Analysis of Electromechanical Swings of a Turbogenerator Based on a Fractional-Order Circuit Model

Abstract

This paper addresses the issue of rotor swings in a high-power synchronous generator during stable operation with a stiff power grid. The analysis of electromechanical swings was conducted using a circuit model incorporating fractional-order derivatives. Assuming that variations in the load angle under small disturbances from a stable equilibrium are minor, a linearized differential equation describing the electrodynamic state of the synchronous machine was derived. Based on this linearized equation of motion and the identified parameters of the equivalent circuit, calculations were performed for a 200 MW turbogenerator. The results indicate that the electromechanical swings are characterized by a constant pulsation and a low damping factor. Calculations were also carried out using a lumped-parameter equivalent circuit model. Based on the obtained results, it can be stated that the fractional-order model provides a more accurate fit of the frequency characteristics compared with the classical model with the same number of rotor equivalent circuits. The relative approximation errors for the fractional-order model are, for the d-axis (one rotor equivalent circuit), relative magnitude error δ_m = 1.53% and relative phase error δ_φ = 6.32%, and for the q-axis (two rotor equivalent circuits), δ_m = 3.2% and δ_φ = 8.3%. To achieve comparable approximation accuracy for the classical model, the rotor electrical circuit must be replaced with two equivalent circuits in the d-axis and four equivalent circuits in the q-axis, yielding relative errors of δ_m = 2.85% and δ_φ = 6.51% for the d-axis, and δ_m = 1.86% and δ_φ = 5.49% for the q-axis.

1. Introduction

During the operation of a synchronous generator connected to a stiff power grid, disturbances may arise that cause oscillatory changes in rotor speed, referred to as electromechanical swings. These speed variations also induce electromechanical swings in the generator’s power angle and electromagnetic torque. A distinctive feature of these swings is their low frequency (typically in the range of 1–2 Hz) and low damping ratio. Such electromechanical swings limit the capacity for electric power transmission within the power system and, under adverse conditions, may lead to system instability [1]. Changes in the operating state of a synchronous generator may result from variations in the excitation current, voltage, and line reactance, or from sudden changes in the driving torque.

This study investigates the electromechanical swings of a turbogenerator subjected to a step change in driving torque while maintaining a constant excitation voltage. The assumption of a sudden increase in turbine torque provides a straightforward means of illustrating transient phenomena in a synchronous generator.

The transient states of synchronous machines are described by nonlinear differential equations. To solve these equations under specified initial conditions, numerical methods are typically employed [2,3]. However, these methods have a limitation: when control and disturbance signals are known, they yield only numerical results at selected time steps. Consequently, they do not provide analytical solutions suitable for assessing the stability of a synchronous machine. By linearizing the nonlinear differential equations around a chosen operating point, a system of linear differential equations with constant coefficients is obtained. Applying operational calculus to these equations transforms them into algebraic equations, enabling analytical solutions. In this study, the linearization of the differential equations enabled the determination of the damping ratio and the frequency of electromechanical swings of a synchronous generator under small disturbances from equilibrium. The analysis of electromechanical swings is typically based on classical circuit models [4], which assume a single damper circuit in the rotor along the d-axis and another along the q-axis. Due to their low computational complexity, classical models are commonly employed in real-time simulations [5,6]. However, classical models often result in significant discrepancies between measured responses and those obtained through computer simulations. This necessitates the use of higher-order mathematical models [7,8,9] incorporating additional lumped-parameter RL circuits into the equivalent model of the synchronous machine. Such models more accurately reflect the skin effect occurring in the solid rotor and in conductive slot wedges. An alternative approach involves employing equivalent circuits with fractional-order operator impedances [10,11]. In the frequency domain, fractional-order operator inductance represents an impedance whose resistance and inductance vary with the frequency of eddy currents induced in the machine’s solid rotor. To model eddy currents in conductive components more precisely, hybrid elements should be used [12,13,14]. These consist of a series connection of lumped-parameter RL elements and fractional-order operator inductance. In [15], a simulation of a three-phase short-circuit in a synchronous generator was performed, whereas in [16], the dynamic states of an induction motor with a solid rotor were analyzed. Both studies employed fractional-order differential calculus, and the nonlinear differential equations were solved using the Grünwald–Letnikov method [17]. Although fractional-order calculus is still rarely applied in the modeling of synchronous machines, its use in power system control research has been steadily increasing [18,19].

In this study, the electric circuit of a solid rotor along the d-axis is represented in the equivalent model by a single hybrid element. In the q-axis, the rotor damper circuit is modeled as a parallel connection of two branches: one containing a hybrid element and the other comprising lumped-parameter RL elements.

The parameters of the synchronous machine’s equivalent circuit can be identified either through measurements [7,20] or based on design data [21]. Among the various methods for parameter identification, the SSFR (standstill frequency response) method [7] is particularly noteworthy. This method involves determining the frequency characteristics of spectral inductances, either from measurements or design data, for a stationary machine. These characteristics are then approximated to determine the parameters of the machine’s equivalent circuit.

The article is structured into four chapters, organized as follows. The introduction outlines the representation of rotor electrical circuits in the equivalent model of a synchronous machine. Chapter Two presents the mathematical model of the synchronous machine. In this model, the rotor damper circuit is replaced by a single fractional-order equivalent circuit along the d-axis and two equivalent circuits along the q-axis. The corresponding differential equations describing the electrodynamic state of the synchronous generator are also provided. Chapter Three describes the procedure for determining the parameters of the synchronous machine’s equivalent circuit. The equivalent circuit parameters along the d-axis and q-axis were derived by approximating spectral inductances obtained through finite element analysis of the electromagnetic field distribution. Chapter Four presents an analysis of the electromechanical swings of the turbogenerator under small disturbances from equilibrium caused by a step change in the turbine driving torque. Through linearization of the differential equations, a fractional-order operational expression was obtained for the increments in the power angle Δδ(p) and angular velocity Δω. The damping ratio and frequency of the electromechanical swings of the high-power turbogenerator were then determined.

2. Mathematical Model of a Synchronous Machine

The analysis of transient states in synchronous machines is based on equivalent circuit models. To accurately represent the phenomena occurring within the solid rotor, the equivalent rotor electric circuit should be replaced with one or two lumped-parameter RL elements in the d-axis, and three or four RL elements in the q-axis, each characterized by different time constants (Figure 1).

An alternative approach is to represent the damper circuit of the solid rotor using a fractional-order impedance composed of hybrid elements [16], which are series combinations of lumped integer-order RL elements and a fractional-order element (Figure 2).

In the equivalent circuits, R_s and R_f are the resistances of the armature winding and field winding, L_σs and L_σf the leakage inductances of the armature winding and field winding, L_md and L_mq the magnetizing inductances in the d- and q-axes, T_ed and T_eq the equivalent time constants of the rotor damper circuit (solid rotor, conducting wedges in the slots) in the d- and q-axes, and R_kd and L_σkd in the d-axis and R_kq and L_σkq in the q-axis are the resistance and leakage inductance with lumped parameters of the solid rotor damper circuit.

The analysis of electromechanical swings of a synchronous machine is conveniently carried out in Park’s d–q coordinate system [22,23] in the rotor reference frame (Figure 3). In this coordinate system, the rotational voltage components present in the equivalent circuits of Figure 1 and Figure 2 are eliminated.

The electrodynamic state of a synchronous machine, where the damper circuit of the solid rotor is modeled by a fractional-order impedance, in Park’s d–q coordinate system, applying the source-oriented arrow system for the stator and the load-oriented arrow system for the rotor, is described by the following equations.

The differential voltage equations for the stator in the d- and q-axes are

$$\left.\begin{array}{c}{u}_{sd}=-R_s{i}_{sd}+{\displaystyle \frac{d{\Psi }_{sd}}{dt}} -\omega {\Psi }_{sq} \\ u_{sq}=-R_s{i}_{sq}+{\displaystyle \frac{d{\Psi }_{sq}}{dt}}+\omega {\Psi }_{sd}\end{array}\right\}$$

(1)

where u_sd and u_sq are the components of the armature voltage in the d- and q-axes, i_sd and i_sq are the components of the armature current in the d- and q-axes, Ψ_sd and Ψ_sq are the components of the flux linkage of the armature winding in the d- and q-axes, and ω is the electrical angular speed.

The differential voltage equation for the excitation circuit is

$$u_f=R_f{i}_f+{\displaystyle \frac{d{\Psi }_f}{dt}}$$

(2)

where u_f, i_f, and Ψ_f are the instantaneous values of the field voltage, field current, and flux linkage of the field winding, referred to as the armature winding side.

The differential voltage equations for the damper circuits of the solid rotor in the d- and q-axes, including fractional-order derivatives, are

$$\left.\begin{array}{c}{e}_{md}={\displaystyle \frac{d{\Psi }_{md}}{dt}}=-\left(R_{kd}{i}_{kd}+L_{\sigma kd}{\displaystyle \frac{d{i}_{kd}}{dt}}+{\displaystyle \frac{L_m}{T_{ed}^{\beta }}}{D}_{\alpha }(i_{kd})\right)\\ e_{mq}={\displaystyle \frac{d{\Psi }_{mq}}{dt}}=-\left(R_{kq}{i}_{kq}+L_{\sigma kq}{\displaystyle \frac{d{i}_{kq}}{dt}}+{\displaystyle \frac{L_m}{T_{eq}^{\beta }}}{D}_{\alpha }(i_{kq})\right) \end{array}\right\}$$

(3)

where Ψ_md and Ψ_mq are the components of the main flux linkage in the d- and q-axes, i_kd and i_kq are the components of the damper winding current in the d- and q-axes, α is the order of the fractional derivative, and β = 1 − α, where 0 < α < 1.

The fractional-order derivatives appearing in Equation (3) are defined as follows [17]:

$$D_{\alpha }\left(i_{kd}\right)={\displaystyle \frac{d^{\alpha }{i}_{kd}}{\mathrm{d}{t}^{\alpha }}}\hspace{1em}\hspace{1em}\hspace{1em}{D}_{\alpha }\left(i_{kq}\right)={\displaystyle \frac{{\mathrm{d}}^{\alpha }{i}_{kq}}{\mathrm{d}{t}^{\alpha }}}$$

The equations for the flux linkages related to the stator windings in the d- and q-axes and the excitation winding are

$$\left.\begin{array}{l}{\Psi }_{sd }=-L_{sd}{i}_{sd}+L_{md}{i}_f+L_{md}{i}_{kd} \\ {\Psi }_{sq}=-L_{sq}{i}_{sq}+L_{mq}{i}_{kq}\\ {\Psi }_{md}=L_{md}(-i_{sd}+i_f+i_{kd})\\ {\Psi }_{mq}={ L}_{mq}(-i_{sq}+i_{kq})\\ {\Psi }_f={ L}_{md}{i}_{sd}+{ L}_{md}{i}_{kd}+L_f{i}_f\end{array}\right\}$$

(4)

The equation of motion is

$${\displaystyle \frac{J}{p_b}}{\displaystyle \frac{d\omega }{dt}}=M_m-M_e-{\displaystyle \frac{D}{p_b}}\omega$$

(5)

These equations must be supplemented by the equation for the electromagnetic torque:

$$M_e = {\displaystyle \frac{3}{2}}{p}_b({\Psi }_{sd}{i}_{sq} -{ \Psi }_{sq}{i}_{sd})$$

(6)

It is assumed that the turbogenerator operates on a symmetrical three-phase stiff grid. In this case, the voltages applied to the stator phase windings are

$$\left.\begin{array}{l}{u}_{sa}=U_{sm}\mathit{cos}({\omega }_{s}t+{\psi }_u)\\ u_{sb}=U_{sm}\mathit{cos}({\omega }_{s}t+{ \psi }_u-{\displaystyle \frac{2}{3}}) \\ u_{sc}=U_{sm}\mathit{cos}({\omega }_{s}t+{\psi }_u-{\displaystyle \frac{4}{3}})\end{array}\right\}$$

(7)

where u_sa, u_sb, and u_sc are instantaneous values of phase voltages a, b, and c, respectively; ω_s is the angular frequency of the power grid; U_sm is the amplitude of the phase voltage; and ψ_u is the initial phase angle of the voltage.

Applying the Park transformation [22],

$$\left.\begin{array}{l}{u}_{sd}=\hspace{0.33em}\hspace{0.33em}{\displaystyle \frac{2}{3}}\left(u_{sa}\mathit{cos}(\vartheta )+u_{sb}\mathit{cos}(\vartheta -{\displaystyle \frac{2}{3}}\pi )+u_{sc}\mathit{cos}(\vartheta -{\displaystyle \frac{4}{3}}\pi )\right) \\ u_{sq}=\hspace{0.33em}-{\displaystyle \frac{2}{3}}\left(u_{sa}\mathit{sin}(\vartheta )+u_{sb}\mathit{sin}(\vartheta -{\displaystyle \frac{2}{3}}\pi )+u_{sc}\mathit{sin}(\vartheta -{\displaystyle \frac{4}{3}}\pi )\right)\end{array}\right\}$$

(8)

to the phase voltages from Equation (7), the voltage components in the stator winding d- and q-axes are obtained as follows:

$$\left.\begin{array}{l}{u}_{sd} = \hspace{0.33em}{U}_{sm}\mathit{sin}({\omega }_{s}t + {\psi }_u - \vartheta )\\ u_{sq} = U_{sm}\mathit{cos}({\omega }_{s}t +{ \psi }_u - \vartheta )\end{array}\right\}$$

(9)

where ϑ is the angle between the axis of phase a of the armature winding and the rotor d-axis (Figure 3).

From the vector diagram in Figure 3, it follows that

$$\left.\begin{array}{l}{u}_{sd }=\hspace{0.33em}{U}_{sm}\mathit{sin}\delta =U_{sm}\mathit{cos}({\displaystyle \frac{\pi }{2}}-\delta )\\ u_{sq}={ U}_{sm}\mathit{cos}\delta =U_{sm}\mathit{sin}({\displaystyle \frac{\pi }{2}}-\delta )\end{array}\right\}$$

(10)

where δ is the load angle.

By comparing Equations (9) and (10), we obtain

$${\omega }_{s}t+{\psi }_u-\vartheta ={\displaystyle \frac{\pi }{2}}-\delta$$

Thus,

$$\vartheta ={\omega }_{s}t+{\psi }_u+\delta -{\displaystyle \frac{\pi }{2}}$$

(11)

Differentiating Equation (11) with respect to time yields the following relationship for the electrical angular velocity of the turbogenerator rotor:

$$\omega ={\displaystyle \frac{d\vartheta }{dt}}={\omega }_s+{\displaystyle \frac{d\delta }{dt}}$$

(12)

The time derivative of angular velocity is

$${\displaystyle \frac{d\omega }{dt}}={\displaystyle \frac{d^2\vartheta }{d{t}^2}}={\displaystyle \frac{d^2\delta }{d{t}^2}}$$

(13)

The equation of motion in Equation (5) can, therefore, be expressed as

$${\displaystyle \frac{J}{p_b}}{\displaystyle \frac{d^2\delta }{d{t}^2}}+{\displaystyle \frac{D}{p_b}}{\displaystyle \frac{d\delta }{dt}}+{\displaystyle \frac{D}{p_b}}{\omega }_s+M_e=M_m$$

(14)

where J and D are the moment of inertia and the viscous damping coefficient of the generator–turbine set, respectively; M_e and M_m are the electromagnetic torque and the turbine driving torque, respectively; and p_b is the number of pole pairs.

3. Identification of Equivalent Circuit Parameters

The equivalent circuit parameters in the d- and q-axes of the synchronous machine (Figure 2) were determined by approximating spectral inductances derived from the analysis of electromagnetic field distributions using the finite element method (FEM) [24], with the machine at a standstill. These inductances are determined from the relations in Equation (A21).

$$\left.\begin{array}{l}{L}_{sd}\left(j\omega \right)=-{{\displaystyle \frac{{\Psi }_{sd}\left(j\omega \right)}{I_{sd}\left(j\omega \right)}}}_{\left|U_f=0\right.}=L_{sd0}(j\omega ) -{\displaystyle \frac{p{L}_{fd}\left(j\omega \right)L_{df}\left(j\omega \right)}{R_f+p{L}_{f0}\left(p\right)}} \\ G_{fd}\left(j\omega \right)={ {\displaystyle \frac{{\Psi }_{sd}\left(j\omega \right)}{U_f\left(j\omega \right)}}}_{\left|I_{sd}=0\right.}={\displaystyle \frac{L_{df}\left(j\omega \right)}{R_f+p{L}_{f0}\left(j\omega \right)}}\end{array}\right\}$$

(15)

$$L_{sq}\left(j\omega \right)=-{\displaystyle \frac{{\Psi }_{sd}\left(j\omega \right)}{I_{sq}\left(j\omega \right)}}$$

(16)

From relation Equation (15), it follows that the spectral transfer functions L_sd and G_fd can be determined based on the inductances L_d0, L_f0, and L_df, obtained with the armature winding supplied and the field winding open, as well as with the field winding supplied and the armature winding open (Appendix A). The calculations were carried out in the FEMM 4.2 [25] software, under sinusoidal current excitation of either the armature or field winding in the frequency range from 0 to 1000 Hz.

Based on the structure of the synchronous machine’s equivalent circuit diagrams (Figure 2), the operator inductances L_sd(p) in the d-axis and L_sq(p) in the q-axis, as well as the transfer function G_fd(p), can be represented as ratios of polynomials with real coefficients and fractional-order exponents.

$$\left.\begin{array}{c}{L}_{sd}\left(p\right)=L_{sd}{\displaystyle \frac{p^2{b}_{1d}+p^{\alpha +1}{b}_{2d}+p{b}_{3d}+p^{\alpha }{b}_{4d}+b_{5d}}{p^2{a}_{1d}+p^{\alpha +1}{a}_{2d}+p{a}_{3d}+p^{\alpha }{a}_{4d}+a_{5d}}} \\ G_{fd}\left(p\right)={\displaystyle \frac{L_{md}}{R_f}}{\displaystyle \frac{p{b}_{1k}+p^{\alpha }{b}_{2k}+b_{3k}}{p^2{a}_{1d}+p^{\alpha +1}{a}_{2d}+p{a}_{3d}+p^{\alpha }{a}_{4d}+a_{5d}}}\end{array}\right\}$$

(17)

$$L_{sq}\left(p\right)=L_{sq}{\displaystyle \frac{p{b}_{1q}+p^{\alpha }{b}_{2q}+b_{3q}}{p{a}_{1q}+p^{\alpha }{a}_{2q}+a_{3q}}}$$

(18)

The numerator and denominator coefficients of the polynomials of the inductance L_sd(p) and the transfer function G_fd(p) are functions of the parameter vector λ_d of the sought parameters of the equivalent circuit in the d-axis. Similarly, the numerator and denominator coefficients of the polynomials of the armature inductance L_sq(p) in the q-axis are functions of the vector λ_q of the sought parameters of the equivalent circuit in the q-axis.

$$\left.\begin{array}{l}{\mathit{\lambda }}_{\mathit{d}}=\hspace{0.33em}{\left[L_{\sigma s}\hspace{0.33em}{L}_{kd\sigma }\hspace{0.33em}{R}_{kd}\hspace{0.33em}{T}_{ed}\right]}^T \\ {\mathit{\lambda }}_{\mathit{q}}=\hspace{0.33em}{\left[L_{\sigma s}\hspace{0.33em}{L}_{kq\sigma }\hspace{0.33em}{R}_{kq}\hspace{0.33em}{T}_{eq}\right]}^T\end{array}\right\}$$

(19)

The parameters of the equivalent circuit λ_d in the d-axis of the turbogenerator, and, consequently, of the approximating functions in Equation (17), are selected so that the magnitude of these functions approximates, with minimal error, the magnitude of the functions defined in Equations (15) and (16), which were obtained from electromagnetic field analysis. Similarly, the parameters of the equivalent circuit in the q-axis are determined. The sought values of the parameter vectors λ_d and λ_q are obtained by minimizing the sum of squared deviations.

For the d-axis,

$${\epsilon }_d\left({\mathit{\lambda }}_{\mathit{d}}\right)=\sum _{i=1}^N\left\{\hspace{0.33em}{\left(\left|L_{sd}\left(f_i\right)\right|-\left|L_{sd}^{\ast }(f_i,{\mathit{\lambda }}_{\mathit{d}})\right|\right)}^2\hspace{0.33em}+\hspace{0.33em}{\left(\left|G_{fd}\left(f_i\right)\right|-\left|G_{fd}^{\ast }(f_i,{\mathit{\lambda }}_{\mathit{d}})\right|\right)}^2\right\}$$

(20)

For the q-axis,

$${\epsilon }_q\left({\mathit{\lambda }}_{\mathit{q}}\right)=\sum _{i=1}^N\left\{\hspace{0.33em}{\left(\left|L_{sq}\left(f_i\right)\right|-\left|L_{sq}^{\ast }(f_i,{\mathit{\lambda }}_{\mathit{q}})\right|\right)}^2\hspace{0.33em}\right\}$$

(21)

where N is the number of measurement points, f is the frequency; L_sd, L_sq, and G_fd are the spectral quantities obtained from FEMM 4.2; and L*_sd, L*_sq, and G*_fd are the spectral quantities from the approximating functions, determined from Equations (17) and (18) by substituting p = jω, where ω = 2πf.

To solve the non-linear Equations (20) and (21), the Levenberg–Marquardt algorithm implemented in Matlab R2010a [26] was used.

Figure 4, Figure 5 and Figure 6 present the frequency characteristics of the spectral inductances in the d- and q-axes of the TWW-200-2 turbogenerator, calculated using the FEM method and obtained based on the approximation with fractional-order rational functions Equations (17) and (18). The magnitude on the frequency characteristics is expressed in p.u. (per unit, i.e., relative to the selected base values).

Figure 7 and Figure 8 show the frequency characteristics of the spectral inductances, calculated using the FEM method and obtained from the approximation for the lumped-parameter model (Figure 1). The approximating functions of the operational inductances in the d- and q-axes were assumed in the form of Equation (22)

$$\begin{gathered} L_{sd}\left(p\right)=L_{sd}{\displaystyle \frac{{\prod }_{i=1}^{n_d}(1+p{T}_{di})}{{\prod }_{i=1}^{n_d}(1+p{T}_{d0i})}} \\ {L}_{sq}\left(p\right)=L_{sq}{\displaystyle \frac{{\prod }_{i=1}^{n_q}(1+p{T}_{qi})}{{\prod }_{i=1}^{n_q}(1+p{T}_{q0i})}} \end{gathered}$$

(22)

where L_sd and L_sq are the self-inductances of the armature winding in the d- and q-axes, respectively; T_di and T_d0i are the time constants in the d-axis; T_qi and T_q0i are the time constants in the q-axis; and n_d and n_q are the order of the operational inductance in the d- and q-axes, respectively.

Two equivalent circuits (n_d = 3) were adopted in the rotor d-axis and four equivalent circuits (n_q = 4) in the rotor q-axis.

To compare the amplitude and phase frequency characteristics of the spectral inductances calculated by the FEM 4.2 software with those derived from fractional-order approximations (Equations (17) and (18)), the relative error was calculated using the following definition [27]:

$${\epsilon }_y = 100{\displaystyle \frac{\sqrt{\frac{1}{n}{\sum }_{i=1}^n(Y_i-Y_i^{\ast }{)}^2}}{\frac{1}{n}{\sum }_{i=1}^n{Y}_i}}$$

(23)

where Y is the actual value, and Y* is the value obtained from approximation.

The values of the equivalent circuit parameters for the fractional-order model and the time constants for the lumped-parameter model are presented in

Table 1. Equivalent circuit parameters for the fractional-order model.

Parameter	d-Axis Figure 2a	Parameter	q-Axis
			Figure 2b	Figure 2c
Lsd [p.u.]	1.812	Lsq [p.u.]	1.775	1.775
Lσs [p.u.]	0.186	Lσs [p.u.]	0.186	0.186
Rkd [p.u.]	0.001083	Rkq [p.u.]	0.001457
Lσkd [p.u.]	0.000012	Lσkq [p.u.]	0.01029
Lσf [p.u.]	0.1197	Teq [s]	5.952	0.97
Rf [p.u.]	0.00128	Rkq1 [p.u.]		0.001593
Ted [s]	11.708	Lσkq1 [p.u.]		0.000011
α	0.5	Rkq2 [p.u.]		0.00735
εm [%]	1.53	Lσkq2 [p.u.]		0.4722
εφ [%]	6.32	α	0.5	0.5
R2m	0.9998	εm [%]	2.98	3.2
R2φ	0.9958	εφ [%]	21.69	8.31
		R2m	0.9987	0.9985
		R2φ	0.8901	0.9839

and

Table 2. Time constants for the lumped-parameter generator model.

Parameter	d-Axis	Parameter	q-Axis
	Figure 1a		Figure 1b
Lsd [p.u.]	1.812	Lsq [p.u.]	1.775
Lσs [p.u.]	0.186	Lσs [p.u.]	0.186
Td1 [p.u.]	386.560	Tq1 [p.u.]	1362.9
Td2 [p.u.]	18.047	Tq2 [p.u.]	109.288
Td3 [p.u.]	1.301	Tq3 [p.u.]	10.3458
Td01 [p.u.]	2450.7	Tq4 [p.u.]	0.6357
Td02 [p.u.]	22.872	Tq01 [p.u.]	1912.3
Td03 [p.u.]	1.565	Tq02 [p.u.]	352.0454
εm [%]	2.85	Tq03 [p.u.]	14.5075
εφ [%]	6.51	Tq04 [p.u.]	0.9744
R2m	0.9992	εm [%]	1.87
R2φ	0.9955	εφ [%]	5.49
		R2m	0.9995
		R2φ	0.9930

(quantities marked as p.u. are expressed in per unit, i.e., relative to the selected base values). These tables also include the values of the relative errors for the magnitude (ε_m) and phase (ε_φ), as well as the coefficient of determination R².

The resulting relative errors for both magnitude and phase indicate good accuracy in approximating the amplitude and phase characteristics of the spectral inductance in the d-axis. However, significant discrepancies were observed in the phase characteristics in the q-axis when the damper circuit was modeled using a single equivalent circuit (Figure 2b). To achieve a high-accuracy approximation of these characteristics, it is necessary to use two equivalent circuits in the q-axis (Figure 2c).

4. Analysis of Electromechanical Swings of a Turbogenerator Under Small Disturbances Around the Equilibrium Point

4.1. Linearization of Differential Equations

The analysis of electromechanical swings addresses small disturbances from the stable equilibrium of a synchronous generator connected to an infinite bus, caused by variations in the turbine driving torque from the steady-state condition. A variation in the driving torque ΔM_m is accompanied by variations in the voltage components Δu_sd and Δu_sq, the current components Δi_sd and Δi_sq, and the flux linkages ΔΨ_sd and ΔΨ_sq in the d- and q-axes, as well as variations in the electromagnetic torque ΔM_e, the angular speed Δω, and the load angle Δδ. To obtain a system of linear differential equations, the nonlinear differential Equations (1) and (5) were linearized [28]. The assumption is made that, during stable operation of the generator under small disturbances, such as those that occur during normal operation of the power system [8], the electromechanical quantities are equal to the sum of the pre-disturbance values (denoted by subscript 0) and small deviations from the equilibrium point (denoted by the delta symbol). The following system of equations is then obtained:

$$\left.\begin{array}{l}{u}_{sd} = U_{sd0}+\Delta u_{sd} \hspace{0.33em}{i}_{sd} = I_{sd0}+\Delta i_{sd}\\ u_{sq} = U_{sq0}+\Delta u_{sq} \hspace{0.33em}{i}_{sq} = I_{sq0}+\Delta i_{sq}\\ {\Psi }_{sd} = {\Psi }_{sd0}+\Delta {\Psi }_{sd} M_e = M_{e0}+\Delta M_e\\ {\Psi }_{sq} = {\Psi }_{sq0}+\Delta {\Psi }_{sq} M_m = M_{m0}+\Delta M_m\\ \omega = {\omega }_s+\Delta \omega \hspace{0.33em}\hspace{0.33em}\delta = {\delta }_0+\Delta \delta \end{array}\right\}$$

(24)

By substituting the relations in Equation (24) into Equations (1), (6), (12) and (14), and subtracting the equations for the steady states, while neglecting the armature resistance and higher-order terms, the equations in the operator form are obtained:

$$\left.\begin{array}{l}\Delta u_{sd}\left(p\right)= p\Delta {\Psi }_{sd}\left(p\right)-{\omega }_s\Delta {\Psi }_{sq}\left(p\right)-\Delta \omega \left(p\right){\Psi }_{sq0} \\ \Delta u_{sq} = p\Delta {\Psi }_{sq}\left(p\right)+{\omega }_s\Delta {\Psi }_{sd}\left(p\right)+\Delta \omega \left(p\right){\Psi }_{sd0}\end{array}\right\}$$

(25)

$$\Delta \omega \left(p\right)=p\Delta \delta \left(p\right)$$

(26)

$$\begin{gathered} \Delta M_e\left(p\right)= {\displaystyle \frac{3}{2}}{\displaystyle \frac{p_b}{{\omega }_s}}\left[{\omega }_s{\Psi }_{sd0}\Delta i_{sq}\left(p\right)-{ \omega }_s{\Psi }_{sq0}\Delta i_{sd}\left(p\right)+I_{sq0}{\omega }_s\Delta {\Psi }_{sd}\left(p\right) \\ -I_{sd0}{\omega }_s\Delta {\Psi }_{sq}\left(p\right)\right] \end{gathered}$$

(27)

$${\displaystyle \frac{J}{p_b}}{p}^2\Delta \delta \left(p\right)+{\displaystyle \frac{D}{p_b}}p\Delta \delta \left(p\right)+\Delta M_e\left(p\right)=\Delta M_m(p)$$

(28)

Taking into account the expression for the electromagnetic moment ΔM_e(p) (A31) in Equation (28), we obtain the relationship for the load angle Δδ(p):

$$\Delta \delta \left(p\right)= {\displaystyle \frac{\Delta M_m\left(p\right)}{\begin{array}{l}\frac{J{\omega }_s^2}{p_b}{\left(\frac{p}{{\omega }_s}\right)}^2 + \frac{D{\omega }_s}{p_b}\frac{p}{{\omega }_s} + \\ +\frac{p_b}{{\omega }_s}\left\{\begin{array}{l} Q_{e0}+\frac{3}{2}{U}_{sm}^2\left(\frac{{\mathit{cos}}^2{\delta }_0}{X_{sq}\left(p\right)}+\frac{{\mathit{sin}}^2{\delta }_0}{X_{sd}\left(p\right)}\right)+\\ -\frac{p}{{\omega }_s}\left[P_{e0}+\frac{3}{4}{U}_{sm}^2\left(\frac{1}{X_{sq}\left(p\right)}-\frac{1}{X_{sd}\left(p\right)}\right)\mathit{sin}2{\delta }_0\right]\end{array}\right\}\end{array}}}$$

(29)

Equation (29) allows for determining the variation in the load angle as the driving torque changes. Using polynomial calculus in Matlab, Equation (29) can be expressed in the following form:

$$\Delta \delta \left(p\right)= \Delta M_m\left(p\right)W\left(p\right)= \Delta M_m(p){\displaystyle \frac{N\left(p\right)}{M\left(p\right)}}$$

(30)

Both the numerator N(p) and the denominator M(p) of the algebraic fraction W(p) in Formula (30) are polynomials of the fractional order. This poses challenges for decomposing the function into partial fractions and, consequently, for converting the operator form into the time domain. The order of the fractional derivative of α in the expressions for the operator inductance Equations (17) and (18) is a decimal fraction in the range 0 < α <1. The exponent of α can be represented as a proper irreducible fraction:

$$\begin{gathered} \alpha = {\displaystyle \frac{n}{m}} \\ \alpha + 1 = {\displaystyle \frac{n}{m}}+1 = {\displaystyle \frac{m+n}{m}} \end{gathered}$$

(31)

where n and m are integers, and n < m.

In such a case, the exponents of the Laplace operator p of the inductances L_sd(p) and L_sq(p) in Equations (17) and (18) will be

$$\begin{gathered} p^2={\left(p^{{\displaystyle \frac{1}{m}}}\right)}^{2m} \\ {p}^{\alpha +1}={\left(p^{{\displaystyle \frac{1}{m}}}\right)}^{m+n} \\ {p}^1={\left(p^{{\displaystyle \frac{1}{m}}}\right)}^m \\ {p}^{\alpha }={\left(p^{{\displaystyle \frac{1}{m}}}\right)}^n \end{gathered}$$

(32)

Using the substitution

$$q=p^{{\displaystyle \frac{1}{m}}}$$

(33)

the operator inductances L_sd(p) and L_sq(p) in Equations (17) and (18) are transformed into

$$\left.\begin{array}{l}{L}_{sd}\left(p\right)=L_{sd}{\displaystyle \frac{q^{2m}{b}_{1d}+q^{m+n}{b}_{2d}+q^m{b}_{3d}+q^n{b}_{4d}+b_{5d}}{q^{2m}{a}_{1d}+q^{m+n}{a}_{2d}+q^m{a}_{3d}+q^n{a}_{4d}+a_{5d}}} \\ L_{sq}\left(p\right)=L_{sq}{\displaystyle \frac{q^m{b}_{1q}+q^n{b}_{2q}+b_{3q}}{q^m{a}_{1q}+q^n{a}_{2q}+a_{3q}}}\end{array}\right\}$$

(34)

The order of the fractional derivative should be selected so that the polynomial degree of both the numerator and denominator of L_sd(q) is as low as possible. When α = 1/2, the resulting polynomial order is 4. When α = 2/5 or 3/5, the polynomial order is 10. Therefore, from the perspective of numerical computation, the most favorable order of the fractional derivative is α = 0.5. Furthermore, for fractional-order derivatives with α = 0.5, analytical expressions can be derived for the damping factor and the swing frequency.

4.2. Calculation of Transients

To simplify the calculations, it is convenient to use a relative units system. The following reference units are adopted:

$$\begin{array}{ll}{I}_b = & I_{smN} U_b = U_{smN} S_b = {\displaystyle \frac{3}{2}}{U}_{smN}{I}_{smN} M_b = {\displaystyle \frac{p_b}{{\omega }_s}}{S}_b {\omega }_b = {\omega }_s\\ Z_b = & {\displaystyle \frac{U_{smN}}{I_{smN}}} L_b = {\displaystyle \frac{Z_b}{{\omega }_s}} {\Psi }_b = {\displaystyle \frac{U_b}{{\omega }_s}}\hspace{1em}\hspace{1em}\hspace{1em}{t}_b = {\displaystyle \frac{1}{{\omega }_s}}\end{array}$$

where U_smN and I_smN are the rated maximum phase values of the armature voltage and current, ω_s is the synchronous speed, and p_b is the number of pole pairs.

Introducing the relevant notations,

$$\left.\begin{array}{l}{\tau }_m={\displaystyle \frac{J{\omega }_s^2}{p_b{M}_b}} {\tau }_D={\displaystyle \frac{D{\omega }_s}{p_b{M}_b}} p_{e0}={\displaystyle \frac{P_{e0}}{S_b}} q_{e0}={\displaystyle \frac{Q_{e0}}{S_b}}\hspace{0.33em}\\ b=u_{sm}^2{\mathit{sin}}^2{\delta }_0 c=u_{sm}^2{\mathit{cos}}^2{\delta }_0 d={\displaystyle \frac{u_{sm}^2}{2}}\mathit{sin}2{\delta }_0\end{array}\right\}$$

(35)

and using a relative units system, Equation (29) can be rewritten in the following form:

$$\Delta \delta \left(p\right)= {\displaystyle \frac{\Delta m_m\left(p\right)}{{\left(\frac{p}{{\omega }_s}\right)}^2{\tau }_m+\frac{p}{{\omega }_s}{\tau }_D+q_{e0}+\frac{c}{x_{sq}\left(p\right)}+\frac{b}{x_{sd}\left(p\right)}-\frac{p}{{\omega }_s}\left(p_{e0}+\frac{d}{x_{sq}\left(p\right)}-\frac{d}{x_{sd}\left(p\right)}\right)}}$$

(36)

where τ_m and τ_D are the starting time constant and the mechanical time constant of the turbogenerator set, respectively; x_sd(p) and x_sq(p) are the d- and q-axis operator reactance in relative units; and Δm_m is the mechanical torque in relative units.

Considering relationships Equations (32)–(34), Equation (36) can be represented as a rational algebraic function of the variable q, with the numerator and denominator being polynomials with real coefficients of the integer order:

$$\Delta \delta \left(p\right)= \Delta m_m(p){\displaystyle \frac{d_1{q}^{r-4}+d_2{q}^{r-5}+\cdots +d_{r-4}q+d_{r-3}}{c_1{q}^r+c_2{q}^{r-1}+\cdots +c_{r}q+c_{r+1}}}=\Delta m_m(p){\displaystyle \frac{N\left(q\right)}{M\left(q\right)}}$$

(37)

For an equivalent circuit comprising two rotor circuits in the d-axis and one rotor circuit in the q-axis, the polynomial degree is given by r = 3m + 4, where m is the denominator of coefficient α in the relationship in Equation (31). Thus, for α = 1/2, r = 10, and for α = 2/5, r = 19. For two rotor equivalent circuits in the d- and q-axes, the order of the polynomial is r = 4m + 4, and for α = 1/2, r = 12, and for α = 2/5, r = 24.

The transition from operator form to time form requires determining the roots of the denominator polynomial in Equation (37). Since the coefficients of the denominator polynomial M(p) of function Equation (37) are real, it has r roots, which may be real or occur as complex conjugate pairs. Some roots may have positive real parts. For α = 0.5 and two equivalent rotor circuits in the d- and q-axes (order of polynomial r = 12), the roots are presented in

Table 3. Roots of the denominator polynomial of the load angle transfer function.

i	Fractional-Order Model	Lumped-Parameter Model
	Figure 2c	Figure 1
	qi	qi
1	−1.1904∙100	−1.5723∙100
2	−5.3160∙10−1	−7.6868∙10−1
3	−1.2412∙10−1	−9.3459∙10−2
4	−6.5951∙10−4	−5.3735∙10−2
5	−9.9400∙10−2 + j1.2725∙10−1	−6.4067∙10−3
6	−9.9400∙10−2 − j1.2725∙10−1	−1.4487∙10−3
7	−1.9889∙10−2 + j4.550∙10−2	−6.2823∙10−4
8	−1.9889∙10−2 − j4.550∙10−2	−4.6353∙10−3 + j4.0803∙10−2
9	−8.2872∙10−3 + j9.3595∙10−2	−4.6353∙10−3 − j4.0803∙10−2
10	−8.2872∙10−3 − j9.3595∙10−2
11	1.3273∙10−1 + 1.4535∙10−1
12	1.3273∙10−1 + 1.4535∙10−1

.

Thus, there are three negative real roots, six complex conjugate roots with negative real parts, and two complex conjugate roots with positive real parts, where the imaginary component exceeds the real component.

By decomposing Equation (37) into partial fractions, the operator equation for the increment in the power angle Δδ(p) can be written as

$$\Delta \delta \left(p\right)=\Delta m_m\left(p\right)\sum _{i=1}^r{\displaystyle \frac{Q_i}{p^{\frac{1}{m}}-q_i}}=\Delta m_m(p)\sum _{i=1}^r\hspace{0.33em}{\displaystyle \frac{Q_i}{p^{\lambda }+b_i}}$$

(38)

where q_i represents the roots of the denominator polynomial in expression (37), b_i = −q_i, Q_i represents the coefficients from the partial fraction decomposition, and λ = 1/m, where for α = 1/2 m = 2, λ = 0.5.

For a step change in the mechanical torque, Δm_m(t) = Δm_m1(t), and Δm_m(p) = Δm_m/p, and the variation in the load angle is determined by

$$\Delta \delta \left(p\right)={\displaystyle \frac{\Delta m_m}{p}}\sum _{i=1}^r\hspace{0.33em}{\displaystyle \frac{Q_i}{p^{\lambda }+b_i}}=\Delta m_m\sum _{i=1}^r\hspace{0.33em}{\displaystyle \frac{Q_i}{p(p^{\lambda }+b_i)}}$$

(39)

From the relationship in Equation (26), it follows that the increment in the rotor angular velocity Δω(p), with a step change in the mechanical torque, is given by

$$\Delta \omega \left(p\right)=p\Delta \delta \left(p\right)=\Delta m_m\sum _{i=1}^r\hspace{0.33em}{\displaystyle \frac{Q_i}{p^{\lambda }+b_i}}$$

(40)

Applying the inverse Laplace transform (A33) yields the time form of the functions in Equations (39) and (40):

$$\left.\begin{array}{l}\Delta \delta \left(t\right)= \Delta m_m{\displaystyle \sum _{i=1}^r}\hspace{0.33em}{\displaystyle \frac{Q_i}{b_i}}\left[1-E_{\lambda }(-b_i{t}^{\lambda })\right] \\ \Delta \omega \left(t\right)= \Delta m_m{\displaystyle \sum _{i=1}^r}{Q}_i{t}^{\lambda -1}{E}_{\lambda ,\lambda }(-b_i{t}^{\lambda })\end{array}\right\}$$

(41)

where E_α,β (z) is the two-parameter Mittag-Leffler function in Equation (A34).

Further analysis of the electromechanical swings of the synchronous generator is carried out for fractional-order derivatives, where α = 0.5. In this case, the order of the denominator polynomial in Equation (37) is minimal, which improves the numerical stability of the partial fraction decomposition.

From the relationships in Equations (A35) and (A36), the load angle Δδ can be expressed as the sum of three components: the steady-state component Δδ_s, the aperiodic component Δδ_a, and the periodic component Δδ_p.

$$\Delta \delta \left(t\right)=\Delta {\delta }_s\left(t\right)+\Delta {\delta }_a\left(t\right)+\Delta {\delta }_p\left(t\right)$$

(42)

The corresponding equations describe each component:

$$\left.\begin{array}{l}\Delta {\delta }_s\left(t\right)= \Delta m_m\left({\displaystyle \sum _{i=1}^{r-2}}{\displaystyle \frac{Q_i}{b_i}}-{\displaystyle \sum _{i=r-1}^r}{\displaystyle \frac{Q_i}{b_i}}\right)\\ \Delta {\delta }_a\left(t\right)= -\Delta m_m{\displaystyle \sum _{i=1}^r}{\displaystyle \frac{Q_i}{b_i}}{e}^{b_i^2}erfc(b_i\sqrt{t}\hspace{0.33em})\hspace{0.33em}\\ \Delta {\delta }_p\left(t\right)= 2\Delta m_m{\displaystyle \sum _{i=r-1}^r}{\displaystyle \frac{Q_i}{b_i}}{e}^{b_i^{2}t}\end{array}\right\}$$

(43)

When the fractional-order derivative in Equation (40) is of the order λ = 0.5, the angular velocity response Δω is given by

$$\Delta \omega \left(t\right)=\Delta m_m{\displaystyle \frac{1}{\sqrt{\mathsf{\pi }t}}}\sum _{i=1}^r{Q}_i\hspace{0.33em}-\Delta m_m\sum _{i=1}^r{Q}_i{b}_i{e}^{b_i^2}erfc(b_i\sqrt{t}\hspace{0.33em})+2\Delta m_m\sum _{i=1}^r{Q}_i{b}_i{e}^{b_i^2}$$

(44)

From the partial fraction decomposition [2], if the degree of the numerator polynomial N(q) in Equation (37) is at least two orders lower than that of the denominator M(q), the sum ΣQ_i in Equation (44) equals zero. Thus, Equation (44) becomes

$$\Delta \omega \left(t\right)=-\Delta m_m\sum _{i=1}^r{Q}_i{b}_i{e}^{b_i^2}erfc(b_i\sqrt{t}\hspace{0.33em})+2\Delta m_m\sum _{i=r-1}^r{Q}_i{b}_i{e}^{b_i^2}$$

(45)

The first term of Equation (45) represents the sum of the aperiodic components of the angular velocity:

$$\Delta {\omega }_a = -\Delta m_m\sum _{i=1}^r{Q}_i{b}_i{e}^{b_i^2}erfc(b_i\sqrt{t}\hspace{0.33em})$$

(46)

Meanwhile, the second term of Equation (45) represents the periodic component of the angular velocity:

$$\Delta {\omega }_p\left(t\right)= 2\Delta m_m\sum _{i=r-1}^r{Q}_i{b}_i{e}^{b_i^{2}t}$$

(47)

If b_r = x + jy, then exp(b_r²t) in Formulas (43) and (47) can be written as

$$e^{b_r^{2}t}=e^{(x^2-y^2)t}{e}^{j2xy}=e^{{\alpha }_{h}t}(\mathit{cos}{\omega }_{h}t+j\mathit{sin}{\omega }_{h}t)$$

(48)

where α_h is the damping factor of the electromechanical swings, and ω_h is the angular frequency of the electromechanical swings, defined as

$$\left.\begin{array}{l}{\alpha }_h=x^2-y^2\\ {\omega }_h=2xy\end{array}\right\}$$

(49)

Given the relationship in Equation (48) and considering that b_r−1 = b_r* and Q_r−1 = Q_r*, the periodic components of the load angle Δδ_p and angular velocity Δω_p can be expressed as

$$\left.\begin{array}{l}\Delta {\delta }_p\left(t\right)=4\Delta m_m\mathit{Re}\left[{\displaystyle \frac{Q_r}{b_r}}{e}^{{\alpha }_{h}t}(\mathit{cos}{\omega }_{h}t+j\mathit{sin}{\omega }_{h}t)\right] \\ \Delta {\omega }_p\left(t\right)=4\Delta m_m\mathit{Re}\left[Q_r{b}_r{e}^{{\alpha }_{h}t}(\mathit{cos}{\omega }_{h}t+j\mathit{sin}{\omega }_{h}t)\right]\end{array}\right\}$$

(50)

From Equations (43) and (45), it follows that during a transient following a step change in the driving torque, the steady-state component of the power angle Δδ_s and the aperiodic components of the power angle Δδ_a and angular velocity Δω_a are determined by all the roots of the denominator polynomial in Expression (37). In contrast, the periodic components of the power angle Δδ_p and angular velocity Δω_p are defined by two complex conjugate roots with positive real parts. From the relationship in Equation (49), the electromechanical swing frequency is equal to twice the product of the real and imaginary parts of the complex root with a positive real part, and the damping coefficient equals the difference in the squares of its real and imaginary parts. The periodic components of the responses of the power angle and the angular velocity will decay if the damping coefficient in Equation (49) is negative, which occurs when the real part of the root with a positive real part is smaller than its imaginary part.

The calculations of the power angle waveform were also performed for the lumped-parameter circuit model using the data presented in Table 2. Taking into account the relation in Equation (22), the operational expression of the power angle in Equation (36) for the lumped-parameter model, under a step change in the turbine torque, takes the following form:

$$\Delta \delta \left(p\right) ={\displaystyle \frac{\Delta m_m}{p}}{\displaystyle \frac{d_1{p}^{r-2}+c_2{p}^{r-3}+\cdots +c_{r-2}p+c_{r-1}}{c_1{p}^r+c_2{p}^{r-1}+\cdots +c_{r}p+c_{r+1}}} = \Delta m_m\sum _{i=1}^r{\displaystyle \frac{Q_i}{p(p-q_i)}}$$

(51)

where r is the order of the denominator polynomial, and r = n_d + n_q + 2.

The roots of the polynomial of Equation (51) are presented in Table 3. The time-domain form of Equation (51) is defined by the relation

$$\Delta \delta \left(t\right)=-\Delta m\sum _{i=1}^r{\displaystyle \frac{Q_i}{q_i}}+\Delta m\sum _{i=1}^{r-2}{\displaystyle \frac{Q_i}{q_i}}{e}^{q_{i}t}+\Delta m\sum _{i=r-1}^r{\displaystyle \frac{{\underset{\_}{Q}}_i}{{\underset{\_}{q}}_i}}{e}^{{\underset{\_}{q}}_{i}t}$$

(52)

The first term of Equation (52) represents the steady-state component. The second term represents the sum of the aperiodic components associated with the real roots. The third term of Equation (52) defines the periodic component of the power angle and is associated with a pair of complex conjugate roots q_r = q*_r–1. The real part of these roots determines the damping factor, while the imaginary part determines the frequency of the electromechanical oscillations. The values of the damping factor and the oscillation frequency are presented in

Table 4. Values of the damping factor and the pulsation of electromechanical swings.

Parameter	Fractional-Order Model		Lumped-Parameter Model (Figure 1)
	One Circuit in the q-Axis (Figure 2b)	Two Circuits in the q-Axis (Figure 2c)
αh [p.u.]	−0.00565	−0.0035094	−0.0046353
ωh [p.u.]	0.04168	0.038585	0.040803
Th [s]	0.563	0.907	0.687
fh [Hz]	2.08	1.93	2.04

.

The waveforms of the load angle and angular velocity, as well as the damping coefficient and electromechanical swing frequency, were calculated for the TWW_200 turbogenerator with the following rated parameters: P_N = 200 MW, U_N = 15.75 kV, I_N = 8625 A, and cosφ_N = 0.85.

It was assumed that the time constant of the turbine generator unit T_m = 6.8 s, the induced voltage in the armature winding e_fm = 2.46 p.u., the load angle δ₀ = 34⁰, and the step change in the driving torque Δm_m = 0.2.

For the fractional-order model, Figure 9 and Figure 10 show the load angle and angular velocity waveforms, while Figure 11 presents the angular velocity–load angle trajectory.

From the waveforms presented in Figure 9, it follows that the number of equivalent circuits (Figure 2) used in the rotor affects the load angle waveform, as well as the values of the damping factor and the frequency of electromechanical swings (Table 4).

The values of the damping coefficient and the pulsation of electromechanical swings, calculated from Equation (49), are α_h = 0.00351 pu and ω_h = 0.03858 p.u, which correspond to the damping time constant and the electromechanical swing frequency of the turbogenerator: T_h = 0.907 s and f_h = 1.929 Hz.

Figure 12 compares the power angle waveforms calculated based on the fractional-order model and those determined from the lumped-parameter circuit model.

Table 4 presents the values of the damping factor α_h and the pulsation ω_h, as well as the time constant of damping T_h and the frequency f_h of the electromechanical swings (quantities marked as p.u. are expressed in per unit, i.e., relative to the selected base values).

The values of the swing frequency f_h determined from the fractional-order model and the lumped-parameter model are similar.

5. Conclusions

From the conducted considerations and performed calculations, it follows that it is possible to determine the variation in the load angle of a synchronous generator connected to an infinite bus under small disturbances from the equilibrium state by applying fractional derivatives of order 0.5. In this case, analytical relations are obtained, on the basis of which the damping coefficient and the frequency of electromechanical swings can be determined.

The presented fractional-order model of the synchronous generator can be used to analyze small disturbances, such as those that normally occur in a power system. These disturbances are sufficiently small to allow the linearization of the differential equations around the equilibrium point. Therefore, this model cannot be applied to simulations of large disturbances, such as faults in the transmission system. Due to its computational complexity, it is also not suitable for real-time simulation.

Commonly used models of synchronous generators for power system analysis are based on equivalent circuit models with lumped parameters. They typically include one damper circuit in the d-axis and one in the q-axis, or two damper circuits in the d-axis and the q-axis. A drawback of lumped-parameter models is the difficulty of measurement-based identification of more than two equivalent circuits in the rotor in the d-axis and q-axis.

A key advantage of the fractional-order model is its accuracy in the frequency domain. The results indicate that the fractional-order model achieves a more precise fit of the frequency characteristics compared with the classical model with the same number of equivalent circuits in the rotor. Specifically, using one equivalent circuit in the rotor d-axis and two circuits in the q-axis for the fractional-order model corresponds to the need for two equivalent circuits in the d-axis and four circuits in the q-axis in the lumped-parameter model to attain comparable approximation accuracy. The correctness of the obtained results can be verified by calculating the steady-state value of the load angle using Equation (29) based on the final value theorem in the Laplace domain, and by calculating it from Equation (43) in the time domain. In both cases, the same values were obtained.

Future work will involve developing a cylindrical rotor synchronous generator model and performing both simulation and experimental studies under large disturbances, employing nonlinear integer-order and fractional-order models. The test setup will consist of a synchronous generator with a power of approximately 10 kW driven by a separately excited DC motor and a high-power single-phase sinusoidal voltage generator for measuring frequency characteristics of spectral transmittances in the range of 0.001–200 Hz.