Nonlinear Algebraic Parameter Estimation of Doubly Fed Induction Machine Based on Rotor Current Falling Curves

Abstract

Currently, wind turbines utilize doubly fed induction machines that incorporate a frequency converter in the rotor circuit to manage slip energy. This setup ensures a stable voltage amplitude and frequency that align with the alternating current. It is crucial to accurately determine the parameters of the equivalent circuit from the rotor side of the vector control system of the frequency converter. The objective of this study is to develop a method for the preliminary identification of the doubly fed induction machines parameters by analyzing the rotor current decay curves using Newton’s method. The numerical estimates of the equivalent circuit parameters a doubly fed induction machines with a fixed short-circuited rotor are obtained during the validation of the results on a real plant. It is along with the integral errors of deviation between the experimental rotor current decay curve and the response of the adaptive regression model. The integral errors do not exceed 4% in nearly all sections of the curves. It is considered acceptable in engineering practice. The developed algorithm for the preliminary identification for the parameters of the doubly fed induction machines substitution scheme can be applied with the configuring machines control systems, including a vector control system.

1. Introduction

The relatively low share of wind power stations in the overall structure of electricity generation enterprises can be attributed to the stochastic nature of electricity production based on wind energy conversion [1,2,3]. One method for mitigating this effect is the use of doubly fed induction machines (DFIMs) [4]. A doubly fed induction machine is a type of induction machine with a wound rotor and a voltage and frequency converter (VFC) installed in the rotor circuit to regulate slip energy as wind speed changes [5,6,7,8,9]. This approach allows for stabilized voltage amplitude and frequency at the stator winding terminals [10,11,12]. Additionally, the DFIM functions as a low-pass filter, filtering out the carrier frequency of the pulse width modulation from the self-commutated voltage inverter, which further enhances system performance.

The field-oriented control of a doubly fed induction machine is employed to shape and manage the orthogonal projection contours of the state variable vector, which is directly influenced by the three-phase inverter. In motoring mode, the DFIM vector control features two external control loops: one for speed and another for flux linkage. The stator current vector is split into two controllable projections: one indirectly governs the rotor flux linkage (magnetization process), while the other is responsible for generating electromagnetic torque and subsequently controlling speed [13]. In generator mode, vector control enables independent management of the active and reactive power of the induction generator [14,15,16].

A well-calibrated vector automatic control system for an induction machine, whether operating in the motor or generator mode, offers significant advantages, such as a rapid response to control commands and external disturbances [17]. However, the primary drawback of the field-oriented control system for a doubly fed induction machine is its sensitivity to machine parameter estimation accuracy [18,19]. If the parameters are inaccurately estimated, it can not only degrade the quality of state variable regulation but also potentially lead the system into instability.

The estimation of induction machine parameters can be categorized into two primary methods: dynamic identification [11,18,19,20,21,22] and preliminary identification [23]. During dynamic identification, the testing processes for the induction machine must not adversely affect the quality of electromechanical energy conversion across all operational modes [13]. Preliminary identification of induction machine parameters is characterized by its execution while the electric machine is in a non-operational state. Typically, this procedure is conducted using the voltage and frequency converter before the induction machine is commissioned and is repeated as necessary.

Methods of preliminary identification provide the determination of electrical parameter values for doubly fed induction machines under conditions close to idle (no temperature increase in the DFIM active part over ambient temperature, minimal residual magnetization). These parameters are necessary for implementing control during heavy startups, using systems with IR compensation and slip compensation. Moreover, these values are sufficient for setting up a vector control system, ensuring its correct functioning during initial startups and in short-term or intermittent operating modes of the machine.

However, it should be noted that because of machine heating and changes in winding resistances, which are characteristic of prolonged operating modes, the parameters obtained through preliminary identification methods require refinement in real-time operation. For this purpose, dynamic identification methods can be effectively applied. In this context, the values obtained through preliminary identification methods can serve as initial conditions for dynamic identification methods, enhancing the quality and speed of their operation.

Various methods exist for the preliminary identification of induction machine parameters, employing different approaches. One effective approach involves identifying parameters in generator mode for wind power systems through the software and hardware of the VFC, which manages the rotor circuits [24,25].

In this paper, the identification process is considered as a nonlinear optimization problem. Among the various optimization techniques, the least squares method (LSM) and recursive LSM are particularly notable for their effectiveness in the online identification of linear models. Additionally, gradient-based methods—including fast and stochastic gradient descent, Nesterov accelerated gradients, and Adam-type algorithms—are widely employed in this context. While these methods exhibit relatively slow convergence, their low computational costs render them suitable for online identification tasks.

Moreover, a range of metaheuristic strategies, such as differential evolution and ant colony algorithms, add another dimension to the optimization landscape [18,26,27,28,29]. These methodologies are significant for their capacity to optimize non-differentiable functions, though it is important to note that their implementation on local microcontrollers often incurs substantial computational costs.

In [18], a novel approach for identifying the parameters of doubly fed induction generators is introduced, utilizing the adaptive gray wolf optimization algorithm with an information sharing search strategy (ISIAGWO). Experimental results demonstrate that ISIAGWO markedly enhances identification accuracy, stability, and convergence compared to the conventional gray wolf metaheuristic. However, similar to other metaheuristic algorithms, it suffers from significant computational demands when implemented on local microcontrollers.

In this article, we investigate a method for identifying the parameters of an induction generator by analyzing the decay curves of stationary rotor currents through Newton’s method. The choice of Newton’s method is predicated on its fast convergence, acceptable accuracy in engineering applications, and effectiveness in addressing smooth optimization problems characterized by well-defined differentiability.

The structure of this paper is outlined as follows: Section 2 presents the preliminary identification method for the DFIM. Section 3 discusses the mathematical model of the DFIM in the u and v axes. Section 4 details the parameter identification algorithm for the DFIM. The formulation of the optimization problem for the adaptive model of the DFIM is presented in Section 5. Section 6 presents simulation results. Section 6 presents the experimental research results. Section 7 contains the conclusions of the work.

2. Concept

A wind turbine drives the shaft of a doubly fed induction machine through a coupling and step-up gearbox in a wind power system that operates under normal conditions (Figure 1). The stator of the DFIM is connected to a three-phase transformer via a “Stator Network Breaker” contactor, which provides galvanic isolation and aligns the generated voltage with the grid voltage. To excite the DFIM, a voltage and frequency converter is connected to the rotor circuit, powered through a separate transformer winding from the mains voltage. The inverter’s power section is conventional, comprising an input three-phase diode bridge, a capacitive low-pass filter, and a three-phase inverter. Feedback signals, including rotor currents and, if needed, information about the rotor’s speed and wind speed, are fed into the VFC control system.

The preliminary identification method proposed in the article for the wind power system must be conducted with the DFIM rotor held stationary. It is achieved by locking the mechanical coupling or securing the DFIM shaft with a mechanical brake. The “Stator Network Breaker” contactor is then opened, disconnecting the DFIM stator from the electrical network. Simultaneously, to ensure the accuracy of the preliminary identification experiment, the DFIM stator windings must be short-circuited, which is accomplished by closing the “Stator Short Circuit Breaker” contactor.

The method for preliminary identification of DFIM parameters proposed in this article can be implemented using a frequency converter that utilizes only MOSFETs operating in active mode, characterized by a relatively low drain-source on-resistance. This choice is motivated by the simplification of the mathematical model, as the subsequent calculations do not account for the specific characteristics of the transistors used in three-phase inverters. This simplification streamlines the mathematical description of the system.

Figure 2 depicts the stages of the DFIM parameter preliminary identification experiment. The three-phase inverter consists of transistors VT1 through VT6, and connections to the DFIM rotor windings’ terminals A2, B2, and C2 are established via slip rings P1, P2, and P3. The DFIM stator windings’ terminals A2, B2, and C2 are short-circuited using the “Stator Short Circuit Breaker” contactor. The instantaneous current values for phase A, denoted as iA, are measured through the appropriate measurement channel.

In the initial stage of the preliminary identification experiment (as shown in Figure 2a), the rotor windings of the doubly fed induction machine are energized with a current (depicted in the blue contour) until the transient response is fully settled. Once this state is achieved, the current value is recorded by the microprocessor. To establish the required current level, transistors VT4 and VT6 are activated, and a pulse-width modulated signal is applied to VT1, while transistors VT2, VT3, and VT5 remain deactivated. This specific switching sequence for the three-phase inverter transistors helps minimize switching losses, although it is not the only method available for current generation.

The direct current component of the rotor circuits during the pumping phase should match the rated current value of the induction machine. It ensures that the doubly fed induction machine operates under conditions similar to its nominal mode, enabling accurate modeling of saturation processes and enhancing the signal-to-noise ratio in the VFC’s measuring channel. Consequently, this setup facilitates more precise parameter identification.

In the second stage of the preliminary identification experiment (illustrated in Figure 2b), the rotor windings of the doubly fed induction machine are disconnected from the direct current supply, allowing them to form a closed loop by activating transistors VT1, VT3, and VT5 while keeping VT2, VT4, and VT6 open. Once the transistors switch, the rotor currents begin a smooth decay, sustained by the energy stored in the magnetic field of the doubly fed induction machine. These decaying currents are discretely recorded (represented by the pink contour) using the measurement channel of the VFC. The sampling frequency of the instantaneous current values and the total number of recorded values are contingent on the gradient of the current decay curve and are determined by the specifications of the microprocessor employed in the control system.

3. DFIM Mathematical Model in Axes u, v with a Frequency Converter Connected to a Stationary Rotor

We will utilize the calculations outlined in reference [30] to accurately characterize the processes occurring within an induction machine when a frequency converter is connected to its rotor circuit. We present a system of differential equations that articulates the behavior of the state variables of a doubly fed induction machine, building upon the methodologies established in [25,30]. This system is expressed in vector form for the u, ν and α, β axes, as depicted in (1).

$$\left\{\begin{array}{c}{\overline{u}}_{2u\nu }(t)=R_2\cdot {\overline{i}}_{2u\nu }(t)+{\displaystyle \frac{d{\overline{\mathsf{\psi }}}_{2u\nu }(t)}{dt}}\\ {\overline{u}}_{1\mathsf{\alpha }\mathsf{\beta }}(t)=R_1\cdot {\overline{i}}_{1\mathsf{\alpha }\mathsf{\beta }}(t)+{\displaystyle \frac{d{\overline{\mathsf{\psi }}}_{1\mathsf{\alpha }\mathsf{\beta }}(t)}{dt}}\end{array}\right.$$

(1)

where R₁ and R₂ represent the active resistances of the stator and rotor windings, respectively, measured in ohms; ${\overline{u}}_{2u\nu }\left(t\right)$ denotes the state of the voltage vector projecting into the rotor circuit of the doubly fed induction machine at time t in the stationary orthogonal coordinate system firmly attached to the rotor u, ν, V; ${\overline{u}}_{1\mathsf{\alpha }\mathsf{\beta }}\left(t\right)$ indicates the state of the voltage vector supplying the stator circuit at time t in the orthogonal coordinate system rigidly attached to the stator α, β, V; ${\overline{i}}_{2u\nu }\left(t\right)$ represents the rotor current vector at time t in the coordinate system u, ν, A; ${\overline{i}}_{1\mathsf{\alpha }\mathsf{\beta }}\left(t\right)$ denotes the stator current vector at time t in the coordinate system α, β, A; ${\overline{\mathsf{\psi }}}_{2u\nu }\left(t\right)$ refers to the rotor flux linkage vector, induced by the current flow that constitutes the vector ${\overline{i}}_{2u\nu }\left(t\right)$, at time t in the coordinate system u, ν, Wb; ${\overline{\mathsf{\psi }}}_{1\mathsf{\alpha }\mathsf{\beta }}\left(t\right)$ is the stator flux linkage vector, induced by the current flow ${\overline{i}}_{1\mathsf{\alpha }\mathsf{\beta }}\left(t\right)$, at time t in the coordinate system α, β, Wb.

To transition from the original differential Equation (1) to its corresponding image functions, we apply the unilateral Laplace transform while assuming zero initial conditions. This approach is commonly used in the theory of automatic control to construct structural diagrams of dynamic systems based on differential Equation (2).

$$\left\{\begin{array}{c}{\overline{U}}_{2u\nu }(p)=R_2\cdot {\overline{I}}_{2u\nu }(p)+p\cdot {\overline{\Psi }}_{2u\nu }(p)\\ {\overline{U}}_{1\mathsf{\alpha }\mathsf{\beta }}(p)=R_1\cdot {\overline{I}}_{1\mathsf{\alpha }\mathsf{\beta }}(p)+p\cdot {\overline{\Psi }}_{1\mathsf{\alpha }\mathsf{\beta }}(p)\end{array}\right.$$

(2)

where ${\overline{U}}_{2u\nu }\left(p\right)$ represents the Laplace transform image of the voltage vector supplying the rotor circuit of the doubly fed induction machine in the fixed orthogonal coordinate system u, ν that is rigidly connected to the rotor, measured in volts per second, V⋅s; ${\overline{U}}_{1\mathsf{\alpha }\mathsf{\beta }}\left(p\right)$ denotes the Laplace transform image of the voltage vector feeding the stator circuit in the fixed orthogonal coordinate system α,β directly associated with the rotor, measured in volts per second, V⋅s; ${\overline{I}}_{2u\nu }\left(p\right)$ is the image of the rotor current vector in the coordinate system u, ν, expressed in amperes per second, A⋅s; ${\overline{I}}_{1\mathsf{\alpha }\mathsf{\beta }}\left(p\right)$ indicates the image of the stator current vector in the coordinate system α, β, expressed in amperes per second, A⋅s; ${\overline{\Psi }}_{2u\nu }\left(p\right)$ signifies the image of the flux linkage vector, induced by the magnitude of the current vector ${\overline{i}}_{2u\nu }\left(p\right)$, in the coordinate system u, ν, measured in weber per seconds, Wb⋅s; ${\overline{\Psi }}_{1\mathsf{\alpha }\mathsf{\beta }}\left(p\right)$ describes the flux linkage vector state, induced by the magnitude of the current vector ${\overline{i}}_{1\mathsf{\alpha }\mathsf{\beta }}\left(p\right)$, in the coordinate system α, β, also measured in weber per seconds, Wb⋅s.

Since capturing the damping curves on the rotor circuits of the doubly fed induction machine necessitates the short-circuiting of stator circuits, there is ${\overline{U}}_{2\mathsf{\alpha }\mathsf{\beta }}\left(p\right)=0$.

Once the stator variables are transformed into the fixed two-phase orthogonal coordinate system u, ν, rigidly connected to the rotor:

$$\left\{\begin{array}{l}{\overline{U}}_{2u\nu }(p)=R_2\cdot {\overline{I}}_{2u\nu }(p)+p\cdot {\overline{\Psi }}_{2u\nu }(p)\\ 0={R_1}^{\prime }\cdot {\overline{I}}_{1u\nu }(p)+p\cdot {\overline{\Psi }}_{1u\nu }(p)-j\cdot {\mathsf{\omega }}_{e2}\cdot {\overline{\Psi }}_{1u\nu }(p)\end{array}\right.$$

(3)

where ${R_1}^{\prime }$ is the equivalent active resistance of the stator winding referred to the rotor, measured in ohms; j is the operator that induces a 90-degree phase shift.

The process of referring the active resistance of the stator winding to the rotor involves [31] is described as follows:

$${R_1}^{\prime }={\displaystyle \frac{R_1}{k_e\cdot k_i}}$$

(4)

where $k_e$ is the voltage transformation coefficient; $k_i$ is the current transformation coefficient.

For the particular doubly fed induction machine under consideration, we have $k_e=k_i$, thus

$${R_1}^{\prime }=R_1\cdot k_{conv}$$

(5)

where $k_{conv}=1/k_e^2$ is the coefficient of conversion of stator winding parameters to rotor winding.

It is more practical to employ the rotor current vector as the descriptor of the rotor electric circuit’s dynamic state ${\overline{\Psi }}_{2\mathrm{u}\mathsf{\nu }}$, rather than using the rotor circuit current vector ${\overline{I}}_{2\mathrm{u}\mathsf{\nu }}$, due to the measurement channels of the voltage and frequency converter track currents in the rotor circuits of the doubly fed induction machine.

To achieve it, we will utilize the equilibrium equation of the magnetic circuits for the doubly fed induction machine [31,32]:

$$\left\{\begin{array}{c}{\overline{\Psi }}_{2u\nu }(p)=L_2\cdot {\overline{I}}_{2u\nu }(p)+L_m\cdot {\overline{I}}_{1u\nu }(p)\\ {\overline{\Psi }}_{1u\nu }(p)=L_1\cdot {\overline{I}}_{1u\nu }(p)+L_m\cdot {\overline{I}}_{2u\nu }(p)\end{array}\right.$$

(6)

where $L_2=L_{2\mathsf{\sigma }}+L_m$ represents the equivalent inductance of the rotor winding, measured in henries, H; $L_1={L^{\prime }}_{1\mathsf{\sigma }}+L_m$ denotes the equivalent inductance of the stator winding, also in henries, H; $L_{2\mathsf{\sigma }}$ is the leakage inductance of the stator winding, given in henries, H; ${L^{\prime }}_{1\mathsf{\sigma }}$ is the stator winding leakage inductance referred to the rotor, adjusted by the transformation coefficient k_conv, in henries, H; $L_m$ is the resultant inductance attributed to the magnetic flux within the air gap of the electric machine, also in henries, H.

By substituting ${\overline{\Psi }}_{2\mathrm{u}\mathsf{\nu }}$ from (6) into (3), we obtain the following:

$$\left\{\begin{array}{l}{\overline{U}}_{2u\nu }(p)=R_2{\overline{I}}_{2u\nu }(p)+L_2\cdot p\cdot {\overline{I}}_{2u\nu }(p)+L_m\cdot p\cdot {\overline{I}}_{1u\nu }(p)\\ 0={R_1}^{\prime }{\overline{I}}_{1u\nu }(p)+L_1\cdot p\cdot {\overline{I}}_{1u\nu }(p)+L_m\cdot p\cdot {\overline{I}}_{2u\nu }(p)-j\cdot {\mathsf{\omega }}_{\mathrm{e}2}\cdot {\overline{\Psi }}_{1u\nu }(p)\end{array}\right.$$

(7)

The vectors featured in (7), namely ${\overline{U}}_{2\mathrm{u}\mathsf{\nu }}$, ${\overline{\Psi }}_{1\mathrm{u}\mathsf{\nu }}$, ${\overline{I}}_{1\mathrm{u}\mathsf{\nu }}$, ${\overline{I}}_{2\mathrm{u}\mathsf{\nu }}$, can be broken down into orthogonal components within the u, ν coordinate system of the doubly fed induction machine:

$$\left\{\begin{array}{l}{\overline{U}}_{2u\nu }(p)=U_{2u}(p)+j\cdot U_{2\nu }(p)\\ {\overline{\Psi }}_{1u\nu }(p)={\Psi }_{1u}(p)+j\cdot {\Psi }_{1\nu }(p)\\ {\overline{I}}_{1u\nu }(p)=I_{1u}(p)+j\cdot I_{1\nu }(p)\\ {\overline{I}}_{2u\nu }(p)=I_{2u}(p)+j\cdot I_{2\nu }(p)\end{array}\right.$$

(8)

Projecting the vectors outlined in (7) using the framework of (8) onto the orthogonal coordinate system u, ν, subsequently formulates differential equations for the corresponding projections, as follows:

$$\left\{\begin{array}{l}{U}_{2u}(p)=R_2{I}_{2u}(p)+L_2\cdot p\cdot I_{2u}(p)+L_m\cdot p\cdot I_{1u}(p)\\ U_{2\nu }(p)=R_2{I}_{2\nu }(p)+L_2\cdot p\cdot I_{2\nu }(p)+L_m\cdot p\cdot I_{1\nu }(p)\\ 0={R^{\prime }}_1{I}_{1u}(p)+L_1\cdot p\cdot I_{1u}(p)+L_m\cdot p\cdot I_{2u}(p)+{\mathsf{\omega }}_r{\Psi }_{1\nu }(p)\\ 0={R^{\prime }}_1{I}_{1\nu }(p)+L_1\cdot p\cdot I_{1\nu }(p)+L_m\cdot p\cdot I_{2\nu }(p)-{\mathsf{\omega }}_r{\Psi }_{1u}(p)\end{array}\right.$$

(9)

We formulate differential equations by the aforementioned considerations and establish Formulas (1)–(9). It captures the dynamics of the doubly fed induction machine within the u, ν axes across all dynamic modes, assuming the mechanical system operates as a single-mass system:

$$\left\{\begin{array}{l}{\displaystyle \frac{d{i}_{2u}\left(t\right)}{dt}}={\displaystyle \frac{1}{\mathsf{\sigma }{L}_2}}{u}_{2u}\left(t\right)-{\displaystyle \frac{R_{2e}}{\mathsf{\sigma }{L}_2}}{i}_{2u}\left(t\right)+{\displaystyle \frac{{R^{\prime }}_1{L}_m}{\mathsf{\sigma }{L}_2{L}_1^2}}{\mathsf{\psi }}_{1u}\left(t\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{L_m}{\mathsf{\sigma }{L}_2{L}_1}}{z}_p{\mathsf{\omega }}_r\left(t\right){\mathsf{\psi }}_{1\nu }\left(t\right)\\ {\displaystyle \frac{d{i}_{2\nu }\left(t\right)}{dt}}={\displaystyle \frac{1}{\mathsf{\sigma }{L}_2}}{u}_{2\nu }\left(t\right)-{\displaystyle \frac{R_{2e}}{\mathsf{\sigma }{L}_2}}{i}_{2\nu }\left(t\right)+{\displaystyle \frac{{R^{\prime }}_1{L}_m}{\mathsf{\sigma }{L}_2{L}_1^2}}{\mathsf{\psi }}_{1\nu }\left(t\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \frac{L_m}{\mathsf{\sigma }{L}_2{L}_1}}{z}_p{\mathsf{\omega }}_r\left(t\right){\mathsf{\psi }}_{1u}\left(t\right)\\ {\displaystyle \frac{d{\mathsf{\psi }}_{1\mathrm{u}}\left(t\right)}{dt}}=-{\displaystyle \frac{R^{\prime }}{L_1}}{\mathsf{\psi }}_{1u}\left(t\right)+{\displaystyle \frac{{R^{\prime }}_1{L}_m}{L_1}}{i}_{2u}\left(t\right)-z_p{\mathsf{\omega }}_r\left(t\right){\mathsf{\psi }}_{1\nu }\left(t\right)\\ {\displaystyle \frac{d{\mathsf{\psi }}_{1\mathsf{\nu }}\left(t\right)}{dt}}=-{\displaystyle \frac{R^{\prime }}{L_1}}{\mathsf{\psi }}_{1\nu }\left(t\right)+{\displaystyle \frac{{R^{\prime }}_1{L}_m}{L_1}}{i}_{2\nu }\left(t\right)-z_p{\mathsf{\omega }}_r\left(t\right){\mathsf{\psi }}_{1u}\left(t\right)\\ {\displaystyle \frac{d{\mathsf{\omega }}_r\left(t\right)}{dt}}={\displaystyle \frac{1}{J}}\left({\displaystyle \frac{3L_m{z}_p}{2L_1}}\left({\mathsf{\psi }}_{1u}\left(t\right)i_{2\nu }\left(t\right)-{\mathsf{\psi }}_{1\nu }\left(t\right)i_{2u}\left(t\right)\right)-M_s\left(t\right)\right)\end{array}\right.$$

(10)

where $\mathsf{\sigma }=1-{L_m}^2/\left(L_2{L}_1\right)$ is the leakage coefficient, per unit; $R_{2\mathrm{e}}=R_2+{R^{\prime }}_1{L_m}^2/L_1^2$ equivalent active resistance of the rotor circuits of the doubly fed induction machine, measured in ohms; J moment of inertia of the single mass mechanism, in kg·m²; M_s(t) instantaneous value of the static resistance torque on the motor shaft, in N·m; z_p is the number of pole pairs of the DFIM; ω_r(t) instantaneous speed of the DFIM rotor, in rad/s.

The system of differential Equation (10) corresponds to the equivalent circuits of the DFIM shown in Figure 3a,b in the stationary coordinate system u, v, which is rigidly connected to the rotor.

In the context of the experiment measuring current attenuation curves performed with the rotor stationary (ω_r(t) = 0), the processes occurring in the equivalent windings of the two-phase doubly fed induction machine along the u, ν components become decoupled and can be likened to those found in two distinct single-phase loaded transformers. Each of these transformers represents one of the axes, u or ν. The current sensor captures the instantaneous current value in the base winding and relays these values to the analog-to-digital converter (ADC), as shown in Figure 2. Owing to this decoupling along the u, ν axes with the rotor remaining stationary, the differential equations required to describe the state variables $i_{2\nu }\left(t\right)$ and ${\mathsf{\psi }}_{1\nu }\left(t\right)$ in the differential Equation (10) can be removed. These results in a universal system of differential equations are applicable to both the current injection mode (Figure 2a) and the current dampening mode (Figure 2b) in the base winding A2.

$$\left\{\begin{array}{l}{\displaystyle \frac{d{i}_{2u}\left(t\right)}{dt}}={\displaystyle \frac{1}{\mathsf{\sigma }{L}_2}}{u}_{2u}\left(t\right)-{\displaystyle \frac{R_{2e}}{\mathsf{\sigma }{L}_2}}{i}_{2u}\left(t\right)+{\displaystyle \frac{{R^{\prime }}_1{L}_m}{\mathsf{\sigma }{L}_2{L}_1^2}}{\mathsf{\psi }}_{1u}\left(t\right)+0\\ {\displaystyle \frac{d{\mathsf{\psi }}_{1u}\left(t\right)}{dt}}=-{\displaystyle \frac{{R^{\prime }}_1}{L_1}}{\mathsf{\psi }}_{1u}\left(t\right)+{\displaystyle \frac{{R^{\prime }}_1{L}_m}{L_1}}{i}_{2u}\left(t\right)-0\end{array}\right.$$

(11)

The T-shaped equivalent circuit of the DFIM with a stationary rotor and closed secondary windings, corresponding to the differential equations, is shown in Figure 4.

It is feasible to determine the parameters of the doubly fed induction machine by analyzing the current growth curves in the base winding A2 (depicted in Figure 2) using Equation (11). A limitation of this methodology is the requirement to measure not only the current in the base winding but also the voltage applied to it. Typically, in engineering practice, the voltage measured is the direct current link voltage, which markedly differs from the voltage directly applied to the base winding. Therefore, conducting a pre-identification experiment based on the current decay curves in the base winding (shown in Figure 2b) is more advantageous, as it eliminates the need for voltage measurement, with $u_{2u}\left(t\right)$ zero equated:

$$\left\{\begin{array}{l}{\displaystyle \frac{d{i}_{2u}\left(t\right)}{dt}}=-{\displaystyle \frac{R_{2e}}{\mathsf{\sigma }{L}_2}}{i}_{2u}\left(t\right)+{\displaystyle \frac{{R^{\prime }}_1{L}_m}{\mathsf{\sigma }{L}_2{L}_1^2}}{\mathsf{\psi }}_{1u}\left(t\right)\\ {\displaystyle \frac{d{\mathsf{\psi }}_{1\mathrm{u}}\left(t\right)}{dt}}=-{\displaystyle \frac{{R^{\prime }}_1}{L_1}}{\mathsf{\psi }}_{1u}\left(t\right)+{\displaystyle \frac{{R^{\prime }}_1{L}_m}{L_1}}{i}_{2u}\left(t\right)\end{array}\right.$$

(12)

To achieve an accurate current decay curve in the base winding (as shown in Figure 2b), it is imperative to ensure full damping of currents in the shorted stator windings during the initial current injection phase (illustrated in Figure 2a). It can be accomplished by one of two methods. The first method involves allowing the current damping process to proceed naturally, without employing pulse width modulation, until the transient response fully diminishes the free component of current in the base winding. However, this approach necessitates applying an external reduced voltage that is independent of the main power supply to the converter. The second method involves actively driving the base winding current to a specific level and maintaining this level with electronic switches for multiple pulse width modulation cycles. With this approach and at sufficiently high pulse width modulation frequencies, guaranteed current damping with acceptably high accuracy can be expected in the shorted stator windings. The ensuing calculations are applicable for implementing both the first and second methods of current injection into the rotor windings of a doubly fed induction machine.

Using the Laplace transform [32] and the original differentiation theorem [33], we derive the direct Laplace transform under non-zero initial conditions for $i_{2\mathrm{u}}\left(t\right)$ and ${\mathsf{\psi }}_{1\mathrm{u}}\left(t\right)$:

$$\left\{\begin{array}{l}{\left.{\displaystyle \frac{d{i}_{2u}\left(t\right)}{dt}}\right|}_{i_{2u}(0+)\ne 0}\Rightarrow p{I}_{2u}\left(p\right)-i_{2u}\left(0+\right)\\ {\left.{\displaystyle \frac{d{\mathsf{\psi }}_{1u}\left(t\right)}{dt}}\right|}_{{\mathsf{\psi }}_{1u}(0+)\ne 0}\Rightarrow p{\Psi }_{1u}\left(p\right)-{\mathsf{\psi }}_{1u}\left(0+\right)\end{array}\right.$$

(13)

where $i_{2\mathrm{u}}(0+)=i_{2\mathrm{u}0}\ne 0$ the projection of the rotor current vector along the u-axis at the onset of the current damping curve (Figure 2b), A; ${\mathsf{\psi }}_{1\mathrm{u}}(0+)={\mathsf{\psi }}_{1\mathrm{u}0}\ne 0$ the projection of the stator current vector along the u-axis at the onset of the current damping curve (Figure 2b), Wb.

It is assumed that the switching of the voltage and frequency converter keys is instantaneous in Equation (13). This assumption is valid since the switching duration of the VFC keys is much shorter than the smallest time constant of the DFIM. The current is recorded in the microprocessor memory before commutation begins (Figure 2b).

According to the equilibrium equations of the magnetic circuits of a doubly fed induction machine (6) at the moment of VFC transistor switching (Figure 2b).

$${\mathsf{\psi }}_{1u}\left(0+\right)=L_1{i}_{1u}\left(0+\right)+L_m{i}_{2u}\left(0+\right)$$

(14)

where $i_{1u}\left(0+\right)$—stator current of the induction machine along the u-axis in the moment of voltage and frequency converter key switching (Figure 2b).

As mentioned earlier, during the experiment to identify model parameters of the doubly fed induction machine, the stator current is necessarily brought to zero at the moment of the voltage and frequency converter key switching $i_{1u}\left(0+\right)=0$. Consequently, expression (14) takes the form: ${\mathsf{\psi }}_{1u}\left(0+\right)=0+L_m\cdot i_{2u}\left(0+\right)$.

Pass to the operator form of notation, counting the above non-zero initial conditions:

$$\left\{\begin{array}{l}p{I}_{2u}\left(p\right)-i_{2u0}(0+)=-{\displaystyle \frac{R_{2e}}{\mathsf{\sigma }{L}_2}}{I}_{1u}\left(p\right)+{\displaystyle \frac{{R_1}^{\prime }{L}_m}{\mathsf{\sigma }{L}_2{L_1}^2}}{\Psi }_{2u}\left(p\right)\\ p{\Psi }_{1u}\left(p\right)-L_m{i}_{2u0}=-{\displaystyle \frac{{R_1}^{\prime }}{L_1}}{\Psi }_{1u}\left(p\right)+{\displaystyle \frac{{R_1}^{\prime }{L}_m}{L_1}}{I}_{2u}\left(p\right)\end{array}\right.$$

(15)

Express ${\Psi }_{1\mathrm{u}}\left(p\right)$ from the second equation of the system (15) we obtain the following:

$${\Psi }_{1\mathrm{u}}\left(p\right)={\displaystyle \frac{\left(\frac{{R_1}^{\prime }}{L_1}{I}_{2\mathrm{u}}\left(p\right)+L_m{i}_{1\alpha 0}\right)}{\left(p+\frac{{R_1}^{\prime }}{L_1}\right)}}$$

Substitution of ${\Psi }_{1\mathrm{u}}\left(p\right)$ in the first equation of the system (15) and expression of $I_{2\mathrm{u}}\left(p\right)$ produces the following:

$$I_{2u}\left(p\right)={\displaystyle \frac{p+\frac{{R_1}^{\prime }}{L_1}+\frac{{R_1}^{\prime }{L_m}^2}{\mathsf{\sigma }{L}_2{L_1}^2}}{p\left(p+\frac{{R_1}^{\prime }}{L_1}\right)+\frac{R_{2e}}{\mathsf{\sigma }{L}_2}\left(p+\frac{{R_1}^{\prime }}{L_1}\right)-\frac{{{R_1}^{\prime }}^2{L_m}^2}{\mathsf{\sigma }{L}_2{L_1}^3}}}{i}_{2u0}$$

(16)

Let us perform the inverse substitution of the values σ, R_e2, L₁, and L₂ in (16). We can suppose that L_1σ = L_2σ = L_σ. After simplifying the expression, we obtain the final Laplace representation for the transient component of the rotor current during the damping process:

$$I_{2u}\left(p\right)={\displaystyle \frac{p+\left(\frac{{R_1}^{\prime }}{L_{\mathsf{\sigma }}}\left(\frac{L_m+L_{\mathsf{\sigma }}}{2L_m+L_{\mathsf{\sigma }}}\right)\right)}{p^2+p\left(\frac{\left(R_2+{R_1}^{\prime }\right)\cdot \left(L_m+L_{\mathsf{\sigma }}\right)}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}\right)+\frac{R_2{R_1}^{\prime }}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}}{i}_{2u0}$$

(17)

In the denominator of expression (17), we obtain the characteristic equation by replacing statement p by variable γ and equating the resulting polynomial to zero as follows:

$${\mathsf{\gamma }}^2+\mathsf{\gamma }\left({\displaystyle \frac{\left(R_2+{R_1}^{\prime }\right)\cdot \left(L_m+L_{\mathsf{\sigma }}\right)}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}\right)+{\displaystyle \frac{R_2{R_1}^{\prime }}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}=0$$

(18)

We obtain the poles of the denominator polynomial (19) by solving the characteristic Equation (18). The following are the roots of this characteristic equation:

$$\left\{\begin{array}{l}{\mathsf{\gamma }}_1={\displaystyle \frac{-\left(\frac{\left(R_2+{R_1}^{\prime }\right)\cdot \left(L_m+L_{\mathsf{\sigma }}\right)}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}\right)}{2}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \frac{\sqrt{{\left(\frac{\left(R_2+{R_1}^{\prime }\right)\cdot \left(L_m+L_{\mathsf{\sigma }}\right)}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}\right)}^2-4\frac{R_2{R_1}^{\prime }}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}}{2}}\\ {\mathsf{\gamma }}_2={\displaystyle \frac{-\left(\frac{\left(R_2+{R_1}^{\prime }\right)\cdot \left(L_m+L_{\mathsf{\sigma }}\right)}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}\right)}{2}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \frac{\sqrt{{\left(\frac{\left(R_2+{R_1}^{\prime }\right)\cdot \left(L_m+L_{\mathsf{\sigma }}\right)}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}\right)}^2-4\frac{R_2{R_1}^{\prime }}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}}{2}}\end{array}\right.$$

(19)

Designing an efficient doubly fed induction machine suitable for engineering practice requires the stator and rotor windings to have distinctly different designs, resulting in varying electrical parameters. This difference in the winding parameters of the stator and rotor ensures that the roots of the denominator for the transfer function (18) are not repeated.

The multiple roots can arise only if the discriminant of Equation (18) is equal to zero. It is possible only under the following condition:

$${\displaystyle \frac{(R_2+{R_1}^{\prime })}{2}}{\displaystyle \frac{\left(L_m+L_{\mathsf{\sigma }}\right)}{\sqrt{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}}=\sqrt{R_2{R_1}^{\prime }}$$

However, this equality is not satisfied in any case, as the arithmetic mean of the resistances of the rotor and stator windings is always greater than or equal to the geometric mean.

$${\displaystyle \frac{(R_2+{R_1}^{\prime })}{2}}\ge \sqrt{R_2{R_1}^{\prime }}$$

Equality is under the condition R’₁ = R₂. However, even in this case, since L_σ is much smaller than L_m, the factor is always greater than one.

$${\displaystyle \frac{\left(L_m+L_{\mathsf{\sigma }}\right)}{\sqrt{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}}$$

The discriminant does not become zero.

The absence of multiple roots in the characteristic Equation (18) allows us to perform partial fraction decomposition on the Laplace transform and recover the original current from the known image:

$$f\left(t\right)={\displaystyle \sum }_{k=1}^2{\displaystyle \frac{P_m\left({\mathsf{\gamma }}_k\right)}{Q_n^{\prime }\left({\mathsf{\gamma }}_k\right)}}\cdot e^{{\mathsf{\gamma }}_k\cdot t}$$

(20)

where k is the number of the root for the characteristic equation; $P_m\left(\mathsf{\gamma }\right)$ is the numerator of the transfer function (17) at the corresponding root γ; $Q_n^{\prime }\left(\mathsf{\gamma }\right)$ is the derivative for the denominator of the transfer function (17) at the corresponding root γ.

The expression $P_m\left(\mathsf{\gamma }\right)$ is defined as the numerator of (20) as follows:

$$P_m\left(\mathsf{\gamma }\right)=\left[\mathsf{\gamma }+\left({\displaystyle \frac{{R_1}^{\prime }}{L_{\mathsf{\sigma }}}}\cdot \left({\displaystyle \frac{L_m+L_{\mathsf{\sigma }}}{2\cdot L_m+L_{\mathsf{\sigma }}}}\right)\right)\right]$$

The expression $Q_n^{\prime }\left(\mathsf{\gamma }\right)$ is defined as the denominator of (20) as follows:

$$Q_n^{\prime }\left(\mathsf{\gamma }\right)=2\mathsf{\gamma }+{\displaystyle \frac{\left(R_2+{R^{\prime }}_1\right)\cdot \left(L_m+L_{\mathsf{\sigma }}\right)}{L_{\mathsf{\sigma }}\cdot \left(2L_m+L_{\mathsf{\sigma }}\right)}}.$$

Thus, the original function can be expressed as a sum of two exponential terms by substituting the distinct (which are non-repeated, as established above) roots γ₁ and γ₂ from the characteristic Equation (18) into the partial fraction decomposition (20):

$$i_{2u}\left(t\right)={\displaystyle \frac{P_m\left({\mathsf{\gamma }}_1\right)}{Q_n^{\prime }\left({\mathsf{\gamma }}_1\right)}}{e}^{{\mathsf{\gamma }}_{1}t}+{\displaystyle \frac{P_m\left({\mathsf{\gamma }}_2\right)}{Q_n^{\prime }\left({\mathsf{\gamma }}_2\right)}}{e}^{{\mathsf{\gamma }}_{2}t}$$

(21)

Expression (21) is an analytical description of the rotor current damping process of the model in continuous time.

The expression $P_m\left({\mathsf{\gamma }}_1\right)$ is equal to the following:

$$P_m\left({\mathsf{\gamma }}_1\right)=\left[\begin{array}{l}{\displaystyle \frac{\sqrt{{\left(\frac{\left(R_2+{R_1}^{\prime }\right)\cdot \left(L_m+L_{\mathsf{\sigma }}\right)}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}\right)}^2-4\frac{R_2{R_1}^{\prime }}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}}{2}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \frac{\left(R_2-{R_1}^{\prime }\right)\cdot \left(L_m+L_{\mathsf{\sigma }}\right)}{2L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}\end{array}\right]i_{2u0}$$

The expression $P_m\left({\mathsf{\gamma }}_2\right)$ is equal to the following:

$$P_m\left({\mathsf{\gamma }}_2\right)=\left[\begin{array}{l}-{\displaystyle \frac{\sqrt{{\left(\frac{\left(R_2+{R_1}^{\prime }\right)\cdot \left(L_m+L_{\mathsf{\sigma }}\right)}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}\right)}^2-4\frac{R_2{R_1}^{\prime }}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}}{2}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \frac{\left(R_2-{R_1}^{\prime }\right)\cdot \left(L_m+L_{\mathsf{\sigma }}\right)}{2L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}\end{array}\right]i_{2u0}$$

The expression $Q_n^{\prime }\left({\mathsf{\gamma }}_1\right)$ is equal to the following:

$$Q_n^{\prime }\left({\mathsf{\gamma }}_1\right)=\sqrt{{\left({\displaystyle \frac{\left(R_2+{R_1}^{\prime }\right)\cdot \left(L_m+L_{\mathsf{\sigma }}\right)}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}\right)}^2-4{\displaystyle \frac{R_2{R^{\prime }}_1}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}}$$

The expression ${Q^{\prime }}_n\left({\mathsf{\gamma }}_2\right)$ is equal to the following:

$$Q_n^{\prime }\left({\mathsf{\gamma }}_2\right)=-\sqrt{{\left({\displaystyle \frac{\left(R_2+{R_1}^{\prime }\right)\cdot \left(L_m+L_{\mathsf{\sigma }}\right)}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}\right)}^2-4{\displaystyle \frac{R_2{R^{\prime }}_1}{L_{\mathsf{\sigma }}\left(2L_m+L_{\mathsf{\sigma }}\right)}}}$$

We place determined $P_m\left({\mathsf{\gamma }}_1\right)$, $P_m\left({\mathsf{\gamma }}_2\right)$, ${Q^{\prime }}_n\left({\mathsf{\gamma }}_1\right)$, ${Q^{\prime }}_n\left({\mathsf{\gamma }}_2\right)$ to Equation (21). When we suppose that $\mathsf{\gamma }=\mathsf{\gamma }\left({\widehat{L}}_{\sigma },{\widehat{L}}_m\right)$ we get an adaptive regression model in continuous time:

$$\begin{array}{l}{i}_{2u}\left(t,{\widehat{L}}_{\mathsf{\sigma }},{\widehat{L}}_m\right)={\displaystyle \frac{P_m\left({\mathsf{\gamma }}_1\left({\widehat{L}}_{\mathsf{\sigma }},{\widehat{L}}_m\right)\right)}{Q_n^{\prime }\left({\mathsf{\gamma }}_1\left({\widehat{L}}_{\mathsf{\sigma }},{\widehat{L}}_m\right)\right)}}{e}^{{\mathsf{\gamma }}_1\left({\widehat{L}}_{\mathsf{\sigma }},{\widehat{L}}_m\right)t}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{P_m\left({\mathsf{\gamma }}_2\left({\widehat{L}}_{\mathsf{\sigma }},{\widehat{L}}_m\right)\right)}{Q_n^{\prime }\left({\mathsf{\gamma }}_2\left({\widehat{L}}_{\mathsf{\sigma }},{\widehat{L}}_m\right)\right)}}{e}^{{\mathsf{\gamma }}_2\left({\widehat{L}}_{\mathsf{\sigma }},{\widehat{L}}_m\right)t}\end{array}$$

(22)

where ${\widehat{L}}_{\sigma }$ is parasitic inductance value; ${\widehat{L}}_m$ is the resultant inductance value of the magnetic flux within the air gap.

4. Parameters Identification Algorithm of the T-Shaped Equivalent Circuit of DFIM in Discrete Time

The identification algorithm based on the experimentally obtained doubly fed induction machine current attenuation curve, based on the analytical expression describing the behavior of the dynamic object, is shown in Figure 5.

Where “_ô” is a superscript indicating an estimate of an unmeasured value. The procedure for parameter identification of a dynamic system using current decay curves with a modern microprocessor device involves two main stages. During the first stage, depicted in the functional diagram (Figure 5), a step control input u(t) is applied, originating from the VFC, to the dynamic system, where t is the time. It is performed until the transient response finishes completely and assuredly, a timing that depends on the characteristics of the dynamic system. In the second stage, the system is disconnected from the power source, and its terminals are short-circuited through the VFC, causing a gradual decay in the current i_init(t).

In theoretical scenarios, the initial current signal i_init(t) is directly observable; however, in real-world systems, this signal is mixed with measurement system noise ξ₁(t). This noise ξ₁(t) includes interference resulting from measurement channel inefficiencies and sensor regulation imperfections, such as insensitivity, saturation, and hysteresis. When the signal $i_{in.ADC}(t)$ passes through an analog-to-digital converter, two transformations occur: temporal sampling with period Δt; and level quantization, introducing additive quantization noise ξ₂(t), which hinges on the ADC’s resolution. For later computations, it is assumed that total noise ξ(t) = ξ₁(t) + ξ₂(t) is analogized to Gaussian-distributed white noise. The measured signal from the system’s output, the dynamic object’s current i_out.ADC(n·Δt) (where n indicates the nth sampling step from the outset of identification), is obtained using a microcontroller. From this signal i_out.ADC(n·Δt), we extract the steady-state current value i(0+) captured during the initial identification phase, as well as the free current component i_meas(n·Δt). Note that n also denotes the instants beyond which the current decay monitoring begins. The microcontroller performs an iterative procedure to estimate the parameters of the adaptive dynamic object model. This approach calculates a target function $S(x_1,x_2)$ based on the difference $\Delta \widehat{i}(n\cdot \Delta t)$ between the measured current value i_meas(n Δt) and the response of the adaptive regression model $\Delta \widehat{i}(n\cdot \Delta t)$ to estimate the necessary parameters of the adaptive model ($\widehat{X}$). An iterative calculation $\widehat{X}(k)$ within the target function’s minimization block refines the parameter estimates are refined until the predefined minimum of the target function is achieved, denoted by the index k of the iterative identification process. Initial approximations for the estimates $\widehat{X}(0)$ are established before computations begin. The iterative process concludes once the specified condition for the target function’s minimum is fulfilled or if the maximum iteration count is surpassed, at which point the final evaluations ${\widehat{X}}_{total}$ are provided. Notably, the accuracy of the resulting estimates ${\widehat{X}}_{total}$ correlates with the quantization error of the ADC used in the measurement system.

5. Formulation of the Parameter Optimization Problem for a DFIM Adaptive Model

The parameter optimization problem $x={\left[x_1,x_2\right]}^T={\left[{\widehat{L}}_{\mathsf{\sigma }},{\widehat{L}}_m\right]}^T$ for an adaptive model of a doubly fed induction machine is formulated as a search for the minimum of a function $S\left({\widehat{L}}_{\mathsf{\sigma }},{\widehat{L}}_m\right)=S\left(x_1,x_2\right)$ in n-dimensional Euclidean space ℝⁿ, where n = 2, based on the previous assumptions. The target function is as follows:

$$S({\widehat{L}}_{\mathsf{\sigma }},{\widehat{L}}_m)=S(x_1,x_2)={\displaystyle \sum _{n=1}^m{\left[i_{2u}((n\cdot \Delta t),{\widehat{L}}_{\mathsf{\sigma }},{\widehat{L}}_m)-i_n\right]}^2}$$

(23)

where m is the number of digitized values of the rotor current obtained experimentally.

The value of the target function $S\left({\widehat{L}}_{\mathsf{\sigma }},{\widehat{L}}_m\right)=S\left(x_1,x_2\right)$ indicates the level to which the optimization objective has been achieved. This optimization problem is solved within a set of permissible solutions for the target function $X\subseteq$ℝⁿ, identifying the vector $x^{*}$ within the feasible set $X\subseteq$ℝⁿ that yields the minimum value of the objective function across this set:

$$S\left(x^{*}\right)= \underset{x\in X}{\min }S\left(x\right)$$

(24)

The Newton method (tangent method) is applied to solve the problem of the target function minimization (24) and finding the inductances desired estimates [34,35]. Newton’s method for finding two unknowns x₁, x₂ is applicable to a system of nonlinear equations of the following form:

$$\left\{\begin{array}{c}{F}_1\left(x_1,x_2\right)=0\\ F_2\left(x_1,x_2\right)=0\end{array}\right.$$

(25)

where $F_1\left(x_1,x_2\right)$ first function of the left part of the system of nonlinear algebraic equations; $F_2\left(x_1,x_2\right)$ The second function of the left part of the system of nonlinear algebraic equations.

The system of nonlinear Equation (25) requires two partial derivatives of the target function $S\left(x_1,x_2\right)$ relatively with x₁ and x₂ as follows:

$$\left\{\begin{array}{c}{F}_1\left(x_1,x_2\right)={\displaystyle \frac{\partial S\left(x_1,x_2\right)}{\partial x_1}}\\ F_2\left(x_1,x_2\right)={\displaystyle \frac{\partial S\left(x_1,x_2\right)}{\partial x_2}}\end{array}\right.$$

(26)

The solution will be values x₁, x₂, at which each of the equations of the system turns into a true numerical equality, which, for a two-dimensional problem, corresponds to the intersection of two curves in the plane.

Applying Taylor series expansion for functions of two variables $F_1\left(x_1,x_2\right)$ and $F_2\left(x_1,x_2\right)$ of the system of Equation (26) in the neighborhood of some point n with coordinates x_1n, x_2n with rejection of all components of the series except the value of the functions $F_1\left(x_1,x_2\right)$, $F_2\left(x_1,x_2\right)$, at point n and the tangent to this point, allows us to construct a linear approximation in the form of the system:

$$\left\{\begin{array}{c}\begin{array}{l}{F}_1\left(x_1,x_2\right)\approx F_1\left(x_{1n},x_{2n}\right)+\left(x_1-x_{1n}\right)\cdot {\displaystyle \frac{\partial F_1\left(x_{1n},x_{2n}\right)}{\partial x_1}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(x_2-x_{2n}\right)\cdot {\displaystyle \frac{\partial F_1\left(x_{1n},x_{2n}\right)}{\partial x_2}}+0\end{array}\\ \begin{array}{l}{F}_2\left(x_1,x_2\right)\approx F_2\left(x_{1n},x_{2n}\right)+\left(x_1-x_{1n}\right)\cdot {\displaystyle \frac{\partial F_2\left(x_{1n},x_{2n}\right)}{\partial x_1}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(x_2-x_{2n}\right)\cdot {\displaystyle \frac{\partial F_2\left(x_{1n},x_{2n}\right)}{\partial x_2}}+0\end{array}\end{array}\right.$$

(27)

After reducing the system (27) to a matrix form, we form an iterative procedure by Newton’s method to find x₁, x₂ as follows:

$$\left[\begin{array}{c}{x}_{1\hspace{0.17em}\left(k\right)}\\ x_{2\hspace{0.17em}\left(k\right)}\end{array}\right]=\left[\begin{array}{c}{x}_{1\hspace{0.17em}\left(k-1\right)}\\ x_{2\hspace{0.17em}\left(k,-,1\right)}\end{array}\right]-W^{-1}\left(x_{1\hspace{0.17em}\left(k-1\right)},x_{2\hspace{0.17em}\left(k-1\right)}\right)\cdot \left[\begin{array}{c}{F}_1\left(x_{1\hspace{0.17em}\left(k-1\right)},x_{2\hspace{0.17em}\left(k-1\right)}\right)\\ F_2\left(x_{1\hspace{0.17em}\left(k-1\right)},x_{2\hspace{0.17em}\left(k-1\right)}\right)\end{array}\right]$$

(28)

k calculation step of the iteration procedure, $W^{-1}\left(x_{1\hspace{0.17em}\left(k-1\right)},x_{2\hspace{0.17em}\left(k-1\right)}\right)$—inverse Jacobi matrix.

The inverse Jacobi matrix is formed on the basis of the Jacobi matrix, which requires determining the partial derivatives of functions $F_1\left(x_1,x_2\right)$, $F_2\left(x_1,x_2\right)$, for each of the desired variables x₁, x₂ as follows:

$$W\left(x_1,x_2\right)=\left[\begin{array}{cc}{\displaystyle \frac{\partial F_1\left(x_1,x_2\right)}{\partial x_1}}& {\displaystyle \frac{\partial F_1\left(x_1,x_2\right)}{\partial x_2}}\\ {\displaystyle \frac{\partial F_2\left(x_1,x_2\right)}{\partial x_1}}& {\displaystyle \frac{\partial F_2\left(x_1,x_2\right)}{\partial x_2}}\end{array}\right]$$

(29)

Since the functions $F_1\left(x_1,x_2\right)$, $F_2\left(x_1,x_2\right)$ are partial derivatives of the target function by x₁, x₂ (29), the problem of the Jacobi matrix requires finding second-order partial derivatives.

After inversion (29), an inverse Jacobi matrix is formed in the following form:

$$\begin{gathered} \begin{array}{l}{W}^{-1}\left(x_1,x_2\right)=\left[\begin{array}{cc}{\displaystyle \frac{\frac{\partial F_2\left(x_1,x_2\right)}{\partial x_2}}{G\left(x_1,x_2\right)}}& {\displaystyle \frac{-\frac{\partial F_1\left(x_1,x_2\right)}{\partial x_2}}{G\left(x_1,x_2\right)}}\\ {\displaystyle \frac{-\frac{\partial F_2\left(x_1,x_2\right)}{\partial x_1}}{G\left(x_1,x_2\right)}}& {\displaystyle \frac{\frac{\partial F_1\left(x_1,x_2\right)}{\partial x_1}}{G\left(x_1,x_2\right)}}\end{array}\right]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\left[\begin{array}{cc}{\displaystyle \frac{\frac{\partial }{\partial x_2}\left(\frac{\partial S\left(x_1,x_2\right)}{\partial x_2}\right)}{G\left(x_1,x_2\right)}}& {\displaystyle \frac{-\frac{\partial }{\partial x_2}\left(\frac{\partial S\left(x_1,x_2\right)}{\partial x_1}\right)}{G\left(x_1,x_2\right)}}\\ {\displaystyle \frac{-\frac{\partial }{\partial x_1}\left(\frac{\partial S\left(x_1,x_2\right)}{\partial x_2}\right)}{G\left(x_1,x_2\right)}}& {\displaystyle \frac{\frac{\partial }{\partial x_1}\left(\frac{\partial S\left(x_1,x_2\right)}{\partial x_1}\right)}{G\left(x_1,x_2\right)}}\end{array}\right]\end{array}, \\ \begin{array}{l}G\left(x_1,x_2\right)={\displaystyle \frac{\partial F_1\left(x_1,x_2\right)}{\partial x_1}}\cdot {\displaystyle \frac{\partial F_2\left(x_1,x_2\right)}{\partial x_2}}-{\displaystyle \frac{\partial F_1\left(x_1,x_2\right)}{\partial x_2}}\cdot {\displaystyle \frac{\partial F_2\left(x_1,x_2\right)}{\partial x_1}}\\ ={\displaystyle \frac{\partial }{\partial x_1}}\left({\displaystyle \frac{\partial S\left(x_1, x_2\right)}{\partial x_1}}\right)\cdot {\displaystyle \frac{\partial }{\partial x_2}}\left({\displaystyle \frac{\partial S\left(x_1, x_2\right)}{\partial x_2}}\right)-{\displaystyle \frac{\partial }{\partial x_2}}\left({\displaystyle \frac{\partial S\left(x_1, x_2\right)}{\partial x_1}}\right)\cdot {\displaystyle \frac{\partial }{\partial x_1}}\left({\displaystyle \frac{\partial S\left(x_1, x_2\right)}{\partial x_2}}\right)\end{array} \end{gathered}$$

The adaptive model describing the process of current damping in the doubly fed induction machine rotor consists of two terms, each of which is a transcendental elementary continuously differentiable function. This makes it possible to obtain analytical expressions of first- and second-order partial derivatives to form the Jacobi matrix. Formation of the inverse Jacobi matrix in the analytical form allows the implementation of the algorithm of evaluation of electromagnetic parameters of the induction machine of a different design on a modern digital element base (programmable logic device or digital signal processor) based on Newton’s method.

To start the calculation of the iteration procedure, the initial approximations of the values of x₁₍₀₎, x₂₍₀₎, at k = 0 are required. The iterative procedure for estimating the parameters x₁, x₂ may have a divergent character when given incorrect initial approximations x₁₍₀₎, x₂₍₀₎. To ensure a guaranteed convergence of the method of preliminary identification of the parameters of the T-shaped scheme of substitution of an induction machine on the basis of empirical experience, we should recommend setting as initial approximations the values of inductances ${\widehat{L}}_{\mathsf{\sigma }}$, ${\widehat{L}}_m$ which are known to be an order of magnitude smaller than known reference data for similar electric machines, which are close in their totality of parameters.

The iterative algorithm based on Newton’s method is shown in Figure 6.

The criterion for the end of the iteration process is the condition $‖x_{\hspace{0.17em}\left(k-1\right)}-x_{\hspace{0.17em}\left(k\right)}‖<\mathsf{\epsilon }$ (ε—allowable accuracy of value estimation x₁, x₂).

6. Simulation Results

We assess the performance of the proposed identification method on model data for a DFIM, whose mathematical model parameters are known and shown in

Table 1. Mathematical model parameters.

i (0+), (A)	R’1, (Ohm)	R2, (Ohm)	${\mathit{L}}_{\mathit{\sigma }}$, (mH)	${\mathit{L}}_{\mathit{m}}$, (H)
10	1.15	1.012	3	0.105

.

To visualize the form of the target function (23), which is the basis for the further construction of the identification algorithm by Newton’s method, we construct it with a three-dimensional graph (Figure 7). Figure 7 characteristics of a typical horizontal function.

A convergent iterative process to find estimates ${\widehat{L}}_{\mathsf{\sigma }}, {\widehat{L}}_m$ of a doubly fed induction machine with a stationary rotor is shown in Figure 8 and Figure 9.

Figure 8 illustrates the iterative convergence process of the solution using Newton’s method in the coefficient space. The obtained results (Figure 8) for estimating the parameters ${\widehat{L}}_{\mathsf{\sigma }}=3.04$ mH, ${\widehat{L}}_m=0.105$ H, compared with the model values (Table 1), have an error of no more than 2%. The preliminary identification algorithm, implemented using the Newton method, exhibits a rapid convergence rate (Figure 9).

7. Experimental Research Results

The dynamic object selected for the experimental evaluation of algorithm performance in the preliminary identification of a substitution model, and for validating the correctness of the initial assumptions, is a DFIM. Its specifications are detailed in

Table 2. Datasheet information of the DFIM.

Parameters	Value
Rated power, (kW)	1.4
Rated rotor speed, (rpm)	890
Rated current of the stator, (A)	8.5/4.9
Rated voltage of the stator, (V)	220/380
Rated rotor current, (A)	8.8
Rated voltage of the rotor, (V)	114
Energy conversion efficiency, (%)	65
Power factor	0.67

.

The identification and assessment of the machine’s equivalent circuit parameters are based on the analysis of its current decay curves. They are recorded by connecting the DFIM to a setup designed for capturing current decay profiles and powered by a laboratory source (Figure 10).

As the power switch of the standalone voltage inverter unit, an N-type MOSFET IRFP7530 (JSMicro Semiconductor Co., Ltd., Xi’an, China) with an induced channel is used, with the following main characteristics:

• When carrying a current of 1 A in the open state, the transistor has a low voltage value (1A·2mOhm = 2 mV) on the source-drain power channel;
• The maximum source-drain voltage of the power channel is 60 V;
• The limit current flowing through the transistor’s power channel is 195 A;
• The closing and opening times do not exceed 1 µs, which is an order of magnitude smaller than the electromagnetic time constant of any electrical machines.

Thus, the low voltage drop and high switching speed allow one to neglect the influence of power switches on the shape of the recorded decaying current curves during the preliminary identification procedure.

To acquire the voltage signal from current shunts and provide galvanic isolation, the HCPL-7510 (Broadcom Limited, San Jose, CA, USA) integrated circuit is used. This HCPL-7510 has a high linearity output characteristic and is specialized for organizing the measurement of the motor’s phase current based on a current shunt. The degree of the nonlinearity of the output characteristic is 0.06%, and the distortion of the output characteristic with temperature change is 60 ppm/°C. The optical channel of the HCPL-7510 IC, using sigma-delta modulation, ensures a high frequency of data transmission about the measured voltage drop on the current shunt, as well as immunity to electromagnetic interference. The maximum signal transmission delay is 9.9 µs, which is an order of magnitude smaller than the electromagnetic time constant of any electrical machine.

To manage the power switches of the standalone voltage inverter, digitize signals with the current measurement channel, and conduct the identification procedure based on the obtained phase current decay curves, the STM32F407VGT6 microcontroller (ST Microelectronics, Geneva, Switzerland) is utilized, with a maximum operating frequency of 168 MHz. The microcontroller is equipped with a 12-bit ADC of successive approximation, for which, with an external 3.3 V reference voltage source connected, the limit sensitivity of voltage digitization is 0.8mV, sufficient for solving tasks of preliminary identification of parameters and DFIM control. Sampling frequency 8 kHz.

The results of the identification algorithm execution and the values of the experimental parameters of the equivalent circuit are presented in

Table 3. Experimental and estimated parameters in the DFIM equivalent circuit.

i (0+), (A)	R’1, (Ohm)	R2, (Ohm)	${\widehat{\mathit{L}}}_{\mathit{\sigma }}$, (mH)	${\widehat{\mathit{L}}}_{\mathit{m}}$, (H)
2	0.45	0.545	1.1	0.184

and in Figure 11. Figure 11 illustrates the iterative convergence process of the solution using Newton’s method in the coefficient space.

Figure 12 illustrates the experimentally acquired curve of DFIM’s current decay, recorded by the analog-to-digital converter of the microcontroller in the laboratory setup, starting from the switching moment of the VFC keys after the current injection phase.

The visual examination of the experimental rotor current damping curve for the DFIM, with the secondary stator windings closed (Figure 12), highlights the complexity of the decay profile. Initially, there is a rapid decrease in current characterized by a small time constant, transitioning to a smoother decay represented by a much larger time constant.

Figure 12 presents a comparative graph of the experimentally obtained rotor current damping dependence of the DFIM with closed secondary stator windings, i₂(t), against the current responses predicted by the adaptive regression model ${\widehat{i}}_2(t)$ (22).

Figure 12 demonstrates that the reconstructed current curve of the doubly fed induction machine closely aligns with the experimentally obtained data across most of its length. This alignment validates the assumptions made during the DFIM modeling under closed rotor winding conditions. The most significant discrepancy appears at the curve’s “break point” linked to the initial assumption that the stator’s leakage inductance is equivalent to that of the rotor.

For assessing the dynamics of the restoration error of the current curve i₂(t), the integral error is calculated as the ratio of the residual integral to the integral of the reference trajectory (30), both for each characteristic segment of the transient process curve and for the entire duration of the experiment. The choice of integration limits for calculating the integral error is made based on the following considerations: in the interval from 0 s to 0.1 s, the greatest deviation of the restored curve from the experimental one is observed, and the curve segment up to 0.4 s is taken as the duration of the transient process. From 0.4 s to 1 s, the integral error is calculated for the steady-state period.

$${\mathsf{\sigma }}_{i2}={\displaystyle \frac{{\displaystyle \underset{t_{st}}{\overset{t_{end}}{\int }}\left|{\widehat{i}}_2(t)-i_2(t)\right|dt}}{{\displaystyle \underset{t_{st}}{\overset{t_{end}}{\int }}\left|i_2(t)\right|dt}}}\cdot 100\%$$

(30)

where t_st, t_end are the initial time and ending time of integration, consequently.

The errors σ_i2 between the reconstructed and experimental current curves within various sections are detailed in

Table 4. Error of the reconstructed and experimental current curves.

	Time Interval, s
	Whole Plot	From 0 to 0.1	From 0.1 to 0.4	From 0.4 to 1
σi2, (%)	3.79	13.64	1.32	0.9

.

Given that σ_i2 across the entire section is 3.79%, which falls within acceptable limits in engineering practice, the estimates for the leakage inductances of the stator and rotor windings, as well as the inductance resulting from the magnetic flux in the air gap of the electric machine, are deemed accurate.

It should be noted that the errors σ_i2 between the reconstructed and experimental current curves depend more on the assumptions made when creating the adaptive model of the doubly fed induction machine, as the setup presented in the article, with the components used, allows obtaining the decaying current curve i₂(t) with an error of no more than 1%, as shown in Figure 11. When using other equipment, the authors recommend employing simple digital filtering methods, such as moving average or median filters, to improve the quality of the obtained current curve i₂(t).

8. Conclusions

Using differential equations, analytical expressions are derived to describe the rotor current damping curve of a doubly fed induction machine with closed secondary windings. It facilitates a parameter identification method under stationary rotor conditions with non-zero initial states.

The necessity of estimating DFIM parameters for constructing a control system with high responsiveness to control and disturbance inputs, especially when the machine operates as a generator within a wind energy system, is demonstrated.

An algorithm for the preliminary identification of two parameters related to the substitution circuit of an adaptive DFIM based on rotor circuits is developed, employing Newton’s method. Numerical estimates of these parameters are successfully determined.

The integral errors depicting the divergence between the experimental DFIM stator current damping curves and the corresponding responses from the adaptive regression model are calculated. In most sections of the curves, the integral error remains below 4%, which is considered acceptable for engineering standards. This analysis affirms the correctness of the assumptions made in the adaptive regression model crafted for the observed DFIM operating mode and indicates the stable convergence of the developed preliminary identification algorithm. This algorithm can be applied to configure various DFIM control systems, including vector control systems, particularly considering the machine’s role as part of a wind power system.

The proposed identification method is designed for engineering practice applications. Compared to other known identification methods, the proposed method, which utilizes Newton’s method, stands out under equal conditions due to its simplicity, transparency, and clarity, while simultaneously accounting for all technical characteristics of the computational device used.