Model-Based Mechanical State Parameter Estimation for High-Voltage Circuit Breakers

Abstract

This paper presents a model-based parameter estimation approach for high-voltage circuit breaker operating mechanisms using the dynamic model and stroke curves. First, an equivalent dynamic model of the mechanism is established, and its fidelity is validated through comparison with a virtual prototype model built in ADAMS, confirming the validity of the modeling assumptions. Then, the parameter estimation task is formulated as an optimization problem that minimizes the discrepancy between model predictions and actual stroke curves, and the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm is adopted to search for the optimal parameter set. Subsequently, to ensure estimation reliability, a thorough identifiability analysis is conducted, including sensitivity and correlation analyses, which collectively indicate that the key parameters are well identifiable. Afterward, experimental tests are performed on a 252 kV circuit breaker under various operating conditions, and the results demonstrate that the parameter identification errors are consistently below 6.3%. Finally, we compare the proposed method with three representative data-driven classifiers on the same dataset for a classification task. The comparison shows that our method achieves higher accuracy than others. Furthermore, the proposed method yields physically interpretable parameters directly linked to mechanical components, providing valuable information for condition assessment and maintenance planning in engineering applications.

1. Introduction

The control and protection roles of HVCBs in power systems make their reliability crucial for stable grid operation [1]. Investigation statistics show that mechanical operating faults constitute the primary failure type for in-service circuit breakers [2,3]. The reliability of the mechanical operating system is therefore a vital component of overall breaker reliability and key to achieving condition assessment and intelligent maintenance.

As one of the main types of operating mechanisms for HVCBs, spring operating mechanisms are widely used at voltage levels of 252 kV and below [4]. However, their complex mechanical structure and high-impact operating conditions pose challenges to mechanical reliability. The reliability of the spring mechanism primarily involves the trip device and the mechanical transmission system. Mechanical condition assessment methods in trip devices are addressed in references [5,6]. This paper concentrates on mechanical state parameter estimation for the mechanical transmission system.

The travel curve, describing the motion characteristics of the moving contact during the opening and closing operations, directly reflects changes in mechanical behavior and is, thus, an ideal signal type for mechanical defect identification. Both references [7] and [8] simplify the operating mechanism of circuit breakers into a mass–spring–damper second-order system, yet with different emphases in model handling and diagnostic application. Reference [7] models the mechanism with adjustable mass, damping, and spring parameters to generate coil current and contact travel waveforms under healthy and various faulty conditions (e.g., spring constant change, damper failure), and then extracts features to train classifiers such as SVM. Reference [8] simplifies the mechanism into a second-order system with a time-varying damping ratio, which is linearized every 100 μs, and obtains features such as velocity and over-travel by fitting the measured travel curve; fault prediction is then achieved by combining maximum likelihood classification with interacting multiple model (IMM) estimation. Neither model considers the influence of cam profiles or critical structural dimensions on modeling. By building a precise dynamic model of a spring operating mechanism for a 72.5 kV SF6 circuit breaker in SOLIDWORKS and ADAMS, travel curves under both healthy and various faulty conditions are generated, from which key features are extracted to train and compare multiple machine learning classifiers [9,10]. Dou et al. [11] established a mathematical model of the energy storage system and investigated an online monitoring method for closing spring fatigue in HVCBs. Related research on dynamic modeling methods and condition assessment techniques can also be found in [12,13]. The above model-based circuit breaker mechanical condition assessment methods use the model to generate defect data and train a state classification model. The practical function of the established model is to replace defect simulation experiments.

Vibration signals contain rich mechanical state information about the opening and closing operations. The nonintrusive measurement nature has attracted significant interest from researchers in condition assessment, leading to numerous research outcomes [14,15,16], primarily focused on time–frequency domain feature extraction of vibration signals. Furthermore, the mechanical fault diagnosis method based on time-domain feature extraction and entropy features from vibration signals is investigated in reference [17], where the combination of time-domain segmentation and multiple entropy features significantly improves feature extraction efficiency and completeness. Yang et al. [18] investigates a condition assessment approach for opening buffers that utilizes vibration time–frequency images and convolutional neural networks, where the method converts one-dimensional vibration signals into three-dimensional time–frequency images via Hilbert–Huang Transform to enhance buffer condition characteristics, and subsequently develops an end-to-end image recognition model. While substantial achievements have been made in vibration signal research [16,19], the strong object-dependency of signal features limits the practical engineering application of related methods.

Building on this, researchers have explored mechanical condition assessment methods for circuit breakers using multisource signal fusion, including feature fusion of coil current and vibration signal [20], feature fusion of travel curve and coil current [21,22], and feature fusion of coil current, travel curve, and vibration signal [23].

While significant progress has been made in research on mechanical condition assessment methods for HVCBs, certain limitations remain in the following aspects:

(1)

Current research primarily focuses on feature extraction from operating signals and intelligent classification of operating conditions, yet remains limited in achieving parameter estimation of key dynamic parameters.

(2)

State classification methods based on signal features rely heavily on extensive complete-state data, which is often difficult to obtain in practical engineering applications for HVCBs.

To address these challenges, this paper proposes a dynamic model parameter estimation method based on the travel curve and CMA-ES, providing physically interpretable parameters for condition assessment of circuit breaker operating mechanisms. The main contributions of this paper are as follows:

(1)

A relatively precise dynamic model for the opening and closing operations of HVCBs is established. Compared to the existing models, the proposed model fully accounts for key factors such as the contour of the closing cam and the damping effect of the opening pneumatic reaction force.

(2)

A model parameter estimation method for HVCBs is proposed to overcome the limitations of existing approaches, which rely heavily on extensive simulation data.

(3)

By eliminating the dependency on defect-specific data from circuit breakers, the proposed method offers strong feasibility for engineering applications. Furthermore, it provides a new perspective for the model-based parameter identification in circuit breaker tripping systems and energy storage systems.

This paper is organized as follows: Section 2 establishes the dynamic model for the opening and closing operations. Section 3 presents the framework and implementation workflow of the model parameter estimation method. Section 4 validates and analyzes the proposed method. Finally, Section 5 summarizes the research conclusions.

2. Operating Principle and Dynamic Model

2.1. Operating Principle

Although the structural configuration of spring mechanisms may vary among different types, their fundamental principle remains the same, as follows. (a) The opening and closing springs are utilized as energy storage elements to drive the mechanical transmission system, enabling the movement of the contacts; (b) cam transmission is employed to regulate and control the mechanical characteristics during the closing process; (c) a buffer is used to ensure a smooth stop at the end of the opening operation.

A typical spring mechanism mainly consists of a closing spring and its transmission unit, an opening spring and its transmission unit (including the buffer), a camshaft, a cam, a roller, and an output shaft. The mechanical power is transmitted via connecting rods to a circuit breaker body to execute the opening and closing of the contacts. The structure is illustrated in Figure 1.

The opening and closing operating system comprises two subsystems: the closing spring–camshaft system (CS–CS) and the opening spring–output shaft system (OS–OS). The former includes the closing spring with its transmission assembly, the cam, and the camshaft. The latter consists of the opening spring, roller, output shaft, cranks, connecting rods, and the arc-extinguishing chamber. Force and motion transmission between the two subsystems are achieved through a cam mechanism formed by the interaction of the cam and the roller.

To improve transmission efficiency, the rotation angle of the crank in the four-bar linkage of the OS–OS is typically limited to no more than 60°. Under this condition, the following assumptions are made to simplify calculations:

(1)

Linear motion assumption: During the opening and closing operations, the motion of the OS–OS is considered linear. The transmission ratios among the output shaft of the mechanism, the input shaft of the circuit breaker body, and the moving contact remain constant and do not change with the rotation of the crank.

(2)

Constant inertia assumption: During the opening and closing operations, the equivalent inertia of the OS–OS remains constant and does not vary with the rotation of the crank. Consequently, the total inertia of the OS–OS can be equivalently concentrated onto the output shaft of the operation mechanism using the lumped parameter method, thereby simplifying the dynamic model.

The influence of the above linear motion and constant inertia assumptions on the accuracy of the dynamic model and parameter estimation will be discussed later in conjunction with experimental results in the sections on parameter estimation accuracy analysis. It should be noted that when the crank rotation angle becomes large, these assumptions may no longer hold. For such cases, one of the following two approaches is recommended: (i) a refined analytical model, i.e., establishing a rotation-angle-dependent variable transmission ratio model based on specific crank and connecting rod dimensions; (ii) a virtual prototype model, i.e., developing a detailed virtual prototype of the circuit breaker using multibody dynamics software such as ADAMS. In this case, the model will exhibit significantly increased dependence on structural dimensions.

2.2. Dynamic Model of the Closing Operation

The dynamic model of the closing operation incorporates three key aspects: the CS–CS, the OS–OS, and the force and motion transmission relationship between them. The schematic diagram with key geometric parameters is provided in Figure 2.

The CS–CS moves under the combined action of the driving force from the closing spring and the resistance from the roller. With the angular displacement of the camshaft as the state variable, its dynamic model is established as follows:

$$k_{\mathrm{C}}(x_{\mathrm{C}0}-(d+r_{\mathrm{C}}-{\displaystyle \frac{r_{\mathrm{C}}\sin {\theta }_1}{\sin \phi }}))d\sin \phi -T_{21}=J_1{\displaystyle \frac{{\mathrm{d}}^2{\theta }_1}{\mathrm{d}{t}^2}}$$

(1)

where k_C is the stiffness of the closing spring, x_C0 is the compression of the closing spring after energizing, d is the distance from the camshaft to the axis of the closing spring, r_C is the length of the crank actuated by the closing spring, φ is the angle between the closing chain and the vertical direction, θ₁ is the angular displacement of the camshaft, T₂₁ is the resisting torque generated by the roller, J₁ is the equivalent moment of inertia of the CS–CS, and t is the operating time of the camshaft. Under the condition of a small φ, Equation (1) can be transformed into the following form:

$$k_{\mathrm{C}}{r}_{\mathrm{C}}d{\displaystyle \frac{x_{\mathrm{C}0}-r_{\mathrm{C}}(1-\cos {\theta }_1)}{d+r_{\mathrm{C}}\cos {\theta }_1}}\sin {\theta }_1-T_{21}=J_1{\displaystyle \frac{{\mathrm{d}}^2{\theta }_1}{\mathrm{d}{t}^2}}$$

(2)

The motion of the OS–OS is governed by the driving force from the cam on the roller, the force from the opening spring, the transmission resistance, and the pneumatic resistance. With the angular displacement of the mechanism’s output shaft as the state variable, its dynamic model is given by the following:

$$T_{12}-T_{\mathrm{Cf}}-k_{\mathrm{O}}(x_{\mathrm{O}0}+{\theta }_2{r}_{\mathrm{O}})r_{\mathrm{O}}=J_2{\displaystyle \frac{{\mathrm{d}}^2{\theta }_2}{\mathrm{d}{t}^2}}$$

(3)

where T₁₂ is the driving torque from the cam acting on the output shaft (note that T₁₂ T₂₁); T_Cf is the equivalent resistance torque, which includes the transmission and pneumatic resistances and is considered constant during the closing process; k_O is the stiffness of the opening spring; x_O0 is the pre-compression of the opening spring before energy storage; θ₂ is the angular displacement of the output shaft; r_O is the length of the crank actuated by the closing spring; and J₂ is the equivalent moment of inertia of the OS–OS.

The kinematic relationship between the camshaft and the mechanism’s output shaft is described as follows:

$${\theta }_2=h({\theta }_1)$$

(4)

where h denotes the mapping relationship between the angular displacement of the camshaft and that of the output shaft, determined by the cam profile curve, which is the key factor governing the closing travel curve.

The cam transmission in spring mechanisms typically employs rolling contact. Neglecting energy loss in the cam transmission, the relationship between the resisting torque on the camshaft and the driving torque on the output shaft is derived from power conservation, as follows:

$$T_{21}=T_{12}{\displaystyle \frac{\mathrm{d}{\theta }_2}{\mathrm{d}{\theta }_1}}$$

(5)

In conclusion, the complete dynamic model of a mechanical transmission system during the closing process is comprehensively given below:

$$\left\{\begin{array}{l}{k}_{\mathrm{C}}{r}_{\mathrm{C}}d{\displaystyle \frac{x_{\mathrm{C}0}-r_{\mathrm{C}}(1-\cos {\theta }_1)}{d+r_{\mathrm{C}}\cos {\theta }_1}}\sin {\theta }_1-T_{21}=J_1{\displaystyle \frac{{\mathrm{d}}^2{\theta }_1}{\mathrm{d}{t}^2}}\hfill \\ T_{12}-T_{\mathrm{Cf}}-k_{\mathrm{O}}(x_{\mathrm{O}0}+{\theta }_2{r}_{\mathrm{O}})r_{\mathrm{O}}=J_2{\displaystyle \frac{{\mathrm{d}}^2{\theta }_2}{\mathrm{d}{t}^2}}\hfill \\ T_{21}=T_{12}{\displaystyle \frac{\mathrm{d}{\theta }_2}{\mathrm{d}{\theta }_1}}\hfill \\ {\theta }_2=h({\theta }_1)\hfill \end{array}\right.$$

(6)

The aforementioned dynamic model describes the dynamic process of the closing operation of the circuit breaker driven by the spring mechanism. Here, θ₁ and θ₂ are the state variables, while k_C, k_O, T_Cf, and J₂ may change with system degradation and can thus be used to characterize the system state, and are referred to as state parameters. The remaining parameters, which are determined by the mechanical structure, are termed structural parameters.

To reduce the dependence of the parameter identification process on structural parameters and enhance the engineering applicability of the method, parameter reorganization is performed on the dynamic model. The structural parameters and state parameters are collectively treated as integrated model parameters for estimation. As structural parameters remain invariant during system degradation, variations in the estimated model parameters directly reflect changes in the corresponding state parameters. The direction and magnitude of deviations in the estimated model parameters from their normal-state values indicate variations in the state parameters. The reorganized parameter set for the closing dynamic model is as follows:

$$\left\{\begin{array}{l}{A}_{\mathrm{C}}{\displaystyle \frac{x_{\mathrm{C}0}-r_{\mathrm{C}}(1-\cos {\theta }_1)}{d+r_{\mathrm{C}}\cos {\theta }_1}}\sin {\theta }_1-B_{\mathrm{C}}{\displaystyle \frac{T_{12}}{J_2}}{\displaystyle \frac{\mathrm{d}{\theta }_2}{\mathrm{d}{\theta }_1}}={\displaystyle \frac{{\mathrm{d}}^2{\theta }_1}{\mathrm{d}{t}^2}}\hfill \\ {\displaystyle \frac{T_{12}}{J_2}}-C_{\mathrm{C}}-D_{\mathrm{C}}{\theta }_2={\displaystyle \frac{{\mathrm{d}}^2{\theta }_2}{\mathrm{d}{t}^2}}\hfill \end{array}\right.$$

(7)

The defining equations for the parameters of the dynamic model, which consist of structural parameters and state parameters, are as follows:

$$\begin{array}{cc}\left\{\begin{array}{c}\hfill A_{\mathrm{C}}={\displaystyle \frac{k_{\mathrm{C}}{r}_{\mathrm{C}}d}{J_1}}\\ B_{\mathrm{C}}={\displaystyle \frac{J_2}{J_1}}\hfill \end{array}\right.& \left\{\begin{array}{l}{C}_{\mathrm{C}}={\displaystyle \frac{T_{\mathrm{Cf}}+k_{\mathrm{O}}{x}_{\mathrm{O}0}{r}_{\mathrm{O}}}{J_2}}\hfill \\ D_{\mathrm{C}}={\displaystyle \frac{k_{\mathrm{O}}{r_{\mathrm{O}}}^2}{J_2}}\hfill \end{array}\right.\end{array}$$

(8)

The model parameters A_C, B_C, C_C, and D_C are termed closing spring stiffness factor, inertia factor, closing resistance factor, and opening spring stiffness factor, respectively. These parameters collectively determine the dynamic response of the mechanical system, which represents the closing mechanical characteristics of the circuit breaker. Any variation in the closing mechanical characteristics implies changes in these model parameters. Therefore, by analyzing the closing mechanical characteristics, these model parameters can be estimated.

2.3. Dynamic Model of the Opening Operation

The opening process involves only the motion of the OS–OS and is independent of the CS–CS. However, the complexity lies in the engagement of the buffer.

Prior to buffer engagement, the OS–OS moves under the action of the opening spring force, transmission resistance, and the gas compression resistance from the arc-extinguishing chamber. This gas compression resistance acts through the puffer cylinder and exhibits damping characteristics, equivalently modeled as a linear damping force in this study. Thus, the dynamic model of the OS–OS at this stage is given by the following:

$$k_{\mathrm{O}}(x_{\mathrm{O}1}-{\theta }_2{r}_{\mathrm{O}})r_{\mathrm{O}}-T_{\mathrm{Of}}-c_{\mathrm{aec}}{r_{\mathrm{aec}}}^2{\displaystyle \frac{\mathrm{d}{\theta }_2}{\mathrm{d}t}}=J_2{\displaystyle \frac{{\mathrm{d}}^2{\theta }_2}{\mathrm{d}{t}^2}}$$

(9)

where x_O1 is the compression of the opening spring after energization, T_Of denotes the transmission resistance torque during the opening process, c_aec represents the damping coefficient of the arc-extinguishing chamber during opening process, and r_aec is the force arm of the arc-extinguishing chamber resistance relative to the output shaft.

Upon buffer engagement, the force exerted by the buffer is superimposed on the aforementioned forces acting on the OS–OS. Spring mechanisms commonly employ a hydraulic buffer, and a rigid mechanical stop is used at the final stage of the opening motion to ensure accurate breaking position. Therefore, the buffer resistance is equivalently modeled as a damper–spring system. The dynamic model of the OS–OS at this stage is, thus, given by the following:

$$k_{\mathrm{O}}(x_{\mathrm{O}1}-{\theta }_2{r}_{\mathrm{O}})r_{\mathrm{O}}-T_{\mathrm{Of}}-c_{\mathrm{aec}}{r_{\mathrm{aec}}}^2{\displaystyle \frac{\mathrm{d}{\theta }_2}{\mathrm{d}t}}=J_2{\displaystyle \frac{{\mathrm{d}}^2{\theta }_2}{\mathrm{d}{t}^2}}+c_{\mathrm{b}}{r_{\mathrm{b}}}^2{\displaystyle \frac{\mathrm{d}{\theta }_2}{\mathrm{d}t}}+k_{\mathrm{b}}{r_{\mathrm{b}}}^2({\theta }_2-{\alpha }_0)$$

(10)

where c_b is the damping coefficient of the buffer, r_b is the force arm of the buffer resistance relative to the output shaft, k_b is the stiffness coefficient of the buffer, and α₀ denotes the rotational angle of the mechanism’s output shaft at the instant of buffer engagement. In the opening dynamic model, θ₂ is the state variable, while k_O, T_Of, J₂, c_b and k_b are taken as the state parameters. The remaining parameters are structural parameters. The parameters of the opening dynamic model are reorganized as follows:

$$\left\{\begin{array}{cc}-A_{\mathrm{O}}{\theta }_2+B_{\mathrm{O}}-C_{\mathrm{O}}{\displaystyle \frac{\mathrm{d}{\theta }_2}{\mathrm{d}t}}={\displaystyle \frac{{\mathrm{d}}^2{\theta }_2}{\mathrm{d}{t}^2}}\hfill & ({\theta }_2<{\alpha }_0)\hfill \\ -A_{\mathrm{O}}{\theta }_2+B_{\mathrm{O}}-C_{\mathrm{O}}{\displaystyle \frac{\mathrm{d}{\theta }_2}{\mathrm{d}t}}={\displaystyle \frac{{\mathrm{d}}^2{\theta }_2}{\mathrm{d}{t}^2}}+D_{\mathrm{O}}{\displaystyle \frac{\mathrm{d}{\theta }_2}{\mathrm{d}t}}+E_{\mathrm{O}}({\theta }_2-{\alpha }_0)\hfill & ({\theta }_2>{\alpha }_0)\hfill \end{array}\right.$$

(11)

The defining equations for the dynamic model parameters are as follows:

$$\begin{array}{cc}\left\{\begin{array}{l}{A}_{\mathrm{O}}={\displaystyle \frac{k_{\mathrm{O}}{r_{\mathrm{O}}}^2}{J_2}}\hfill \\ B_{\mathrm{O}}={\displaystyle \frac{k_{\mathrm{O}}{x}_{\mathrm{O}1}{r}_{\mathrm{O}}-T_{\mathrm{Of}}}{J_2}}\hfill \\ C_{\mathrm{O}}={\displaystyle \frac{c_{\mathrm{aec}}{r_{\mathrm{aec}}}^2}{J_2}}\hfill \end{array}\right.& \left\{\begin{array}{l}{D}_{\mathrm{O}}={\displaystyle \frac{c_{\mathrm{b}}{r_{\mathrm{b}}}^2}{J_2}}\hfill \\ E_{\mathrm{O}}={\displaystyle \frac{k_{\mathrm{b}}{r_{\mathrm{b}}}^2}{J_2}}\hfill \\ \hfill \end{array}\right.\end{array}$$

(12)

The dynamic model parameters A_O, B_O, C_O, D_O, and E_O are referred to as opening spring stiffness factor, opening resistance factor, arc-extinction chamber damping factor, buffer damping factor, and buffer stiffness factor, respectively. These parameters collectively determine the state response θ₂ of the model, which represents the opening mechanical characteristics of the circuit breaker. By analyzing the opening mechanical characteristics, these parameters can be estimated.

2.4. Dynamic Model Validation

To verify the rationality of the model assumptions and the accuracy of the established dynamic model, a virtual prototype model of a 252 kV circuit breaker is built using ADAMS, as shown in Figure 3. In addition, dynamic simulations are carried out based on the established dynamic model, and the corresponding parameters are obtained from the virtual prototype model, as listed in

Table 1. Parameters of the virtual prototype model.

Symbol	Value	Symbol	Value	Symbol	Value
kC (kN/m)	90	kO (kN/m)	130	caec	1000
xC0 (m)	0.21	xO0 (m)	0.09	raec (m)	0.15
d (m)	0.29	rO (m)	0.08	rb (m)	0.08
rC (m)	0.086	J2 (kgm2)	0.243	cb (Ns/m)	1500
J1 (kgm2)	0.095	xO1 (m)	0.17	kb (kN/m)	150
TCf (Nm)	1600	TOf (Nm)	600	α0 (rad)	0.698

. The comparison between the simulated stroke curve from the virtual prototype model and that from the established dynamic model are shown in Figure 4.

The dynamic model calculations agree well with the virtual prototype simulations. The RMSE values are 0.0132 rad and 0.0087 rad for the opening and closing processes, respectively. These results confirm the reasonableness of the modeling assumptions and the accuracy of the developed dynamic model.

3. Parameter Estimation Method

3.1. Basic Idea

The travel curve characterizes the motion pattern of the moving contact during the opening and closing operations of HVCBs. The parameters of the dynamic model uniquely determine this motion pattern. In other words, a mapping exists between the travel curve and the dynamic model parameters. Therefore, the parameters of the dynamic model can be estimated from the measured travel curve.

For a given set of model parameters, substituting them into the dynamic model yields a corresponding simulated travel curve. An optimization problem is formulated with the objective of minimizing the deviation between the simulated travel curve and the measured travel curve. The optimal parameter set identified through an optimization algorithm represents the estimated parameters of the dynamic model that correspond to the measured travel curve.

This paper employs the CMA-ES to solve the optimization problem. The system described by the opening and closing dynamic models exhibits strong nonlinearity and high parameter coupling. Moreover, the experimental-based parameter estimation problem faces challenges including the difficulty in obtaining gradient information for the objective function. The core strength of CMA-ES lies in its gradient-free operation. By adaptively updating the covariance matrix of the search distribution, it can effectively learn the intrinsic geometric structure of the parameter space and automatically adjust the search direction and step size, endowing it with powerful global search capability. This makes it particularly suitable for solving such complex engineering inverse problems.

3.2. CMA-ES Method

The specific implementation steps of the CMA-ES method are as follows:

(1)

Set the population size N_c, parameter dimensionality M_c, maximum number of iterations G_max, and number of offspring λ. Denote the evolution generation by g, and initialize the evolution parameters at g = 0, including the mean vector μ⁽⁰⁾, step size σ⁽⁰⁾, and covariance matrix C⁽⁰⁾.

(2)

Based on the current mean vector μ^(g), step size σ^(g), and covariance matrix C^(g), generate N_c individuals that follow the multivariate normal distribution as follows:

$${{\mathsf{\theta }}_p}^{(g)}~{\mathsf{\mu }}^{(g)}+{\sigma }^{(g)}N(\mathbf{0},{\mathit{C}}^{(g)}) p=1,\dots ,N_c$$

(13)

where θ_p^(g) represents the p-th individual in the g-th generation population. Each individual is an M_c-dimensional vector, corresponding to a specific parameter set.

(3)

Construct a fitness function for the specific optimization problem at hand, compute the corresponding fitness value for each individual, and rank the individuals in ascending order based on their fitness values.

(4)

Select the top λ individuals with the highest fitness values as offspring, and update the mean vector for the next generation μ^(g+1) according to the following expression:

$${\mathsf{\mu }}^{(g+1)}={\displaystyle {\displaystyle \sum _{i=1}^{\lambda }}{\omega }_i}{{\mathsf{\theta }}_i}^{(g) } ({\displaystyle {\displaystyle \sum _{i=1}^{\lambda }}{\omega }_i}=1)$$

(14)

where ω_i denotes the individual weight, generated according to an exponential decay rule such that higher-fitness individuals are assigned larger weight coefficients.

(5)

Update the covariance matrix C^(g+1) using the following expression:

$${\mathit{C}}^{(g+1)}=(1-c_1-c_{\mathsf{\mu }}){\mathit{C}}^{(g)}+c_1{{\mathit{p}}_{\mathrm{c}}}^{(g+1)}{({{\mathit{p}}_{\mathrm{c}}}^{(g+1)})}^{\mathrm{T}}+c_{\mathsf{\mu }}{\displaystyle \sum _{i=1}^{\lambda }{\omega }_i{{\mathit{y}}_i}^{(g+1)}}{({{\mathit{y}}_i}^{(g+1)})}^{\mathrm{T}}$$

(15)

where c₁ is the global trend learning rate of the covariance matrix, which accumulates long-term search directions to enhance global trend learning capability; c_μ is the local structure learning rate of the covariance matrix, which utilizes information from the current elite offspring to capture local structures; and p_c^(g+1) is the updated evolution path, expressed as follows:

$${{\mathit{p}}_{\mathrm{c}}}^{(g+1)}=(1-c_{\mathrm{c}}){{\mathit{p}}_{\mathrm{c}}}^{(g)}+\sqrt{c_{\mathrm{c}}(2-c_{\mathrm{c}})N_{\mathrm{c}}}{\displaystyle \frac{{\mathsf{\mu }}^{(g+1)}-{\mathsf{\mu }}^{(g)}}{{\sigma }^{(g)}}}$$

(16)

where c_c is the learning rate for the evolution path. In Equation (15), y_i^(g+1) denotes the i-th evolution path deviation vector in the (g + 1)-th generation, which is calculated as follows:

$${{\mathit{y}}_i}^{(g+1)}={\displaystyle \frac{{{\mathsf{\theta }}_i}^{(g)}-{\mathsf{\mu }}^{(g+1)}}{{\sigma }^{(g)}}}$$

(17)

(6)

Calculate the updated step size σ^(g+1) using the following expression:

$${\sigma }^{(g+1)}={\sigma }^{(g)}\exp ({\displaystyle \frac{c_{\mathsf{\sigma }}(‖{\mathit{p}}_{\mathsf{\sigma }}‖-\sqrt{M_{\mathrm{c}}})}{d_{\mathsf{\sigma }}\sqrt{M_{\mathrm{c}}}}})$$

(18)

where d_σ is the damping coefficient, which controls the convergence rate of the step size; c_σ is the learning rate for the evolution path of the step size; and p_σ is the evolution path of the step size, which satisfies the following relation:

$${{\mathit{p}}_{\mathsf{\sigma }}}^{(g+1)}=(1-c_{\mathsf{\sigma }}){{\mathit{p}}_{\mathsf{\sigma }}}^{(g)}+\sqrt{c_{\mathsf{\sigma }}(2-c_{\mathsf{\sigma }})N_{\mathrm{c}}}{({\mathit{C}}^{(g+1)})}^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{{\mathit{\mu }}^{(g+1)}-{\mathit{\mu }}^{(g)}}{{\sigma }^{(g)}}}$$

(19)

(7)

Stopping criterion check: If the computed results satisfy the stopping criterion, the iteration is terminated. Otherwise, the algorithm returns to Step 2 to proceed with parameter updates and continue the iterative process.

3.3. Parameter Estimation Process

The CMA-ES method is introduced into the parameter estimation for mechanical transmission system of HVCBs. The corresponding parameter estimation process is shown in Figure 5.

(1)

Based on the travel curve and vibration signal monitored online, the start of the valid time interval of the travel curve is determined by the shock signature in the vibration signal at the completion of the trip operation, while the end of this interval is identified as the time instant at which the travel curve reaches its maximum value. The time-series segment of the travel curve between these two points is then used as the input for parameter estimation.

(2)

Preliminary determination of the variation range of the parameters to be estimated is performed based on design documents or offline test results, followed by normalization.

(3)

Initialization of CMA-ES parameters (μ, σ, C) is carried out, and a population following a multivariate normal distribution is generated in the normalized parameter space. Each individual in the population is denormalized back to the physical parameter space and substituted into the dynamic model to obtain the corresponding simulated travel curve time series {θ_2,j}. A fitness function is then constructed based on the root mean square error (RMSE) between the simulated travel curve and the monitored travel curve time series {θ_2,j^*}, as follows:

$$F={\displaystyle \frac{1}{\sqrt{\frac{1}{L}{\displaystyle \sum _{j=1}^L{({{\theta }_{2,\mathrm{j}}}^{*}-{\theta }_{2,\mathrm{j}})}^2}}}}$$

(20)

where L is the length of the valid time interval of the monitored travel curve.

(4)

The fitness values of all individuals in the population are sorted, and elite individuals are selected. The CMA-ES parameters are then updated in the normalized parameter space to generate a new candidate population, for which the fitness values are recalculated. This process is repeated iteratively. After each fitness evaluation, the stopping criterion is checked. If the criterion is not met, the iteration continues. Otherwise, the individual with the highest fitness value is denormalized to obtain the parameter estimation result for this phase.

(5)

The estimated parameters are compared with the variation ranges defined in Step 2. If any estimated parameter lies at the boundary of its variation range, it indicates that the predefined parameter range may be unreasonable. In this case, the parameter variation ranges are adjusted and normalized again, and the process returns to Step 3 for further iteration.

(6)

Steps 3, 4, and 5 are executed iteratively until all estimated parameters fall within the interior of their respective variation ranges. At this point, the parameter estimation is considered complete, and the final results are output.

4. Method Validation and Analysis

4.1. Experimental Platform

An experimental test platform for the opening and closing operations of a ZF11C-252(L) HVCB, manufactured by Henan Pinggao Electric Co., Ltd. (Pingdingshan, China), was established, as illustrated in Figure 6. The platform comprises a 252 kV circuit breaker body, a spring operating mechanism, switch characteristic tester, a vibration sensor, two resistive angular displacement sensors, a data acquisition card, and an upper computer. The vibration sensor, with a measurement range of ±1000 g, is employed to capture vibration signal during the circuit breaker’s operation, which serves as the basis for determining the starting point of the valid time interval of the travel curve. Two angular displacement sensors, each with a range of 0–180 degrees, are used to measure the angular displacement of the mechanism’s output shaft and the camshaft, respectively. The data acquisition card, with a sampling rate of 128 kHz, collects signals from all sensors and transmits them to the upper computer for display and storage.

4.2. Verification of Model Parameter Identifiability

This paper adopts a statistical data-driven approach to verify the identifiability of the dynamic model parameters. The specific methodology is as follows: multiple sets of model parameters (serving as the true parameters) are randomly generated within a specified range. These parameter sets are then substituted into the dynamic model to obtain the corresponding travel curves. Subsequently, the model parameter estimation method proposed in this paper is applied to each generated travel curve to derive the estimated parameter values. Finally, the identifiability of the model parameters is verified based on statistical measures of the deviation between the estimated and the true parameter values.

Based on the design documents of the circuit breaker in the experimental platform, the ranges of the model parameters are configured as listed in

Table 2. Parameter range settings for the dynamic model.

Operation Type	Parameter Name	Symbol	Range	Unit
Opening operation	Opening spring stiffness factor	AO	[2000, 3500]	S−2
	Opening resistance factor	BO	[4000, 6000]	S−2
	Buffer damping factor	DO	[10, 50]	S−1
	Buffer stiffness factor	EO	[4000, 6000]	S−2
Closing operation	Closing spring stiffness factor	Ac	[18,000, 23,000]	S−2
	Inertia factor	Bc	[1, 3]	--
	Closing resistance factor	Cc	[8000, 12,000]	S−2

. It should be noted that the model parameter C_O, which characterizes the damping effect of the arc-extinguishing chamber, is determined by the chamber’s structure and is treated as a constant in the parameter estimation. The parameter D_C of the closing dynamic model is equal to the parameter A_O of the opening model. To improve the overall accuracy of the dynamic model parameter estimation in closing operation, the estimated value of A_O from the opening operation is used as a known value for D_C in the closing operation.

For both the opening and closing operations, 100 sets of parameter combinations were generated, from which the corresponding travel curves were derived. Model parameter estimation was then conducted on the travel curves after superimposing 1% random noise. A comparison between the estimated parameter values and the true values is presented in Figure 7 and Figure 8, which demonstrates good agreement between the two.

The evaluation metrics for the deviations between the estimated and true values of the model parameters are presented in

Table 3. Deviation index between estimated parameters and true values.

Indicators	AO	BO	DO	EO	AC	BC	CC
R2	1.00	0.997	0.997	0.992	0.995	0.990	0.991
RMSE	0.0463	29.1	0.605	56.3	90.0	0.060	104.5
MAE	0.0214	25.1	0.206	46.8	77.4	0.046	88.2
MRE	0.010%	0.498%	1.780%	0.992%	0.375%	2.640%	0.925%

.

Here, R² denotes the coefficient of determination, RMSE represents the root mean square error, MAE is the mean absolute error, and MRE stands for the mean relative error. The data demonstrate strong agreement between the estimated model parameters and the true values under ideal noise-free condition. Together with the experimental validations presented in the following sections, they support the practical identifiability of the model parameters under realistic conditions.

For further analysis of identifiability, to assess robustness under measurement noise, Gaussian white noise (SNR = 50 dB) was added to the synthetic travel curves. The CMA-ES estimator was rerun on 100 noisy curves. The R² and MRE for all parameters are summarized in

Table 4. Parameter estimation errors under noisy conditions.

Indicators	AO	BO	DO	EO	AC	BC	CC
R2	0.977	0.965	0.972	0.975	0.969	0.974	0.959
MRE	0.522%	0.885%	2.869%	1.954%	0.776%	3.553%	1.882%

. The estimation accuracy remained well within acceptable limits, indicating that the method is robust to typical noise levels.

On this basis, sensitivity analysis and parameter correlation analysis were performed. The former quantifies the influence of each parameter on the model output, while the latter assesses the coupling among parameters.

(1)

Sensitivity analysis: Each parameter was perturbed individually by ±5% around its nominal value while keeping the others fixed. The resulting output variation is recorded, and the sensitivity coefficient S_i is defined as

$$S_i={\displaystyle \frac{1}{N}}{\displaystyle \sum _{j=1}^N\left|\frac{y_j(p_i^0+\Delta p_i)-y_j(p_i^0)}{y_j(p_i^0)}\cdot \frac{p_i^0}{\Delta p_i}\right|}$$

(21)

where N is the number of sampling points in the output trajectory, y_j is the output at the j-th sampling point, and Δp is a 5% perturbation around the nominal value $p_i^0$. A larger S_i implies higher sensitivity and better identifiability.

The sensitivity analysis results are presented in

Table 5. Table of sensitivity coefficients of the stroke curve to each parameter.

Opening Process				Closing Process
AO	BO	DO	EO	AC	BC	CC
1.21	1.13	0.94	0.89	1.26	1.17	1.09

. It can be observed that, for the opening process, the sensitivity coefficients of the opening spring stiffness factor and the transmission resistance factor exceed 1, indicating high sensitivity, while the buffer damping factor and the stiffness factor yield values around 0.9, also demonstrating relatively high sensitivity. For the closing process, the sensitivity coefficients of all three parameters are greater than 1, indicating high sensitivity.

(2)

Parameter correlation analysis: To examine the coupling among the estimated parameters, the Pearson correlation coefficients among all estimated parameters were computed over 30 independent runs, and the correlation matrix is presented in Figure 9. Coefficients with an absolute value close to 1 are considered strongly correlated, indicating potential coupling that may hinder separate identification. The results show that the correlation coefficients between all parameter pairs are below 0.3, indicating weak coupling and good identifiability.

The combined results of the sensitivity and correlation analyses confirm that the key dynamic parameters in both the opening and closing processes are well identifiable and can be reliably estimated from the measured displacement curves.

Additionally, to demonstrate the advantage of CMA-ES, we compared it with particle swarm optimization (PSO), genetic algorithm (GA), differential evolution (DE), and nonlinear least-squares (NLS) methods. All algorithms were tested on the same synthetic travel curve (opening process) with 50 dB noise, and each algorithm was run 30 times. To ensure a fair and comprehensive comparison, all optimization algorithms were implemented with a unified stopping criterion: the iteration process terminates when the maximum number of iterations reaches 2000 or the change in the fitness function value falls below 1 × 10⁻⁶. The statistical indicators of each method are summarized in

Table 6. Comparison of parameter estimation errors among different methods.

Method	MRE	Runtime (s)	Function Evaluations	Standard Deviation of Fitness Function
CMA-ES	2.63%	0.69	740	1.05
PSO	4.92%	0.75	2240	1.16
GA	3.88%	0.75	2120	1.11
DE	2.79%	0.96	2160	1.09
NLS	4.61%	2.42	4615	1.13

, and the fitness convergence curves during the optimization process are presented in Figure 10, from which it can be seen that CMA-ES achieves the lowest mean relative error. Notably, CMA-ES not only provides the best estimation accuracy but also requires the fewest function evaluations and a competitive runtime. All methods exhibit comparably low standard deviations of fitness, indicating robust convergence across multiple runs. These findings demonstrate that CMA-ES outperforms the other methods in both accuracy and efficiency, making it the most suitable choice for the present parameter estimation problem.

4.3. Opening Operation

During the opening operation, the vibration signal and the angular displacement of the mechanism’s output shaft during the opening process were measured, as shown in Figure 11. The valid time interval of the travel curve was identified and is marked by the blue shaded area.

Based on the design parameters, the preliminary ranges for the model parameters to be estimated are listed in Table 1. If the initially determined range is unsuitable, it will be adaptively adjusted during the iterative process, as illustrated in Figure 3. The iteration stopping criteria were defined as follows: (a) The change in the estimated parameters between two consecutive iterations in the normalized parameter space is less than 10⁻⁸; (b) the change in the best fitness value remains below 10⁻⁸ for 20 consecutive iterations; (c) the step size falls below the minimum threshold of 10⁻⁸; (d) the maximum number of iterations is set 1000. The process terminates if any one of the above four conditions is met.

Prior to buffer engagement, the travel curve during the opening process is determined by A_O and B_O, and is independent of D_O and E_O. To enhance the focus of the model parameter estimation, the estimation process is performed in two sequential stages:

Stage 1: The time-series segment of the travel curve before buffer engagement is used as the input for parameter estimation, during which A_O and B_O are estimated.

Stage 2: Using the results from Stage 1 as known conditions, the entire valid time series of the travel curve is taken as the input to estimate D_O and E_O.

The iterative process of dynamic parameter estimation in both stages is illustrated in Figure 12. In Stage 1, the iteration terminated at the 36th step when the change in estimated parameters between two consecutive iterations reached 5.9 × 10⁻⁹, meeting the stopping criterion. In Stage 2, the process stopped at the 35th iteration with a parameter change of 6.9 × 10⁻⁹. Throughout the iterations, the progressive increase in the fitness value reflects the ongoing optimization of the parameters. The step size factor exhibited a gradual decrease, indicating a transition from global exploration to local refinement in the parameter space. The variation in parameter estimates further confirms the refinement and convergence of the optimization process.

The estimated parameters for the opening dynamic model are listed in

Table 7. Estimation results of dynamic parameters for the opening process.

Parameter Name	Symbol	Estimated Result	Unit
Opening spring stiffness factor	AO	3378	S−2
Opening resistance factor	BO	4794	S−2
Buffer damping factor	DO	43.8	S−1
Buffer stiffness factor	EO	3832	S−2

. Substituting these estimated parameters into the opening dynamic model yields the corresponding travel curve. A comparison between this simulated curve and the measured curve is shown in Figure 13.

The maximum error between the two curves is less than 0.007 rad, and the RMSE is 0.0028 rad. Building upon the “one-to-one correspondence” relationship between model parameter sets and travel curves demonstrated in Section 4.2, the consistency between the two travel curves indicates a strong agreement between the estimated parameter values and their true values.

4.4. Closing Operation

The vibration signal, camshaft angular displacement, and output shaft angular displacement of the mechanism during the closing operation of the circuit breaker were measured, as shown in Figure 14. Based on these signals, the valid time interval of the travel curve was determined, indicated by the blue shaded area in the figure. The motion transmission relationship of the cam mechanism, derived from the camshaft angular displacement and the output shaft angular displacement of the mechanism, is illustrated in Figure 15.

It should be noted that the simultaneous measurement of camshaft angular displacement and mechanism output shaft angular displacement is intended to establish the kinematic relationship of the cam mechanism, thereby enabling the solution of the closing dynamic model. In practical engineering applications, only the output shaft angular displacement needs to be monitored online. Combined with the offline-established cam kinematic relationship, the method proposed in this paper can be effectively implemented in practice.

The preliminary ranges of the model parameters to be estimated, determined based on design parameters and engineering experience, are listed in

Table 8. Estimation results of dynamic parameters for the closing process.

Parameter Name	Symbol	Estimated Result	Unit
Closing spring stiffness factor	AC	22,650	S−2
Inertia factor	BC	2.46	--
Closing resistance factor	CC	11,731	S−1

. The setting of the optimization stopping criteria is the same as the opening process. The iterative process of parameter estimation for the closing dynamic model is shown in Figure 16.

The convergence metrics of the parameter estimation for the closing dynamic model are as follows: the iteration terminated at step 143 when the variation range of the estimated parameters between two consecutive iterations in the normalized space reached 3.0 × 10⁻⁹; the fitness value gradually increased and eventually stabilized at 108; the step size factor progressively decreased and converged to 6.81 × 10⁻⁸; the evolution of estimated parameters in the parameter space further reflects the refinement and convergence of the optimization process.

The estimated parameters for the closing dynamic model are summarized in Table 6. The simulated travel curve obtained using these estimated parameters is compared with the measured travel curve in Figure 17. The maximum error between the two curves is below 0.031 rad, with an RMSE of 0.0083 rad. Similarly, the consistency between the two travel curves indicates a strong agreement between the estimated parameter values and their true values.

4.5. Multicondition Validation

To further validate the effectiveness of the parameter estimation method for the opening and closing dynamic model, experiments under multiple operating conditions were conducted.

The opening operation conditions were configured as follows: six distinct conditions were established by adjusting the pre-compression of the opening spring, with a 3.5 mm increase in pre-compression between adjacent conditions. This adjustment was achieved by rotating the spring pressure plate fixing nut by two turns. According to Equation (12), parameter B_O should theoretically increase by 150 s⁻² between adjacent conditions, while the other parameters are expected to remain unchanged. The travel curves for each condition were tested separately, and parameter estimation was performed. The results are shown in Figure 18.

Based on the above comparison, the following conclusions can be drawn:

(1)

As the spring pre-compression increases, the model parameter B_O increases—a trend that aligns with theoretical expectations. As the transmission resistance in B_O is difficult to measure accurately, the estimated value under condition 1 is taken as the baseline. A reference vector for B_O is constructed with a step size of 150 s⁻². The maximum deviation of the estimated B_O from the reference vector is 2.7%.

(2)

Using the average of the estimated values of A_O across different conditions as the reference value, the maximum deviation of the estimated A_O relative to the reference value is 3.6%. Similarly, the estimated value of D_O exhibits a maximum deviation of 6.3% from the reference value.

(3)

The buffer stiffness factor E_O shows a clear and regular trend: it gradually increases with the rise in spring pre-compression. This behavior can be explained as follows: under high-speed impact motion, the stiffness effect of the buffer is primarily reflected in the compressibility of the oil. In the experiment, as the spring pre-compression increases, the motion speed of the transmission system rises, enhancing the oil compression effect. This, in turn, leads to a more pronounced stiffness behavior of the buffer, manifesting as an increase in E_O.

In practical engineering, the main failure mode of the buffer is oil leakage, which leads to a reduction in damping performance. Therefore, the variation in D_O can be used for diagnosing oil leakage in the buffer, while E_O is not considered as an indicator for buffer diagnosis.

The closing operation conditions were set as follows: using the same method, the pre-compression of the closing spring was adjusted to establish a total of five conditions. According to Equations (7) and (8), the increase in compression is reflected only in the model calculation process but not in the extracted model parameters, meaning that all parameters should theoretically remain unchanged. The travel curves for each condition were tested separately, and parameter estimation was performed. The results are shown in Figure 19.

Similarly, using the average of the estimated values across different conditions as the reference value, the estimated values of A_C, B_C, and C_C exhibit maximum deviations of 2.1%, 3.0%, and 4.3% from their reference values, respectively.

The multicondition parameter estimation results indicate that the maximum deviation between the estimated and reference values for the opening model parameters is 6.3%, observed in the buffer damping factor D_O, while the maximum deviation for the closing model parameters is 4.3%, observed in the transmission resistance factor C_C.

It should be noted that the synthetic identifiability study in Section 4.2 primarily demonstrates internal consistency under ideal model conditions, while the experimental results provide evidence of practical identifiability under realistic conditions including measurement noise and minor model imperfections. Additionally, the parameter estimation error is the combined result of actual measurement noise, algorithmic optimization error, and model assumption approximation error. This deviation is acceptable in engineering practice, which further demonstrates that the influence of the linear motion and constant inertia assumptions on the parameter estimation accuracy is limited within the applicable scope of this study, and that the model assumptions are reasonable. These results further validate the effectiveness of the proposed method for estimating dynamic parameters of the mechanical transmission system in the HVCB.

To provide a direct baseline comparison, the proposed method is evaluated against three representative feature-based classifiers: support vector machine (SVM), back propagation neural network (BPNN), and convolutional neural network (CNN). The dataset used for comparison is obtained from experimental tests under five operating conditions, corresponding to different compression levels of the opening spring. For each condition, 40 stroke curves are recorded, yielding a total of 200 samples. To ensure fair comparison and statistical robustness, fivefold cross-validation is adopted for all classifiers due to the limited dataset size. The dataset was randomly partitioned into five equal subsets; in each fold, four subsets were used for training and the remaining one for testing.

For the baseline methods, the raw sampling points of the stroke curve are used as input features. For the proposed method, the classification is performed based on the estimated parameter vector. For each of the five operating conditions, a nominal parameter vector is predefined to represent the typical mechanical characteristics of that condition. Given an estimated parameter vector obtained from a test sample, the normalized Euclidean distance between this vector and each nominal parameter vector is computed. The normalization is applied to eliminate the influence of different units and magnitudes among parameters, ensuring that each parameter contributes equally to the distance. The test sample is then assigned to the operating condition whose nominal parameter vector is closest to the estimated one. This classification rule is physically interpretable, and, thus, the distance-based decision is both meaningful and transparent. The average accuracy over the five folds is summarized in

Table 9. Classification accuracy comparison of different methods.

Method	Proposed Method	SVM	BPNN	CNN
Accuracy	98.3%	95.0%	93.3%	95.0%

. The proposed method achieves an accuracy of 98.3%, outperforming SVM, BPNN and CNN, further validating its effectiveness and interpretability advantage over black-box classifiers.

5. Conclusions

This study investigates the dynamic modeling and parameter estimation method for the mechanical transmission system of HVCBs, and demonstrates its effectiveness through experimental validation and comparative analysis. The main conclusions are as follows:

(1)

The dynamic model of the opening and closing system was established, constructing a physical description between model parameters and the travel curve. The idea of using model parameters to characterize mechanical system states was proposed, enhancing the physical interpretability of the estimated parameters.

(2)

A model parameter estimation method for HVCBs was developed. By estimating the model parameters using the travel curve and CMA-ES, accurate estimation of key dynamic parameters (e.g., stiffness, damping, inertia) was achieved. In the studied cases, the maximum estimation error of the model parameters was 6.3%.

Compared with conventional methods, the proposed approach provides accurate estimates of key dynamic parameters without requiring extensive complete-state data, thereby offering substantial practical value for engineering applications.