A generalized dynamical model for transformers with saturation and hysteresis effects

Abstract

This article presents a new generalized non-linear dynamical model of a transformer with saturation and hysteresis effects. The structure of this new model is based on a magnetic equivalent circuit with non-linear reluctance elements. Using this method, accurate models of three- and five-legged transformers with magnetic saturation and hysteresis phenomena of iron core are introduced and inrush current in normal and sequential phase energization techniques are evaluated. Moreover, the effects of neutral resistance, magnetic structure and winding connections on inrush current are determined. In addition, for validation of the proposed model, some numerical simulations are made and compared with other techniques. Simulation results illustrate that the proposed model has high accuracy and efficiency for non-linear dynamical modelling of a transformer.

Keywords:

• core saturation
• hysteresis loss
• inrush current
• magnetic equivalent circuit
• non-linear dynamical model
• three-five-legged transformers

1. Introduction

Ferromagnetic materials are a major component of electromagnetic equipment in power systems that have non-linear characteristics such as core saturation and hysteresis loops. Structures of magnetic and electric circuits in these types of equipment cause different parts of the magnetic core to operate at different points of the B(H), thus, a full non-linear dynamical model of such a system becomes too complicated [1]. In these cases, the non-linear dynamical model of a system based on non-linear inductances are not validated.

However, power transformers are one of the most important and common magnetic devices in power systems. According to magnetic and electric circuits, different models have been developed for analysis of transformers with saturation and hysteresis loss. In Reference [2], using a magnetic equivalent circuit, a geometrical model for linear transformers was developed and sequence inductance matrices of the system were determined. In addition, saturation and hysteresis characteristics of individual magnetic circuit branches can be approximated by B-spline curve fitting techniques [3]. In this method, reluctance of magnetic paths in a transformer depends on flux paths. Consequently, the flux linkage in a transformer system cannot be separately determined in a non-linear condition. In [4], a PSpice model of a three-phase transformer with three legs has been developed using a magnetic equivalent circuit and three constant resistances in parallel with the induced primary voltages illustrating total core losses in the transformer. Although the proposed model is simple and reliable, it cannot consider the saturation and loss of different parts of a magnetic core, individually. Moreover, transient analysis softwares such as the Alternative Transients Program (ATP) and ElectroMagnetic Transient Program (EMTP) have not determined an exact model of a transformer with linkage flux among different phases [5].

Recently, a dynamic model of hysteresis loops with non-linear magnetization characteristics has been developed for the prediction of iron loss in ferromagnetic materials [6]. In small signal applications, a circuit model of a magnetic core with non-linearity, hysteresis and eddy currents were presented [7]. In power applications, a dynamic hysteresis method [8] has often been used for the hysteresis loop in a non-linear magnetic core. These methods have higher accuracy but lower computing speed in comparison to other methods and require specific accurate characteristics of the magnetic material.

In this article, using connection matrices in magnetic and electric circuits, a systematic method is introduced to determine the magnetic and electrical models of transformers with saturation and hysteresis effects. In this method, the hysteresis loss for each magnetic path of a transformer is individually included by a fictional coil. In addition, a non-linear reluctance, which is dependent on the flux path, is used for considering the saturation effect of a magnetic path in a system. Furthermore, accurate models of transformers with three and five legs are developed and effects of neutral resistance in reduction of inrush current in the sequential phase energizing technique are evaluated [9,10]. Simulation results show that the generalized non-linear dynamical model has high accuracy and efficiency for analysis of the non-linear phenomena in electromagnetic system.

The outline of this article is as follows: in Section 2, we introduce a Fundamental Magnetic Path (FMP) and utilize it to determine electrical equations of an electromagnetic system. In Section 3, we describe Amper's law to obtain a non-linear algebraical–differential equation in the system. In Section 4, the proposed method is applied to three- and five-legged transformers. In Sections 5 and 6, some simulations results, comparing and concluding remarks are given. Finally, a summary of main symbols is given in Table A1 in the Appendix.

2. Electrical model of electromagnetic systems

Magnetic structure of an electromagnetic device is formed by several magnetic paths where the saturation and loss of the magnetic core depend on the density of flux, (B – H) characteristic and source frequency. The magnetic topology can be divided into some magnetic tubes with constant flux density, thus an FMP, which is shown in Figure 1(a), is used for modelling a magnetic circuit with constants of cross area (A_k ), length (l_k ) and magnetic permeability of a material . In this figure, FMP is excited with m_k by separated coils, where n_k,j , L_k,j and r_k,j denote turn, leakage inductance and series resistance, respectively, of the kth FMP in the jth coil. In addition, ϕ _k (t) and B_k (t) are magnetic flux and density of the kth FMP, simultaneously. Moreover, Figure 1(b) shows the magnetic equivalent circuit of the FMP considering non-linear magnetic reluctance and magnetomotive force (MMF) in ampere-turns.

Figure 1. Magnetic structure and equivalent circuit of the FMP. (a) Fundamental path and (b) magnetic equivalent circuit.

The flux density of the kth FMP (B_k (t)) can be expressed in the -order of Fourier series as

(1)

where ω and B_kj are the main frequency and the jth components of flux density at kth FMP, respectively. In low-frequency operations and neglecting minor loop in the hysteresis loss, the total hysteresis loss (P_k ) for the kth FMP with mass (M_k ) and flux density (B_k ) can be determined by [13]:

(2)

where C_k and s are two constants, which are dependent on the characteristics of a material. For the modern transformer and B_k < 2.1 T and   f < 550 Hz, s is between 1.5 and 2.5 (s ≈ 2) [11]. Equation (2)(2) shows that hysteresis loss in the kth FMP depends on the magnetic flux of a system. Thus, we can consider by a parallel resistor with a fictional coil as power loss in equals the hysteresis power loss in sinusoidal condition. Thus, we have

(3)

where is number of turns of kth fictional coil. For specific frequency, all parameters in this relation are constant, therefore, a fictional perfect coil without any flux leakage, which is connected to resistor , is utilized for modelling the hysteresis power loss in an FMP.

If a magnetic circuit can be formed by number of n FMPs, the number of total actual coils (m) in an electromagnetic system will be equal to

(4)

In Figure 1(a), voltage equations of the actual coils at the kth FMP (without fictional coil) in vector form can be written as

(5)

where N _k (t), i _k (t), e _k (t) and v _k (t) are m_k column vectors that denote the number of coil turns, current, exciting voltage and terminal voltage of the m_k coils in the kth FMP, respectively. These vectors can be given by

(6)

where v_{k ,i} (t) and i_{k ,i} (t) are the voltage and current of the ith coils in the kth FMP. Furthermore, R _k and L _k are m_k  × m_k diagonal matrices that present resistor and leakage inductances of the m_k coils in the kth FMP. These diagonal matrices are given as

(7)

The sign of n_k,i for right(left)-oriented coil is assumed positive (negative). Core loss of the kth FMP can be introduced by a fictional coil with turns where the voltage equation for this coil can be written as

(8)

where is the current of a fictional coil. For a magnetic system with number of n FMPs, the voltage equation of the system can be augmented by using Equations (5)(5) and (7)(7) as

(9)

where i _s (t), e _s (t) and v _s (t) are m × 1 vectors of current, exciting voltage and terminal voltage of actual coils, respectively. In addition, ψ _s (t) is an n × 1 vector of flux of the main coils. These vectors can be presented as

(10)

Using Equations (5)(5) and (7)(7), inductance, resistance and turn matrices in Equation (9)(9) can be obtained as

(11)

Similarly, the voltage relations of the fictional coils in Equation (8)(8) can be augmented as

(12)

where i _f (t) is an n × 1 current vector of fictional coil of FMPs. This vector is assumed as

(13)

Using Equation (8)(8), resistance and turn matrices in Equation (12)(12) can be obtained by

(14)

Because coil currents depend on the excitation voltages and circuit connections, we consider m main coils in the system that are connected in the p meshes. Using the connection matrix C _v, we can write the Kirchhoff's Voltage Law (KVL) equations for the m meshes in the system. For this purpose, the matrix relation between the current vector of the actual coils ( i _s (t)) with the vector of mesh currents ( i _m (t)) can be written as

(15)

where i _m (t) is given by

(16)

where is current of the jth mesh of the connected circuits by some actual coils. In addition, the KVL for loop voltage can be written using connection matrix C _v as

(17)

The (jth, kth) element of the matrix C _v is 1 when the jth coil of the system is connected to the kth mesh current in the same reference direction, is –1 for opposite reference direction and is otherwise 0. By multiplying Equation (9)(9) by C _v in the left-hand side and combining it with Equations (15)(15) and (17)(17), the electrical voltage relations of a magnetic system can be expressed as

(18)

where R _m, L _m and e _m are resistance, leakage inductance and exciting voltage matrices, respectively, which can be expressed by

(19)

In some conditions, the actual coils in a system may be connected by external impedances without any dependence on a system's fluxes. For example, the star connection of three primary coils in a three-phase transformer with neutral impedance (r_n ) is a case in that the neutral impedance is connected to three-phase primary coils (Figure 3). In this case, the three-phase primary coils depend on FMP fluxes; however, the neutral impedance does not depend on any FMP fluxes. In these conditions, turn number of the external impedance in Equation (18)(18) is set to zero.

Figure 3. Construction of a transformer with three legs.

For determining currents of actual coils ( i _m (t)) in Equation (18)(18), we need to obtain a relation between electrical currents ( i _m (t)) and magnetic fluxes ψ_s (t). In the next section, non-linear algebraic relations are introduced using a magnetic circuit as shown in Figure 1(b).

3. MMF in magnetic circuit

By applying Ampere's law to the kth FMP in Figure 1(b), we can write

(20)

where is reluctance of the kth FMP that is dependent on the density of flux of its magnetic path. This reluctance can be determined by

(21)

where is the kth FMP permeability that can be obtained from slope of H(B) curve of the magnetic material in the operation point. Thus, we can write

(22)

where B_k and H_k are magnetic flux density and field intensity of the kth FMP, respectively. Because a magnetic path in an FMP is a reluctance with a constant area, B_k and H_k do not vary along the path, thus using Equation (20)(20), augmented relation of Ampere's law without connection FMP in a magnetic topology can be yielded as

(23)

where F _s(t) is n × 1 MMF vector of FMP and is n × n diagonal matrix, which presents reluctance of the FMPs. These matrices are given by

(24)

The non-linear (B – H) curve and hysteresis loop in magnetic paths are introduced by variable reluctances in Equation (23)(23) and fictional coil currents in Equation (12)(12), which are dependent on magnetic flux. Tangent, exponential, polynomial and piecewise linear functions have been proposed to approximate the B – H curve [12,13]. In this article, the permeability of the FMP is given as

(25)

where c ₁, c ₂ and c ₃ are three constants that can be chosen by cure fitting technique for the best approximation of the magnetizing characteristic. Figure 2 shows saturation curve B – H and permeability FMP for Armco M5 steel core, where c ₁, c ₂ and c ₃ in Equation (25)(25) are [14]

(26)

Figure 2. B – H and μ _r (H) characteristics of the Armco M5 Steel [14].

According to the FMP connection in the magnetic circuit and using the continuity of flux with Ampere's law, a relation between the magnetic flux vector of the FMP (ψ_s (t)) and the mesh flux of the magnetic circuit (ψ_m (t)) can be expressed as

(27)

where C _M is a q × n connection matrix, which can be determined using magnetic circuit topology. The (jth, kth) element of a magnetic connection matrix is 1 or –1 if the jth FMP connects to the kth mesh flux in the same or opposite reference direction, respectively. In addition, _m (t) is a q × 1 vector of mesh flux that can be written as

(28)

where is the jth mesh flux of a magnetic circuit. Substituting Equations (15)(15) and (27)(27) in (23)(23), the non-linear model of magnetic flux in a system is

(29)

where is the q × q matrix of the mesh reluctance. This matrix can be obtained by

(30)

Substituting Equation (27)(27) in Equations (12)(12) and (18)(18) and using Equation (29)(29), we can obtain a generalized and augmented model of an electromagnetic system as

(31)

where

(32)

This matrix relation presents non-linear singular differential equations with p + q + n unknown variables ( i _m (t), _m (t) and i _f (t)). Table A1 provides an overview and a succinct definition of all important quantities used in the model. Due to a non-linear magnetic characteristic, the non-linear algebraical–differential equations in (31) have to be solved simultaneously. In the following section, this model is applied to transformers with three and five legs for transient analysis of non-linear conditions, such as inrush current phenomena in the energizing process.

4. Non-linear model of transformers with three and five legs

Magnetic structure and electric connection of transformers with three and five legs are shown in Figures 3 and 4. The magnetic circuit of the three-legged transformer consists of five non-linear cores with non-linear reluctances ( to ). Three linear reluctances ( to ) are provided to present air gap fluxes for each leg, individually. In asymmetrical exciting, coil currents and fluxes of core paths can vary independently. In these conditions, a non-linear dynamic model of a transformer based on flux, current and core losses is necessary. Figure 5 shows the magnetic equivalent circuit for Figures 3 and 4. In these circuits, linear reluctances , and illustrate the air path for each internal leg. In addition, in Figures 3 and 4, the primary and secondary actual coils are connected in star with neutral resistance r_n and delta configurations, respectively.

Figure 5. Magnetic equivalent circuit of (a) three- and (b) five-legged transformers.

Figure 4. Construction of a transformer with five legs.

The worst-case condition for analysis of the inrush current of a transformer is a no-load situation. Thus, in this condition, and are secondary currents for Y/Δ and Y/Y connections, respectively. Based on Figures 3 and 4, we can write three (p = 3) and four meshes (p = 4) for Y/Y and Y/Δ connections, respectively. Similarly, using Figure 5, the independent mesh fluxes to , for a magnetic circuit can be written. Numbers of independent variables for a magnetic circuit are q = 5 for three and q = 7 for five leg configurations. In addition, n and m for transformers with three and five legs are 9 and 12 and 11 and 14, respectively. Thus with attention to Figures 3 and 5(a), the system variables in Equation (31)(31) for the three-legged transformer are chosen as

(33)

Due to the reluctance of the , and as linear and without any losses, we ignore , and in i _f (t). In addition, using Equations (17)(17) and (27)(27), connection matrices C _v and C _M can be obtained as

(34)

(35)

In addition, using Equations (19)(19), (30), (34) and (35), mesh resistance, inductance and reluctance matrices in Equation (31)(31) can be written as

(36)

(37)

(38)

in which

(39)

two other sub-matrices in Equation (31)(31) can be obtained as

(40)

(41)

The relations (31)–(41) present non-linear algebraical–differential models of a three-legged transformer in that unknown variables are completely coupled together. Similarly, we can obtain a non-linear algebraical–differential model for a five-legged configuration with non-linearity and hysteresis phenomena. Some simulation results can be obtained using advance numerical methods for transient analysis of asymmetrical conditions. Decreasing the inrush current of transformers by a sequential phase energization method is an interesting case in an asymmetrical condition [9,10]. In this case, the transient inrush current in a three-phase transformer will decrease energizing when the primary coils energize sequentially. By sequential phased energization of each coil, the neutral resistor can be connected as a series with an excited coil; thus, the transient inrush current is significantly reduced. In the next section, the performance and characteristics of this method for three- and five-legged transformers are evaluated.

5. Simulation results

For evaluation of two types of three-phase transformers, three switches shown in Figures 3 and 4 are simultaneously closed and the three primary coils of the transformers are energized by three-phase sinusoidal voltage. The geometrical and electrical parameters of two transformers are introduced in Tables 1 and 2 [13,15].

Table 1. Dimensions of transformers.

Symbol	Quantity	Values
	Inter legs area	43.33 cm^2
	Yoke area	61.82 cm^2
	Outer legs area	43.33 cm^2
	Inter legs length	35 cm
	Yoke length	29.2 cm
	Outer legs length	93.4 cm

Table 2. Electrical parameters of the transformers with three and five legs [13].

Quantity	Symbol	Values
Power	S	10 KVA
Frequency	f	50 Hz
Primary exciting voltage		220V
Secondary exciting voltage		0
Turn of primary coils		128 turns
Turn of secondary coils		128 turns
Primary coil resistors		2.5Ω
Secondary coil resistors		0.5Ω
Primary coil leakages		8.5 mh
Secondary coil leakages		Negligible
Reluctance of air path		2.20 × 10^6 h^−1
Leg hysteresis loss for rating values		6.82 KΩ
Yoke hysteresis loss for rating values		13.64 KΩ

Primary and secondary currents of the transformer with three legs are shown in some different configurations (Figure 6(a)–(c)). The maximum transient current is obtained by Y/Y connection with r_n  = 0. For validation of the proposed model with other techniques, the comparison of experimental and simulation results of the maximum magnitude of currents in Y/Y connection with r_n  = 0 are shown in Table 3. Similar simulation results for a transformer with five legs are illustrated in Figure 6(d)–(f). By comparing Figure 6, transient inrush currents in transformers with five legs are slightly higher than other in winding connections.

Table 3. Maximum inrush currents for three legs, Y/Y, r _n = 0.

Method	ia (A)	ib (A)	ic (A)	in (A)
Experimental [13]	54.1	27.2	32.1	26.3
Simulation [13]	55.6	29.6	26.7	26.6
Proposed model	56.7992	29.9407	33.8394	29.7132

Figure 6. Inrush currents of the three- and five-legged transformers, scale: . (a) Three legs, Y/Δ, r_n  = 0; (b) three legs, Y/Y, r_n  = 0; (c) three legs, Y/Y, r_n  = 2.5Ω; (d) five legs, Y/Δ, r_n  = 0; (e) five legs, Y/Y, r_n  = 0; and (f) five legs, Y/Y, r_n  = 2.5Ω.

In addition, the maximum current in a coil coincides with the flux saturation in its magnetic path. This phenomenon for Figure 6(c) is shown in Figure 7(a). Due to the cross area of the yoke being higher from leg in this transformer, contrary to leg paths operated in the saturation zone, the yoke paths are in the linear region. Figure 7(b) shows the dynamic hysteresis loops in these cases. Because of transient characteristics of inrush currents, the hysteresis loops are asymmetric. Moreover, areas of the hysteresis loops for each leg are higher than those of yoke paths.

Figure 7. Flux and hysteresis in a three-leg Y/Y transformer with r_n  = 2.5Ω. (a) Flux density, scale: 1 T/div. and (b) hysteresis loops.

The phase sequential energization method was introduced for limiting magnitude of an inrush current [9,10]. For evaluating the ability of this method, the phase sequential energization technique in two types of transformers is implemented. Three switches (s ₁, s ₂ and s ₃) in two types of transformers are sequentially closed. First, s ₁ is closed and after reaching i _1,1 (t) to a steady condition, s ₂ is closed. This sequential operation is continued for s ₃. Figure 8 shows the three-phase currents when the sequential switching is applied to transformers. The results show that the phase sequential energization method in the three-legged transformer can effectively reduce the maximum magnitude inrush current. The maximum amplitude of the inrush currents for these transformers is included in Table 4. By comparing results in this table, we can find out the maximum magnitude ratio of a phase current in a phase sequential to simultaneous energization methods. The table shows reductions of about 75% for i_a and 15% for i_b and i_c .

Table 4. Maximum inrush currents for different types of transformer.

Configuration	rn (Ω)	Legs	ia (A)	ib (A)	ic (A)	in (A)
Y/Y	0	3	56.7992	29.9407	33.8394	29.7132
Y/Δ	0	3	55.3527	25.7496	33.8720	7.2734
Y/Y	0	5	58.3500	33.9328	33.9864	55.3121
Y/Δ	0	5	55.3684	25.8851	33.8942	7.8952
Y/Y	2.5	3	51.4095	29.9196	31.0695	19.0604
Y/Y	2.5	5	51.9941	38.2087	28.5391	35.0519
Y/Y ^a	2.5	3	39.9214	4.5770	4.3302	39.8928
Y/Δ^a	2.5	5	39.9361	4.4844	5.7075	39.8963
Y/Y ^a	2.5	5	39.9512	16.1405	18.5808	39.9287
Note: ^aEnergize sequentially.

Figure 8. Inrush currents in the sequential phase energization method with r_n  = 2.5Ω, scale: . (a) Five legs, Y/Δ, (b) three legs, Y/Y and (c) five legs, Y/Y.

These results show that the proposed method can be used to analyse complex structures of magnetic and electrical circuits with different hysteresis and saturation characteristics, separately.

6. Conclusion

In this article, a generalized non-linear dynamical model of electromagnetic systems with saturation and hysteresis effects was presented. Based on the electric and magnetic connection matrices, the magnetic and electrical relations of transformers were introduced and an algebraic and a dynamic model of the system was presented. This model was very effective for modelling and analysing of asymmetric non-linear conditions such as inrush current and hysteresis phenomena in a power transformer. Finally, this model was applied to both three- and five-legged transformers, and it was demonstrated that a sequential phase energization technique in a transformer is a suitable method for reduction of maximum magnitude of inrush current.

7. Appendix

A list of the main symbols is presented in the following table.

Table A1. List of main symbols.

Symbol	Quantity
n	Number of FMP
m	Number of total coils
p	Number of electrical meshes
q	Number of magnetic meshes
mk	Number of actual coils in the kth FMP
nk ,j	Turn numbers of jth actual coil in the kth FMP
	Turn numbers of the fictional coilonal coil in the kth FMP
μ k	Magnetic permeability of the kth FMP
Lk ,j	Leakage inductance of jth actual coil in the kth FMP
	Paralleled resistor of the fictional coil in the kth FMP
rk ,j	Series resistance of jth actual coil in the kth FMP
Bk (t)	Magnetic density of the kth FMP
ϕ k (t)	Magnetic flux of the kth FMP
Fk (t)	MMF of the of the kth FMP
	Reluctance of the kth FMP
ik ,j (t)	Current of the jth actual coils in the kth FMP
	Current of the fictional coil in the kth FMP
	Current of the jth mesh
vk ,j (t)	Terminal voltage of the jth actual coils in the kth FMP
ek ,j (t)	Exciting voltage of the jth actual coils in the kth FMP
Note: .