In recent years, power electronic converters have become integral components in various fields such as industrial applications, aerospace, and renewable energy systems. The reliability and efficiency of these systems heavily depend on the health of their components, particularly in three-phase inverters, which are widely used for AC motor drives, grid-tied systems, and uninterruptible power supplies. Parameter identification in power electronic converters plays a crucial role in enhancing control performance, fault prediction, and condition monitoring. Traditional methods for parameter identification, such as time-domain and frequency-domain approaches, often face limitations in simultaneously identifying multiple parameters with high accuracy. For instance, least squares methods require complex hybrid system modeling and may involve intrusive signal injection, while frequency-domain techniques struggle with precise parameter estimation due to noise and component interdependencies. To address these challenges, we propose a novel parameter identification method for three-phase inverters based on digital twin technology. This approach leverages a virtual replica of the physical system, enabling non-intrusive, real-time monitoring and accurate estimation of circuit parameters, including inductances, parasitic resistances, and internal resistances of power switches.
The digital twin concept involves creating a high-fidelity mathematical model of the physical three-phase inverter, which mirrors its behavior under various operating conditions. In our study, we develop the digital twin by formulating the mathematical model of the main circuit and the discrete controller model. The main circuit of a three-phase inverter typically consists of a DC-link voltage source, power switches, filter inductors, and parasitic resistances. The dynamic behavior of the three-phase inverter can be described by differential equations derived from Kirchhoff’s voltage laws (KVL). For a three-phase system, the output phase voltages and currents are interrelated through these equations, accounting for the switching states of the inverter bridges. The mathematical model is essential for accurately simulating the inverter’s response and facilitating parameter identification.
To solve the differential equations governing the three-phase inverter, we employ the fourth-order Runge-Kutta method, a numerical technique that provides high accuracy with minimal discretization error. This method discretizes the continuous-time equations into iterative steps, allowing us to compute the inductor currents at each time instant. The general form of the differential equation for each phase is given by:
$$ \frac{di_{Li}}{dt} = \frac{1}{L_i} \left( V_i – i_{Li} r_i – i_{Li} R_{Li} \right) $$
where \( i_{Li} \) represents the inductor current for phase \( i \) (where \( i \) denotes phases a, b, or c), \( L_i \) is the inductance, \( r_i \) is the internal resistance of the power switches, \( R_{Li} \) is the parasitic resistance of the inductor, and \( V_i \) is the output voltage of the inverter phase. The Runge-Kutta method approximates the solution by calculating intermediate values, as shown below:
$$ i_{Li,n+1} = i_{Li,n} + \frac{h}{6}(k_{1i} + 2k_{2i} + 2k_{3i} + k_{4i}) $$
Here, \( h \) is the time step, and \( k_{1i} \) to \( k_{4i} \) are the slopes computed at different points within the step. This numerical approach ensures that the digital twin model closely matches the physical system’s dynamics, even under transient conditions.
In addition to the main circuit model, the digital twin incorporates a digital controller to simulate closed-loop operation. The controller uses a proportional-integral (PI) regulator to maintain the output currents at desired reference values. The control process involves transforming the three-phase currents into the dq0 reference frame using Park and Clark transformations, adjusting the errors via PI control, and generating modulation signals for pulse-width modulation (PWM). The discrete-time PI controller equations are as follows:
$$ i_{ed,n} = i_{ref} – i_{od,n} $$
$$ i_{eq,n} = – i_{oq,n} $$
$$ i_{md,n} = i_{md,n-1} + k_i h i_{ed,n} + k_p (i_{ed,n} – i_{ed,n-1}) + i_f $$
$$ i_{mq,n} = i_{mq,n-1} + k_i h i_{eq,n} + k_p (i_{eq,n} – i_{eq,n-1}) $$
where \( i_{ed} \) and \( i_{eq} \) are the d-axis and q-axis current errors, \( i_{md} \) and \( i_{mq} \) are the modulated outputs, \( k_p \) and \( k_i \) are the proportional and integral gains, and \( i_f \) is a fixed bias current. The modulated signals are then inverse-transformed to obtain the three-phase modulation indices, which are used to generate the PWM signals for the switches. This controller model ensures that the digital twin replicates the regulatory behavior of the physical three-phase inverter.
For parameter identification, we utilize an adaptive particle swarm optimization (APSO) algorithm to minimize the discrepancy between the digital twin and physical system outputs. The APSO algorithm is chosen for its ability to adaptively adjust learning parameters, preventing premature convergence and enhancing search efficiency. The objective function for each phase is defined as the mean squared error between the digital twin’s inductor currents and the measured currents from the physical three-phase inverter:
$$ f_{obj,i} = \frac{1}{N} \sum_{j=1}^{N} (i_{Li,j} – i_{Lm,i,j})^2 $$
where \( i_{Li,j} \) is the inductor current from the digital twin, \( i_{Lm,i,j} \) is the measured current from the physical inverter, and \( N \) is the number of samples. The APSO algorithm iteratively updates the parameter set \( P = (L_i, R_{Li}, r_i) \) for each phase until the objective function falls below a predefined threshold. The particle update equations in APSO are:
$$ V_{g,k} = \omega_{g-1} V_{g-1,k} + 2 r_{1,g-1} (P_{G} – P_{g-1,k}) + 2 r_{2,g-1} (P_{H,g-1,k} – P_{g-1,k}) $$
$$ P_{g,k} = P_{g-1,k} + V_{g,k} $$
Here, \( V_{g,k} \) is the velocity of particle \( k \) at generation \( g \), \( P_{g,k} \) is the particle position, \( P_G \) is the global best position, \( P_H \) is the individual best position, and \( \omega \), \( r_1 \), and \( r_2 \) are adaptive parameters adjusted based on the evolutionary state. The APSO process involves initializing a population of particles, evaluating the objective function, updating personal and global bests, and adaptively tuning the parameters to guide the search. This method enables simultaneous identification of multiple parameters in the three-phase inverter with high precision.
To validate the proposed method, we conducted simulation studies under steady-state and dynamic conditions. The simulation parameters for the three-phase inverter are summarized in the table below:
| Parameter | Value |
|---|---|
| DC-link voltage \( V_{dc} \) | 350 V |
| Inductance \( L_i \) | 5 mH |
| Parasitic resistance \( R_{Li} \) | 0.5 Ω |
| Internal resistance \( r_i \) | 0.2 Ω |
| Switching frequency \( f_s \) | 2.5 kHz |
| Sampling frequency \( f_t \) | 200 kHz |
In steady-state conditions, the APSO algorithm successfully identified the parameters with relative errors below 2%. The following table presents the identification results for the inductances and resistances:
| Parameter | Phase A | Phase B | Phase C | Target Value |
|---|---|---|---|---|
| \( L_i \) (mH) | 4.9915 | 5.0021 | 4.9792 | 5 |
| \( R_{Li} \) (Ω) | 0.5055 | 0.4992 | 0.5046 | 0.5 |
| \( r_i \) (upper, Ω) | 0.2034 | 0.1996 | 0.1984 | 0.2 |
| \( r_i \) (lower, Ω) | 0.2025 | 0.2011 | 0.1989 | 0.2 |
The relative errors for these parameters are calculated as:
$$ \text{Relative Error} = \frac{|\text{Identified Value} – \text{Target Value}|}{\text{Target Value}} \times 100\% $$
For instance, the relative error for phase A inductance is -0.17%, demonstrating high accuracy. The convergence of the objective functions for each phase is achieved within seconds, indicating the efficiency of the APSO algorithm. Under dynamic conditions, we introduced parameter changes at 20 seconds, reducing the inductance to 4.5 mH and parasitic resistance to 0.45 Ω. The digital twin quickly adapted to these changes, with identification results shown below:
| Parameter | Phase A | Phase B | Phase C | Target Value |
|---|---|---|---|---|
| \( L_i \) before change (mH) | 5.0017 | 4.9638 | 4.9926 | 5 |
| \( L_i \) after change (mH) | 4.4996 | 4.5242 | 4.5107 | 4.5 |
| \( R_{Li} \) before change (Ω) | 0.5037 | 0.5030 | 0.5032 | 0.5 |
| \( R_{Li} \) after change (Ω) | 0.4560 | 0.4511 | 0.4528 | 0.45 |
The relative errors in dynamic conditions remain below 1.5%, confirming the method’s robustness in tracking parameter variations. This capability is vital for real-time condition monitoring of three-phase inverters in practical applications, where component degradation over time can lead to performance issues.
Experimental validation was performed using a hardware platform of a three-phase inverter with a DSP controller for PWM generation. The physical parameters were measured with an LCR meter, and current data were acquired using current probes and a data acquisition card. The experimental parameters are as follows:
| Parameter | Value |
|---|---|
| DC-link voltage \( V_{dc} \) | 50 V |
| Inductance \( L_i \) | 3.1968 mH, 3.1976 mH, 3.1995 mH |
| Parasitic resistance \( R_{Li} \) | 0.1138 Ω, 0.1145 Ω, 0.1142 Ω |
| Switching frequency \( f_s \) | 5 kHz |
The identification results for the experimental setup are summarized in the table below, showing close agreement with the measured values:
| Parameter | Phase A | Phase B | Phase C | Target Value |
|---|---|---|---|---|
| \( L_i \) (mH) | 3.1938 | 3.2097 | 3.2224 | 3.1968, 3.1976, 3.1995 |
| \( R_{Li} \) (Ω) | 0.1140 | 0.1159 | 0.1152 | 0.1138, 0.1145, 0.1142 |
The relative errors for inductances are within 1%, and for resistances, within 2%, validating the method’s applicability to real-world three-phase inverters. The digital twin model effectively captures the behavior of the physical system, enabling accurate parameter estimation without intrusive measurements.
To assess the impact of noise on parameter identification, we injected Gaussian white noise into the current signals of the physical inverter in simulation. The noise had an average amplitude of 0.32 dB, and the sampling frequency was maintained at 200 kHz. Despite the noise, the identification results for inductances and parasitic resistances remained accurate, as shown in the following tables:
| Parameter | Phase A | Phase B | Phase C | Target Value |
|---|---|---|---|---|
| \( L_i \) (mH) | 4.9855 | 4.9879 | 4.9954 | 5 |
| \( R_{Li} \) (Ω) | 0.5053 | 0.5074 | 0.4952 | 0.5 |
Additional identification runs under noise conditions yielded consistent results, with inductance errors below 1% and resistance errors below 2%. This demonstrates the robustness of the proposed method in noisy environments, which is common in industrial settings for three-phase inverters.
In conclusion, our study presents a digital twin-based parameter identification method for three-phase inverters that achieves high accuracy in estimating inductances, parasitic resistances, and internal resistances of power switches. The use of the Runge-Kutta method for modeling and APSO for optimization ensures efficient and non-intrusive identification under both steady-state and dynamic conditions. Simulation and experimental results confirm the method’s effectiveness, with relative errors within 2%. This approach provides a practical solution for real-time condition monitoring and predictive maintenance of three-phase inverters, enhancing system reliability and reducing operational costs. Future work will focus on extending this method to other types of power electronic converters and integrating it with cloud-based digital twin platforms for large-scale applications.
The mathematical foundation of the three-phase inverter model is further elaborated through the state-space representation. The system can be expressed in matrix form as:
$$ \frac{d}{dt} \begin{bmatrix} i_{La} \\ i_{Lb} \\ i_{Lc} \end{bmatrix} = \mathbf{A} \begin{bmatrix} i_{La} \\ i_{Lb} \\ i_{Lc} \end{bmatrix} + \mathbf{B} \mathbf{V} $$
where \( \mathbf{A} \) is a matrix containing the parameters \( L_i \), \( R_{Li} \), and \( r_i \), and \( \mathbf{B} \) relates the input voltages. For a balanced three-phase system, this simplifies the analysis, but our method handles unbalanced conditions as well. The digital twin continuously updates the parameters based on the APSO output, ensuring alignment with the physical three-phase inverter.
Moreover, the adaptive nature of APSO allows it to handle nonlinearities and coupling effects in the three-phase inverter, which are common in practical applications. The evolutionary factor in APSO is computed as:
$$ f = \frac{l_g – l_{\min}}{l_{\max} – l_{\min}} $$
where \( l_g \) is the average distance of the global best particle, and \( l_{\min} \) and \( l_{\max} \) are the minimum and maximum distances in the population. This factor guides the adjustment of \( \omega \), \( r_1 \), and \( r_2 \), optimizing the search process. The fuzzy classification of evolutionary states—exploration, exploitation, convergence, and jumping—enhances the algorithm’s performance, making it suitable for complex parameter identification tasks in three-phase inverters.
In summary, the integration of digital twin technology with advanced optimization algorithms offers a powerful tool for the health management of power electronic systems. Our method not only identifies parameters accurately but also provides insights into the dynamic behavior of three-phase inverters, facilitating proactive maintenance and improved system performance. As the demand for reliable power conversion grows, such approaches will play a pivotal role in the advancement of smart grids and renewable energy integration.