), and an ammeter (A) are connected to the transformer as shown in Figure 1. 

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Figure 1. Open circuit test arrangement for 1-phase transformer

As per Figure 1 in this test, high voltage winding is kept open to discovery the rated output OC voltage, minimum cause for core loss and core loss typical result as tabulated in Table 1 for 2 kVA transformer. 

Table 1. Open Circuit Test Readings 

It is known that, W_oc = Iron losses and W_oc = V_oI_o cos Ø_oc (watt-meter reading); cos Ø_oc = W_oc / (V_oc I_oc) = no load power factor and once cos Ø_oc is obtained, then as per  phasor diagram of Figure 2 under  no-load condition of transformer.

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Figure 2. Phasor diagram under no-load condition of 1-phase transformer

Conventional method to find excitation components under OC-SC tests is as follows:

$I_w=I_o \cos \phi_{o c}$                                                     $I_\mu=I_o \sin \phi_c$

$R_0=V_o / I_{\mathrm{w}}$                                                        $X_0=V_o / I_\mu$

where, I_w = working component of the current; Iµ = magnetizing component of the current.  To start V_oc and W_oc need to be known to determine parameters like, X₀ and R₀ of transformer found in the conventional (in non-matrix form) away. However, to find R₀ and X₀  arrange relation in matrix form using identity matrix as shown in equation (1). 

$\left[\begin{array}{l}R_o \\ X_o\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{l}V_{o c} / I_w \\ V_{o c} / I_\mu\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{c}V_{o c} / I_{o c} \cos \phi_{o c} \\ V_{o c} / I_{o c} \sin \phi_{o c}\end{array}\right]$                 (1)

Then equation (1) can be further simplified in matrix form Table 1 OC Test readings as below: 

$\left[\begin{array}{l}R_o \\ X_o\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{c}V_{o c}{ }^2 / W_{o c} \\ \mathrm{I}_{o c} \frac{V_{o c}}{\sqrt{\left(\mathrm{I}_{o c} V_{o c}\right)^2-W_{o c}^2}} \\ \mathrm{I}_{o c} V_{o c}\end{array}\right]$                        (2)

Furthermore, the equation (2) is simplified to equation (3) in matrix form to find R₀ & X₀    

$\left[\begin{array}{l}R_o \\ X_o\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{c}V_{o c}^2 / W_{o c} \\ V_{o c}^2 / \sqrt{\left(\mathrm{I}_{o c} V_{o c}\right)^2-W_{o c}^2}\end{array}\right]$           (3)

Now, exciting components R₀ and X₀ are directly computed using equation (3) of order 2*2 in MATLAB/Calculator and results are given below: 

$\left[\begin{array}{c}R_o \\ X_o\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{c}115^2 / 22.5 \\ 115^2 / \sqrt{\left(0.8^* 115\right)^2-22.5^2}\end{array}\right]$                 (4)

$\left[\begin{array}{c}R_O \\ X_O\end{array}\right]=\left[\begin{array}{l}587.77 \\ 148.25\end{array}\right]$             (5)

3.2 Modified equation for short circuit test 

When an experimental set-up is made as per Figure.3, then rated current flowing through the windings of transformer and creates copper loss in it. However, iron losses are neglected due to small amount of voltage is sufficient to drive the secondary short circuited condition. Where Table 2 shows the wattmeter reading as W_sc (full load copper losses) and ammeter reading as I_sc and voltmeter reading as V_sc  and these values  can be used to find primary equivalent resistance R₀₁ and X₀₁. 

Table 2. Short circuit (SC) test readings

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Figure 3. SC Test arrangement for 1-phase transformer

From SC test readings, the power factor, cos Ø_oc = V_scI_sc / W_sc ; R₀₁=W_sc /I²_sc and X₀₁ = √(Z₀₁ – R₀₁) and then modified realtions, exlusively for R₀₁ & X₀₁ with k=transmission ratio is expressed in equation (6). 

$\left[\begin{array}{l}R_{01} \\ X_{01}\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{c}R_{0 2} / k^2 \\ \sqrt{\frac{V_{s c}^2}{k^4}-\frac{W_{s c}^2}{k^4 I_{s c}^4}}\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{l}W_{s c} / k^2 I_{s c}^2 \\ \sqrt{\frac{Z_{02}^2-R_{02}^2}{k^4}}\end{array}\right]$  (6)

Then SC Test related equation (6) can be further re-arranged as: 

$\left[\begin{array}{l}R_{01} \\ X_{01}\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{c}W_{s c} / k^2 I_{s c}^2 \\ \sqrt{\left(I_{s c}{ }^2 V_{s c}{ }^2-W_{s c}{ }^2\right) /\left(k^4 I_{s c}{ }^4\right)}\end{array}\right]$

$\left[\begin{array}{l}R_{01} \\ X_{01}\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{c}W_{s c} / k^2 I_{s c}^2 \\ \sqrt{\left(I_{s c}{ }^2 V_{s c}{ }^2-W_{s c}{ }^2\right) / k^4 I_{s c}{ }^4}\end{array}\right]$           (7)

Now, equivalent core components R₀₁ and X₀₁ are directly computed using equation (7) in MATLAB/Calculator and results are given below: 

$\left[\begin{array}{l}\mathrm{R}_{01} \\ \mathrm{X}_{01}\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{c}80 / 2^2 * 8.696^2 \\ \sqrt{\left(8.696^2 * 15^2-80^2\right) / 2^4 * 8.696^4}\end{array}\right]$                     

$\left[\begin{array}{l}R_{01} \\ X_{01}\end{array}\right]=\left[\begin{array}{l}0.264 \\ 0.343\end{array}\right]$

3.3 Single equation for both OC-SC tests 

Now, transformer core components (R₀₁ plus X₀) and equivalent winding parameters (R₀₁ plus X₀₁) can be directly calculated using equation (8) and OC-SC Tests reading tabulated in Table 1 and Table 2, respectively. 

$\left[\begin{array}{l}R_o \\ X_o \\ R_{01} \\ X_{01}\end{array}\right]=\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right]\left[\begin{array}{c}V_{o c}{ }^2 / W_{o c} \\ V_{o c}^2 / \sqrt{\left(\mathrm{I}_{o c} V_{o c}\right)^2-W_{o c}^2} \\ W_{s c} / k^2 I_{s c}^2 \\ \sqrt{\left(I_{s c}{ }^2 V_{s c}{ }^2-W_{s c}^2\right) / k^4 I_{s c}{ }^4}\end{array}\right]$             (8)

Single equation (8) based calculation for parameters of a single-phase transformer in MATLAB. 

$\left[\begin{array}{l}R_o \\ X_o \\ R_{01} \\ X_{01}\end{array}\right]=\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right]\left[\begin{array}{c}115^2 / 22.5 \\ 115^2 / \sqrt{(0.8 * 115)^2-22.5^2} \\ 80 / 2^2 * 8.696^2 \\ \sqrt{\left(8.696^2 * 15^2-80^2\right) / 2^4 * 8.696^4}\end{array}\right]=\left[\begin{array}{c}587.77 \\ 148.25 \\ 0.264 \\ 0.343\end{array}\right]$