An Approach to Estimate the Temperature of an Induction Motor under Nonlinear Parameter Perturbations Using a Data-Driven Digital Twin Technique

Abstract

To monitor temperature as a function of varying inductance and resistance, we propose a data-driven digital twin approach for the rapid and efficient real-time estimation of the rotor temperature in an induction motor. By integrating differential equations with online signal processing, the proposed data-driven digital twin approach is structured into three key stages: (1) transforming the nonlinear differential equations into discrete algebraic equations by substituting the differential operator with the difference quotient based on the sampled voltage and current; (2) deriving approximate analytical solutions for rotor resistance and stator inductance, which can be utilized to estimate the rotor temperature; and (3) developing a general procedure for obtaining approximate analytical solutions to nonlinear differential equations. The feasibility and validity of the proposed method were demonstrated by comparing the test results with a 1.5 kW AC motor. The experimental results indicate that our method achieves a minimum estimation error that falls within the standards set by IEC 60034-2-1. This work provides a valuable reference for the overheating protection of induction motors where direct temperature measurement is challenging.

1. Introduction

Accurate temperature estimation is very important for the overheating protection of an induction motor, which is difficult to measure directly [1,2]. Considering nonlinear parameter perturbation, such as the inductance and resistance, which is an implicit function of the temperature of the induction motor [3,4], how to increase the estimation accuracy of the temperature is still a serious problem that it is urgent to solve. On the other hand, the inductance and resistance of an induction motor cannot be measured directly due to the restriction of the mounting sensor or sampling signal transmission. Also, the influence of the harmonic current produced by the inverter on identifying the temperature of the induction motor should be considered (as shown in

Table 1. Influence of harmonic current on estimating the temperature of the induction motor.

Order	Source of the Thermal	Negative Effect Analysis	Influence on Estimating Temperature	Reference
1	Harmonic current	Additional copper loss of stator and rotor	1. Influence on the normal change rule of temperature with the stator current. 2. Causing irregular changes in the inductance and resistance of the induction motor.	[5]
		Stray loss of stator and rotor		[6]
		Additional iron loss of stator		[7]
		Increase the impedance of the rotor		[8]

). As discussed above, it is necessary to develop an approach for estimating the temperature that can overcome the influence of the varying inductance and resistance of an induction motor.

Investigations show that there are many methods that are suitable for estimating the temperature of the induction motor. As shown in

Table 2. Summarized temperature estimation algorithms of the induction motor.

Order	Methods	Advantages	Disadvantages	References
1	Recursive Least Squares (RLS)	The algorithm is simple	the problem of data saturation	[9,10]
2	Stator windings resistance (SWR) measurement	It is entirely accurate and is used as a reference for measuring the precision.	1. SWR measurement is highly intrusive. 2. SWR measurement requires turning off the IM and disconnecting the IM supply connections.	[11]
3	IM dynamic model-based SWR	1. The method is non-intrusive. 2. Can be used in thermal monitoring and IM protection programs.	1. The need to know the values of IM dynamic model parameters for SWR estimation. 2. Less accuracy in SWR estimation at speeds other than low speed. 3. Intense sensitivity to changes in IM dynamic model parameters.	[12]
4	DC signal injection-based SWR	1. For SWR estimation, there is no need to know the values of the IM dynamic model parameters. 2. Can be used in thermal monitoring and IM protection programs. 3. The method has a high level of accuracy.	1. The need for an additional circuit to inject the DC signal in the line-fed IMs. 2. Unbalance in IM feeding due to additional signal injection. 3. The presence of ripple in the IM torque.	[13,14]
5	Model Reference Adaptive System (MRAS)	The algorithm is simple	It is suitable for the problem of poor order	[15]
6	Extended Kalman Filter (EKF) algorithm	Noise impact is small	The algorithm is complex, with poor robustness	[16]
7	Intelligent algorithms	Good search ability, strong global search capability	The algorithm is complex and easy to fall into local optimum	[17]
8	Proximal Policy Optimization-Reinforcement Learning (PPO-RL) algorithm	The algorithm is stable and computationally efficient	The algorithm is sample-inefficient and cannot be run online	[18]
9	Machine learning method	The algorithm can effectively predict motor temperature.	The algorithm is offline, with poor applicability	[19,20]
10	Combination parameter identification method	This approach combines the advantages of Recursive Least Squares (RLS) and the Model Reference Adaptive System (MRAS).	The algorithm is complex and hard to realize online.	[21]

, several representative methods are Recursive Least Squares (RLS), Stator Windings Resistance (SWR) measurement method, Model Reference Adaptive System (MRAS), Extended Kalman Filter (EKF) algorithm, Proximal Policy Optimization-Reinforcement Learning (PPO-RL) algorithm and Intelligent algorithms. The above-mentioned methods have been used to estimate the motor temperature directly or indirectly already, which can serve as the research base of this paper. But, considering the rapidity and effectiveness requirements for the overheating protection in an induction motor, there are still many issues that should be solved: (1) it is generally difficult to obtain the closed-form solution of the temperature that is associated with the sampling voltage and sampling current of the induction motor; (2) the harmonics current caused by the inverter influences the estimation accuracy of the motor’s temperature; and (3) the inductance and resistance of the stator and the rotor are nonlinearly changing with the temperature of the motor. Therefore, deriving an analytical expression of the temperature to the varying inductance and varying impedance may be a suitable method of estimating the temperature of the induction motor. The difficulty of this work is to research a method for solving the general differential equations with varied coefficients associated with the varying inductance and resistance of the induction motor based on online sampling voltage and current.

As discussed above, the main goal of this paper is to develop an approach that can be used to find analytical solutions to nonlinear differential equations. Empirical studies show that the classical methods make it difficult to solve the nonlinear differential equations in a real-time-based on-chip system (as shown in

Table 3. Summarized classical methods for solving the differential equations.

Order	Methods		Advantages	Disadvantages	References
1	Numerical methods	Gradient-based differential neural solutions	The essence of these methods is iterative calculation, which is suitable to solve any differential equations.	It can be derived from the analytical solution of a differential equation directly and online.	[22]
		Nonlinear fuzzy method			[23]
		Highly efficient numerical scheme			[24]
		Finite element method			[25]
		Polynomial particular solutions method			[26]
		Adomian decomposition method			[27]
2	Semi-analytical methods	semi-analytical solution approach	By these methods, approximate analytical solutions can be obtained.	It is difficult to obtain an online analytical solution to a differential equation in on-chip system.	[28,29]
		semi-analytical scheme			[30]
		computational semi-numerical technique			[31]

). As Table 3 shows, there are successful methods for solving differential equations. But, for obtaining the analytical solution of a differential equation in real time, there are still some problems that should be solved. For example, among all the proposed algorithms to approximate the influence maximization, some are time-consuming, some inaccurate, and some include too many assumptions. However, the accurate monitoring temperature of the induction motor requires an analytical expression of the temperature due to the following reasons: (1) it needs to be executed rapidly and precisely; and (2) this approach should be realized in an on-chip system. Also, considering that the overwhelming majority of nonlinear differential equations are not solvable analytically, we should explore a suitable path to realize this research work. Therefore, it is necessary to develop an approach to obtaining the analytical solution for the motor temperature corresponding to Table 3.

As mentioned above, this paper aims to explore the use of the data-driven digital twin approach for temperature estimation in induction motors. Here, we propose a data-driven digital twin approach [32,33] that can effectively be used for the online identification of the varying inductance and resistance of induction motors, addressing the limitations of the traditional methods in terms of the accuracy and speed of deriving expressions. The structure of the paper includes the following: (1) establishing simplified differential equations related to the induction motor; (2) transforming the differential equations into time-difference algebraic equations by analyzing the sampled voltage and current of the induction motor; (3) deriving an analytical expression for the motor temperature by solving the difference algebraic equations; and (4) developing a data-driven digital twin processing procedure that can be implemented in an on-chip system. The feasibility and validity of the proposed method were demonstrated by comparing the test results with a 1.5 kW AC motor. As the induction motor is powered by an inverter, this results in a high harmonic current, which is one of the causes of the increased temperature in the motor. To estimate the current, it is necessary to extract the fundamental component of the current. To ensure the accuracy of voltage acquisition, inductors are used for filtering. The experimental results show that the estimation accuracy of the rotor temperature achieved by this method is below $1.6$ °C and meets the expected standards.

2. Modelling and Analyzing the Induction Motor

The differential equations that describe the temperature of an induction motor are taken as an example to demonstrate the proposed approach. The state equation of an induction motor in $\alpha \beta$ coordinate is shown as follows [34]

$${\left[\begin{array}{cccc}{\dot{i}}_{s\alpha }& {\dot{i}}_{s\beta }& {\dot{\psi }}_{r\alpha }& {\dot{\psi }}_{r\beta }\end{array}\right]}^T=A_1{\left[\begin{array}{cccc}{i}_{s\alpha }& i_{s\beta }& {\psi }_{r\alpha }& {\psi }_{r\beta }\end{array}\right]}^T+A_2{\left[\begin{array}{cc}{u}_{s\alpha }& u_{s\beta }\end{array}\right]}^T$$

(1)

where the detailed parameters of (1) are shown in

Table 4. Corresponding parameters of Equations (1)–(29).

Order	Parameters	Unit	Note
1	$i_{s\alpha }$, $i_{s\beta }$	A	Stator currents of the induction motor in $\alpha \beta$ coordinate system
2	$u_{s\alpha }$, $u_{s\beta }$	V	Stator voltage of the induction motor in $\alpha \beta$ coordinate system
3	${\psi }_{r\alpha }, {\psi }_{r\beta }$	Wb	Stator flux of the induction motor in $\alpha \beta$ coordinate system
4	$R_s$/$L_s$	Ω/H	Stator impedance/inductance of the induction motor
5	$R_r$/$L_r$	Ω/H	Rotor impedance/inductance of the induction motor
6	$L_m$	H	Mutual/leakage inductance of the induction motor
7	$\sigma$	H	Leakage inductance of the motor, $\sigma =1-L_m^2/\left(L_s{L}_r\right)$
8	$\tau$	s	Time constant of the motor rotor, $\tau =L_r/R_r$
9	${\omega }_e$	rad/s	Power supply angular frequency
10	$\xi ,\eta$		Variable $\xi =R_s/\sigma L_s+{L_m}^2/\sigma L_s{L}_r\tau$, $\eta =L_m/\sigma L_s{L}_r$
11	$t_s$	s	Sampling period
12	$x_{s\alpha }(k)$, $x_{s\beta }(k)$	V/A	Sampling voltage and current at kth period in $\alpha$ axis and $\beta$ axis, respectively, where $x\in [u,i]$
13	$T_0$	°C	The ambient temperature of the induction motor
14	$R_{s0}$	Ω	Initial impedance of the stator of the motor under T0
15	${\alpha }_s$		The impedance coefficient of the induction motor
16	$T_r$	°C	The rotor temperature of the induction motor
17	${\tilde{V}}_s/{\tilde{I}}_s$	V/A	The sampling voltage and current of the motor stator
18	$\varphi$	°	Phase difference between the sampling ${\tilde{V}}_s$ and the ${\tilde{I}}_s$
19	$V_{sx}/V_{sy}$	V	Positive sequence voltage $V_{sx}={\tilde{V}}_s\cos \varphi$, $V_{sy}={\tilde{V}}_s\sin \varphi$
20	$k_r$		The resistance constant of the induction motor
21	$x_a,x_b,x_c$		Voltage and current in $abc$ coordinate system, where $x\in [u,i]$
22	${\beta }^{\prime }$		${\beta }^{\prime }=\sqrt{L_m^4{\omega }_e^2{\tilde{I}}_s^2+4L_r^2{V}_{sy}^2-4L_r^2{R}_s^2{\tilde{I}}_s^2+8L_r^2{R}_s{V}_{sx}{\tilde{I}}_s-4L_r^2{\tilde{V}}_s^2}$
23	$\delta$		The coefficient of correction

. The coefficient matrices $A_1$ and $A_2$ are shown in (2) and (3). It means that, since the parameters $R_s$, $L_s$, $R_r$, and $L_r$ are functions of the temperature, system (1) can be viewed as a time-varying nonlinear system of differential equations.

$$A_1=\left[\begin{array}{cccc}-{\displaystyle \frac{R_s}{\sigma L_s}}-{\displaystyle \frac{{L_m}^2}{\sigma L_s{L}_r\tau }}& 0& {\displaystyle \frac{L_m}{\sigma L_s{L}_r\tau }}& {\displaystyle \frac{L_m{\omega }_e}{\sigma L_s{L}_r}}\\ 0& -{\displaystyle \frac{R_s}{\sigma L_s}}-{\displaystyle \frac{{L_m}^2}{\sigma L_s{L}_r\tau }}& -{\displaystyle \frac{L_m{\omega }_e}{\sigma L_s{L}_r}}& {\displaystyle \frac{L_m}{\sigma L_s{L}_r\tau }}\\ {\displaystyle \frac{L_m}{\tau }}& 0& -{\displaystyle \frac{1}{\tau }}& -{\omega }_e\\ 0& {\displaystyle \frac{L_m}{\tau }}& {\omega }_e& -{\displaystyle \frac{1}{\tau }}\end{array}\right]$$

(2)

$$A_2={\left[\begin{array}{llll}1/\left(\sigma L_s\right)\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1/\left(\sigma L_s\right)\hfill & 0\hfill & 0\hfill \end{array}\right]}^T$$

(3)

Let us rewrite system (1) as follows

$${\displaystyle \frac{d{i}_{s\alpha }}{dt}}=-\xi i_{s\alpha }+{\displaystyle \frac{\eta }{\tau }}{\psi }_{r\alpha }+{\omega }_e\eta {\psi }_{r\beta }+{\displaystyle \frac{1}{\sigma L_s}}{u}_{s\alpha }$$

(4)

$${\displaystyle \frac{d{i}_{s\beta }}{dt}}=-\xi i_{s\beta }-{\omega }_e\eta {\psi }_{r\alpha }+{\displaystyle \frac{\eta }{\tau }}{\psi }_{r\beta }+{\displaystyle \frac{1}{\sigma L_s}}{u}_{s\beta }$$

(5)

$${\displaystyle \frac{d{\psi }_{r\alpha }}{dt}}={\displaystyle \frac{L_m}{\tau }}{i}_{s\alpha }-{\displaystyle \frac{1}{\tau }}{\psi }_{r\alpha }-{\omega }_e{\psi }_{r\beta }$$

(6)

$${\displaystyle \frac{d{\psi }_{r\beta }}{dt}}={\displaystyle \frac{L_m}{\tau }}{i}_{s\beta }+{\omega }_e{\psi }_{r\alpha }-{\displaystyle \frac{1}{\tau }}{\psi }_{r\beta }$$

(7)

Corresponding parameters of (1–7) are shown in Table 4. System (4–7) illustrates the relationship between the leakage inductance and the stator current of the induction motor. In order to eliminate the flux leakage vectors ${\psi }_{r\alpha }$ and ${\psi }_{r\beta }$ that cannot be measured directly, we differentiate (4) and (5) with respect to time. We obtain

$${\displaystyle \frac{d^2{i}_{s\alpha }}{dt}}=-\xi {\displaystyle \frac{d{i}_{s\alpha }}{dt}}+{\displaystyle \frac{\eta }{\tau }}{\displaystyle \frac{d{\psi }_{r\alpha }}{dt}}+{\omega }_e\eta {\displaystyle \frac{d{\psi }_{r\beta }}{dt}}+\eta {\psi }_{r\beta }{\displaystyle \frac{d{\omega }_e}{dt}}+{\displaystyle \frac{1}{\sigma L_s}}{\displaystyle \frac{d{u}_{s\alpha }}{dt}}$$

(8)

$${\displaystyle \frac{d^2{i}_{s\beta }}{dt}}=-\xi {\displaystyle \frac{d{i}_{s\beta }}{dt}}+{\displaystyle \frac{\eta }{\tau }}{\displaystyle \frac{d{\psi }_{r\beta }}{dt}}-{\omega }_e\eta {\displaystyle \frac{d{\psi }_{r\alpha }}{dt}}-\eta {\psi }_{r\alpha }{\displaystyle \frac{d{\omega }_e}{dt}}+{\displaystyle \frac{1}{\sigma L_s}}{\displaystyle \frac{d{u}_{s\beta }}{dt}}$$

(9)

By substituting (6) and (7) into (8) and (9), we eliminate $d{\psi }_{r\alpha }/dt$ and $d{\psi }_{r\beta }/dt$. Then, (8) and (9) can be rewritten as

$${\displaystyle \frac{d^2{i}_{s\alpha }}{d{t}^2}}=-\xi {\displaystyle \frac{d{i}_{s\alpha }}{dt}}+{\displaystyle \frac{L_m\eta }{{\tau }^2}}{i}_{s\alpha }-{\displaystyle \frac{\eta }{{\tau }^2}}{\psi }_{r\alpha }-{\displaystyle \frac{{\omega }_e\eta }{\tau }}{\psi }_{r\beta }+{\displaystyle \frac{L_m\eta {\omega }_e}{\tau }}{i}_{s\beta }+{{\omega }_e}^2\eta {\psi }_{r\alpha }-{\displaystyle \frac{{\omega }_e\eta }{\tau }}{\psi }_{r\beta }+\eta {\psi }_{r\beta }{\displaystyle \frac{d{\omega }_e}{dt}}+{\displaystyle \frac{1}{\sigma L_s}}{\displaystyle \frac{d{u}_{s\alpha }}{dt}}$$

(10)

$${\displaystyle \frac{d^2{i}_{s\beta }}{d{t}^2}}=-\xi {\displaystyle \frac{d{i}_{s\beta }}{dt}}-{\displaystyle \frac{{\omega }_e{L}_m\eta }{\tau }}{i}_{s\alpha }+{\displaystyle \frac{\eta {\omega }_e}{\tau }}{\psi }_{r\alpha }+{{\omega }_e}^2\eta {\psi }_{r\beta }+{\displaystyle \frac{L_m\eta }{{\tau }^2}}{i}_{s\beta }+{\displaystyle \frac{\eta {\omega }_e}{\tau }}{\psi }_{r\alpha }-{\displaystyle \frac{\eta }{{\tau }^2}}{\psi }_{r\beta }-\eta {\psi }_{r\alpha }{\displaystyle \frac{d{\omega }_e}{dt}}+{\displaystyle \frac{1}{\sigma L_s}}{\displaystyle \frac{d{u}_{s\beta }}{dt}}$$

(11)

Although (10) and (11) incorporate second-order differential equations, we can still eliminate the flux leakage vectors ${\psi }_{r\alpha }$ and ${\psi }_{r\beta }$ by adding $(4)\times {\omega }_e+\left(10\right)$ and $(5)\times {\omega }_e+\left(11\right)$ to obtain new functions and substituting (4) and (5) into them as follows

$${\displaystyle \frac{d^2{i}_{s\alpha }}{d{t}^2}}+{\omega }_e{\displaystyle \frac{d{i}_{s\beta }}{dt}}=\left({\displaystyle \frac{L_m\eta }{{\tau }^2}}-{\displaystyle \frac{\xi }{\tau }}\right)i_{s\alpha }+\left({\displaystyle \frac{L_m\eta }{\tau }}-\xi \right){\omega }_e{i}_{s\beta }+{\displaystyle \frac{1}{\sigma L_s\tau }}{u}_{s\alpha }+{\displaystyle \frac{1}{\sigma L_s}}\left({\displaystyle \frac{d{u}_{s\alpha }}{dt}}+{\omega }_e{u}_{s\beta }\right)-\left(\xi +{\displaystyle \frac{1}{\tau }}\right){\displaystyle \frac{d{i}_{s\alpha }}{dt}}+\eta {\psi }_{r\beta }{\displaystyle \frac{d{\omega }_e}{dt}}$$

(12)

$${\displaystyle \frac{d^2{i}_{s\beta }}{d{t}^2}}-{\omega }_e{\displaystyle \frac{d{i}_{s\alpha }}{dt}}=\left({\displaystyle \frac{L_m\eta }{{\tau }^2}}-{\displaystyle \frac{\xi }{\tau }}\right)i_{s\beta }+\left({\displaystyle \frac{L_m\eta }{\tau }}-\xi \right){\omega }_e{i}_{s\alpha }+{\displaystyle \frac{1}{\sigma L_s\tau }}{u}_{s\beta }+{\displaystyle \frac{1}{\sigma L_s}}\left({\displaystyle \frac{d{u}_{s\beta }}{dt}}+{\omega }_e{u}_{s\alpha }\right)-\left(\xi +{\displaystyle \frac{1}{\tau }}\right){\displaystyle \frac{d{i}_{s\beta }}{dt}}-\eta {\psi }_{r\alpha }{\displaystyle \frac{d{\omega }_e}{dt}}$$

(13)

When the motor is in a stable working state or the motor speed has a small variation range, it can be assumed that $d{\omega }_e/dt\approx 0$. Rewrite (12) and (13) in a matrix form:

$$\left[\begin{array}{c}{\displaystyle \frac{d^2{i}_{s\alpha }}{d{t}^2}}+{\omega }_e{\displaystyle \frac{d{i}_{s\beta }}{dt}}\\ {\displaystyle \frac{d^2{i}_{s\beta }}{d{t}^2}}-{\omega }_e{\displaystyle \frac{d{i}_{s\alpha }}{dt}}\end{array}\right]=\left[\begin{array}{ccccc}{i}_{s\alpha }& {\omega }_e{i}_{s\beta }& u_{s\alpha }& {\displaystyle \frac{d{u}_{s\alpha }}{dt}}+{\omega }_e{u}_{s\beta }& {\displaystyle \frac{d{i}_{s\alpha }}{dt}}\\ i_{s\beta }& -{\omega }_e{i}_{s\alpha }& u_{s\beta }& {\displaystyle \frac{d{u}_{s\beta }}{dt}}-{\omega }_e{u}_{s\alpha }& {\displaystyle \frac{d{i}_{s\beta }}{dt}}\end{array}\right]\left[\begin{array}{c}{k}_1\\ k_2\\ k_3\\ k_4\\ k_5\end{array}\right]$$

(14)

Here, $k_i\left(i=1,2\cdots 5\right)$ are the induction motor system parameters:

$$\begin{array}{c}{k}_1=-R_s/\left(\delta L_s\tau \right)\\ k_2=-R_s/\left(\delta L_s\right)\\ k_3=1.0/\left(\delta L_s\tau \right)\\ k_4=1.0/\left(\delta L_s\right)\\ k_5=-[R_s/(\delta L_s)+1.0/(\delta \tau )]\end{array}$$

(15)

Since $k_1=-R_s{k}_3$ and $k_4=\tau k_3$, there are only two independent variables, $k_3$ and $k_5$, in (15). We should find $k_3$ and $k_5$ from (15) using the mutual inductance, $L_m$, whereas the rotor inductance, $R_r$, will be obtained and employed below to calculate the rotor temperature of the induction motor.

Since the motor temperature is a function of the parameters $R_s$, $L_s$, $L_m$, $R_r$, and $L_r$, it is necessary to derive the analytical expression of the above-mentioned parameters from (14). The parameters $R_s$, $L_s$, $L_m$, $R_r$, and $L_r$ are varying with the current $i_s$ in real time, and this computational process needs to be performed in an on-chip system; therefore, the numerical methods, as well as semi-analytical methods, do not quite meet this computing demand.

3. Data-Driven Digital Twin Model

In order to use the sampling current and voltage to estimate the temperature of the induction motor, a data-driven digital twin model has been derived. The core work is to transform differential Equation (14) into algebraic difference equations by replacing the differential operator with the differential quotient, which is described by the derivative of two adjacent sampling data to the sampling period.

Moreover, for obtaining the analytic expressions for variables $k_3$ and $k_5$, the stator current $i_{s\alpha }$ and $i_{s\beta }$ and the stator voltage $u_{s\alpha }$ and $u_{s\beta }$ can be rewritten as follows based on the discretization of the differential operator:

$$\begin{array}{l}{\displaystyle \frac{d^2{i}_{sx}}{d{t}^2}}\approx {\displaystyle \frac{i_{sx}\left(m+2\right)-2i_{sx}\left(m+1\right)+i_{sx}\left(m\right)}{t_s^2}}\begin{array}{l},\hfill \end{array}\left(x=\alpha ,\beta \right)\\ {\displaystyle \frac{d{i}_{sx}}{dt}}\approx {\displaystyle \frac{i_{sx}\left(m+1\right)-i_{sx}\left(m\right)}{t_s}}\begin{array}{l},\hfill \end{array}\left(x=\alpha ,\beta \right)\\ {\displaystyle \frac{d{u}_{sx}}{dt}}\approx {\displaystyle \frac{u_{sx}\left(m+1\right)-u_{sx}\left(m\right)}{t_s}}\begin{array}{l},\hfill \end{array}\left(x=\alpha ,\beta \right)\end{array}$$

(16)

where m, (m + 1) and (m + 2) denote mth, (m + 1)th and (m + 2)th sampling periods, respectively. In order to avoid measurement errors caused by numerical differentiation, it is necessary to use a filter to obtain a smooth waveform before measurement. Considering the measurement error level for the current, after the current $i\left(m\right)$ has been measured by the sensor, the $i\left(m\right)$ should be delivered to the CPU, which takes some time –$\Delta t$. Due to this delay time, the true current becomes $i\left(m+\Delta t\right)$. So, the measurement error is equal to $\left|i\left(m\right)-i(m+\Delta t)\right|/i(m+\Delta t)\times 100\%$. The voltage is the same; the measurement errors can be $\left|u\left(m\right)-u(m+\Delta t)\right|/u(m+\Delta t)\times 100\%$.

By substituting (16) into (14), the following system of algebraic equations can be obtained

$$B_2=B_1{\left[\begin{array}{lllll}{k}_1\hfill & k_2\hfill & k_3\hfill & k_4\hfill & k_5\hfill \end{array}\right]}^T$$

(17)

$$\begin{array}{l}{B}_1={\left[\begin{array}{ll}{i}_{s\alpha }\left(m\right)\hfill & i_{s\beta }\left(m\right)\hfill \\ {\omega }_e{i}_{s\beta }\left(m\right)\hfill & -{\omega }_e{i}_{s\alpha }\left(m\right)\hfill \\ u_{s\alpha }\left(m\right)\hfill & u_{s\beta }\left(m\right)\hfill \\ {\displaystyle \frac{u_{s\alpha }\left(m+1\right)-u_{s\alpha }\left(m\right)}{t_s}}+{\omega }_e{u}_{s\beta }\left(m\right)\hfill & {\displaystyle \frac{u_{s\beta }\left(m+1\right)-u_{s\beta }\left(m\right)}{t_s}}-{\omega }_e{u}_{s\alpha }\left(m\right)\hfill \\ {\displaystyle \frac{i_{s\alpha }\left(m+1\right)-i_{s\alpha }\left(m\right)}{t_s}}\hfill & {\displaystyle \frac{i_{s\beta }\left(m+1\right)-i_{s\beta }\left(m\right)}{t_s}}\hfill \end{array}\right]}^T\\ B_2=\left[\begin{array}{c}{\displaystyle \frac{i_{s\alpha }\left(m+2\right)-2i_{s\alpha }\left(m+1\right)+i_{s\alpha }\left(m\right)}{t_s^2}}+{\omega }_e{\displaystyle \frac{i_{s\beta }\left(m+1\right)-i_{s\beta }\left(m\right)}{t_s}}\\ {\displaystyle \frac{i_{s\beta }\left(m+2\right)-2i_{s\beta }\left(m+1\right)+i_{s\beta }\left(m\right)}{t_s^2}}-{\omega }_e{\displaystyle \frac{i_{s\alpha }\left(m+1\right)-i_{s\alpha }\left(m\right)}{t_s}}\end{array}\right]\end{array}$$

(18)

Equation (18) includes two characteristic sets associated with the variables $k_3$ and $k_5$. Based on the above analysis, the expressions of $k_3$ and $k_5$ can be obtained.

Analytic expressions of $k_3$ and $k_5$ are provided in (A2) to (A3a,b) in Appendix A. Then, we can find variables $\delta$ and $\tau$ as follows

$$\begin{array}{l}\delta =(T_s(-i_{s\beta }(m)R_s{T}_s+{\omega }_e{i}_{s\alpha }(m)R_s\tau T_s+u_{s\beta }(m)T_s+\tau u_{s\beta }(m+1)\\ -\tau u_{s\beta }(m)-{\omega }_e{u}_{s\alpha }(m)\tau T_s+i_{s\beta }(m)\tau R_s-i_{s\beta }(m+1)\tau R_s\\ +i_{s\beta }(m)L_s-i_{s\beta }(m+1)L_s))/(\tau L_s({\omega }_e{T}_s{i}_{s\alpha }(m)-{\omega }_e{T}_s{i}_{s\alpha }(m)\\ +1)+i_{s\beta }(m+2)+i_{s\beta }(m)-2i_{s\beta }(m+1)))\end{array}$$

(19)

$$\tau ={\tau }_1/{\tau }_2$$

(20)

where ${\tau }_1$ and ${\tau }_2$ can be written as the following expressions

$$\begin{array}{l}{\tau }_1=i_{s\beta }(m)u_{s\beta }(m)T_s^2{\omega }_e-u_{s\beta }(m)i_{s\beta }(m+1)T_s^2{\omega }_e+{i_{s\beta }}^2(m)L_s{T}_s{\omega }_e+{i_{s\beta }}^2(m+1)L_s{T}_s{\omega }_e+i_{s\beta }(m)i_{s\beta }(m+1)R_s{T}_s^2{\omega }_e\\ -i_{s\beta }(m+2)i_{s\alpha }(m)R_s{T}_s+2i_{s\beta }(m+1)i_{s\alpha }(m)R_s{T}_s-{\omega }_e{T}_s^2{i}_{s\alpha }(m+1)u_{s\alpha }(m)+{\omega }_e{T}_s^2{i}_{s\alpha }(m)u_{s\alpha }(m)\\ -{i_{s\beta }}^2(m)R_s{T}_s^2{\omega }_e-2i_{s\beta }(m)i_{s\alpha }(m+1)R_s{T}_s-i_{s\alpha }(m)i_{s\beta }(m+1)L_s+i_{s\beta }(m)i_{s\alpha }(m+1)L_s-i_{s\beta }(m)i_{s\alpha }(m+2)L_s-{\omega }_e{T}_s^2{i}_{s\alpha }(m)R_s\\ -i_{s\alpha }(m+1)i_{s\beta }(m+2)L_s-2i_{s\beta }(m+1)T_s{u}_{s\alpha }(m)+i_{s\beta }(m)T_s{u}_{s\alpha }(m)-i_{s\alpha }(m)u_{s\beta }(m)T_s+2u_{s\beta }(m)T_s{i}_{s\alpha }(m+1)\\ -u_{s\beta }(m)i_{s\alpha }(m+2)T_s-2i_{s\beta }(m)i_{s\beta }(m+1)L_s{T}_s{\omega }_e+{\omega }_e{T}_s^2{i}_{s\alpha }(m+1)i_{s\alpha }(m)R_s-{\omega }_e{T}_s{i}_{s\alpha }(m+1)L_s{i}_{s\alpha }(m)\\ +{\omega }_e{T}_s{i}_{s\alpha }(m+1)L_s{i}_{s\alpha }(m+1)+i_{s\beta }(m)i_{s\alpha }(m+2)R_s{T}_s+{\omega }_e{T}_s{i}_{s\alpha }(m)L_s{i}_{s\alpha }(m)-{\omega }_e{T}_s{i}_{s\alpha }(m)L_s{i}_{s\alpha }(m+1)\end{array}$$

(21)

$$\begin{array}{l}{\tau }_2=-i_{s\beta }(m+2){\omega }_e{u}_{s\beta }(m)T_s-{\omega }_e{T}_s{i}_{s\alpha }(m+1)u_{s\alpha }(m)+{\omega }_e{T}_s{i}_{s\alpha }(m+1)u_{s\alpha }(m+1)+{{\omega }_e}^2{T_s}^2{i}_{s\alpha }(m+1)u_{s\beta }(m)\\ +{\omega }_e{T}_s{i}_{s\alpha }(m)u_{s\alpha }(m)-{{\omega }_e}^2{T_s}^2{i}_{s\alpha }(m)u_{s\beta }(m)+i_{s\beta }(m)u_{s\alpha }(m)+u_{s\alpha }(m)i_{s\beta }(m){T_s}^2{{\omega }_e}^2-u_{s\alpha }(m)i_{s\beta }(m+1){T_s}^2{{\omega }_e}^2\\ -u_{s\alpha }(m)i_{s\alpha }(m)T_s{\omega }_e+2u_{s\alpha }(m)i_{s\alpha }(m+1)T_s{\omega }_e-u_{s\alpha }(m)i_{s\alpha }(m+2)T_s{\omega }_e-u_{s\beta }(m+1)i_{s\beta }(m)T_s{\omega }_e\\ +u_{s\beta }(m+1)i_{s\beta }(m+1)T_s{\omega }_e+u_{s\beta }(m)i_{s\beta }(m+1)T_s{\omega }_e-{i_{s\beta }}^2(m+1)R_s{T}_s{\omega }_e+i_{s\alpha }(m)i_{s\beta }(m+1)R_s-2i_{s\beta }(m+1)u_{s\alpha }(m)\\ -i_{s\beta }(m)i_{s\alpha }(m+1)R_s+i_{s\beta }(m)i_{s\alpha }(m+2)R_s-u_{s\beta }(m)i_{s\alpha }(m+2)-i_{s\beta }(m+1)i_{s\alpha }(m+2)R_s-i_{s\beta }(m+2)i_{s\alpha }(m)R_s\\ +i_{s\beta }(m+2)i_{s\alpha }(m+1)R_s+i_{s\beta }(m+2){\omega }_e{i}_{s\beta }(m)R_s{T}_s-{{\omega }_e}^2{T_s}^2{i}_{s\alpha }(m+1)i_{s\beta }(m)R_s+{\omega }_e{T}_s{i}_{s\alpha }(m+1)i_{s\alpha }(m)R_s\\ -{\omega }_e{i}_{s\alpha }(m)R_s{T}_s{i}_{s\alpha }(m+1)+{\omega }_e{i}_{s\alpha }(m)R_s{T}_s{i}_{s\alpha }(m+2)+u_{s\beta }(m+1)i_{s\alpha }(m)-2u_{s\beta }(m+1)i_{s\alpha }(m+1)\\ +u_{s\beta }(m+1)i_{s\alpha }(m+2)-i_{s\alpha }(m)u_{s\beta }(m)-i_{s\beta }(m)u_{s\alpha }(m+1)+2u_{s\beta }(m)i_{s\alpha }(m+1)+2i_{s\beta }(m+1)u_{s\alpha }(m+1)\\ +i_{s\beta }(m+2)u_{s\alpha }(m)-i_{s\beta }(m+2)u_{s\alpha }(m+1)\end{array}$$

(22)

As a supplement, the impedance $R_s$ and the inductance $L_s$ of the motor can be provided by the previous research [34,35]

$$R_s=R_{s0}+\left[\left(T_r-1.0\right)-T_0\right]{\alpha }_s{R}_{s0}$$

(23)

$$L_s=0.5\left(L_m^2{\omega }_e{\tilde{I}}_s+2L_r{V}_{sy}+{\beta }^{\prime }\right)/\left(L_r{\omega }_e{\tilde{I}}_s\right)$$

(24)

where $R_{s0}$ denotes the initial value of the stator resistance at the ambient temperature $T_0$; ${\alpha }_s$ denotes the resistance coefficient, and $T_r$ is the rotor temperature. The middle variable ${\beta }^{\prime }$ is

$${\beta }^{\prime }=\sqrt{L_m^4{\omega }_e^2{\tilde{I}}_s^2+4L_r^2{V}_{sy}^2-4L_r^2{R}_s^2{\tilde{I}}_s^2+8L_r^2{R}_s{V}_{sx}{\tilde{I}}_s-4L_r^2{\tilde{V}}_s^2}$$

(25)

where ${\tilde{I}}_s$ and ${\tilde{V}}_s$ are the sampling current and the sampling voltage of the motor stator, respectively; $\varphi$ is the phase difference between the stator voltage ${\tilde{V}}_s$ and the stator current ${\tilde{I}}_s$. As shown in Figure 1, the x-axis is aligned with the current phasor ${\tilde{I}}_s$. The positive sequence voltage can be described by $V_{sx}={\tilde{V}}_s\cos \varphi$ and $V_{sy}={\tilde{V}}_s\sin \varphi$.

According to (19) and (24), the rotor impedance $R_r$, the rotor inductance $L_r$, and the rotor temperature $T_r$ can be derived and expressed as follows

$$R_r=L_r/\tau$$

(26)

$$L_r=\sqrt{{\left[\left(k_2-k_5\right)/k_3\right]}^2-\left[\left(k_2-k_5\right)/\left(k_3{k}_4\right)\right]}$$

(27)

$$T_r=(R_r/R_{r0})\left(T_0+k_r\right)-k_r$$

(28)

where $R_{r0}$ denotes the initial impedance of the rotor at ambient temperature $T_0$ and the sign $k_r$ denotes the resistance constant.

Assuming the voltage vector and the current vector correspond to the stator voltage $u_a$ and the stator current $i_a$ in the $abc$ coordinate system, the transformation of the stator voltage and the stator current in the $\alpha \beta$ coordinate system to the $abc$ coordinate system can be described as

$$\left[\begin{array}{l}{x}_{s\alpha }\hfill \\ x_{s\beta }\hfill \end{array}\right]=\sqrt{{\displaystyle \frac{2}{3}}}\left[\begin{array}{lll}1\hfill & -1/2\hfill & -1/2\hfill \\ 0\hfill & \sqrt{3}/2\hfill & -\sqrt{3}/2\hfill \end{array}\right]\left[\begin{array}{c}{x}_a\\ x_b\\ x_c\end{array}\right]\left(x=i,u\right)$$

(29)

As discussed above, we have developed a data-driven digital twin model of an induction motor. The topology analysis and algorithmic procedure for this model are summarized in Figure 2 and Algorithm 1 respectively.

As shown in Algorithm 1, The main steps of the analytic solution of the rotor temperature is summarized in

Table 5. Main steps to find the analytic solution of the rotor temperature of an induction motor.

Order	Main Steps
1	Algebraization of the differential equations: substitute the differential operator by the quotient of two adjacent sampling data and the sampling period.
2	Solving the equivalent algebraic equations: to derive the expression regarding the rotor temperature by solving approximate algebraic difference equations.
3	Repeat the above procedures at every sampling period.

.

Algorithm 1: Algorithm of proposed data-driven digital twin model

0: Modeling an induction motor by the differential equations, providing $\mathrm{initial}\begin{array}{c}\end{array}\mathrm{value}:t_s,R_{s0}/T_0,R_{r0}/T_0,L_r,L_{m0},{\omega }_e$

1: Call (14), and replace the differential operator with (16)

2: Differential equations were transformed into algebraic equations (17)

3.1: Solving characteristic algebraic equation (17)

3.2: Extract accuracy fundamental voltage and current and eliminate intermediate variable based on sampling data

3.3: Obtain analytic solutions $L_r$, $R_r$ based on (21) and (22)

4.1: k = 1

4.2: $\left(I_{sk},V_{sk}\right)$, calculate power factor angle $\varphi$, $V_{skx}=V_{sk}\cos \varphi$, $V_{sky}=V_{sk}\sin \varphi$

4.3: $L_{sk}=f\left(R_{sk}\right)=0.5\left(L_m^2{\omega }_e{I}_{sk}+2L_{rk}{V}_{syk}+{\beta }^{\prime }\right)/\left(L_{rk}{\omega }_e{I}_{sk}\right)$, $T_{rk}=\left(R_{rk}/R_{r0}\right)\left(T_0+k_r\right)-k_r$, $R_{sk}=R_{s0}+\left[\left(T_{rk}-1.0\right)-T_0\right]{\alpha }_s{R}_{s0}$

4.4: Call (18) derive $y_k=\left(i_{s\alpha k},i_{s{\beta }_k},u_{s\alpha k},u_{s{\beta }_k}\right)$

4.6: Call (17) and (20) obtain $k_j,\left(j=1,2,\cdots ,5\right)$

4.7: Call (26) and (27) calculate $R_{rk}$

4.8: Obtain $T_{rk}$ by calling (28)

4.9: k = k + 1 and repeat above process 4.2–4.8

5.0: End

4. Experiment Validation

As shown in Figure 3, in order to verify the estimation accuracy and efficiency of the proposed approach, we have designed an experimental system that consists of a 1.5 kW induction motor (as the power source) and a 2.0 Hp DC motor (as the load). The electric–dynamic phenomena and the rotor temperature of the induction motor can be illustrated in (1) to (14) and (28). The parameters of this AC motor are as follows: $k_r=225$, ${\alpha }_s=0.01/$°C, $T_0=23$ °C, $R_{s0}=4.3 \mathsf{\Omega }$, $R_{r0}=7.2 \mathsf{\Omega }$, $L_{s0}=18 \mathrm{mH}$, $t_s=0.2 \mathrm{ms}$.

Figure 3. Experiment device of 1.5 kW induction motor, where the original cooling fan of the AC motor has been removed in order to obtain and analyze the extreme temperatures of the stator and rotor. The detailed parameters are listed in Table 6.

Figure 3. Experiment device of 1.5 kW induction motor, where the original cooling fan of the AC motor has been removed in order to obtain and analyze the extreme temperatures of the stator and rotor. The detailed parameters are listed in Table 6.

Table 6. Introduction of the experiment system.

Order	Part of the Experiment	Parameters	Note
1	(1) AC motor	1.5 kW/AC380V	We should estimate the rotor temperature of this motor due to it being used to drive a coaxial DC motor.
		$R_{s0}=4.3 \mathsf{\Omega }$
		$R_{r0}=7.2 \mathsf{\Omega }$
		$L_{s0}=18 \mathrm{mH}$
2	(2) DC motor	2.0 Hp/DC180V	It is a load generator for the AC motor.
3	(3) Inverter	5 kW/AC380V	It is used to power the AC motor.
4	(4) Signal conditioning circuit	0–3.3 V output voltage	6 channels for sampling; 3 phases of voltage and 3 phases of voltage
5	(5) Power Analyzer	3 current sensors and 3 voltage sensors	It is used to test the stator voltage and current of the AC motor
6	(6) TMP-A temperature instrument	4 thermocouples	It is used to measure the temperature between the stator and rotor of the AC motor.

Table 6. Introduction of the experiment system.

Order	Part of the Experiment	Parameters	Note
1	(1) AC motor	1.5 kW/AC380V	We should estimate the rotor temperature of this motor due to it being used to drive a coaxial DC motor.
		$R_{s0}=4.3 \mathsf{\Omega }$
		$R_{r0}=7.2 \mathsf{\Omega }$
		$L_{s0}=18 \mathrm{mH}$
2	(2) DC motor	2.0 Hp/DC180V	It is a load generator for the AC motor.
3	(3) Inverter	5 kW/AC380V	It is used to power the AC motor.
4	(4) Signal conditioning circuit	0–3.3 V output voltage	6 channels for sampling; 3 phases of voltage and 3 phases of voltage
5	(5) Power Analyzer	3 current sensors and 3 voltage sensors	It is used to test the stator voltage and current of the AC motor
6	(6) TMP-A temperature instrument	4 thermocouples	It is used to measure the temperature between the stator and rotor of the AC motor.

Figure 4 shows the diagram of the experiment system. The temperature $T_r$ can be obtained by the voltage $u_{s\alpha }$, $u_{s\beta }$ and the current $i_{s\alpha }$, $i_{s\beta }$, which is derived by transforming the sampling voltage $(u_a,u_b,u_c)$ and sampling current $(i_a,i_b,i_c)$ based on (29).

Figure 5 shows the measured instantaneous voltage and current of the induction motor. It denotes that, due to the control errors of the inverter that is chosen as the power supply of the induction motor, there are considerable aberrance and distortions in the stator current. It can also be proven in Figure 6 that the total harmonic distortion of the stator current (THD_i) exceeds standard [36].

Considering that only the fundamental current is useful for identifying the temperature of the motor, the FFT (Fast Fourier Transform) algorithm has been used to extract the fundamental current and its spectrum, which are shown in Figure 6 and Figure 7, respectively. The correctness of Figure 7 can be illustrated by comparing Figure 8 with Figure 9.

Based on the above sampling voltage and current of the induction motor, the estimated stator inductance, estimated rotor impedance, and estimated rotor temperature are shown in Figure 10, Figure 11 and Figure 12. Figure 10 and Figure 11 show that the stator inductance and the rotor impedance increase with the running time, which influences the temperature of the induction motor.

Figure 12 and Figure 13 show the estimated rotor temperature and measured rotor temperature, respectively. Comparison of Figure 12 and Figure 13 shows that the estimated rotor temperature conforms exactly to the measured temperature of the induction motor. Both the estimated and measured values significantly increased from $23$ °C to $90$ °C within 35 min.

It should be noted that, in order to emphasize the estimation effect, the cooling fan has been removed, which is the reason why the rotor temperature has increased to about $90$ °C within 35 min. In addition, because the rotor is located on the central axis of the stator coil, its temperature is slightly higher than the stator coil. It is the reason why the measured stator temperature is lower, $1$ °C, than the estimated rotor temperature of the induction motor. Moreover, Figure 14 shows the temperature obtained by the infrared camera, which is very close to the estimated rotor temperature of the induction motor. The motor is composed of iron, and its emissivity used for the thermographic picture is 0.6.

The error $\epsilon$ between the measured temperature and the estimated temperature is shown in

Table 7. Error of proposed approach regarding measured temperature (°C).

	Data	$\mathbf{Estimation}\mathbf{Temperature}{\mathit{T}}_{\mathit{r}}$	$\mathbf{Measured}\mathbf{Temperature}{\mathit{T}}_{\mathit{r}}^{*}$	Error $\mathit{\epsilon }$
Term
End time		89.6	88.0	1.8%
Start time		23.3	23.0	1.3%

. It is defined as $\epsilon =\left|T_r^{*}-T_r\right|/T_r^{*}\times 100\%$.

Table 7 shows that the estimation error of the temperature was lower, $1.6$ °C, which can satisfy the demand for the overheating protection of the induction motor. It means that we can use (20) to monitor the temperature of the induction motor based on sampling the stator voltage and current.

For this work, we have proposed and demonstrated the configuration of a data-driven digital twin model for an induction motor, as shown in Figure 15.

5. Conclusions

This paper proposes a data-driven digital twin approach to estimate the real-time temperature of an induction motor. The main contribution of this work is to derive the analytical expressions of the rotor temperature from the nonlinear differential equations used to describe the response phenomena of the induction motor. Due to our work, as shown in the experimental results, when the induction motor is driven by a variable-frequency drive (VFD), approximately 25% THD_i is generated, and these harmonics are a significant cause of motor heating. Additionally, the inductance and resistance also increase. By identifying the resistance and inductance, the temperature is indirectly estimated, resulting in a final estimation error of 1.8%, and the minimum error for testing the temperature of the induction motor is less than $1$ °C, which has reached the standard of standard [37].

At the algorithm level, an approach that can be used to obtain the analytical solution to the nonlinear differential equations has been proposed based on modeling the data-driven digital twin. The procedure for substituting the differential operator by the differential quotient was provided and used to obtain the discretization algebraic equations. This work is suitable for deriving an analytical solution to differential equations with varying coefficients, which is the basis of the proposed data-driven digital twin approach.

At the engineering level, in order to suppress the interturn short circuit in an induction motor caused by overheating, we have developed a protection technology that efficiently and rapidly estimates the rotor temperature of an induction motor. So far, this technology has been used to monitor the temperature of the wind power generator and electrical submersible motor. This work can provide a reference for generating a solution for increasing the security and reliability of a nonlinear dynamic system. Furthermore, if different operating conditions or motor types can be mathematically transformed into similar differential equation problems, relevant algorithms can be developed. After the proper validation of the corresponding models, these algorithms would enable rapid and accurate temperature estimation.