The two parts of a SM's structure are magnetically interdependent. Thus, the formulation of its behavior requires the definition of a coordinate axis of reference. Among the various possibilities for the reference, only two have certain advantages: a) the stator-fixed axis (referred to as abc) in which the main axis aligns with the magnetic axis of phase a, and a d axis coincides with the magnetic axis of the rotor rotating at speed ω_s relative to the fixed axis a; b) the rotor-located axes (referred to as 0dq) in which the main axis aligns with the magnetic axis of the field winding, d, but shifted at an angle δ depending on the load so that θ in Figure 1 is ωt+δ+π⁄2 (refer to ^[15]). Additionally, Figure 1 shows the windings of the SM and the two reference axis systems.

Fig. 1
Representation of the windings in the synchronous machine.

1) Model Structure of the Machine in the ABC Domain. The formal theoretical model of the SM assumes v_F and i_F as constants in steady state, while setting i_D and i_Q to zero for the damper windings. Figure 2 displays six windings, each with a resistor (R_i ), self-inductance (L_i ), and mutual inductances (dots in Figure 2) for characterization. The structure of the model comprises mutually coupled equivalent circuits in either phase-domain or ABC domain.

Fig. 2
Mutually coupled circuits of a SM in the abc domain.

According to the structure in Figure 2, the following expression represents the voltage general equation.

(1)

The self and mutual inductances of the machine determine the flux linkages between the stator and rotor. Thus, in (1)

(2)

the vectors ψ_abc and ψ_FDQ consist of the instantaneous values of fluxes in the windings, while I_abc and I_FDQ represent the current's instantaneous values in those windings. The subscripts s and r in the inductances L indicate stator and rotor, respectively. The submatrix L_sr contains the mutual inductances between the stator and rotor, and L sr = 𝐋 𝑟𝑠 T .

When the reference axes are fixed in the stator, as illustrated in Figure 1, the magnetic circuit of the machine varies depending on the rotor's position. Most inductances rely on the angle θ, representing the angle between the magnetic axis of phase a in the stator and axis d in the rotor. Equations (3), (4), and (5) display L_ss , L_sr , and L_rr for the salient pole generator.

(3)

Equation (3) presents L_s as the constant component of L_a , which does not depend on the angle θ, and L_m as the coefficient of L_a 's variable component. L_m relies on θ and, therefore, varies with time. The self-inductance for phases b and c account for a 120º offset between the windings concerning phase a. M_s is the constant component of the mutual inductance L_ba , independent of θ. By appropriately considering the 120º offset, we can determine L_ca and L_bc . Moreover, L_ab equals L_ba , L_bc equals L_cb , and L_ac equals L_ca .

In a cylindrical poles machine, L_m equals zero, meaning L_a , L_b , and L_c are identical to L_s . Therefore, the self-inductances are constant and independent of the rotor's angular position, and the mutual inductances are equivalent to -M_s .

(4)

Equation (4) describes 𝑀 𝐹 = 𝑁 𝐹 𝑁 𝑠 ?? 𝑑 ; 𝑀 𝐷 = 𝑁 𝐷 𝑁 𝑠 𝜆 𝑑 y 𝑀 𝑄 = 𝑁 𝑄 𝑁 𝑠 𝜆 𝑞 , where N_F , N_D , N_Q , and N_s correspond to the number of turns in the field winding, damping winding in the d-axis, damping winding in the q-axis, and stator winding, respectively. The symbol λ represents the permeance of the medium in which the flux circulates.

(5)

For both types of machines, the magnetic fluxes of the windings in the rotor are independent of the angular position, so all components of L_rr are constants. Nonetheless, there is a flux linkage between the field winding F and the direct axis damper winding D as they share the same axis (Figure 1). Therefore, L_FD = L_DF = M_r . There is no magnetic coupling between any of the windings in the direct axis (F and D) and the damping winding in the q axis. Consequently, L_FQ = L_QF = L_DQ = L_QD = 0.

Based on (3)-(5) in the general voltage equation (1) and (2), it can be concluded that the terminal voltages of the SM are described by non-linear trigonometric ODEs with time-varying coefficients dependent on the rotor position, where θ=ωt+δ+π⁄2. This ODE structure is commonly referred to as the machine model in the time domain, also known as "phase domain" or "abc domain".

2) Model Structure of the Machine in the DQ Domain. The variables in the stator (phases a, b, and c) can be transformed into new variables with a reference frame moving with the rotor using the "theory of two axes" ^[22-23]. This theory allows for the transformation of the equations expressed in the static axis abc to the equations in the reference system 0dq, which rotates at a constant speed ω_s. If we assume that the rotor speed is constant, and given a balanced system, the transformed equations for the inductances are linear. However, if one of these conditions is not met, the transformed equations become non-linear, and the model must be solved by numerical discretization. Park's work has shown that the variables in the abc axes are related to the variables in the dq axes through the normalized matrix P, given by the following equation.

(6)

Park's transformation matrix is orthogonal, meaning that its inverse is equal to its transpose, i.e., P^-1 = P^T. This property ensures that the total power is conserved during the transformation. The angle θ represents a fixed phase shift between the abc and dq reference frames at the time of transformation at t = t_x at which the transformation is made. This angle is given by θ_x = ωt_x and is constant because ω is constant. The Park transformation for the fluxes, tensions and currents of the machine are

(7)

Given (7) is also deduced that

(8)

Equation (2) in the dq axes is obtained by substituting (8) for the fluxes ψ_abc and currents I_abc , while omitting the terms ψ_FDQ and I_FDQ . By operating P and P^-1, as defined in (6), on the submatrix of inductances from (3) to (5), we can express (2) as shown below, noting that L_rs P^-1 = L_rs P^T = (PL_sr )^T.

(9)

where L₀ = L_s - 2M_s ; L_d = L_s + M_s + 1.5L_m ; L_q = L_s + M_s - 1.5L_m y k = √1.5. From this equalities L_s = ⅓(L₀ + L_d + L_q ) and L_m = ⅓(L_d - L_q ). Equation (10) comes from the substitution of (8) and the flux matrix in (9) in (1).

(10)

If the reference axes are fixed in the rotor, the voltage equation for the SM expressed in (1) and (2) takes the form of a linear ODE with constant coefficients given by (10). This is because the inductance matrix elements, which correspond to the flux linkages in (9), are independent of time. The model structure of the SM as an ODE in (10) defines its model in dq domain, which is represented in the form of mutually coupled equivalent circuits, as shown in Figure 3. Equation (10) can be expressed in a compact form as follows.

(11)

Three-phase short circuits, also known as symmetric failures, account for approximately 5% of electrical failures. These failures generate the highest current intensity, making it crucial for power systems to be protected against them. Therefore, short-circuit studies prioritize the analysis of this type of failure.

Fig. 3
Mutually coupled circuits of the SM in the dq domain [15].

Transient-stability studies require observation of the power angle of the SM since it is related to torque. Therefore, it becomes important to compute torque in three scenarios: 1) during the design stage of the SM and power system, 2) during a failure to ensure the current does not exceed operational limits, and 3) for a few seconds after the failure has been cleared. To compute torque, the current and magnetic flux in the SM must be simulated using models. The parameters of these models are determined using the Frequency Response test.

Tests are conducted to obtain measurements from the SM to determine its parameters. Over the years, a wide range of methods have been used to characterize the SM, which can be classified into three groups: steady-state tests, transient tests, and frequency response tests. The following is a reference to the last two categories.

1) Sudden Three-Phase Short-Circuit Test, SSC. To characterize the behavior of a SM in transient state, the SSC test is performed. The SM, initially in a steady-state and open-circuit condition, is suddenly switched to an operational condition by applying a short circuit to the armature terminals, as detailed in section 11.4 of ^[24].

The structure of the model used for the SSC test is called the "constant voltage behind reactance" model. The equivalent network of this model is based on the constant field flux linkage theorem, which is derived from (10), as shown in Chapter 6 of ^[15]. The theorem's deductions involve approximations that lead to the representation of the SM in its stationary, sub-transient, and transient states using phasor diagrams and equivalent circuits to obtain the constant voltage behind impedance. In this model, when the SSC test occurs at t=0, the effective value of the maximum current is computed based on the envelope, with the dc component subtracted, as shown in Figure 4.

Fig. 4
Envelope of the armature current I_ac for the SSC fault [15].

The symmetric component I_ac in Figure 4, as described in (12), can be obtained from the model of constant voltage behind impedance with three simplifications: 1) neglecting the saliency and resistances of the armature windings, 2) approximating I_a ≈ I_d , and 3) assuming a single voltage 𝐸= 𝐸 𝑞 > 𝐸 𝑞 ′ > 𝐸 𝑞 ′′ which leads to the assumption of constant field voltage.

(12)

Section 11 of ^[24] presents (12), which describes the structure of the model of constant voltage behind reactance shown in Figure 5.

Fig. 5
Model of constant voltage behind reactance of the SM. (a) Steady state, (b) Sub transient state, and (c) Transient state.

In ^[24], the SSC test procedure explains how to determine the characteristic quantities 𝑋 𝑑 ′ , 𝑋 𝑑 ′′ , 𝜏 𝑑 ′ and 𝜏 𝑑 ′′ . However, these values are only obtained for the direct axis, according to (12) and its model structure shown in Figure 5. Although the values for the quadrature axis can be computed, they cannot be directly measured using these tests.

Section 11.5 of ^[24] describes methods for obtaining parameter values from the SSC test, which involve graphical techniques such as polynomial or spline interpolation. The advances in SM theory introduced many constants ^[45], and the initial approximation of the parameters were subsequently eliminated ^[46-47].

2) Load Rejection Test. To determine the transient state of the SM, load rejection tests require sudden changes in the stator and field windings, which interrupt the armature currents of the machine under the conditions of i_d = 0 and i_q = 0. These conditions can be achieved by sub- or over-exciting the machine at a percentage of the nominal charge. Regardless, precise measurement of the rotor's angular position is necessary.

Two types of tests exist: stator-decrement tests and rotor-decrement tests ^[24]. The methodology for obtaining the SM parameters from these tests varies, as demonstrated in ^[38,48]. However, some experts believe that the stator-decrement test is more acceptable than the rotor-decrement test ^[2].

3) Standstill Frequency-Response Tests, SSFR. The SSFR tests, which were introduced in the previous century, are valuable for determining characteristic quantities and the SM parameters using models of higher complexity than the voltage behind reactance. These tests are an alternative to the short-circuit test ^[49-51], because, unlike the model in (12), they allow identification of the field windings. Moreover, in comparison to the SSC test, they do not require electromagnetic efforts in the SM.

Fig. 6
Equivalent circuits referred to the stator for the 2.1 model of the SM [21].

The lumped-parameter equivalent circuit characterizes the SM’s model structure in this test. It comprises a two-port network for the d-axis, which corresponds to the equivalent armature and field windings, and a single-port network for the q-axis due to the absence of rotor winding in that axis. The model structure’s order is determined by the number of rotor windings in each circuit ^[17]. Figure 6 displays the structure of the 2.1 model, which has 2 rotor windings in the d-axis and 1 in the q-axis, among the six model structures listed in Table 1^[2]. The concept of two-port networks is expressed analytically as follows:

(13)

The transfer functions of the two-port network equation are referred to as “operational” parameters. For the direct axis, the transfer functions are as follows:

(14)

The transfer function for the quadrature axis is

(15)

Although obtaining measurements for the test while the machine is online is possible ^[2], the usual approach for this test involves isolating the generator and bringing it to a standstill while applying steady-state sinusoidal voltages and currents to the armature and/or field winding terminals. The resulting data, corresponding to transfer functions (14) and (15), is saved as relations between the magnitudes and phases of voltages and currents. Specific procedures for the SSFR test are presented in ^[24]. The SSFR test is important for the SM because it allows the study of the machine using data for q and d axis. Making the test online implies that the generator does not get out of operation, but the required instrumentation should be more robust.

The Bode diagrams obtained from the SSFR test are used to estimate the parameters of the SM model through an optimization process to adjust the curves. Determining the parameters in terms of reactance and time constants allows for the evaluation of the model parameters in (12). Conversely, determining the parameters in terms of resistors and inductances indirectly provides the numerical values of the elements in the matrix for the model in (10).

Initially, the EMTP focused on proposing models that accurately reproduced transient behavior during short-circuits with improved computational efficiency, as described in ^[5-6], such as EMTP-RV, ATP, MicroTran, EMTDC, and NETOMAC. A comparative study of simulation techniques, such as ^[7] and specifically ^[8], reexamined the three modeling techniques for SMs using EMTP: dq model, phase-domain model, and voltage-behind-reactance model. It concluded that the voltage-behind-reactance model is the most precise. Currently, the development of EMTP focuses on the use of finite element algorithms ^[9-12], which achieve better approximations in computing the SM ^[3-4].

Large generators with salient poles can be accurately tested using the SSFR method ^[29]. The precision of the obtained parameters is verified by comparing them with two values: 1) factory values during machine design and 2) values obtained using the traditional SSC test with no load. Additionally, ^[29] reports comparing the measured short-circuit current waveform with the curves obtained from simulation using the SSFR machine model in an EMT software.

The SSC test is currently the preferred method of testing SMs due to three main reasons: 1) commercially, it is a classical test used for acceptance, performance testing, and parameter determination for dynamic analysis, and is performed by the factory as part of the technical information ^[24]. 2) The test provides sufficient information for the model with lower parametrization ( 𝑋 𝑑 ′ , 𝑋 𝑑 ′′ , 𝜏 𝑑 ′ and 𝜏 𝑑 ′′ ) as given by (12). 3) It provides accurate results for the behavior of the first electromagnetic oscillation. In ^[24], presents the short circuit function (built empirically) for the computation of the three-phase short-circuit failure of the synchronous generator with no load, which is expressed as:

(16)

The computed results are only as accurate as the estimated parameter values in the equation. It is worth noting that this equation is based on simplifications and approximations, and the value for the quadrature axis is not obtained from test measurements but is instead determined by fitting the equation.

The deterministic solution of the structure model in the dq domain given by (10) is known. Thomas A. Lipo applied the modal theory technique, which uses eigenvalues and eigenvectors, to solve the transient condition of the SM under a three-phase failure ^[44]. In contrast, Gibson H. M. Sianipar presented a solution procedure in ^[13] that differs from the model in (10) by using the eigenvalues of the state-space system ^[52]. Although Sianipar's approach presents an equation of the current for the system in dq in terms of sums, there is no evidence of an explicit example.