FAULT DIAGNOSIS ALGORITHM OF ELECTRIC VEHICLE GEARBOX BASED ON SDEA-BI GRU

This paper suggests a hybrid method that combines the strengths of a bidirectional gated recurrent unit with a stacked denoising autoencoder to enhance the precision and effectiveness of diagnosing transmission faults in electric vehicles. The bidirectional gated recurrent unit makes advantage of these deep features for efficient fault pattern identification and classification. The results revealed that the hybrid algorithm had the best feature extraction ability for gear fault signals, and the signal features extracted by the algorithm were more concentrated and crossed each other less. The neurons in the hidden layer of the stacked denoising autoencoder was 180, and the number of neurons in the bidirectional gated recurrent unit was 160, and the hybrid algorithm performed best when the neurons in the hidden layer was 180 and the neurons in the bidirectional gated recurrent unit was 160. The hybrid algorithm performed best when the number of neurons was 160. The hybrid algorithm had the highest diagnostic accuracy for the faults, with the highest diagnostic accuracy of 97.98% in the balanced samples and 94.86% in the unbalanced samples. The hybrid algorithm constructed in the study effectively improves the diagnostic accuracy of transmission gear faults in electric vehicles..


INTRODUCTION
Electric vehicles have drawn a lot of attention as an alternative form of transportation in light of the growing severity of the energy crisis and environmental pollution.Because of their clean and efficient energy use, electric vehicles have also emerged as a key trend in the development of the contemporary automotive industry [1].However, electric vehicle as a complex mechatronic product, in which the stability and safety of the electric gearbox, as a key component connecting the motor and the traveling system, are critical to the overall vehicle performance [2].Accurate and timely identification of possible transmission defects has become a must to guarantee the regular functioning of electric vehicles, and fault diagnosis is crucial for enhancing the operational safety and dependability of electric vehicles [3].Currently, fault diagnosis techniques for electric vehicle transmission mainly focus on traditional methods such as vibration and sound analysis, but these methods usually require specialized personnel to operate, and the diagnostic accuracy is insufficient in the changing operating environment [4].The use of SDEA and bidirectional gated recurrent unit (Bi-GRU) in the proposed algorithm enhances the accuracy and efficiency of Zhao L, Wu T: Fault Diagnosis Algorithm of Electric Vehicle Gearbox Based on SDEA-Bi GRU.
2 experimental validation of the SDEA-Bi-GRU model.And finally, an overview of the research

RELATED WORKS
Electric vehicle is currently the most compatible low-carbon and energy-saving vehicle type.To improve the effectiveness of battery fault diagnosis for electric vehicles, Li et al. proposed a new battery fault diagnosis method by combining a long and short-term memory recurrent neural network and an equivalent circuit model.The results showed that the method can achieve accurate diagnosis of potential battery faults and precise localization of thermal runaway batteries [5].In order to prevent the damage caused by over discharge of lithium-ion batteries, Gan et al. proposed a two-layer over discharge fault diagnosis strategy based on machine learning.The first layer detected over discharge by comparing the cell voltage and cutoff voltage, while the second layer used a limit gradient boosting algorithm for previous over discharge detection.The results indicated that their proposed method has good results [6].In order to diagnose switched reluctance motor bearing faults in electric vehicle power systems under variable speed conditions, Wang X et al. proposed a multi-sensor data fusion method.The method fused synchronously sampled current and vibration signals to estimate the cumulative rotor angle.The results demonstrated that the method does not require an additional tachometer and can be used for online fault diagnosis of switched reluctance motors under variable speed conditions [7].To protect the safe operation of electric vehicle batteries, Long et al. proposed a model-based fault diagnosis scheme for current and voltage sensors.By comparing the difference between the true state of charge and the estimated state of charge, the occurrence of faults was determined and faulty sensors were isolated.The results showed that the method can effectively diagnose battery faults [8].To overcome the challenges of battery fault diagnosis for electric vehicles, reduce the dependence on the amount of data and improve the diagnostic accuracy and speed, Deng et al. proposed a multi-classification support vector machine (SVM) based fault diagnosis method.The results revealed that the method improved the training speed and accuracy and achieved satisfactory results in battery fault diagnosis of electric vehicles with small sample training set [9].
Both SDEA and Bi-GRU are one kind of deep learning networks.To solve the problem of SDAE's ability to extract bearing fault feature information under multiple operating conditions and strong noise, Jia et al. proposed a bearing health monitoring and fault diagnosis model based on variational mode decomposition, continuous wavelet transform and sparrow search algorithm optimized for SDAE.The outcomes revealed that the model outperforms other methods in terms of diagnostic accuracy, generalization performance and noise immunity [10].To analyze the fault diagnosis method for rolling bearings, Che et al. addressed a SDEAFD model based on convolutional neural network (CNN).The model can denoise and dimensionalize the raw vibration signal data.The outcomes revealed that the model in improved the accuracy by 3%-13% over the traditional model and a single deep learning model [11].A approach based on data augmentation and Bi-GRU was proposed by Fu B et al.The method utilized the principal factor correlation index to select features and used Mixup technique to enhance the data.The approach enhanced the model's resilience and capacity for generalization while maintaining a high level of prediction accuracy, according to the findings [12].To solve the category imbalance problem, A weakly supervised learningbased classification technique was presented by Liu H et al.The method employed bi-directional gated recursive units to construct the fault diagnosis model, and proposed a new weighted cross-entropy function as the loss function to reduce the effect of noise.The method's efficacy and superiority were demonstrated by the experimental results [13].
In summary, fault detection of electric vehicle components is a current research focus.However, most research has been centered on automotive battery fault detection, with fewer studies on other key components.The gearbox is a crucial component of an automobile.Any failure of this component can directly impact the use of the automobile.Therefore, this study aims to detect faults in the electric vehicle transmission system.Commonly used fault detection methods include SDEA and Bi-GRU, but each has its own shortcomings.This study proposes to combine the two methods for fault detection in the gear system of electric vehicle transmission

FAULT DIAGNOSIS OF ELECTRIC VEHICLE TRANSMISSION BASED ON SDEA-BI-GRU MODELING
Gearbox gear failure is one of the most common failures in electric vehicles.Section 3 mainly focuses on the study of diagnostic methods for transmission gear failure in electric vehicles.And it is developed in two directions, the first part is the SDEA-Bi-GRU hybrid network study, and the second part is the gear fault diagnosis based on SDEA-Bi-GRU

SDEA-Bi-GRU Hybrid Networks
SDEA is a deep learning model that performs feature learning and noise reduction by stacking multiple layers of self-encoders [14][15].Selfencoder is an unsupervised learning model that encodes and decodes the input data and learns a lowdimensional representation of the input data by minimizing the reconstruction error [16].Stacked auto-encoders, on the other hand, stack multiple auto-encoders together, with the hidden layer of each auto-encoder serving as the input to the next auto-encoder.And Figure 1 depicts the particular structure. ̂=   1 () = ( 1  +  1 ) (2) In equation ( 2),   1 () denotes the decoding function and  1 denotes the decoding weights. 1 denotes the decoding bias,  denotes the decoding mapping function, and  1 denotes the decoding set parameters.The fine-tuning part means that the weights in the pre-training are used as initial parameters to adjust the parameters of SDEA, and the fine-tuning of SDEA is a supervised learning process.Recurrent neural network (RNN) is a type of traditional neural network, but traditional neural networks are not able to correlate information while processing it.Therefore, scholars proposed RNN, in common RNN, a neuron needs to receive the information processed is more complex.At the current moment , the neuron needs to receive not only the input   , but also the state ℎ −1 of the previous moment, as shown in equation (3).
ℎ  = (  ) = (  + ℎ −1 ) (3) In equation ( 3),  denotes the activation function, ℎ  denotes the state of the hidden unit at the  moment, and  denotes the weight matrix.Gated recurrent unit (GRU) is a type of RNN unit, which has good results when dealing with sequential data [17].GRU is improved on long short-term memory network (LSTM).GRU adds an update gate and a reset gate to the network.The update gate controls how the network stores data and ignores information from prior inputs, while the reset gate controls how the network merges data from previous inputs with the data that is currently being input [18].The output of the update gate is shown in equation ( 4).
= (  +     +   ℎ −1 ) (4) In equation ( 4),   denotes the update gate output and  denotes the sigmoid activation function.  denotes the input data at the current moment and ℎ −1 denotes the hidden unit information output at the  − 1 moment.The reset gate output is shown in equation (5).
The output of GRU is shown in equation (8).
) GRU is a one-way learning model, in the learning process, the last unit of the state information is missing, and the more forward, the more serious the missing information, to address this problem, scholars introduced a two-way learning mechanism in GRU.After the introduction of the two-way learning mechanism, it is Bi-GRU.Additionally, Figure 2 depicts the network's topology.(9) In equation ( 9), ℎ ⃗  denotes the forward learning result.ℎ ⃖⃗  A denotes the reverse learning result.
The gear system shown in Figure 3 is a dual transmission gear system with two clutches, an inner transmission shaft, an outer transmission shaft, and six different speed gears.In the working process of the gear system, the appearance of certain faults can cause the transmission to stop working, therefore, the detection of faults in the gear system of the transmission is an important research direction for automobile safety.the gears during the gearing process is nonstationary.This non-stationary vibration signal produces a great disturbance to the accurate diagnosis of gear failure.Since the gear's vibration system is nonlinear, study on its dynamics must simplify its dynamics equation.And the simplified dynamics equation is shown in equation (10).  ̈+ ̈+ () = () 1 + () 2  (10) In equation (10),   denotes the equivalent mass of the gear pair and  denotes the relative displacement on the gear meshing line. denotes the damping of the gears when they mesh, () denotes the gear mesh stiffness shape. 1 denotes the average static elastic deformation of the gear after being loaded. 2  denotes the relative displacement between gears that mesh with each other as a result of errors or failures that may occur in the gears.The left part of equation (10) represents the vibration characteristics of the gear in the working state, and the right part represents the dynamic loads of the gear.The dynamic loads of the gear system include the inherent vibration of meshing, and the vibration caused by the gear stiffness and gear failure, and the second vibration can be explained for various failure conditions in the gear.Since the meshing stiffness of the gears in a gear system varies with the meshing rotation of the gears, equation (11) illustrates that the meshing frequency of the gear system is equal to the frequency of change of the meshing stiffness.  =  60 (11) In equation (11),   denotes the meshing frequency of the gear and  denotes the harmonics. denotes the revolutions of the shaft where the gear is located, and  denotes the number of gear teeth.Equation (12) illustrates that the gear's vibration signal is made up of the meshing frequency and the high harmonics.() = ∑   (2  +   )  =0 (12) In equation (12), denotes the total number of gear meshing frequency orders, and   denotes the amplitude of the th order gear meshing frequency.  denotes the initial phase of the th order meshing frequency.The fault produced by the gear is the excitation source of this type of vibration, which is based on the rotation period of the gear shaft as the period.Therefore, the vibration information will contain the rotation frequency and multiplicity of the faulty gear.When the fault occurs, not only vibration will be generated, but also shocks will be generated during the meshing process, causing the signal to generate side bands, at which point, equation (12) will be rewritten as equation (13).
In equation ( 13),   () denotes the amplitude modulation function of the th order engagement frequency.  () denotes the frequency modulation function of the th order meshing frequency.The modulation frequency includes the station east frequency of the shaft where the faulty gear is located and the multiplicity, therefore, equation ( 13) can be transformed into equation (14).
In equation ( 14),  denotes the maximum modulation order.  denotes the amplitude modulation coefficient of the th order rotation frequency modulation, and G m denotes the frequency modulation coefficient of the mth order rotation frequency modulation.f n denotes the rotation frequency of the rotating shaft, and its calculation is shown in equation (15).f z = Zf n (15) In the previous paper, two network structures, SDEA and Bi-GRU, are proposed, and the study combines them to construct a hybrid SDEA-Bi-GRU network for transmission gear fault diagnosis.Model training and model testing comprise the two main components of this network's overall architecture, which is seen in Figure 4.When performing fault diagnosis, the time-domain signal of the transmission gear system is used as the input data to the SDEA structure, in which noise reduction is performed.The signal is then fed into the Bi-GRU in order to extract the signal's fault features.Ultimately, the fault signal is categorized using the Softmax classifier, allowing the diagnosis of the gear failure of electric vehicle transmission to be finished.Figure 5 depicts the precise architecture of the hybrid network.

SDEA-Bi-GRU based gear fault diagnosis
Transmission is a common basic component in automobiles with high ratios and compact structural design, it is very common in industrial applications.A transmission gear system is a mechanical system capable of varying the ratio between the drive wheels and the engine to provide different speeds and torque outputs.The main component of the gear system is a series of gears, which work in conjunction with devices such as drive shafts, main shafts and clutches.The structure of the transmission gear system is shown in Figure 3.

EXPERIMENTAL VALIDATION OF THE EFFECT OF SDEA-BI-GRU MODELING
The study constructed a hybrid SDEA-Bi-GRU neural network and used the network for transmission gear fault diagnosis of electric vehicle.The main content of this chapter is the verification of the practical effect of the model, which is divided into three parts.The model's training results are analyzed in the second part, the experimental environment and parameters are set, and the model's impact on practical application is examined in the third part.

Experimental environment and parameter settings
All the experiments of the study are completed using the equipment at the time of the experiment, the operating system is windows 7 professional, the CPU is Intel(R) Core(TM) i5-4460 CPU @ 3.20GHz 3.20 GHz, the RAM is 16GB, and all the data analyzing software is carried out on the MATLAB platform.In addition to the basic experimental environment, the study also builds a fault simulation test bed, which is shown as a 3D model in Figure 7.

Fig. 7. Gear fault simulation test bench
After completing the construction of the experimental environment, the study conducts training tests on SDEA and Bi-GRU.In the training, the input noise ratio of the SDEA structure is 0.3, the discard rate of the network structure is set to 0.1, the learning rate is set to 0.001, the maximum iterations are 600, and the decay rate is 0.95 and 0.99.The training results are shown in Figure 8.
Figure 8(a) illustrates how the neurons affects SDEA performance.The diagnostic time of SDEA for fault occurrence increases with the neurons; for example, when the neurons rises from 80 to 200, the diagnostic time of SDEA increases by around 50 minutes overall.The highest score of 94.8% is reached by SDEA's diagnostic accuracy when the buried layer has 180 neurons.The SDEA structure's diagnostic accuracy rises with the neurons when there are fewer than 180 neurons.When the neurons exceed 180, the accuracy of the diagnosis declines as the neurons rises.It takes 37 minutes to diagnose SDEA when there are 180 neurons in the brain.6 number of neurons is 160, and at this time, the diagnosis of Bi-GRU takes 45 min.The study determined that there should be 180 neurons in the hidden layer of the SDEA structure and 160 neurons in the hidden layer of the Bi-GRU structure, respectively, based on the training results.

Fault diagnosis results for SDEA-Bi-GRU models
The hybrid model constructed by the study is proposed based on SDEA and Bi-GRU, therefore, the study compares the feature extraction ability of SDEA and SDEA-BiGRU for gear failure signals, and the results are shown in Figure 9.
Figure 9(a) shows the feature extraction capability of the SDEA structure for the gear states in 6.There is a crossover between the scatter region of normal gears and mildly cracked gears, a crossover between mildly cracked and heavily cracked, a crossover between heavily cracked and broken teeth, and a crossover between broken teeth and tooth wear.In the gear failure states extracted from the SDEA structure, there is a serious crossover between the different fault feature signals and the respective scatter regions are more dispersed.Figure 9(b) shows the feature extraction capability of SDEA-Bi-GRU for six gear states.The model has a better extraction effect on gear failure states, and there is only a crossover between normal gears and mild cracks, and there is no crossover for the rest of the various types of faults, and the regional distribution of the various types of faults is more concentrated.The extraction effect of SDEA-Bi-GRU on gear failure signal features is much better than that of SDEA structure.In the simulation experiment, the study controls the speed of the gearbox so that the intensity of the vibration signal is reduced from 70 Hz to 30 Hz.And 12 different sample sets are formulated in accordance with each 10 Hz drop as a sample set, which is divided into four sample sets of D, B, A and C. Table 1 presents the findings of a comparison of the diagnostic procedures' accuracy.for SDEA, and 97.98% for SDEA-Bi-GRU.The proposed algorithm of the study is the most effective for fault diagnosis of gears.To simulate the actual situation as much as possible, the study adjusted the ratio of normal gears to faulty gears in the sample set.And Figure 10 displays the test findings.
Figure 10(a) shows the comparison of the accuracy of DBN and CNN with SDEA-Bi-GRU.CNN's diagnostic accuracy in unbalanced samples is at a better level, with a maximum diagnostic accuracy of 90.27%, while DBN's diagnostic accuracy is at a lower level, with a maximum of only 85.33%.SDEA-Bi-GRU has a diagnostic accuracy of 94.86%, which is much higher than the other two algorithms' accuracy in unbalanced samples.Figure 10(b) shows the accuracy comparison of SDAE and RNN with SDEA-Bi-GRU.The highest diagnosis accuracy of both SDAE and RNN can reach more than 90%, but it is still lower than the fault diagnosis accuracy of SDEA-Bi-GRU.In unbalanced samples, the diagnosis accuracy of SDEA-Bi-GRU model for transmission gear failure is also better than the rest of the algorithms.Because the signal-to-noise ratio (SNR) also has a significant impact on fault diagnosis, the study also examines the diagnosis accuracy of SVM using SDEA-Bi-GRU and back propagation (BP) neural networks under various SNRs.The findings are displayed in Figure 11.In Figure 11, the accuracy of the algorithms increases as the SNR increases, with the BP Neural Network showing the greatest improvement in accuracy and the SDEA Bi-GRU showing the least change in accuracy.The fault diagnosis accuracy of the BP neural network is 72.13%, the fault diagnosis accuracy of the SVM is 81.65%, the fault diagnosis accuracy of the RNN is 90.47%, the fault diagnosis accuracy of the SDEA is 95.43%, and the fault diagnosis accuracy of the SDEA-Bi-GRU is 97.68% when the SNR is 0 dB.At a SNR of 10 dB, the fault diagnosis accuracy for BP neural networks is 93.76%, the fault diagnosis accuracy for SVMs is 95.63%, the fault diagnosis accuracy for RNNs is 97.86%, the fault diagnosis accuracy for SDEA is 98.63%, and the fault diagnosis accuracy for SDEA-Bi-GRU is 99.37%.The fault diagnosis accuracy for BP neural networks increases by a total of 21.63%, the fault diagnosis accuracy for SVMs increases by a total of 13.98%, the fault diagnosis accuracy of RNN is 7.39%, the fault diagnosis accuracy of SDEA is 3.20%, and the fault diagnosis accuracy of SDEA-Bi-GRU is 1.69%.The SNR has the least effect on SDEA-Bi-GRU and the algorithm has the highest fault diagnosis accuracy.

CONCLUSION
Aiming at the fault diagnosis problem of electric vehicle transmission, the study successfully developed a hybrid algorithm based on SDEA and Bi-GRU, which is used to enhance the diagnosis of gear failure of electric vehicle transmission.The SDEA-Bi-GRU algorithm enhances the accuracy and efficiency of the gear fault diagnosis of gearboxes by efficiently combining the feature extraction capability of the denoising selfencoder with the highly efficient time-series analyzing capability of the Bi-GRU.The experimental results demonstrated that the SDEA-Bi-GRU algorithm was able to accurately extract the key features when dealing with complex transmission signals, and the hybrid algorithm extracted the fault signal feature regions that were concentrated and had less crossover between the regions.The hybrid algorithm achieved the highest fault diagnosis accuracy of 97.98% in balanced samples and 99.37% in unbalanced samples, and the remaining algorithms achieved the highest accuracy of 94.65% in balanced samples and 98.63% in unbalanced samples, and the hybrid algorithm had the best diagnostic effect on the faults.Additionally, the SNR of the input signal had minimal effect on the hybrid algorithm.As the SNR increased, the diagnostic accuracy of the hybrid algorithm fluctuated only 1.69%, while the rest of the algorithms fluctuated at least 3.20%.The hybrid algorithm demonstrated significant advantages over traditional diagnostic methods and other machine learning algorithms in terms of fault diagnosis of electric vehicle transmission.The study improves the efficiency and accuracy of fault diagnosis of electric vehicle transmissions and provides technical support for the safe operation and maintenance of electric vehicles.Future work will focus on optimizing the algorithm's performance, expanding its application to more electric vehicle critical components, and exploring its practicality in real industrial environments.

Source of funding:
The research is supported by: Huai'an Science and Technology Support Program (Industry) Project: Design and Analysis of Gear System of Pure Electric Vehicle Transmission (No. HAB202161); Jiangsu Qinglan Project (2021).
Author contributions: research concept and design, L.Z.; Collection and/or assembly of data, L.Z., T.W.; Data analysis and interpretation, T.W.; Writing the article, L.Z.; Critical revision of the article, T.W.; Final approval of the article, T.W.

Declaration of competing interest:
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .
Fig. 1.SDEA network structure In stacked noise-reducing self-coding networks, each self-encoder introduces noise to make the input data more robust, which forces the model to learn more useful and abstract features.Usually Gaussian noise or randomly interrupted input data is used as noise.Pre-training and fine-tuning are the two phases of the SDEA network training process.In the pre-training phase, signal-to-noise samples with additional noise are fed into the SDEA and mapped to the hidden layer in equation (1). =   ( ̃) = ( ̃+ ) (1) In equation (1),  ̃ denotes the input and   (⬚) denotes the encoding function.A denotes the encoding weights and  denotes the bias. denotes the mapping function,  denotes the set parameters and  denotes the output.The mapping result is subsequently decoded and reconstructed in equation (2). ̂=   1 () = ( 1  +  1 )(2) In equation (2),   1 () denotes the decoding function and  1 denotes the decoding weights. 1 denotes the decoding bias,  denotes the decoding mapping function, and  1 denotes the decoding set parameters.The fine-tuning part means that the weights in the pre-training are used as initial parameters to adjust the parameters of SDEA, and the fine-tuning of SDEA is a supervised learning process.Recurrent neural network (RNN) is a type of traditional neural network, but traditional neural networks are not able to correlate information while processing it.Therefore, scholars proposed RNN, in common RNN, a neuron needs to receive the information processed is more complex.At the current moment , the neuron needs to receive not only the input   , but also the state ℎ −1 of the previous moment, as shown in equation(3).ℎ = (  ) = (  + ℎ −1 )(3) In equation (3),  denotes the activation function, ℎ  denotes the state of the hidden unit at the  moment, and  denotes the weight matrix.Gated recurrent unit (GRU) is a type of RNN unit, which has good results when dealing with sequential data[17].GRU is improved on long short-term memory

Fig. 2 .
Fig. 2. Bi GRU structureBi-GRU uses all the information of the unit in the past and the future in a bidirectional learning mechanism to complement the features lost in the training process of the GRU.And equation (9) displays the model's output formula.ℎ = [ℎ ⃗  ⊕ ℎ ⃖⃗  ](9)In equation (9), ℎ ⃗  denotes the forward learning result.ℎ ⃖⃗  A denotes the reverse learning result.The gear system shown in Figure3is a dual transmission gear system with two clutches, an inner transmission shaft, an outer transmission shaft, and six different speed gears.In the working process of the gear system, the appearance of certain faults can cause the transmission to stop working, therefore, the detection of faults in the gear system of the transmission is an important research direction for automobile safety.The vibration signal generated by

Fig. 4 .
Fig. 4. Basic framework of fault diagnosis model Based on the basic framework of the model, the study redesigned the network structure of the SDEA-Bi-GRU model.The hybrid network structure

Fig. 5 .
Fig. 5. Hybrid network structure During the training process of hybrid networks, it is necessary to prevent neural networks from overfitting.Dropout technique is a common regularization technique used to reduce the occurrence of overfitting in neural networks.Dropout has a probability that the output of each neuron will be randomly discarded during the training process.This operation forces the neural network to learn redundant features and reduces the collaborative dependency between neurons, improving the generalization performance of the network.And Dropout can randomly disconnect connections during training and then keep all connections for prediction, reducing the neural network's dependence on colorful neurons thus preventing the hybrid network from overfitting phenomenon during training.Figure 6 depicts the Dropout network's structure.

Figure 8 (
b) illustrates how the quantity of neurons affects Bi-GRU performance.Both the changes in diagnostic time and diagnostic accuracy are similar to Figure 8(a).The diagnosis accuracy of Bi-GRU reaches the maximum value of 98.0% when the DIAGNOSTYKA, Vol. 25, No. 2 (2024) Zhao L, Wu T: Fault Diagnosis Algorithm of Electric Vehicle Gearbox Based on SDEA-Bi GRU.

Fig. 11 .
Fig. 11.The impact of signal-to-noise ratio on algorithm performance

7 Table 1 .
Comparison of diagnostic effects of different methods