It is well known that the insulation of large transformers is one of the fundamental and stringent requirements to ensure the safe and reliable operation of electric power substations. In the past decades, the timely detection and analysis of partial discharge (PD) have been widely investigated for fault detection and diagnosis of the internal insulation deterioration of power transformers (e.g., Tarimoradi and Gharehpetian, 2017; Wang et al., 2017; Chen, 2019; Ganguly et al., 2020; Karami et al., 2020). This enables the determination of fault type and location at the early stage, and hence the field maintenance can be timely carried out to prevent the power transformers from failures or outages.

In general, the partial discharge can make the insulation being destroyed and gradually expand due to the direct bombardment of the discharge particles, resulting in insulation breakdown; In addition, the chemical action of the active gases (e.g., heat, ozone and nitrogen oxide) produced by the discharge can lead to corrosion of the partial insulation, which increases the dielectric loss and finally leads to thermal breakdown (Kallberg, 1980; Naderi et al., 2007). More specifically, the partial discharge can introduce the following impacts:

1) Partial discharge can lead to the separation and cleavage of chemical bonds and the destruction of the insulating material’s molecular structure. This may introduce discharge at the concentration of the electric field and leads to dendritic discharge traces and insulation breakdown.

In addition, the high-energy radiation phenomenon during partial discharge can potentially degrade the insulating materials. The X wax (a waxy substance produced by overheating) deposited on the solid insulation makes it difficult to dissipate heat, resulting in overheating and damages to the solid insulation. The examples of insulation discharge phenomena in the transformers are illustrated in Figure 1.

At present, the partial discharge localization of power transformers is mainly based on electrical (mainly in ultra-high frequency) and ultrasonic detection methods (e.g., (Luo Yongfen et al., 2006; Moore et al., 2006; Markalous et al., 2008; Coenen and Tenbohlen, 2012; Tarimoradi and Gharehpetian, 2017)). The electrical method mainly detects the UHF (Ultra High Frequency) electromagnetic wave generated by partial discharge sources. Considering that the propagation speed of the electrical signal is the speed of light, the electrical method requires a high sampling frequency reaching the nanosecond level or even sub-nanosecond level. Moreover, the electrical wave signal is shielded and attenuated by the power trans-former borne (Kweon et al., 2005).

The ultrasonic detection method detects and analyses the arrival time of the PD pulse signal to determine the location of partial discharge sources. Due to the advantages of non-destructive, robust and high precision, the ultrasonic detection method is one of the most widely used location technologies of power transformers (Howells and Norton, 1978; Han-Lee Song, 1994; Cakir et al., 2013; Hekmati and Hekmati, 2017; Wang et al., 2017; Chen, 2019). The location solutions can be classified into the pure acoustic-based method, the pure electrical-based method and the acoustic electrical joint method (e.g., Markalous et al., 2008; Coenen and Tenbohlen, 2012; Rubio-serrano et al., 2012). The pure acoustic localization method and the pure electrical-based localization method select the arrival time of one ultrasonic sensor or UHF sensor signal as the reference time and then measure the time delay of other signals relative to the reference time for partial discharge localization. The acoustic electrical joint method is considered as the arrival time of the electric pulse signal of the partial discharge as the reference time since the electric signal delay is very tiny and can be ignored compared with the ultrasonic signals. Through using a set of ultrasonic synchronized sensors to measure the ultrasonic time delay and multiplying the equivalent sound velocity, the distance of discharge source to individual sensors can be calculated, as suggested in (Meka et al., 2018). The ultrasonic wave propagation speed within the transformer is close to that of sound wave speed, and the electromagnetic wave speed is about two-thirds of that of lightwave speed. Therefore, the time error of the pure electrical-based localization method is much larger than other methods. The existing study (Coenen and Tenbohlen, 2012) confirmed that the positioning accuracy of the electric acoustic joint method is higher than that of the pure acoustic method. Since the time arrival time of the acoustic signal is not easy to be accurately measured, a larger error can be introduced between the acoustic and acoustic signals. Moreover, for multiple PD sources, the acoustic-electrical joint method is expected to provide improved performance in terms of distinguishing the signals from different PD sources in practical deployment compared with the acoustic-based method.

The main factors affecting the accuracy of the partial dis-charge localization are the nonlinear localization equation solving method and the elimination method of various errors in the localization process. In the existing studies, many optimization algorithmic solutions, e.g., the particle swarm optimization and its extensions (Hooshmand et al., 2013; Wang et al., 2017; Meka et al., 2018), genetic algorithm (Chang et al., 2014; Li and Luan, 2018), fuzzy clustering (Contin et al., 2002; Homaei et al., 2014), have been adopted attempting to obtain the optimal solution of the nonlinear localization problem. However, there are many problems in the practical application, e.g., falling into local optimum, slow convergence speed, and premature nature. It is difficult to ensure the robustness and accuracy of the solved results. Also, the existing literature mainly focuses on the influence of sensor measurement error rather than the error caused by multi-path propagation.

Unlike our previous work (Jia et al., 2021) that presented an acoustic-based method for PD location, this paper proposes an acoustic-electrical joint method for locating the partial discharge sources in a power transformer considering the influence of the multi-path propagation effect. In this paper, the following contributions are made:

1) The proposed acoustic-electrical joint localization method fully considered the impact of multipath propagation errors to accurately describe the phenomenon of PD signal propagation in the power transformer.

The remainder of this work is as follows: the analysis of the partial discharge propagation path in the transformer is presented in Analysis of Partial Discharge Propagation Path in Transformer. SOCP Localization Model formulates the PD location problem and adopts the SOCP to solve the localization model. Performance Evaluation and Numerical Result carries out a range of experiments to validate the proposed method. Finally, the conclusions and future work are discussed in Conclusive Remarks.

When the internal medium of the power transformer is affected by dampness, aging, breakdown, or other reasons, the power transformer may cause insulation weakness points. When the applied voltage exceeds the weakness point thresh-old voltage, it will emit electric and ultrasonic signals. Due to the complexity of the internal structure of the power transformer, the propagation process of electric and ultrasonic signals can be divided into four categories, as shown in Figure 2.

1) Internal reflection process of power transformer: when the ultrasonic wave touches the winding or borne, it will be reflected, thus prolonging the time delay to reach the sensor;

Suppose synchronized one electric PD sensor and n ultrasonic PD sensors are arranged around the power transformer. The time delay between the measurement sensor and the PD source can be formulated as Eq. 1: t i = r i v = 1 v ( d i + n i + e i ) (1)

In Eq. 1, r i is the measured distance from the PD to the i th sensor ( i = 1,2 , ⋯ , n ) . v is the ultrasonic wave propagation velocity within the transformer. d i is the real distance between the i th sensor and the PD source (all variables units in this paper are in meters). n i denotes the sensor measurement error that follows the normal distribution N ( 0 , σ 2 ) (Chan and Ho, 1994) with the mean value of 0 and variance of σ 2 , and | n i | ≪ d i . e i is the error caused by the velocity affected by the composite path in the process of signal propagation. r i = t i v (2) d i = ‖ x − s i ‖ (3)

In Eq. 3, x and s i denotes the three-dimensional coordinates of the PD source and the i th sensor, i.e. ( x x , y x , z x ) and ( x s i , y s i , z s i ) , respectively. Here, Figure 2 illustrates the relationship between measurement distance and real distance.

By square the two sides of Eq. 1 and substituting formulas Eqs 2, 3, r i 2 − 2 r i e i + e i 2 − ‖ x − s i ‖ 2 = 2 n i ‖ x − s i ‖ + n i 2 (4)

Considering | n i | ≪ d i , n i 2 can be omitted as a high-order small quantity, n i ≈ r i 2 − 2 r i e i + e i 2 − ‖ x − s i ‖ 2 2 ‖ x − s i ‖ (5)

So, the localization model can be formulated as min x max e i ∑ i = 1 N ( r i 2 − 2 r i e i + e i 2 − ‖ x − s i ‖ 2 2 ‖ x − s i ‖ ) 2 (6)

Eq. 6 can be reformed as: min x ∑ i = 1 N [ max e i ( r i 2 − 2 r i e i + e i 2 − ‖ x − s i ‖ 2 2 ‖ x − s i ‖ ) ] 2 (7)

Considering 0 ≤ e i ≤ ρ i , which ρ i is the upper bound of e i . max e i ( r i 2 − 2 r i e i + e i 2 − ‖ x − s i ‖ 2 2 ‖ x − s i ‖ ) = max ( r i 2 − ‖ x − s i ‖ 2 2 ‖ x − s i ‖ , r i 2 − 2 r i ρ i + ρ i 2 − ‖ x − s i ‖ 2 2 ‖ x − s i ‖ ) (8)

Define ξ i : ξ i = r i 2 − 2 r i e i + e i 2 − ‖ x − s i ‖ 2 2 ‖ x − s i ‖ (9)

So, according to SCOP, Eq. 7 can be rewritten as min x , ξ ∑ i = 1 N ξ i ( r i 2 − 2 r i ρ i + ρ i 2 − ‖ x − s i ‖ 2 2 ‖ x − s i ‖ ) 2 ≤ ξ i ( r i 2 − ‖ x − s i ‖ 2 2 ‖ x − s i ‖ ) 2 ≤ ξ i (10)

Considering ‖ x ‖ 2 is a non-convex parameter, the second-order relaxing parameter is defined as ‖ x ‖ 2 ≤ y . So ‖ x − s i ‖ 2 can be converted to y − 2 s i x + ‖ s i ‖ 2 and the following can be obtained: min x , ξ ∑ i = 1 N ξ i ( r i 2 − 2 r i ρ i + ρ i 2 − y + 2 s i x − ‖ s i ‖ 2 ) 2 4 ( y − 2 s i x + ‖ s i ‖ 2 ) ≤ ξ i ( r i 2 − y + 2 s i x − ‖ s i ‖ 2 ) 2 4 ( y − 2 s i x + ‖ s i ‖ 2 ) ≤ ξ i ‖ x ‖ 2 ≤ y (11)

Here, Eq. 11 is a convex second-order cone program (SOCP)and the PD source localization can be implemented through solving Eq. 11 using the toolbox (CVX toolbox) in MATLAB (ver. R2019b).

This work presented an acoustic-electrical joint PD source localization solution that fully considered the multi-path propagation effect within the transformers. The SOCP algorithm is exploited and designed for PD source localization. The developed method has been extensively assessed and validated through simulations, testbed and field deployment. The obtained experimental results clearly demonstrated the effectiveness of the developed method and its benefit over the existing CHAN algorithm and PSO-based localization solution with the localization error of about 0.1 m.

For future work, a set of directions are considered worth further research exploitation. The proposed method needs to be evaluated through extensive experiments considering the cases of multiple partial discharge sources. Also, the advanced modeling techniques need to be further investigated for accurate characterization of multi-media refraction and diffraction.