Open AccessArticle

A Semi-Analytical Method for the Identification of DC-Decay Parameters at an Arbitrary Rotor Position in Large Synchronous Machines

Zhenming Lai

^1,2

Haoyu Kang

³,

Demin Liu

^4,*,

Zhichao Wang

⁴,

Yong Yang

⁴ and

Jin Wang

Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China

PowerChina Hydropower Development Group Co., Ltd., Chengdu 610000, China

State Key Laboratory of Advanced Electromagnetic Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

⁴

Dongfang Electric Machinery Co., Ltd., Dongfang Electric Corporation, Deyang 618000, China

Author to whom correspondence should be addressed.

Energies 2025, 18(2), 279; https://doi.org/10.3390/en18020279

Submission received: 5 December 2024 / Revised: 6 January 2025 / Accepted: 7 January 2025 / Published: 10 January 2025

(This article belongs to the Section F: Electrical Engineering)

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Browse Figures

Figure 1
The schematic of d-axis DC-decay test. "> Figure 2
The DC-decay test at an arbitrary rotor position. "> Figure 3
Rotor position angle determination experiment. "> Figure 4
The d-axis and q-axis equivalent circuits of synchronous motors. "> Figure 5
Flowchart for parameter identification. "> Figure 6
Switches’ synchronization check results and the experimental site. "> Figure 7
The results of rotor position angle recognition. "> Figure 8
The d-axis current’s curve fitting results. "> Figure 9
The q-axis current’s curve fitting results. "> Figure 10
Prony analysis results of the TPSSC test: (a) time domain; (b) frequency domain; (c) squared error. ">

Versions Notes

Abstract

Experimental approaches for the identification of dynamic parameters in synchronous machines mainly include two methods, a three-phase sudden short-circuit (TPSSC) test and a standstill frequency response (SSFR) test. However, the former has significant safety risks, while the latter has a complex implementation process, resulting in insufficient adaptability to large-scale units. To overcome the above obstacles, this paper proposes an improved DC-decay test method that can be performed at an arbitrary rotor position so that the rotor pre-positioning process in the conventional DC-decay test can be neglected. Meanwhile, combining the transient analysis theory and particle swarm optimization algorithm, a semi-analytical parameter identification method is proposed. Finally, the proposed method is applied using a 172 MVA large synchronous machine. Compared to the results obtained by the TPSSC test using the Prony algorithm and other conventional type tests, the error of the parameter calculation results obtained with the conventional method reached a maximum of 16.6%, while that of the proposed method was merely 8.6%, and the experimental period could be shortened from 5 days to half a day.

Keywords:

arbitrary rotor position; DC decay; parameter identification; synchronous machine

1. Introduction

The stability and dynamic parameters of synchronous machines, including the transient reactances of all orders and time constants, are crucial factors in determining the operational characteristics of the units [1,2]. The control and protection strategies during the start–stop operation, as well as various short-circuit and low-voltage ride-through processes of the motor, are all dependent on the determination of motor parameters. Thus, the accurate measurement of these parameters is of vital significance for performance evaluation and protection setting.

Up to now, experimental test methods for the identification of synchronous machines’ parameters mainly include the three-phase sudden short-circuit (TPSSC) test, the voltage recovery method, the standstill frequency response (SSFR) test, the DC-decay test, and the standstill time-domain response (SSTR) test. Among them, the TPSSC test belongs to destructive tests. During this test, the motor will experience a transient short-circuit current several times the rated current, which may cause irreversible damage to the circuit breaker and motor body [3,4]. Therefore, for large-capacity units, neither motor manufacturers nor owners are inclined to conduct such tests. The voltage recovery test involves the reverse process of the TPSSC test and therefore has better safety, but similar to the three-phase sudden short-circuit test, it can only measure part of the d-axis parameters [5].

Various static testing methods conducted when the motor is stationary, such as SSFR, SSTR, and DC-decay tests, can significantly improve the safety of the test and effectively identify the q-axis parameters. Compared to the other methods, the equipment requirements for the SSFR test are more significant [6], including a high-power power supply with adjustable frequency and testing at multiple frequency points, making the testing process more complex. The SSTR and DC-decay tests are used in time-domain experiments. Multiple types of voltage excitation signals can be applied in SSTR, including a DC step signal [7], DC pulse signal [8], sinc signal [9], chirp signal [10], etc. In contrast, the DC-decay test only requires the use of the simplest DC voltage signal [11]. Although the implementation process of both methods has been greatly simplified, the problem they have in common with the SSFR test is the need to measure the d-axis and q-axis parameters at specific rotor positions. This requires the pre-positioning of the motor before the experiment, and the specific operation process is provided in [4]. Undoubtedly, this process is very difficult to carry out for large-capacity units, especially for hydro-power units with a large number of pole pairs. Small mechanical angle errors during positioning can cause significant electrical angle errors, which greatly affect identification accuracy.

Some scholars have conducted research on rotor pre-positioning for static tests. For the SSFR test, Edson et al. proposed an arbitrary rotor position test method based on an improved Dalton–Cameron transformation [12]. However, this method requires three tests at different stator winding ports, so the identification process is relatively complex and poor in accuracy. A similar concept was also employed in an SSTR test using a large synchronous condenser by Y. Ma [13]. Oteafy proposed an advanced arbitrary rotor position SSTR test method where the stator windings were excited with a three-phase chirp signal; in this method, the rotor angle had to be determined in advance [14]. Meanwhile, for the DC-decay test, Maurer proposed an arbitrary position test method using state space equations and an experimental identification method for the rotor position angle [15]. In utilizing optimization algorithms, Li rapidly identified the parameters of permanent magnet motors through the proposed chaotic Gaussian–Cauchy RAO (CGCRAO) algorithm [16]. Xie utilized the online particle swarm optimization algorithm for the online parameter identification of permanent magnet motors [17]. Nevertheless, in the above literature, numerical methods are mostly used to identify motor parameters, and the analytical relationship between input/output signals and equivalent circuit parameters is not clear enough. Moreover, most of the methods were validated through small experimental prototypes, lacking application explanations on large-capacity motors.

In this work, an arbitrary rotor position DC-decay test method combined with a semi-analytical parameter identification method is proposed. By using preliminary experiments to calculate the rotor position angle of the motor, this method can identify all dynamic parameters of the d-axis and q-axis, including transient reactance and time constants of each order, through a single test. Finally, the proposed method is applied to 167 MVA large hydro-power units. Compared to the results obtained by the TPSSC test and other conventional tests, it offers equivalent identification accuracy, and the safety and testing cycle can be significantly shortened.

The rest of this paper is organized as follows: In Section 2, the principles of the DC-decay test and the test process of the proposed arbitrary rotor position method are illustrated; then, the semi-analytical parameter identification method is presented in Section 3. Section 4 provides the experimental test results of the proposed method and traditional methods on a 172 MVA large hydro-power synchronous machine; the conclusions are given in the final section.

2. Arbitrary Rotor Position Parameter Identification Method Based on Three-Phase Connection

2.1. Conventional DC-Decay Test

The DC-decay test is a method of identifying motor parameters through the transient decay process of stator winding DC current when the motor is in a stationary state. The d-axis and q-axis parameters need to be independently identified at specific positions. Taking the d-axis case as an example, the main process is visualized in the schematic given in Figure 1.

Firstly, the stator’s two-phase windings are determined to perform the DC-decay test. Then, the axis of the rotor is adjusted at the required position. As shown in Figure 1, at this point, the d-axis of the rotor should be perpendicular to the axis of the C-phase winding. Furthermore, the excitation winding is short-circuited, and the DC current’s decay is determined by simultaneously closing and opening switches K₁ and K₂. Finally, the stator’s and rotor’s winding currents are recorded, and the d-axis parameter is identified. The experimental identification process of q-axis parameters is similar. It is worth noting that the rotor pre-positioning method has been specified in the IEEE standard [4] and related research [18], which involves applying a low AC voltage to the stator end, slowly dragging the rotor, and then measuring the induced voltage of the excitation winding. The maximum position of the induced voltage of the excitation winding is the d-axis position, and the minimum position of the induced voltage of the excitation winding is the q-axis position.

2.2. The DC-Decay Test Using an Arbitrary Rotor Position

In conventional DC-decay tests, rotor pre-positioning must be performed before the test. However, performing rotor pre-positioning on large-capacity motors is difficult. Therefore, an improved DC-decay test using an arbitrary rotor position is proposed in this paper, as shown in Figure 2. The stator winding of phase A is connected to the positive terminal of the DC power source, while the stator winding of phases B and C are connected in parallel and then connected to the negative terminal of the power source.

Under this wiring form, the three-phase voltages on the stator side satisfy the following:

\{\begin{matrix} u_{a} - u_{b} = u_{s} \\ u_{b} = u_{c} \end{matrix}

(1)

By performing the Park transformation on the three-phase voltages on the stator side, the following can be obtained:

u_{d} = \frac{2}{3} u_{s} \cos θ, u_{q} = - \frac{2}{3} u_{s} \sin θ

(2)

where θ indicates the rotor position angle, namely the angle between the axis of stator phase A and the d-axis of the rotor.

During steady-state tests, the magnetic saturation of the core is absent, and the coupling between the d-axis and q-axis can be neglected. Thus, the transient current response caused by u_d and u_q can be considered independently. The d-axis current response is equivalent to the pure d-axis current response induced by the voltage signal 2u_scosθ/3 when the angle between the rotor axis and the A-phase winding axis is zero. A similar consideration applies to the q-axis. By altering the winding connection of the test and knowing the rotor position angle θ, the Park transformation of the transient response currents from the stator’s three-phase windings can simultaneously provide the transient response currents required for the d-axis and q-axis tests when the rotor axis aligns with each axis. On this basis, parameter identification can be carried out using the identification method of the conventional DC-decay test.

2.3. Rotor Position Determination Test

Having clarified the methods and procedures for parameter identification, the current challenge lies in determining the rotor position angle θ. To maximize the use of experimental equipment identical to that used in the DC-decay test, this section presents a straightforward and practical method for identifying the rotor position angle.

The schematic for the rotor position angle determination experiment is shown in Figure 3. During this test, the motor remains stationary with the stator’s three-phase windings short-circuited together. A signal source is connected to the rotor’s excitation winding, applying a DC step voltage signal to the excitation winding port. The transient response currents of the stator’s three-phase windings are then recorded. Since only the d-axis winding is excited at this time, the q-axis current i_q obtained after the Park transformation of the stator’s three-phase currents i_a, i_b, and i_c should be zero, thus satisfying the following condition:

Further, the rotor position angle θ is determined as follows:

θ = \arctan (\frac{\sqrt{3} (i_{b} - i_{c})}{2 i_{a} - i_{b} - i_{c}})

(3)

3. Semi-Analytical Parameter Identification

3.1. Relationship Between Equivalent Circuit Parameters and Current Response

The most commonly used model for synchronous motors is the five-winding model; the schematic of the equivalent circuit is given in Figure 4, where X_sl is the stator leakage reactance, and r_s is the stator winding resistance; X_ad and X_aq denote the d-axis and q-axis armature reaction reactance, respectively; X_fl and r_f denote the excitation winding leakage reactance and resistance; X_Dl and r_D denote the d-axis damper winding leakage reactance and resistance; X_Ql and r_Q denote the q-axis damper winding leakage reactance and resistance.

Next, taking the d-axis as an example, the specific semi-analytical parameter identification process is illustrated. According to the transient analysis theory, if the initial magnetic flux of each damper winding is ignored, the magnetic flux of the d-axis satisfies the following equation:

ψ_{d} (s) = X_{d} (s) I_{d} (s) + G_{f} (s) I_{f} (s)

(4)

where X_d(s) represents the d-axis operational reactance; G_f(s) denotes the transfer function of the excitation winding; and I_d(s) and I_f(s) are the d-axis and excitation winding currents, respectively.

If there is only one damper winding, X_d(s) and G_f(s) satisfy the following:

X_{d} (s) = X_{d} \frac{(1 + s T_{d}^{'}) (1 + s T_{d}^{″})}{(1 + s T_{d 0}^{'}) (1 + s T_{d 0}^{″})}

(5)

s G_{f} (s) = \frac{X_{a d}}{r_{f}} \frac{s (1 + s T_{D l})}{(1 + s T_{d 0}^{'}) (1 + s T_{d 0}^{″})}

(6)

where T′_d and T″_d represent the d-axis short-circuit transient and sub-transient time constant, respectively; T′_d₀ and T″_d₀ represent the d-axis open-circuit transient and sub-transient time constant, respectively; T_Dl denotes the leakage time constant of d-axis damper winding; X_d denotes the d-axis synchronous reactance.

Since the excitation winding is short-circuited, the d-axis voltage equation satisfies the following:

V_{d} (s) = s X_{d} (s) I_{d} (s) + r_{s} I_{d} (s)

(7)

In the DC-decay test, besides the steady-state component, the dynamic process involves applying a reverse d-axis DC-step voltage signal to the stator terminal; therefore, V_d(s) can be expressed as follows:

V_{d} (s) = - \frac{u_{d}}{s} = - \frac{2}{3 s} u_{s} \cos θ

(8)

On this basis, the frequency-domain expressions of the d-axis current I_d(s) can be expressed as follows:

I_{d} (s) = - \frac{2 u_{s} \cos θ}{3 X_{d}^{″}} \frac{s^{2} + s (\frac{1}{T_{d 0}^{'}} + \frac{1}{T_{d 0}^{″}}) + \frac{1}{T_{d 0}^{'} T_{d 0}^{″}}}{\begin{array}{l} s^{4} + s^{3} (\frac{1}{T_{d}^{'}} + \frac{1}{T_{d}^{″}} + \frac{r_{s}}{X_{d}^{″}}) + \\ s^{2} [\frac{1}{T_{d}^{'} T_{d}^{″}} + \frac{r_{s}}{X_{d}^{″}} (\frac{1}{T_{d 0}^{'}} + \frac{1}{T_{d 0}^{″}})] + \frac{s r_{s}}{X_{d}^{″} T_{d 0}^{'} T_{d 0}^{″}} \end{array}}

(9)

I_{f} (s) = - \frac{2 u_{s} \cos θ}{3} \frac{\frac{X_{a d} T_{D l}}{r_{f} X_{d}^{″} T_{d 0}^{'} T_{d 0}^{″}} \cdot (s + \frac{1}{T_{D l}})}{\begin{array}{l} s^{3} + s^{2} (\frac{1}{T_{d}^{'}} + \frac{1}{T_{d}^{″}} + \frac{r_{s}}{X_{d}^{″}}) + \\ s^{1} [\frac{1}{T_{d}^{'} T_{d}^{″}} + \frac{r_{s}}{X_{d}^{″}} (\frac{1}{T_{d 0}^{'}} + \frac{1}{T_{d 0}^{″}})] + \frac{r_{s}}{X_{d}^{″} T_{d 0}^{'} T_{d 0}^{″}} \end{array}}

(10)

According to the equivalent circuit given in Figure 1, the time-domain general solutions of i_d(t) and i_f(t) are determined using (11), and their frequency-domain expression are given in (12) and (13).

\begin{array}{l} i_{d} (t) = a_{1} e^{- c_{1} t} + a_{2} e^{- c_{2} t} + a_{3} e^{- c_{3} t} + I_{d \infty} \\ i_{f} (t) = b_{1} e^{- c_{1} t} + b_{2} e^{- c_{2} t} + b_{3} e^{- c_{3} t} \end{array}

(11)

I_{d} (s) = \frac{α_{2} s^{2} + α_{1} s + α_{0}}{γ_{4} s^{4} + γ_{3} s^{3} + γ_{2} s^{2} + γ_{1} s}

(12)

I_{f} (s) = \frac{β_{2} s^{2} + β_{1} s + α_{0}}{γ_{4} s^{4} + γ_{3} s^{3} + γ_{2} s^{2} + γ_{1} s}

(13)

where a₁~a₃ and b₁~b₃ are the amplitude coefficients of the d-axis current and excitation winding current, respectively; c₁~c₃ are the attenuation coefficients of two currents; I_d_∞ represents the steady-state value of the d-axis current.

Comparing (9) and (10) with (12) and (13), the coefficients of (12) and (13) can be expressed as follows:

\{\begin{array}{l} α_{2} = - \frac{2 u_{s} \cos θ}{3 X_{d}^{″}} \\ α_{1} = - \frac{2 u_{s} \cos θ}{3 X_{d}^{″}} (\frac{1}{T_{d 0}^{'}} + \frac{1}{T_{d 0}^{″}}) \\ α_{0} = - \frac{2 u_{s} \cos θ}{3 X_{d}^{″}} \cdot \frac{1}{T_{d 0}^{″} T_{d 0}^{″}} \end{array}

(14)

\{\begin{array}{l} γ_{4} = 1 \\ γ_{3} = \frac{1}{T_{d}^{'}} + \frac{1}{T_{d}^{″}} + \frac{r_{s}}{X_{d}^{″}} \\ γ_{2} = \frac{1}{T_{d}^{'} T_{d}^{″}} + \frac{r_{s}}{X_{d}^{″}} (\frac{1}{T_{d 0}^{'}} + \frac{1}{T_{d 0}^{″}}) \\ γ_{1} = \frac{r_{s}}{X_{d}^{″} T_{d 0}^{'} T_{d 0}^{″}} \end{array}

(15)

\{\begin{array}{l} β_{2} = 0 \\ β_{1} = - \frac{2 u_{s} \cos θ}{3} \frac{X_{a d} T_{D l}}{r_{f} X_{d}^{″} T_{d 0}^{'} T_{d 0}^{″}} \\ β_{0} = - \frac{2 u_{s} \cos θ}{3 T_{D l}} \frac{X_{a d} T_{D l}}{r_{f} X_{d}^{″} T_{d 0}^{'} T_{d 0}^{″}} \end{array}

(16)

Then, the above coefficients can be solved using an equation construction method, and the following auxiliary items are introduced:

\{\begin{array}{l} A = \frac{1}{T_{d 0}^{'}} + \frac{1}{T_{d 0}^{″}} = \frac{α_{1}}{α_{2}} \\ B = \frac{1}{T_{d 0}^{'} T_{d 0}^{″}} = \frac{α_{0}}{α_{2}} \\ C = \frac{1}{T_{d}^{'}} + \frac{1}{T_{d}^{″}} = γ_{3} - \frac{α_{2}}{I_{d \infty}} \\ D = \frac{1}{T_{d}^{'} T_{d}^{″}} = γ_{2} - \frac{α_{1}}{I_{d \infty}} \end{array}

(17)

Taking the d-axis open-circuit time constants as an example, T′_d₀ and T″_d₀ can be solved using the following quadratic equation:

x^{2} - A x + B = 0

(18)

Therefore, the two time constants can be expressed as follows:

\begin{array}{l} T_{d 0}^{'} = \frac{2}{A - \sqrt{A^{2} - 4 B}} = \frac{2}{α_{1} - \sqrt{α_{1}^{2} - 4 α_{0} α_{2}}} \\ T_{d 0}^{″} = \frac{2}{A + \sqrt{A^{2} - 4 B}} = \frac{2}{α_{1} + \sqrt{α_{1}^{2} - 4 α_{0} α_{2}}} \end{array}

(19)

Similarly, using auxiliary items C and D, another quadratic equation can be formed to solve the d-axis short-circuit time constants T′_d and T″_d, which are expressed as follows:

T_{d}^{'} = \frac{2 α_{2}}{C - \sqrt{C^{2} - 4 D}}, T_{d}^{″} = \frac{2 α_{2}}{C + \sqrt{C^{2} - 4 D}}

(20)

Next, the various order reactances need to be solved. According to (14) and (16), the d-axis sub-transient reactance X″_d, armature reaction reactance X_ad, and damper winding leakage time-constant T_Dl are expressed as follows:

X_{d}^{″} = I_{d \infty} r_{s} / α_{2}

(21)

X_{a d} = r_{f} β_{0} / α_{0}

(22)

T_{D l} = β_{1} / β_{0}

(23)

Then, the d-axis synchronous reactance X_d can be solved using the final-value theorem in (24), and the d-axis stator leakage reactance is calculated using (25).

X_{d} = X_{d}^{″} \frac{T_{d 0}^{'} T_{d 0}^{″}}{T_{d}^{'} T_{d}^{″}}

(24)

X_{s l} = X_{d} - X_{a d}

(25)

Finally, according to the definition of transient reactance X′_d and the d-axis operational reactance (5), X′_d can be expressed as follows:

X_{s l} = X_{d} - X_{a d}

(26)

According to the above derivation process, using the time-domain fitting results of the d-axis current and the excitation winding current, all parameters of the d-axis equivalent circuit are solvable. Similarly, this method can also be used to obtain the equivalent circuit parameters of the q-axis. The above method is also applicable to more complex motor models. For example, when using multiple damping structures such as solid rotors, there will be more than one damper winding [13]. Using the method of constructing equations, a first-order equation can be constructed based on the relationship between the roots and coefficients of the first-order equation to solve the short-circuit and open-circuit time constants. In addition, when considering the mutual leakage reactance between the excitation winding and the damping winding on the rotor side, I/T equivalent circuit transformation can be used to ensure the applicability of the above derivation process [13].

3.2. Time-Domain Curve Fitting of Current Response

During the parameter identification procedure for synchronous motors utilizing the DC-decay voltage test, it is imperative to carry out time-domain fitting for the armature currents along both the d-axis and q-axis. Considering that, in the DC-decay test, the three-phase connection for any rotor position can be equivalently modeled as the combined effects of the rotor d-axis and rotor q-axis tests, the fitting processes for these two currents can be decoupled and executed independently. To accomplish this, the formulation of an optimization objective function is necessary, which can be efficiently addressed through the application of a single-objective optimization algorithm. Equation (27) is used to determine the loss function of the curve fitting process.

\min J = {\sum_{i = 1}^{N} [I (t_{i}) - (x_{1} e^{y_{1} t_{i}} + x_{2} e^{y_{2} t_{i}} + x_{3} e^{y_{3} t_{i}} + x_{4})]}^{2}

(27)

where x₁~x₄ and y₁~y₃ are the amplitude coefficients and attenuation coefficients to be solved, respectively; the subscript i denotes the i-th measurement point; and N is the number of measurement points. When solving the d-axis current, the above formula can be directly applied; when solving the q-axis current, it is necessary to reduce one attenuation term; when solving the excitation winding current, x₄ should be zero.

In this work, the simulated annealing particle swarm optimization (SAPSO) algorithm is adopted to perform the abovementioned single-objective optimization. This hybrid algorithm utilizes the probabilistic mutation ability in the simulated annealing algorithm. Not only does it enhance the flexibility and diversity of the algorithm, but it also enhances the diversity of particles. It has the advantages of high adjustment accuracy and a fast convergence rate. Meanwhile, it avoids the phenomena of “premature convergence” and getting trapped in local optima. On this basis, the whole semi-analytical parameter identification process can be described using the flowchart given in Figure 5.

4. Experimental Test Results

To validate the proposed DC-decay test and its corresponding parameter identification method, an onsite DC-decay test was performed on a synchronous generator with a rated capacity of 172 MVA. The main electrical parameters of the synchronous machine are shown in Table 1.

The synchronization of series and parallel switch actions will greatly affect the transient process of armature current, thereby affecting the accuracy of parameter identification. Therefore, before conducting the experiment, the sensitivity of the two switches was checked to ensure the synchronization of their actions. Figure 6a shows the synchronization check results of two switches, and the action error of the two switches was controlled within 9 ms. Meanwhile, the experimental site of the large-capacity synchronous machine is illustrated in Figure 6b.

Firstly, experiments were conducted to identify the rotor position angle. Considering the experimental process described in Section 2.3, the results of rotor position angle recognition are shown in Figure 7, where the moment when the transient current occurred was taken as the zero time point for ease of comparison and data processing.

As shown in Figure 7, the rotor position angle is within the range of 87 degrees to 90 degrees. Given that the tangent of the rotor position angle (tanθ) approximates infinity, and both tan 90° and tan 270° are similarly near infinity, the inverse tangent function produces values that fluctuate around 90 degrees and 270 degrees. This fluctuation solely influences the sign of the conversions along the d-axis and q-axis but does not alter the decay process or the outcomes of parameter identification. Consequently, for the purpose of our calculations, θ was determined to be 87 degrees.

Utilizing the superposition theorem, the initial values of the experimental data along the d-axis can be manipulated and isolated to derive the experimental results suitable for fitting. This manipulation is exemplified through the DC step response experiment, where a negative voltage was imparted to the stator windings. The aforementioned discussion outlines the procedure for identifying parameters in this experiment. Using (11), the time-domain curve fitting results for the d-axis experiment were obtained, which are detailed in Table 2.

According to Table 2, the coefficients using the frequency-domain expression (12) were determined, and the results are listed in Table 3.

On this basis, the curve fitting results using the d-axis current are given in Figure 8.

A similar method can be applied to fit the q-axis current, and the time-domain curve fitting results of the q-axis current are shown in Figure 9. By analogy with the expression of the d-axis current, it can be inferred that the time-domain general solution is expressed as (28) and (29). Their coefficients are given in Table 4 and Table 5, respectively.

i_{d} (t) = q_{1} e^{- e_{1} t} + q_{2} e^{- e_{2} t} + I_{q \infty}

(28)

I_{d} (s) = \frac{μ_{1} s + μ_{0}}{η_{3} s^{3} + η_{2} s^{2} + η_{1} s}

(29)

The fitting results depicted in Figure 6 and Figure 7, utilizing appropriate algorithms and mathematical models, yield current response waveforms that align well with the experimentally measured waveforms. Using the optimized target variables obtained, the analytical formulas derived in Section 3 were employed to calculate the values of various dynamic parameters in the equivalent circuits of the synchronous machine’s d- and q-axes.

It should be noted that the transient current of the TPSSC test was identified using the Prony algorithm [19], rather than the envelope method, for relatively higher accuracy. The fitting effect and error distribution of the Prony algorithm on short-circuit currents in both time and frequency domains are shown in the Figure 10 below.

The comparison in Table 6 reveals that the transient reactances of the d- and q-axes at different stages, along with the corresponding open-circuit and short-circuit transient time constants, closely resemble the design values of parameters, with discrepancies falling within a 10% margin. This underscores the effectiveness of the test methodology. Notably, during the DC-decay test, the excitation winding was maintained in a short-circuit state, resulting in an unsaturated magnetic circuit of the motor. Consequently, the parameters identified through this method represent unsaturated values. It should be emphasized that using conventional type test methods, the q-axis sub-transient open-circuit and short-circuit time constants cannot be measured, so it is not given in the table below. Meanwhile, the relative error column denotes the relative error between the identification results and the parameters’ design values.

5. Conclusions

To tackle the challenge of dynamic parameter identification in large synchronous machines, this paper introduces a novel approach for determining motor dynamic parameters at arbitrary rotor positions, leveraging the DC-decay voltage test. By incorporating a rotor position angle test to ascertain the rotor’s orientation, any DC-decay test conducted with a three-phase connection at any rotor position can be effectively decomposed into independent tests for the d-axis and q-axis positions. Utilizing the SAPSO algorithm, this method enables the measurement of the machine’s dynamic parameters with only one test. Experimental results demonstrate that the identified dynamic parameters exhibit errors within 10%, validating the efficacy of the proposed approach.

The methodology boasts several advantages: it obviates the need for pre-positioning the rotor prior to experimentation, enhances safety protocols, and minimizes equipment requirements. In practical engineering scenarios, a DC welding machine can serve as an alternative voltage source. Furthermore, the method is highly efficient, requiring just two tests to identify the dynamic parameters of both the d- and q-axes.

Author Contributions

Conceptualization, J.W. and H.K.; methodology, Z.W.; software, Z.L.; validation, J.W.; formal analysis, D.L.; investigation, Y.Y.; resources, Z.W.; writing—original draft preparation, Z.L.; writing—review and editing, D.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Sichuan Science and Technology Program, grant No. 2022ZDZX0041.

Data Availability Statement

The data presented in this study are available on request from thecorresponding author. The data are not publicly available due to privacy reasons and the project funding requirement.

Conflicts of Interest

Author Zhenming Lai was employed by the PowerChina Hydropower Development Group Co., Ltd.; Demin Liu, Zhichao Wang and Yong Yang were employed by the Dongfang Electric Machinery Co., Ltd., Dongfang Electric Corporation. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Figure 1. The schematic of d-axis DC-decay test.

Figure 2. The DC-decay test at an arbitrary rotor position.

Figure 3. Rotor position angle determination experiment.

Figure 4. The d-axis and q-axis equivalent circuits of synchronous motors.

Figure 5. Flowchart for parameter identification.

Figure 6. Switches’ synchronization check results and the experimental site.

Figure 7. The results of rotor position angle recognition.

Figure 8. The d-axis current’s curve fitting results.

Figure 9. The q-axis current’s curve fitting results.

Figure 10. Prony analysis results of the TPSSC test: (a) time domain; (b) frequency domain; (c) squared error.

Table 1. Main electrical parameters of the studied synchronous machine.

Parameter	Value	Parameter	Value
Rated capacity (MVA)	172	Rated power (MW)	150.5
Rated power factor	0.875	Rated speed (r/min)	333.3
Rated voltage (kV)	15.75	Rated Current (A)	6305

Table 2. Curve fitting results of the d-axis current.

Amplitude Coefficients		Attenuation Coefficients
a₁	108.65	c₁	−209.94
a₂	1.1152	c₂	−7.62
a₃	0.2290	c₃	−0.125
I_d_∞	−110

Table 3. Coefficients of the d-axis frequency-domain expression.

Denominator		Numerator
g₁	201	a₁	−22,820
g₂	1626.68	a₂	−178,421
g₃	217.95	a₃	−22,164
g₄	1

Table 4. Curve fitting results of the q-axis current.

Amplitude Coefficients		Attenuation Coefficients
q₁	−571.9	e₁	−215.43
q₂	−28.08	e₂	−3.19
I_q_∞	600

Table 5. Coefficients of the q-axis frequency-domain expression.

Amplitude Coefficients		Attenuation Coefficients
h₁	687.22	m₁	123,298
h₂	218.62	m₂	412,333
h₄	1

Table 6. Parameter identification results.

Parameter	Proposed Method	Type Test Method	Design Value	Relative Error
X_d	1.04 p.u.	1.13 p.u.	1.09 p.u.	−4.6%, +3.7%
$X_{d}^{'}$	0.274 p.u.	0.268 p.u.	0.29 p.u.	−5.5%, −7.6%
$X_{d}^{″}$	0.21 p.u.	0.20 p.u.	0.22 p.u.	−4.5%, −9.0%
$T_{d}^{'}$	2.10 s	2.13 s	2.238 s	−6.1%, −4.8%
$T_{d}^{″}$	0.10 s	0.105 s	0.092 s	+8.6%, +14.1%
$T_{d 0}^{'}$	7.82 s	7.79 s	8.34 s	−6.2%, −6.6%
$T_{d 0}^{″}$	0.13 s	0.14 s	0.12 s	+8.3%, +16.6%
X_q	0.832 p.u.	0.851 p.u.	0.9 p.u.	−7.5%, −5.4%
$X_{q}^{″}$	0.212 p.u.	0.211 p.u.	0.22 p.u.	−3.6%, −4.1%
$T_{q}^{″}$	0.0836 s	/	0.0762 s	+9.7%
$T_{q 0}^{″}$	0.293 s	/	0.312 s	−6%

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Lai, Z.; Kang, H.; Liu, D.; Wang, Z.; Yang, Y.; Wang, J. A Semi-Analytical Method for the Identification of DC-Decay Parameters at an Arbitrary Rotor Position in Large Synchronous Machines. Energies 2025, 18, 279. https://doi.org/10.3390/en18020279

AMA Style

Lai Z, Kang H, Liu D, Wang Z, Yang Y, Wang J. A Semi-Analytical Method for the Identification of DC-Decay Parameters at an Arbitrary Rotor Position in Large Synchronous Machines. Energies. 2025; 18(2):279. https://doi.org/10.3390/en18020279

Chicago/Turabian Style

Lai, Zhenming, Haoyu Kang, Demin Liu, Zhichao Wang, Yong Yang, and Jin Wang. 2025. "A Semi-Analytical Method for the Identification of DC-Decay Parameters at an Arbitrary Rotor Position in Large Synchronous Machines" Energies 18, no. 2: 279. https://doi.org/10.3390/en18020279

APA Style

Lai, Z., Kang, H., Liu, D., Wang, Z., Yang, Y., & Wang, J. (2025). A Semi-Analytical Method for the Identification of DC-Decay Parameters at an Arbitrary Rotor Position in Large Synchronous Machines. Energies, 18(2), 279. https://doi.org/10.3390/en18020279

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