Risk Assessment Model for Converter Transformers Based on Entropy-Weight Analytic Hierarchy Process

Abstract

As a critical component in voltage–current conversion and power transmission within HVDC systems, the risk assessment of converter transformers plays a significant role in ensuring their operational safety and enhancing the reliability of the power supply. To address the issues of the incomplete characteristic parameters and limited fault data used for model training in existing transformer evaluation models, this paper develops a risk assessment model for converter transformers based on the entropy-weighted analytic hierarchy process (AHP). Firstly, in accordance with relevant standards in the power industry and existing experimental research, 14 ‘electrical–thermal–mechanical’ multi-dimensional characteristic parameters, including partial discharge, dissolved gases in oil, and hot spot temperature rise, are selected to effectively reflect the risk state of converter transformers. The risk state is then categorized into four levels. Next, the AHP, which uses a subjective weighting method, is combined with the entropy-weight method, an objective weighting approach, to construct the risk assessment model for converter transformers based on the entropy-weighted AHP. Finally, the effectiveness of the model is validated through four case studies of converter transformers. The results indicate that the risk assessment model proposed in this paper can accurately and effectively reflect the risk state of transformers at different levels, providing valuable guidance for the development of maintenance strategies for converter transformers.

1. Introduction

As a core component in the voltage–current conversion and power transmission of HVDC systems, maintaining the converter transformer in good health is crucial for ensuring the safe operation of the power system and a stable power supply [1,2,3]. During actual operation, the insulation components of the valve-side winding, outlet device, and internal wiring experience irreversible degradation due to factors such as electrical stress, thermal stress, and mechanical stress [4,5,6]. A failure of the converter transformer can severely disrupt the normal operation of the power system and lead to significant economic losses [7]. It is of great importance to extend the transformer’s service life and ensure the safety of the power grid by scientifically acquiring operational parameters and comprehensively evaluating its risk level, detecting anomalies in time, and implementing appropriate maintenance measures [8,9,10].

Currently, maintenance methods in China for transformer risk states are still based on a combination of scheduled and post-maintenance. However, there is a lack of real-time monitoring of the transformer’s risk state, leading to issues such as ‘under-repair’ or ‘over-repair’, which result in insufficient transformer reliability or the excessive waste of human and material resources [11,12]. In recent years, with the rise and development of wireless sensor technology, condition-based maintenance methods for transformers, based on condition monitoring technology, are gradually being implemented in power systems. By monitoring characteristic parameters such as dissolved gas in oil, top oil temperature, and vibration signals, the transformer’s state can be assessed in real time, preventing failures and maintenance risks due to improper maintenance [13,14,15,16]. As a necessary prerequisite for condition-based maintenance, the condition assessment of the converter transformer enables the determination of its operational capacity by evaluating its risk state, thereby helping to formulate a reasonable maintenance plan to ensure its stable operation.

Due to its unique operating characteristics and numerous indicators, the converter transformer requires the development of a comprehensive and scientific index system to ensure accurate risk assessment [17]. As a typical oil-immersed transformer, the converter transformer has been the subject of extensive research, both domestically and internationally, on oil-immersed transformer state evaluation methods. Jiang Tao et al. used characteristic parameters, such as transformer load coefficient, dissolved gas in oil, and partial discharge, combined with health index theory, to construct a three-level health evaluation model based on ontological data, service data, and correction factors, achieving a scientific evaluation of the power transformer’s insulation state [18]. Zou Yang et al. extracted characteristic parameters such as the polarization spectrum peak voltage, insulation resistance, and geometric capacitance, employing a combination weighting method that integrated gray correlation analysis and the improved analytic hierarchy process (AHP). They used the cloud model’s atomization characteristics to account for the fuzziness and randomness of the classification boundaries in the evaluation index, constructing a model for evaluating the transformer oil–paper insulation state, addressing the issue of sensitivity differences in insulation state reflection, and avoiding data loss [19]. Zhou Jian fused state characteristic parameters such as dissolved gas, insulation resistance, and micro-water in oil to build a transformer state evaluation model based on a support vector classification mechanism, enhancing the evaluation accuracy [20]. Based on dissolved gas in oil data, Ming Tao et al. measured the relative proximity of data in different transformer states according to the point density criterion and big data clustering principles, constructing a transformer state evaluation model based on a time series, which could effectively assess transformer state changes over time [21]. P. Zhang et al., using dissolved gas data, constructed the phase space of the model using the delayed coordinate method and calculated phase space parameters through the C-C method. By analyzing the convergence of the correlation integral, they developed an optimal data-mining-based transformer state evaluation model that improved the evaluation speed [22]. T. Manoj et al. combined AHP and gray relational analysis to construct a transformer health status assessment model based on parameters such as dissolved gas in oil, oleic acid value, and breakdown voltage, with high accuracy in the results [10]. Z. Hamed et al. developed transformer health status assessment models using multiple linear regression, adaptive neural networks, and artificial neural networks, based on parameters such as dissolved gas in oil, degree of polymerization, and dielectric loss factor. Actual case verification demonstrated the high evaluation accuracy of all three models [23]. In general, existing evaluation models can provide more accurate assessments of oil-immersed transformers, but the number of characteristic parameters used is limited, and comprehensive evaluation remains difficult with inherent uncertainty. Computationally intensive machine learning approaches fail to meet real-time monitoring requirements. Mathematically driven models over-rely on expert knowledge and are prone to misjudgment. Transformer health status assessment methods based on data-driven approaches and machine learning require sufficient training data. However, due to the complex structure, harsh onsite environment, and strong magnetic fields of converter transformers, it is challenging to collect data from different states, leading to lower accuracy. Furthermore, due to factors such as sensor aging, transformer monitoring systems may exhibit data distribution shifts during prolonged operation, requiring continuous model updates to maintain accuracy.

To address the above issues, this paper integrates the ‘electrical–thermal–mechanical’ multi-dimensional characteristic parameters of electricity, heat, and mechanics, proposing a risk assessment model for converter transformers based on the entropy-weight AHP. Firstly, 14 characteristic parameters that effectively reflect the risk state of the transformer are selected based on relevant standards and experimental studies, and the risk states of the transformer are classified. Then, using the AHP, a judgment matrix is constructed to determine the subjective weights of the index layer. Following the entropy-weight method, the risk decision matrix is developed, and the evaluation model is objectively weighted. Finally, by combining the AHP with the entropy-weight method, the comprehensive weight of the evaluation model is obtained, revealing the internal correlations and importance rankings of the state parameters of the converter transformer. Based on the established evaluation model, the risk state of the transformer is assessed.

2. Model Development and Validation Framework

Due to the varying effects of different ‘electric–heat–machine’ characteristic parameters on reflecting the risk state of converter transformers, it is necessary to assign weights to these parameters to ensure the rationality of the risk assessment model. Based on the weight distribution and assignment methods, existing weighting techniques can be categorized into subjective and objective methods. Common subjective methods include the AHP [24,25], expert evaluation, and Delphi method. These techniques take into account the evolution mechanism and law of each characteristic parameter; however, these methods heavily depend on expert experience, which can lead to issues of contingency and poor stability [26,27]. On the other hand, objective weighting methods include the entropy-weight method [28,29], coefficient of variation method, and principal component analysis. These methods have a solid data and theoretical foundation; however, they rely entirely on existing statistical data and are prone to misjudging index weights [30,31].

To address these issues, this study first establishes classification thresholds for different risk levels of the ‘electrical–thermal–mechanical’ multi-dimensional characteristic parameters in converter transformers based on relevant standards from the power industry and existing experimental research. Subsequently, the analytic hierarchy process is employed to construct a hierarchical judgment matrix based on the relative importance of these multi-dimensional parameters, deriving the maximum eigenvalue λ_max and the corresponding normalized eigenvector ω_j′. Concurrently, using the entropy-weight method and the classification thresholds for different risk levels of multi-dimensional characteristic parameters, a risk decision matrix is developed to obtain the normalized deviation matrix μ_j. Then, the AHP and entropy-weight method are integrated to create a risk assessment matrix model for converter transformers. Finally, the model’s effectiveness is validated using actual characteristic parameter monitoring data from converter transformers. The model development and validation framework are illustrated in Figure 1.

This paper combines the AHP from the subjective weighting method with the entropy-weight method from the objective weighting method to establish a converter transformer risk assessment model, integrating both expert experience and existing test data to comprehensively assess the risk state of transformers and improve the reliability and accuracy of the model.

3. Selection of State Characteristic Parameters and Classification of Risk Grades

The risk state of the converter transformer is influenced by electrical, thermal, mechanical, and other factors, each of which exerts a different impact on the risk state of the transformer. To conduct a more scientific and comprehensive analysis of the risk level of the converter transformer, this study selects 14 ‘electrical–thermal–mechanical’ multi-dimensional characteristic parameters, including partial discharge, power frequency capacitance change rate, dissolved gas in oil, relative gas production rate, hot spot temperature rise, and thermal defect temperature, which effectively reflect the risk state of the converter transformer. These parameters are based on relevant standards from the power industry and existing experimental research [32,33,34,35]. The service life and rate of loading effectively indicate the aging degree and current operational intensity of converter transformers. Hot spot temperature rise and thermal defect temperature serve as critical indicators for identifying overheating faults and cooling system failures. The vibration enhancement factor reflects the mechanical stability of the transformer structure. Insulation safety margin and insulation risk failure level provide comprehensive evaluations of the insulation system’s endurance capabilities. Acetylene content, total hydrocarbon gas content, and the relative gas generation rate are essential parameters for detecting faults and assessing oil–paper insulation degradation. Partial discharge quantity is a vital indicator for internal insulation defects. The oil dielectric loss factor and power frequency capacitance change rate directly reflect the deterioration of the insulation medium’s performance. Winding DC resistance differences can reveal poor electrical contact and winding strand breakage issues. The selected parameters comprehensively cover the characteristics of various indices of the converter transformer under different risk states, making them highly significant for the accurate and thorough assessment of the risk state.

The proposed risk assessment model for converter transformers requires only four operational parameters: service life, vibration enhancement factor, insulation safety margin, and insulation failure risk level, which are to be obtained through testing and threshold classification. The remaining parameters are derived from industry standards, with their risk thresholds determined based on practical transformer maintenance experience. The model minimizes data requirements, significantly reducing acquisition costs.

Given that the influence of different electrical, thermal, and mechanical characteristic parameters on the risk state of the converter transformer varies, and their sensitivity in diagnosing different defective transformers differs, this paper performs a comprehensive evaluation of the weighting values of these parameters. The evaluation is conducted in accordance with the “Guidelines for State Evaluation of Oil-immersed Transformers (Reactors)” [32]. The risk state of the transformer is classified into four levels: normal, attention, abnormal, and serious. The threshold values for the multi-dimensional characteristic parameters of the converter transformer under different risk levels are provided in

Table 1. Classification thresholds for different risk levels of multi-dimensional characteristic parameters.

Characteristic Parameter Index	Normal State	Attention State	Abnormal State	Serious State
Service Life K1	700	500	250	150
Rate of Loading K2	1	1.1	1.3	1.5
Hot Spot Temperature Rise K3/K	60	68	78	90
Vibration Enhancement Factor K4	1	1.2	1.4	1.5
Insulation Safety Margin K5	0.8	0.7	0.6	0.5
Insulation Risk Failure Level K6	1	2	3	4
Thermal Defect Temperature K7/°C	150	300	500	700
Acetylene Content K8/μL/L	2	5	20	50
Total Hydrocarbon Gas Content K9/μL/L	100	150	350	500
The Relative Gas Production Rate K10/%/month	7	10	20	30
Partial Discharge Quantity K11/pC	100	250	700	1000
Oil Dielectric Loss Factor tanδ K12/%	0.5	2	3	4
Power Frequency Capacitance Change Rate K13/%	2	3	4	5
Winding DC Resistance Difference K14/%	1	2	4	5

. The specified thresholds for characteristic parameters used in transformer risk assessment differ between domestic [32,33,34,35] and international standards [36,37,38]. Without modifying the classification thresholds for different risk levels of multi-dimensional characteristic parameters, the model cannot directly evaluate the risk state of converter transformers in different countries. Therefore, to enable the assessment of converter transformer risk state across various countries, it is necessary to redefine the classification thresholds for different risk levels of multi-dimensional characteristic parameters according to the relevant standards.

4. The Converter Transformer Risk Assessment Model Based on the Entropy-Weight AHP

4.1. Construction of Hierarchical Judgment Matrix Based on AHP

Firstly, the hierarchical structure of the evaluation system is established. When analyzing objects with multiple characteristics, it is essential to first construct the logical relationships for evaluating these objects. In the case of a redundant model without embedded features, the hierarchy is typically divided into three levels: the target level, the index level, and the decision level. Based on this logic, the hierarchical structure of the converter transformer risk assessment model is constructed, as shown in Figure 2.

The hierarchical judgment matrix A is then constructed. Based on expert experience, the relative importance and correlation between the characteristic parameters in the index layer are assessed, and the judgment matrix A is formed, as shown in Equation (1).

$$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1j}\\ a_{21}& \cdots & \cdots & \cdots \\ ⋮& ⋮& \ddots & ⋮\\ a_{i1}& \cdots & \cdots & a_{ij}\end{array}\right]$$

(1)

Here, i denotes the feature vector in the ith row, j represents the feature vector in the jth column, and a_ij indicates the relative importance of the feature vector in the ith row compared to the one in the jth column.

Since the evaluation model established in this study involves 14 characteristic parameters, i = j = 14. The 1–5 scale method, similar to the 1–9 scale method, is used to indicate the relative importance between different feature vectors, thereby constructing the judgment matrix A. The meaning of each scale is outlined in

Table 2. Meaning of each scale.

Scale	Implication
1	The two feature vectors have the same importance
2	The ith row feature vector is more important than the jth column feature vector
3	The ith row feature vector is much more important than the jth column feature vector
4	The ith row feature vector is extremely important compared to the jth column feature vector
5	The ith row feature vector is the most important compared to the jth column feature vector
1/aij	The feature vector of column j is more important than the feature vector of row i

. Pairwise comparisons are made for each feature vector to effectively reflect the importance of different indicators.

The eigenvalues of the judgment matrix A are first calculated. Using Equation (2), the maximum eigenvalue λ_max of matrix A and its corresponding eigenvector ω_j are determined and normalized, obtaining the normalized eigenvector ω_j′.

$$A={\lambda }_{max}{\omega }_j$$

(2)

Subsequently, the consistency test of matrix A is performed. The consistency index CI and consistency ratio CR of matrix A are calculated according to Equations (3) and (4).

$$CI={\displaystyle \frac{{\lambda }_{max}-n}{n-1}}$$

(3)

$$CR={\displaystyle \frac{CI}{RI}}$$

(4)

Here, RI denotes the average random consistency index, and n represents the order of the matrix, which corresponds to the number of feature vectors. The values of RI for different values of n are shown in

Table 3. Classification thresholds for different risk levels of ‘electrical–thermal–mechanical’ multi-dimensional eigenparameters.

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14
RI	0	0	0.52	0.89	1.12	1.24	1.36	1.41	1.46	1.49	1.52	1.54	1.56	1.58

[39]. A smaller CR indicates a higher reliability of matrix A. When CR < 0.1, the CR value of matrix A is considered acceptable, suggesting that the logical structure of matrix A meets the required standards. At this point, the initial subjective weight of the matrix is represented by ω after normalization. If CR > 0.1, the logical structure of matrix A is deemed unsatisfactory, and the matrix must be revised until CR < 0.1.

4.2. Construction of Risk Decision Matrix Based on Entropy-Weight Method

To assess the variability of different characteristic parameters in the evaluation model, the degree of influence of each characteristic parameter on the overall outcome is evaluated. First, the primary risk decision matrix Y = x_mj is constructed, as shown in Equation (5).

$$Y=\left(\begin{array}{cccc}{x}_{11}& x_{12}& \cdots & x_{1j}\\ x_{21}& x_{22}& \cdots & x_{2j}\\ ⋮& ⋮& \ddots & ⋮\\ x_{m1}& x_{m2}& \cdots & x_{mj}\end{array}\right)$$

(5)

where m represents the risk level.

Next, the optimal value for each characteristic parameter is defined, and the risk characteristic decision matrix Z = x_mj′ is constructed. The data for each characteristic parameter in matrix Z is normalized. For parameters where higher values are better, such as the insulation safety margin, x_mj* = max x_mj, and x_mj′ = x_mj/x_mj*; for parameters where smaller values are better, such as total hydrocarbon gas content, x_mj* = min x_mj, and x_mj′ = x_mj*/x_mj. The normalized values of each characteristic parameter in matrix Z fall within the range [0, 1]. The higher the value of a characteristic parameter, the better it is, with the optimal value being equal to 1.

Subsequently, the risk characteristic decision matrix Z = x_mj′ is standardized to eliminate any ambiguity in the weight assignment of each characteristic parameter and to improve computational efficiency. Without altering the correlation of the characteristic parameters, the data are transformed into real numbers, b_mj, within the range [0, 1], to reflect the relative importance of each characteristic parameter. The standardized risk characteristic decision matrix B is shown in Equation (6).

$$B=\left(\begin{array}{cccc}{b}_{11}& b_{12}& \cdots & b_{1j}\\ b_{21}& b_{22}& \cdots & b_{2j}\\ ⋮& ⋮& \ddots & ⋮\\ b_{m1}& b_{m2}& \cdots & b_{mj}\end{array}\right)$$

(6)

After the construction of the risk decision matrix is completed, the information entropy matrix E_j for each characteristic parameter is computed, as shown in Equation (7), to determine the influence of different characteristic parameters on the risk state of the converter transformer.

$$E_j={\displaystyle \frac{{\displaystyle \sum _{m=1}^4{b}_{mj}\ln b_{mj}}}{\ln m}}$$

(7)

Here, when b_mj = 0 or b_mj = 1, it is defined that b_mjlnb_mj = 0.

Next, the deviation matrix d_j for each characteristic parameter is calculated, as shown in Equation (8).

$$d_j=1-E_j$$

(8)

Finally, the deviation matrix d_j for each feature parameter is normalized, resulting in the normalized deviation matrix μ_j, as shown in Equation (9). At this point, μ_j represents the objective weight of the decision matrix constructed using the entropy-weight method.

$${\mu }_j={\displaystyle \frac{d_j}{{\displaystyle \sum _{j=1}^n{d}_j}}}$$

(9)

4.3. Comprehensive Weight Calculation Based on Subjective and Objective Fusion

Firstly, the μ_j values of the risk decision matrix B in the entropy-weight method are corrected using the λ_max values from the hierarchical judgment matrix A in the AHP, along with their corresponding ω_j′. The correction deviation θ_j is then computed, as shown in Equation (10).

$${\theta }_j={\displaystyle \frac{{\mu }_j{\omega }_j\text{'}}{{\displaystyle \sum _{j=1}^n{\mu }_j{\omega }_j\text{'}}}}$$

(10)

Subsequently, the weight assignment model is further optimized, and the comprehensive weight ξ_j for the converter transformer risk assessment model using the entropy-weight AHP is derived, as shown in Equation (11).

$${\xi }_j={\displaystyle \frac{\lambda {\theta }_j\times (1-\lambda ){\mu }_j}{{\displaystyle \sum _{i=1}^n\lambda {\theta }_j\times (1-\lambda ){\mu }_j}}}$$

(11)

Since the entropy-weight method is sensitive to the extreme values of characteristic parameters, while the analytic hierarchy process over-relies on expert judgment, λ = 0.5 is set to counteract the respective biases of both methods and ensure their balanced contributions in the integrated assessment model.

5. Research on Risk Assessment of Converter Transformers

Firstly, based on Equation (1), the hierarchical structure of the converter transformer risk assessment system, as depicted in Figure 1, is established. Using the 1–5 scale method, 14 ‘electrical–thermal–mechanical’ multi-dimensional characteristic parameters, including partial discharge, power frequency capacitance change rate, dissolved gas in oil, relative gas production rate, hot spot temperature rise, and thermal defect temperature, which effectively reflect the risk state of the converter transformer, are compared and analyzed. The judgment matrix A is constructed, as shown in Equation (12). In Equation (12), the characteristic parameters K₁–K₁₄ are arranged from left to right and top to bottom. The intersection of the horizontal and vertical parameters indicates the relative importance of the ith row parameter compared to the jth column parameter. For example, the matrix values at the 1st row and 2nd column, 3rd row and 4th column, and 6th row and 1st column illustrate this relationship. Specifically, a value of 1/2 indicates that the rate of loading K₂ is more important than the service life K₁, a value of 3 signifies that the hot spot temperature rise K₃ is much more critical than the vibration enhancement factor K₄, and a value of 5 demonstrates that the insulation risk failure level K₆ is the most important compared to the service life K₁.

$$A=\left(\begin{array}{cccccccccccccc}1& 1/2& 1/3& 1& 1/2& 1/5& 1/5& 1/5& 1/4& 1/4& 1/5& 1& 1/3& 1/3\\ 2& 1& 1/2& 2& 1& 1/4& 1/4& 1/4& 1/3& 1/3& 1/4& 2& 1/2& 1/2\\ 3& 2& 1& 3& 2& 1/3& 1/3& 1/3& 1/2& 1/2& 1/3& 3& 1& 1\\ 1& 1/2& 1/3& 1& 1/2& 1/5& 1/5& 1/5& 1/4& 1/4& 1/5& 1& 1/3& 1/3\\ 2& 1& 1/2& 2& 1& 1/4& 1/4& 1/4& 1/3& 1/4& 1/4& 3& 1/2& 1/2\\ 5& 4& 3& 5& 4& 1& 1& 1& 2& 2& 1& 5& 3& 3\\ 5& 4& 3& 5& 4& 1& 1& 1& 2& 2& 1& 5& 3& 3\\ 5& 4& 3& 5& 4& 1& 1& 1& 2& 2& 1& 5& 3& 3\\ 4& 3& 2& 4& 3& 1/2& 1/2& 1/2& 1& 1& 1/2& 4& 3& 3\\ 4& 3& 2& 4& 3& 1/2& 1/2& 1/2& 1& 1& 1/2& 4& 2& 2\\ 5& 4& 3& 5& 4& 1& 1& 1& 2& 2& 1& 5& 3& 3\\ 1& 1/2& 1/3& 1& 1/2& 1/5& 1/5& 1/5& 1/4& 1/4& 1/5& 1& 1/3& 1/3\\ 3& 2& 1& 3& 2& 1/3& 1/3& 1/3& 1/2& 1/2& 1/3& 3& 1& 1\\ 3& 2& 1& 3& 2& 1/3& 1/3& 1/3& 1/2& 1/2& 1/3& 3& 1& 1\end{array}\right)$$

(12)

The maximum eigenvalue, λ_max, of matrix A is 14.3328, and the corresponding eigenvector is ω_j = (0.0661, 0.1021, 0.1641, 0.0661, 0.1067, 0.4272, 0.4272, 0.4272, 0.2890, 0.2661, 0.4272, 0.0661, 0.1641, 0.1641)^T. The eigenvector is normalized, and the normalized feature eigenvector, ω_j′ = (0.0046, 0.0071, 0.0114, 0.0046, 0.0074, 0.0298, 0.0298, 0.0298, 0.0202, 0.0186, 0.0298, 0.0046, 0.0114, 0.0114)^T, is obtained.

The consistency test of matrix A is then conducted. Given that n = 14, it follows from Table 3 that RI = 1.58 and CR = 0.0162 < 0.1, as calculated using Equations (3) and (4). Therefore, the logical consistency of the judgment matrix A meets the necessary requirements, and ω_j′ can be adopted as the initial weight in the evaluation model.

Next, based on Table 1 and employing the entropy-weight method, the primary risk decision matrix Y is constructed, as shown in Equation (13). Subsequently, the data for each characteristic parameter in matrix Y is processed on a per-unit value basis, resulting in the establishment of the risk characteristic decision matrix Z, as shown in Equation (14). Finally, matrix Z is standardized, producing the standardized risk characteristic decision matrix B, as shown in Equation (15).

$$Y=\left(\begin{array}{cccccccccccccc}0.0014& 1.0& 60& 1.0& 1.2500& 1& 150& 2& 100& 7& 100& 0.0050& 2& 1\\ 0.0020& 1.1& 68& 1.2& 1.4286& 2& 300& 5& 150& 10& 250& 0.0200& 3& 2\\ 0.0040& 1.3& 78& 1.4& 1.6667& 3& 500& 20& 350& 20& 700& 0.0300& 4& 4\\ 0.0067& 1.5& 90& 1.5& 2.0000& 4& 700& 50& 500& 30& 1000& 0.0400& 5& 5\end{array}\right)$$

(13)

$$Z=\left(\begin{array}{cccccccccccccc}1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\ 0.7143& 0.9091& 0.8823& 0.8333& 0.8750& 0.5000& 0.5000& 0.4000& 0.6667& 0.7000& 0.4000& 0.2500& 0.6667& 0.5000\\ 0.3571& 0.7292& 0.7692& 0.7143& 0.7500& 0.3333& 0.3000& 0.1000& 0.2857& 0.3500& 0.1429& 0.1667& 0.5000& 0.2500\\ 0.2143& 0.6667& 0.6667& 0.6667& 0.6250& 0.2500& 0.2143& 0.0400& 0.2000& 0.2333& 0.1000& 0.1250& 0.4000& 0.2000\end{array}\right)$$

(14)

$$B=\left(\begin{array}{cccccccccccccc}0.0714& 0.0714& 0.0714& 0.0714& 0.0714& 0.0714& 0.0714& 0.0714& 0.0714& 0.0714& 0.0714& 0.0714& 0.0714& 0.0714\\ 0.0812& 0.1033& 0.1003& 0.0947& 0.0995& 0.0568& 0.0568& 0.0455& 0.0758& 0.0796& 0.0455& 0.0284& 0.0758& 0.0568\\ 0.0617& 0.1329& 0.1329& 0.1234& 0.1210& 0.0576& 0.0518& 0.0173& 0.0494& 0.0605& 0.0247& 0.0288& 0.0864& 0.0432\\ 0.0466& 0.1449& 0.1449& 0.1449& 0.1358& 0.0543& 0.0466& 0.0087& 0.0434& 0.0507& 0.0217& 0.0272& 0.0869& 0.0435\end{array}\right)$$

(15)

The values of E_j, d_j, and μ_j for each characteristic parameter, as calculated using matrix B, are shown in Equation (16).

$$E_j=\left(\begin{array}{c}-0.5100\\ -0.7005\\ -0.6977\\ -0.6851\\ -0.6882\\ -0.4863\\ -0.4672\\ -0.3177\\ -0.4824\\ -0.5127\\ -0.3627\\ -0.3533\\ -0.5828\\ -0.4497\end{array}\right)\hspace{1em}{d}_j=\left(\begin{array}{c}1.5100\\ 1.7005\\ 1.6977\\ 1.6851\\ 1.6882\\ 1.4863\\ 1.4672\\ 1.3177\\ 1.4824\\ 1.5127\\ 1.3627\\ 1.3533\\ 1.5828\\ 1.4497\end{array}\right)\hspace{1em}{\mu }_j=\left(\begin{array}{c}0.0709\\ 0.0798\\ 0.0797\\ 0.7913\\ 0.7927\\ 0.0698\\ 0.0689\\ 0.0619\\ 0.0696\\ 0.0710\\ 0.0640\\ 0.0635\\ 0.0743\\ 0.0681\end{array}\right)$$

(16)

Subsequently, the AHP and the entropy-weight method are combined to compute ξ_j and θ_j, as shown in Equation (17).

$${\theta }_j=\left(\begin{array}{c}0.0214\\ 0.0372\\ 0.0597\\ 0.0238\\ 0.0386\\ 0.1360\\ 0.1343\\ 0.1206\\ 0.0918\\ 0.0862\\ 0.1247\\ 0.0192\\ 0.0556\\ 0.0510\end{array}\right)\hspace{1em}{\xi }_j=\left(\begin{array}{c}0.0046\\ 0.0123\\ 0.0317\\ 0.0051\\ 0.0133\\ 0.1879\\ 0.1855\\ 0.0167\\ 0.0858\\ 0.0742\\ 0.1723\\ 0.0041\\ 0.0292\\ 0.0270\end{array}\right)$$

(17)

Finally, based on the risk characteristic decision matrix Z and the comprehensive weight ξ_j, the risk assessment matrix model Q of the converter transformer, incorporating all characteristic parameters, is calculated, as shown in (18).

$$Q={\displaystyle \sum _{j=1}^{n}Z}\cdot {\xi }_j$$

(18)

The calculated risk assessment model for the converter transformer is Q = (1, 0.5239, 0.2813, 0.2048)^T. The four values of matrix Q categorize the risk state of the converter transformer into four levels: normal, attention, abnormal, and critical. The values in the matrix Q are [0, 1]. A value of 0 indicates that the characteristic parameters of the converter transformer fall below the critical threshold, implying that the transformer is at a serious risk level and should be immediately shut down for maintenance. A value of 1 indicates that the characteristic parameters of the converter transformer exceed the normal threshold, meaning the transformer is in a completely normal state and can continue normal operations.

6. Verification of Risk Assessment Effects of Converter Transformers

To verify the effectiveness of the risk assessment model for converter transformers based on the entropy-weight analytic hierarchy process proposed in Section 4, four representative converter transformers were selected from converter stations across multiple provincial grids, with their operational data serving as validation case studies. The monitoring data for the characteristic parameters are presented in

Table 4. Characteristic parameter monitoring data.

Parameter	Transformer A	Transformer B	Transformer C	Transformer D
K1	800	667	476	118
K2	1	1	1.2	1.5
K3	53	62	75	98
K4	1	1	1.1	1.3
K5	0.80	0.78	0.63	0.47
K6	1	1	2	4
K7	0	87	462	885
K8	0.53	1.38	15.26	187.05
K9	87.04	134.20	267.42	961.99
K10	4	6	13	45
K11	27.47	74.88	296.57	1332.45
K12	0.5	0.5	2	3
K13	2	2	3	5
K14	1	1	2	5

.

The evaluation criteria of the risk assessment model proposed in this study were applied to score the individual characteristic parameters of the selected transformers. Following the risk characteristic decision matrix value calculation method described in Section 4.2, the normal state values of each characteristic parameter from Table 1 were selected as optimal values. The characteristic parameter monitoring data of the four converter transformers in Table 4 were normalized, yielding the individual characteristic parameter scoring presented in

Table 5. Individual characteristic parameter scoring.

Parameter	Transformer A	Transformer B	Transformer C	Transformer D
K1	1	0.9524	0.6803	0.1681
K2	1	1	0.8333	0.6667
K3	1	0.9677	0.8333	0.6122
K4	1	1	0.9091	0.7692
K5	1	0.9690	0.7911	0.5952
K6	1	1	0.5000	0.2500
K7	1	0.58	0.3247	0.1695
K8	1	1	0.1311	0.0107
K9	1	0.7452	0.3739	0.1040
K10	1	1	0.5385	0.1556
K11	1	1	0.3372	0.0750
K12	1	1	0.2500	0.1667
K13	1	1	0.6667	0.4000
K14	1	1	0.5000	0.2000

.

By combining Equations (17) and (18) and Table 5, the comprehensive scores of the four converter transformers are calculated, as presented in

Table 6. Aggregate scoring.

Transformer Type	Transformer A	Transformer B	Transformer C	Transformer D
Comprehensive score	1	0.8986	0.3953	0.1717

.

The aggregate scoring of the four converter transformers from Table 6 were input into the risk assessment model Q = (1, 0.5239, 0.2813, 0.2048)^T developed in Section 4, enabling the determination of their current risk states. The comprehensive score results of different converter transformers reveal that Transformer A, which had recently been put into operation, had a score of 1, indicating a completely normal risk state. Transformer B, which had been in operation for several years, showed signs of internal insulation degradation, with its score falling between normal and attention, warranting enhanced monitoring. Transformer C, with winding defects, had a score between attention and seriousness, requiring timely testing and repair. Transformer D, exhibiting severe winding deformation, had a score indicating a state below serious risk, necessitating immediate overhauling. The risk assessment model aligns with the actual conditions of converter transformers at varying severity levels. It not only retains expert knowledge but also incorporates data-driven theory, ensuring high accuracy.

7. Conclusions

Considering the limitations of the current risk assessment model for converter transformers, which is based on a relatively simple set of characteristic parameters and limited training data, a comprehensive evaluation of the state of the transformer remains unachieved. Building on existing standards and research, this study analyzes and selects 14 multi-dimensional ‘electrical–thermal–mechanical’ characteristic parameters that effectively reflect the risk state of converter transformers. Using the entropy-weight AHP, a risk assessment model for converter transformers is developed. The main conclusions are as follows:

• This paper selects 14 ‘electrical–thermal–mechanical’ multi-dimensional characteristic parameters, such as partial discharge, power frequency capacitance change rate, dissolved gas in oil, relative gas production rate, hot spot temperature rise, and thermal defect temperature, to comprehensively evaluate the risk state of the converter transformer, ensuring the comprehensiveness of the assessment model.
• By combining the subjective weighting method from the AHP with the objective weighting method from the entropy-weight method, the risk assessment model based on the entropy-weight AHP effectively addresses the issues of over-reliance on expert experience or statistical data in existing models, thereby enhancing the reliability and accuracy of the model, while simultaneously satisfying the requirements for both real-time online monitoring and continuous model updating.
• The effectiveness of the model risk assessment is validated using actual operational data from the converter transformer. The results indicate that the proposed risk assessment model can accurately assess the risk state of the transformer by providing comprehensive protection for HVDC systems, spanning from equipment-level anomaly detection to system-wide stability maintenance. The model exhibits particularly robust capabilities in predicting and preventing risks under complex operating conditions, such as commutation failures and harmonic resonance, enabling the proactive mitigation of potential failures.