Voltage Distribution on Transformer Windings Subjected to Lightning Strike Using State-Space Method

Abstract

Transient analysis in power systems is essential for identifying deficiencies in the system, as well as for the protection and design of equipment. Transients can arise from natural events or network operations; in either case, they have the potential to cause significant damage to transmission lines, protection devices, generators, or transformers. This study examines a 20 kA, 1.2/50 µs lightning strike on a distributed-parameter transmission line connected to a power transformer. The voltage distributions across the winding sections on the neutral grounded high-voltage side of a disc-structured power transformer were obtained using the state-space method. An equivalent circuit for the state-space model was also developed in the Alternative Transients Program–Electromagnetic Transients Program (ATP-EMTP), and the results from both methods were compared. Both approaches revealed that the voltage waveforms in the transformer’s winding sections were consistent, with the voltage distribution decreasing linearly. Additionally, the voltage–current waves reached the transformer with a specific delay, depending on the characteristics of the transmission line and the location of the lightning strike. The impact of an increase in the grounding resistance value on the high-voltage side of the transformer on voltage distribution and peak voltage levels was examined. The proposed method effectively captures the voltage–current behavior of the transmission line and transformer windings during transient conditions. It is concluded that the state-space method serves as a viable alternative for transient analysis in power systems and can enhance the design of protection equipment and winding insulation studies.

1. Introduction

Increasing electrical energy consumption makes a continuous and high-quality energy supply essential for both individual and commercial consumers. Accordingly, various measures are taken to prevent failures that may occur in electricity networks during the production, transmission, and distribution of energy. Maintenance and repair of network elements, such as power lines, protective devices, transformers, and others, are performed at regular intervals to minimize failures. In addition, when designing a power line or electrical equipment, both steady-state and transient analyses are necessary. The main reason for this is that the peak current–voltage values of the system in a steady state may be lower than those in the transient state. If the design relies solely on steady-state conditions, protection and insulation failures may occur in the line and other network elements. Events such as lightning strikes, short circuits, switching, and sudden load changes can induce transients.

When transformer windings are subjected to very high voltage during transients, malfunctions such as insulation breakdown, insulation deterioration, overheating of the windings, and arcing between the windings can occur. Therefore, when designing a power system, it is crucial to ensure that it can withstand the worst conditions to maintain energy continuity, durability, and long-term performance.

Transient analysis in transformers has been a subject of research in recent years. In [1], transformer windings were divided into a finite number of segments to represent an electrical network, and a mathematical model for this network was created. In [2], a 345 kV transmission system was energized from a limited-capacity diesel generator, and the inrush currents of the transformer were analyzed using the Electromagnetic Transients Program (EMTP). Current analyses were performed under different load conditions, and inrush current values were examined. In [3], a power transformer was modeled using the ATP-EMTP. To analyze the performance of the transformer relay modeled in MATLAB, transient waveforms were generated under various operating conditions. In [4], voltage calculations were performed in the frequency domain using a hybrid model comprising a Multi-Conductor Transmission Line (MTL) and a Single Transmission Line (STL). In [5], the voltage distribution across the windings was investigated employing the input terminal impedance method. In [6], a method was proposed that utilized the state variables technique, with inductance and capacitance matrices, as well as resistance matrices, obtained from Finite Element Method (FEM) simulations. In [7], voltage distributions in windings were studied using black-box and vector-fitting techniques, alongside a voltage distribution factor. In [8], the resistance, inductance, and capacitance values of the windings were calculated, and the circuit equations were solved in the frequency domain using Wolfram Mathematics. In [9], a study on a 245 kV voltage transformer was conducted to investigate voltage distribution within the windings when subjected to a 2.5/50 µs lightning strike. The Finite Element Method and analytical formulas were employed to determine the model parameters, and a simulation program was used for analysis. The data obtained were compared with experimental results, revealing that the results were almost consistent and that voltage levels decreased from the first disc to the last disc. In [10], transient analysis of a transformer was performed using a Hybrid Analytical-FEM approach. A high-frequency model was used for the transformer windings, which were modeled as lumped RLC parameters and solved in the time domain using Dommel’s method. The data from the proposed method were compared with measured data, concluding that the hybrid model approach is suitable for transient analysis of transformer windings. In [11], lightning impulse testing was conducted using full-wave (FW) and chopped-wave (CW) impulses in the time domain. The test circuit was developed using the state-space method. The equivalent circuit model of the transformer was obtained through frequency response analysis (FRA) measurements and vector-fitting (VF) methods. Full- and chopped-voltage waveforms were generated using the state-space technique. The signals produced from the simulation were compared with measurement data, demonstrating compatibility between them. It was concluded that the proposed method saves time and effort in lightning impulse testing. In [12], a comparison of initial voltage distributions in the windings of layer-type and disc-type power transformers was conducted. Experimentally, a peak voltage of 50 V and a 1.2/50 µs lightning pulse were applied to the transformer, alongside electrostatic potential simulations. The study revealed that initial voltage distributions differ in transformers with layer-type and disc-type windings due to variations in the lumped capacitance networks of the windings.

Transient analysis can be performed in either the time domain or the frequency domain. Today, various simulation programs are available for this purpose, including the ATP-EMTP (Alternative Transients Program–Electromagnetic Transients Program), PSCAD (Power Systems Computer-Aided Design), MATLAB/Simulink, RTDS (Real-Time Digital Simulator), and PSS^®E (Power System Simulator for Engineering). One method utilized for transient-state analysis in the time domain is the state-space model, which allows for the modeling of time-dependent behaviors of dynamic systems.

The ATP-EMTP models capacitances and inductances as current sources and resistances, employing the trapezoidal rule for the transformation of these parameters. However, in this study, the parameters were modeled analytically, while the trapezoidal rule was utilized for numerical integration in solving the equations. The study, conducted using the state-space method, exhibits certain differences compared to other simulation programs. In the ATP-EMTP, the step size is constrained, making it impossible to achieve arbitrarily small step sizes. This limitation poses a disadvantage in analyzing very-short-duration transient regimes, as smaller step sizes correspond to higher accuracy. In contrast, the state-space method imposes no restrictions on step size.

Lightning impulse tests are an important consideration in the transient analysis of transformer windings. Based on their effects and physical characteristics, lightning strikes are generally divided into two categories: direct lightning strikes and indirect lightning strikes. These two main categories are further divided into various subcategories, including positive and negative lightning strikes, as well as single and multiple strikes. Lightning strikes can be modeled in different ways. The double-exponential model, Bruce–Golde model, CIGRE (Conseil International des Grands Réseaux Electriques) model, Diendorfer–Uman model, and Heidler model are some of the lightning strike models used in the literature [13].

In this study, the time-dependent voltage distributions created in the windings of the high-voltage side of the transformer, due to a lightning strike on the phase conductor of a transmission line, were modeled using the state-space method. The modeling involved a transmission line with distributed parameters, a Heidler lightning model, and the high-voltage side of a disc-structured power transformer. First-order differential equations were derived using capacitor voltages and inductor currents, which were then transformed into state-space form. The equations in state-space form were solved using the trapezoidal integration rule, yielding the voltage and current waves of the transmission line and each disc on the high-voltage side of the power transformer. Additionally, the same circuit was drawn and simulated in the ATP-EMTP, and the results were compared with the data obtained from the state-space method.

When constructing the equivalent circuit of disc-type power transformers, a separate circuit must be drawn to represent each disc. For transformers with a large number of discs, this process becomes time-consuming due to the extensive number of discs. However, such a drawback does not exist in the state-space method. In this approach, it is sufficient to know the number of discs. Once the disc count is input into the prepared code, all equations corresponding to that circuit are generated, and the required matrices are constructed automatically.

This paper is organized into five sections. The Section 1 is the introduction, the Section 2 covers network elements, the Section 3 discusses the materials and methodology, the Section 4 presents the application and results, and the Section 5 is the conclusion.

2. Network Elements

2.1. Lightning Surge Model

The Heidler lightning surge model used in this study features the direct lightning impulse model, which is one of its main categories. The Heidler model is among the most frequently used models for accurately analyzing transient overvoltages caused by lightning, owing to its realistic current profile, its endorsement as a reference model by standards organizations such as the International Electrotechnical Commission (IEC 62305-1), and its wide range of application areas. While the calculations involved in this model are more complex than those in other models and require more computations, this complexity can be viewed as a disadvantage. However, it ultimately yields more realistic results compared to simpler models. The function of the Heidler model is provided in (1) [14,15].

$$I\left(t\right)={\displaystyle \frac{I_{max}}{\eta }}x\left(t\right)y\left(t\right)={\displaystyle \frac{I_{max}}{\eta }}{\displaystyle \frac{{\left(\frac{t}{{\tau }_1}\right)}^n}{1+{\left(\frac{t}{{\tau }_1}\right)}^n}}{e}^{-{\displaystyle \frac{t}{{\tau }_2}}}$$

(1)

where $I_{max}$ is the peak value of the lightning current, η is the correction factor of the peak value, t is the time, ${\tau }_1$ and ${\tau }_2$ are time constants depending on current rise (front) time and fall (decay) time, and n is the current steepness factor (constant of current gradient), respectively. The n value is a parameter that affects how the current rises. A higher n value provides a faster rise. The expression x(t) controls the phase of the current rise. With this expression, the current can reach its peak value quickly. The expression y(t) controls the phase of the current fall. This expression determines how the current decreases after it reaches its peak value. The mathematical expression of the correction factor in (2) is obtained by taking the derivative of (1), equating it to zero, and performing various calculations [16].

$$\eta =e^{-\frac{{\tau }_1}{{\tau }_2}}\sqrt[n+1]{\frac{n·{\tau }_2}{{\tau }_1}}$$

(2)

When modeling the Heidler lightning surge, a current source is connected in parallel with a resistance known as the lightning path impedance, as shown in Figure 1. The value of the lightning path impedance can vary based on factors such as the length of the lightning channel, atmospheric conditions, and the physical properties of lightning. In this study, the lightning path impedance is taken to be 400 Ω [17].

2.2. Power Transmission Lines

Transmission lines can be modeled using either lumped parameters or distributed parameters. Lumped-parameter line models consist of numerous π, T, or Γ sections [18] connected in cascade, each formed by resistance, inductance, capacitance, and conductance per section. However, since the line parameters are distributed uniformly along the line, the distributed-parameter transmission line model is more realistic. The well-known Telegrapher equations represent the distributed structure of the line, and the current and voltage values of a line segment are calculated from the solution of the space and time-dependent partial differential equations.

To model a lossless transmission line, an equivalent circuit consisting of current sources and impedance elements can be drawn [19]. The equivalent circuit of this lossless line is a two-port equivalent circuit and is shown in Figure 2, where $Z_0=\sqrt{l/c}$ is the characteristic impedance of the lossless transmission line with inductance and capacitance per unit length l and c.

The method of characteristics can be used to simulate a lossless or lossy line. Assuming that the total effective resistance of a lossy line is R′ = rd, a lossy transmission line can be represented by adding R′/2 at the beginning and end of the transmission line or by adding R′/4 to the terminals and R′/2 to the middle of the line, where r is the resistance of the transmission line per unit length and d is the line length. In this case, Z₀ in Figure 2 is replaced by the Z = Z₀ + R′/4 relation and the past history currents are determined as follows [20,21]:

$$I_1\left(t-\tau \right)=\left({\displaystyle \frac{1+h}{2}}\right)\left[\left(-1/Z\right)v_2\left(t-\tau \right)-h{i}_{\mathrm{2,1}}\left(t-\tau \right)\right]+\left({\displaystyle \frac{1-h}{2}}\right)\left[\left(-1/Z\right)v_1\left(t-\tau \right)-h{i}_{\mathrm{1,2}}\left(t-\tau \right)\right]$$

(3)

$$I_2\left(t-\tau \right)=\left({\displaystyle \frac{1+h}{2}}\right)\left[\left(-1/Z\right)v_1\left(t-\tau \right)-h{i}_{\mathrm{1,2}}\left(t-\tau \right)\right]+\left({\displaystyle \frac{1-h}{2}}\right)\left[\left(-1/Z\right)v_2\left(t-\tau \right)-h{i}_{\mathrm{2,1}}\left(t-\tau \right)\right]$$

(4)

where $h=(Z_0-{\displaystyle \frac{rd}{4}})/(Z_0+{\displaystyle \frac{rd}{4}})$, $\tau =d\sqrt{lc}$.

2.3. Modeling of Power Transformer Winding

Various winding models are utilized for the transient analysis of power transformers, including Multi-Conductor Transmission Lines (MTLs), Circular Multi-Conductor Transmission Lines (CMTLs), ladder networks, and hybrid models (MTL–ladder networks) [22]. MTLs and CMTLs are turn-based models, while this study employs the disc-based ladder network model. Figure 3 illustrates the double-ladder network model, which incorporates both primary (high-voltage side) and secondary (low-voltage side) windings. In contrast, the single-ladder network model represents only the high-voltage or low-voltage windings of the transformer. For this study, a single-ladder network model was utilized. The ladder network is commonly used in frequency response analysis studies of transformers and is also referred to as the high-frequency power transformer model.

The ladder network model of a power transformer consists of a cascade connection of a large number of sections with the parameters R, L, and C. Each section, which consists of lumped parameters, represents a disc in the equivalent circuit. Consequently, the number of discs in the transformer windings determines the number of sections used in the model. The lumped parameters in the sections R_s, L_s, C_s, C_g, and C_HL represent the series resistance, self-inductance, series capacitance, shunt (ground) capacitance, and the capacitance between the high-voltage and low-voltage windings of the transformer, respectively. By utilizing winding design details, such as the geometry and physical properties of the power transformer, or previously measured data, the parameters of the transformer ladder network model can be estimated or calculated. The Finite Element Method, analytical approximation, and terminal-based measurement are some of the methods used to calculate the model parameters [23].

Transformer parameters are distributed as transmission lines. However, in [24,25,26], ladder circuits composed of lumped parameters are proposed for transformer modeling. As a distributed model of the transformer windings suitable for state-space analysis does not exist, a lumped-parameter model composed of ladder circuits is used for both state-space representation and ATP-EMTP simulations of the transformer winding.

2.3.1. Calculation of Series Resistance of the Winding

Equations (5) and (6) are used to calculate the series resistance value (R_s) of the disc-type winding.

$$R_s=R_{s1}{\mathrm{Ƈ}}_{ir}$$

(5)

where ${\mathrm{Ƈ}}_{ir}$ represents the circumference resistance of the conductor and ${\mathrm{Ƈ}}_{ir}$= 2πr.

$$R_{s1}={\displaystyle \frac{1}{2(h+w)}}\sqrt{{\displaystyle \frac{\pi f\mu }{\sigma }}} \Omega /\mathrm{m}$$

(6)

h, w, µ, σ, and f in (6) represent the height of the turns, width of the turns, permeability, conductivity, and signal frequency of the conductor, respectively. The conductance matrix G is calculated as in (7).

$$G=2\pi f{C}_{s}tan\delta$$

(7)

Here, C_s represents the series capacitance and tanδ represents the insulation dissipation factor [27].

2.3.2. Calculation of Self- and Mutual Inductance of the Winding

Equations (8) and (9) are used to calculate the self-inductance and mutual inductance values between turns [28].

$$L_{km}={\mu }_o{N}_k{N}_m\sqrt{\left(d_1{d}_2\right)}{\displaystyle \frac{2}{k}}\left[\left(1-{\displaystyle \frac{k^2}{2}}\right)M\left(k\right)-E(k)\right]$$

(8)

where M is given by

$$M=\sqrt{{\displaystyle \frac{4d_1{d}_2}{{d_3}^2+{(d_1+d_2)}^2}}}$$

(9)

Here, L_km is the mutual inductance value between discs k and m. ${\mu }_o$ is the magnetic permeability value of the medium, and ${\mu }_o=4\pi \times {10}^{-7} \mathrm{H}/\mathrm{m}$. While N_k and N_m represent the number of the kth section and mth section, d₁ and d₂ represent the mean radii of the discs, and d₃ represents the axial distance between the discs. When k = m for self-inductance, it is calculated as d₃ = 0.02235(h + w). In Figure 4, h₁ and w₁ are similar to h₂ and w₂. M and E represent elliptic integrals of the first and second kind. For the L_km matrix, the diagonal elements represent the self-inductance, while the off-diagonal elements represent the mutual inductance between the discs. Since the average radius of the sections is the same, the d₁ and d₂ values are equal in this calculation [29].

2.3.3. Calculation of Series Capacitance of the Winding

To calculate the series capacitance value according to the transformer winding structure, the capacitance value between the conductors in the same disc (C_t) and the capacitance between the conductors in the adjacent disc, also known as the capacitance between the discs (C_d), are calculated. Structure of single-phase disc-type winding is shown in Figure 5.

The capacitance value between the conductors in the same disc (between two turns) is calculated in (10) [27,31].

$$C_t={\displaystyle \frac{2\pi {\epsilon }_r{\epsilon }_o{d}_w}{ln\frac{b_1}{b_2}}}$$

(10)

Here, b₁ represents the inner radius of the outer conductor, b₂ represents the outer radius of the inner conductor, and d_w represents the length of the conductor. ${\epsilon }_o$ is the permittivity of free space and ${\epsilon }_r$ is the relative permittivity of the insulation paper between the inner and outer conductors, where ${\epsilon }_o=8.854\times {10}^{-12}$ and ${\epsilon }_r=3.85$. Equations (11–14) are used to calculate the capacitance (C_d) value between any two discs [29].

$$C_d=\epsilon {\displaystyle \frac{S}{d}}$$

(11)

$$S=\pi \left({\left({\displaystyle \frac{d_o}{2}}\right)}^2-{\left({\displaystyle \frac{d_i}{2}}\right)}^2\right)$$

(12)

$$S=\pi \left({\displaystyle \frac{d_o^2}{4}}-{\displaystyle \frac{d_i^2}{4}}\right)$$

(13)

where $\epsilon ={\epsilon }_r{\epsilon }_o$.

$$C_d={\epsilon }_r{\epsilon }_o\pi {\displaystyle \frac{{d^{\prime }}_o^2-{d^{\prime }}_i^2}{4d}}$$

(14)

where ${d^{\prime }}_i=d_i-d$ and ${d^{\prime }}_o=d_o+d$.

In the equations, d, d_o, and d_i represent the distance between the turns, the outer diameter of the turn, and the inner diameter of the winding, respectively. S is the surface area of the disc. According to the equivalent circuit of the transformer capacitance in Figure 6, (15) is used to calculate the total C_t value and (16) is used to calculate the total C_d value.

$$C_{t total}={\displaystyle \frac{1}{\frac{1}{C_{t1}}+\frac{1}{C_{t2}}+\frac{1}{C_{t3}}+\frac{1}{C_{tn}}}}$$

(15)

$$C_{d total}=C_{d1}+C_{d2}+C_{d3}+C_{dn}$$

(16)

Using Figure 6 and Equations (15) and (16), the capacitance value of the transformer winding for a disc can be calculated as C_d + C_t. However, (17) is used to calculate the total series capacitance value of the winding.

$$C_s={\displaystyle \frac{1}{N_d}}\left({\displaystyle \frac{C_t}{mt}}+4C_{dt}\right)$$

(17)

$$C_{dt}={\displaystyle \frac{(mt-1)(2mt-1)}{6mt}}{C}_d$$

(18)

where mt is the number of turns per disc.

$$C_g=\left(2\pi {\epsilon }_r{\epsilon }_o{d}_w\right)/\mathrm{l}\mathrm{n}(d_i/d_o)$$

(19)

Here, C_dt is calculated as the result of the capacitance between the discs. Apart from C_t and C_d, there is also a capacitance to ground called C_g in the disc model of the transformer winding. This capacitance value of C_g is calculated by (19) [29].

The voltage distribution factor $\left(a\right)$ is an important factor when analyzing the voltage distributions in transformer windings. This factor is calculated with the formula $a=\sqrt{C_g/C_s}$ [32]. It is desired that the value $a$ is small. Otherwise, transformer insulation may be at risk. Therefore, this factor can be beneficial for winding insulation in transformer design.

2.3.4. Penetration Depth and Ferromagnetic Effects

Ferromagnetic materials, such as iron, exhibit high magnetic permeability, resulting in a very small penetration depth. Consequently, magnetic fields penetrate only the surface region of the iron core and do not reach its deeper layers. This phenomenon is known as the skin effect. At very high frequencies, the penetration depth becomes significantly smaller than the physical dimensions of the iron core. Under these conditions, the magnetic flux is concentrated near the surface, rendering the inner regions of the core ineffective and non-contributory to the magnetic flux. Therefore, magnetization within the inner part of the iron core can be neglected at high frequencies.

The relationship between penetration and the iron core determines how electromagnetic waves or magnetic fields propagate within ferromagnetic materials, such as iron. This relationship is crucial for transformer design and for understanding the effects of core losses in high-frequency applications. The penetration depth is expressed as follows [33,34]:

$$\delta =\sqrt{{\displaystyle \frac{2}{\omega \mu \sigma }}}$$

(20)

where ω = 2πf is the angular frequency, µ is the magnetic permeability of the iron core, and σ is the electrical conductivity.

In this study, the lightning source is in the range of µs, which corresponds to the levels of MHz in the frequency domain. Therefore, it is considered that penetration depth is low and magnetization effect can be neglected in the proposed method and in the ATP-EMTP simulation. This is mainly due to the very low magnetic penetration depth of power transformers at high frequencies [35], as described above. Flux penetrations differ at different frequency values such as 20 Hz, 500 kHz, and 1 MHz. At 500 kHz and 1 MHz, the flux penetration decreases significantly. As the frequency increases, the magnetic field strength decreases and eddy currents are induced in the conductor (core and winding) [36]. In transformer modeling studies, assumptions have been made that the core effect can be neglected, especially at frequencies above 10 kHz [37].

3. Materials and Methodology

3.1. Lightning Source and Transmission Line Parameters

In this study, the Heidler lightning model shown in Figure 1, a single-phase distributed-parameter transmission line model shown in Figure 2, and the high-voltage side of a transformer with a disc-based winding structure were used. The parameters of the lightning model and the lightning path impedance are given in

Table 1. Heidler lightning model parameters.

$I_{max}$	20 kA
${\tau }_1$	1.2 µs
${\tau }_2$	50 µs
$n$	10
$Z_h$	400 Ω

.

The parameters of the distributed transmission line are presented in

Table 2. Transmission line parameters.

d	300 km
$r$	0.02 Ω/km
$l$	1.14 mH/km
$c$	9.8 nF/km
$L$	50 mH

, which are obtained from [21]. The length of the single-phase distributed-parameter transmission line is d, the series resistance of the line is r, the series inductance of the line is l, the capacitance of the line is c, and the series inductance of the source side is L.

3.2. Power Transformer Parameters

In the system considered, it is assumed that the transmission line is terminated by a 40 MVA power transformer. The high-voltage side of the transformer consists of 10 discs. The capacitance between the high-voltage winding and the grounded tank is represented as C_g, the capacitance between the discs is C_tr, the inductance for each series disc is L_tr, the series resistance is R_tr, and the grounding resistance connected to the end of the winding is R_g. The capacitance between the high-voltage and low-voltage windings is C_HL. The transformer parameters obtained from [38] are presented in

Table 3. Transformer parameters.

	High-voltage Side	Low-voltage Side
Cg	61.19 pF	115.53 pF
Ctr	393.4 pF	127.67 pF
Ltr	10 µH	10.5 µH
Rtr	1 Ω	0.25 Ω
Rg	1 Ω	1 Ω
CHL	89.283 pF

.

The number of discs is an important parameter in the high-frequency model of a transformer. When modeling a disc-based power transformer, the number of sections in the equivalent circuit is determined by considering the number of discs in the transformer. The number of sections in the equivalent circuit corresponds directly to the number of discs in the actual transformer. In this study, the high-voltage side of the transformer consists of 10 discs, resulting in an equivalent circuit for the high-voltage winding that includes 10 sections. Each section comprises the circuit elements C_g, C_tr, L_tr, and R_tr. The equivalent circuit, which combines the Heidler lightning strike model, a transmission line with distributed parameters, and the high-frequency model of the power transformer with N discs, is presented in Figure 7. The ATP-EMTP simulation model is illustrated in Figure 8.

3.3. State-Space Modeling and Analysis

The state-space method is used to investigate the effects of a lightning strike on a single-phase transmission line and transformer windings. The equivalent circuit shown in Figure 7 is employed for transient analysis. Based on this model, equations for the inductor currents and capacitor voltages are derived using the graph theory. The general form of the state equations is expressed in (21). However, as presented in Appendix A, the state equations take the form shown in (22) for the system illustrated in Figure 7. To resolve this discrepancy, both sides of the equation were multiplied from the left by F⁻¹, resulting in (23) and (24).

$$\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)$$

(21)

$$F\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)$$

(22)

$$F^{-1}F\dot{x}\left(t\right)=F^{-1}\left(Ax\left(t\right)+Bu\left(t\right)\right)$$

(23)

$$\dot{x}\left(t\right)=F^{-1}Ax\left(t\right)+F^{-1}Bu\left(t\right)$$

(24)

or

$$\dot{x}\left(t\right)=A^{\prime }x\left(t\right)+B^{\prime }u\left(t\right)$$

(25)

where $A^{\prime }=F^{-1}A$ and $B=F^{-1}B$.

To obtain the state equations, first matrices A, B, and F are formed by incorporating the values of the circuit elements of the lightning model, transmission line, and transformer. Next, values for x_k+1 are calculated by updating A′ each time using the trapezoidal integration rule. The current and voltage values of the lossy transmission line are calculated using (3) and (4). The current and voltage values in the transformer windings are determined based on capacitor voltages and inductor currents. The flow diagram of the analysis is illustrated in Figure 9.

While creating matrices A, B, and F, it is important to consider the number of discs used in the disc structure transformer winding. The state-space equations derived from the circuits for N = 1, N = 2, and N = 3 are provided in the Appendix A, in (A1), (A2), and (A3). Here, the value of N represents the number of discs (sections) in the transformer.

As can be seen from (A1), (A2), and (A3), the dimension of state equations obtained increases as the number of discs increases. In the case of a large number of discs, obtaining these equations will both take time and may cause errors in the creation of the equations. When these equations are examined, matrix elements and dimensions change according to a certain pattern rule. In this study, the codes for the simulation were written by including the pattern rule. Using the prepared algorithm, the matrices needed for any N value are automatically obtained. In the case that a transformer consisting of N discs is used in the circuit, F and A matrices are produced as (2N + 2) × (2N + 2) and the B matrix is produced as (2N + 2) × 3. Accordingly, for the model used in this study, the generated F and A matrices are (22 × 22) dimensional, and the B matrix is (22 × 3) dimensional. Integrating the pattern rule into the written codes provides ease of operation. C_2N−1, C_N+1, R_N, L_N, L_s, Z_h, and Z₀ in the equations represent C_g, C_tr, R_tr, L_tr, series inductance, lightning path impedance, and characteristic impedance of the line, respectively. I_h(t) = I_max is the maximum current value in the Heidler lightning model. i_L(t) and V_c(t) are the inductor currents and capacitor voltages, respectively. The transient analysis method used in the study was developed by coding completely in MATLAB without using SIMULINK simulations.

4. Application and Results

The correct operation of the model used in the study first of all requires the correct modeling of the source. The Heidler lightning model used as the source was simulated with codes written in MATLAB and drawn in the ATP-EMPT program. The graphs of Heidler current (I_H) given in Figure 10 and Figure 11 are obtained from the MATLAB model and the ATP-EMTP model, respectively. When the current–time graphs in both figures are examined, it is seen that they are compatible with each other.

The voltage–time graphs of the transmission line sending end (V_s) and receiving end (V_r) obtained using the state-space method are given in Figure 12. The voltage–time graph of the ATP-EMTP model is given in Figure 13.

At t = 0 ms, the lightning strike generates a sharp voltage peak at the sending end, reaching approximately 1.2 MV, followed by a significant negative peak (−1.8 MV) at t = 2 ms due to wave reflections. The delay between V_s and V_r reveals the wave propagation time along the line, with V_r exhibiting lower-amplitude responses due to attenuation and energy dispersion. The lightning wave reaches the receiving end of the transmission line with a delay of τ = 1 ms. The delay in the line is calculated by dividing the length of the line by the characteristic impedance $(\tau =d/\sqrt{lc)}$. The receiving-end voltage demonstrates significant attenuation compared to the sending-end voltage, highlighting the lossy nature of the transmission line and its distributed parameters. A secondary high-amplitude transient at t = 4 ms in V_s indicates potential resonance or multi-reflection effects within the system. It is seen that the waveforms and voltage values given in Figure 12 and Figure 13 are compatible with each other.

The transmission line sending-end (I_s) and receiving-end (I_r) current–time graphs are given in Figure 14. The current–time graph of the ATP-EMTP model is given in Figure 15. The application of a lightning strike at the sending end of the transmission line results in a sudden rise in I_s. The current at the sending end sharply increases following the lightning strike, reaching an initial peak of approximately 3.5 kA. In contrast, the current at the receiving end exhibits significantly higher amplitudes, with the first peak reaching approximately 6.6 kA. This difference represents the transient effects caused by the high-frequency energy introduced by the lightning impulse. The initial peak reflects the rapid rise time of the impulse and the system’s response to this high-energy disturbance. Multiple peaks and troughs are observed in both I_s and I_r, indicating the presence of reflection and damping effects along the transmission line. The difference in impedances between the transmission line and transformer causes some of the waves to reflect back, leading to the transient oscillations observed at both ends. High-frequency transients dominate during the first few milliseconds. Over time, the amplitudes of these transients decay, and the system approaches a more stable state, demonstrating the effectiveness of the line’s damping mechanisms. Lightning impulses typically have a broad-frequency spectrum, and the sharp rises and damped oscillations observed in the graph reflect the transmission line’s response to these high-frequency components. This behavior aligns with the distributed-parameter model of the line.

V₁ represents the winding voltage on the 1st disc, while V₁₀ represents the winding voltage on the 10th disc. Voltage waveforms V₁ and V₁₀ for different grounding resistances (R_g) are given in Figure 16 and Figure 17. When the graphs are examined, the voltage difference between V₁ and V₁₀ of the neutral grounded high-voltage side of transformer is clearly seen. When the grounding resistance value increases, the voltage values V₁ and V₁₀ also increase. The change in grounding resistance caused an increase of approximately 118 kV in the peak voltage value of the first disc and the last disc.

The voltage–time graphs for each disc of the power transformer with disc-type winding are presented in Figure 18 and Figure 19. The simulation period for the state-space method is 6 ms. The discs were subjected to different voltage values. A rapid rise in voltage within the first 1 ms indicates the presence of high-frequency components generated when the lightning strike reaches the transformer. This lightning strike causes excessively high voltages in the transformer windings during the initial period (0–1 ms), a phenomenon attributed to the short rise time of the lightning strike and wave propagation along the transmission line. Around 3 ms, a second significant voltage spike is observed, likely caused by reflection effects between the transmission line and the transformer, with reflected waves moving at a period of approximately 2 ms. The highest peak voltage (67 kV) is observed in the first disc, indicating that the majority of the lightning energy is absorbed by this disc. Voltages gradually decrease across the subsequent discs (e.g., V₂, V₃, V₄, etc.). The high-frequency oscillations observed in the graph, particularly during the first 1 ms, justify neglecting the magnetic core’s influence. At such high frequencies, the inductance and capacitance of the windings dominate the system’s behavior, while the magnetic properties of the core become less significant.

The graphs presenting the peak voltage values on the discs of the neutral grounded high-voltage side of the power transformer are shown in Figure 20 for different R_g values. According to these graphs, the peak voltage values in the disc windings decrease linearly from the first disc to the last disc. Consequently, the windings in the first disc were subjected to the highest voltage, while those in the last disc were subjected to the lowest voltage.

In addition, modeling a power transformer consisting of multiple discs in the ATP-EMTP can be time-consuming. This is because each disc must be drawn multiple times, or a drawn disc must be copied to the end of the circuit. However, the model used in this study does not require extensive drawings; instead, the matrices for a transformer with a large number of discs can be easily created, allowing the simulation to be completed by simply determining the value of N (the number of discs).

5. Conclusions

In this paper, we investigated the voltage distributions in the windings of the discs on the neutral grounded high-voltage side of a disc-structured power transformer, as well as the voltage–current waveforms generated on the transmission line by a lightning strike on a single-phase distributed-parameter transmission line, using the state-space method. Inductor currents and capacitor voltages are selected as state variables. At the sending end of the line, there was a sudden voltage spike at megavolt levels due to a lightning strike. The arrival time of the lightning strike at the transformer windings depends on the characteristic impedance of the line, causing the current–voltage waves to reach the transformer windings with a delay of approximately 1 ms. Therefore, this study also includes the effect of the transmission line on the voltage distributions in the windings due to lightning strikes. Peak voltage values at the transformer discs and current–voltage values at the sending and receiving ends of the transmission line are recorded. The peak voltage values of the first disc, the fifth disc, and the last disc are 67 kV, 40 kV, and 6.5 kV, respectively, for a grounding resistance of 1 Ω. The voltage difference between the interconnected discs is approximately 6.5 kV, while the voltage difference between the first and last discs is 60.5 kV. It was also observed that the peak voltage values on the discs increased as the grounding resistance value increased. Although the peak voltage values increased, the voltage difference between the discs remained constant at approximately 6.5 kV, while the voltage distribution decreased linearly from the first disc to the last. This indicates that insulation in the first discs is more critical than in the last discs. Since the peak voltage level varies in each disc, the winding insulation in the discs may also differ. The windings must be designed to withstand excessive voltages. The first disc, which was subjected to the highest voltage, should be reinforced with additional insulation materials. When transformer windings are subjected to overvoltage conditions, several critical issues can arise, including insulation breakdown, thermal stress, dielectric failure, mechanical stresses, and spark or arc formation. Any deformation that may occur in the windings, or any short-circuit faults that may arise between the discs, between the windings, or between the disc and the transformer tank under operating voltage, can also be analyzed using this method. Additionally, the effect of a lightning strike on one phase and its impact on other phases can be examined with this approach. These issues underscore the critical need for robust overvoltage protection systems to ensure the reliability and longevity of transformers. The results obtained using the state-space method were compared with those from the ATP-EMTP. It was determined that the results from both methods are nearly identical and compatible. The state-space method can be used as an alternative approach for transient-state analysis of the transformer model, also known as a high-frequency power transformer, which is utilized in frequency response analysis. It can be applied to power transformers with varying numbers of discs. This method facilitates the design of transmission lines and transformer protection equipment. Thanks to the state-space method, transient analysis can be performed on transmission lines and transformers without requiring a transient simulation program; a math solver is sufficient.