Wind Turbine Planetary Gearbox Research Articles

An optimized stacked diagnosis structure for fault diagnosis of wind turbine planetary gearbox

Fault diagnosis of the planetary gearbox (PGB) of wind turbines (WTs) plays an important role in the normal operation of WTs. Current studies commonly focus on the diagnosis of fault types of WT PGBs. Nevertheless, in addition to identifying the fault type, the current severity of the fault is also instructive for the maintenance and repair of WT PGBs. Thus, a novel optimized stacked diagnosis structure (OSDS) is proposed for the identification of fault type and severity. Compressed sensing is adopted to implement the compressed sampling of original vibration signals collected by the wireless sensor. Then, the compressed samples are input into first- and second-layer deep belief networks (DBNs) for the separate identification of fault type and severity. In order to realize the best feature extraction performance of DBNs, every single DBN in the OSDS is optimized with the chaotic quantum particle swarm optimization (CQPSO) algorithm. For OSDS, which is a hierarchical diagnosis system, the misdiagnosis results of the first layer will bring irreversible influence to the diagnosis of the second layer. That is to say, an incorrect fault type diagnosis will mean that these signals are wrongly classified, making them unable to judge the severity of the fault. Because the first-layer DBN is optimized with PGB historical data and the CQPSO algorithm, it shows an excellent performance in identifying fault types. Therefore, the diagnostic performance of OSDS has not been affected by the absence of diagnosis, and still shows an excellent recognition performance of fault type and severity in the experiment. This verifies its excellent role in the fault diagnosis of WT PGBs.

Wind Turbine Planetary Gearbox Condition Monitoring Method Based on Wireless Sensor and Deep Learning Approach

Condition monitoring for wind turbine (WT) planetary gearbox is of great significance to the reliable operation of WT. A novel condition monitoring method for WT planetary gearbox is proposed in this article. This method can identify and predict the fault in the initial stage. Raw vibration signals are gathered from WT planetary gearbox through an enhanced piezoelectric self-powered wireless sensor. First, after the data processing back-end receives the original samples transmitted by the wireless sensor, the high-dimensional raw vibration signals are compressed and collected by adopting a random Bernoulli matrix to obtain the compressed samples containing the characteristics of the raw signals by compressed sensing (CS). Moreover, the operation of reducing the signal dimension not only eliminates noise pollution but also greatly reduces the overall calculation. Second, the training part of compressed samples is first exploited to optimize the deep belief network (DBN) by the chaotic quantum particle swarm optimization (CQPSO) algorithm. In the optimization process, the CQPSO algorithm can avoid local optimal problem better compared with traditional quantum particle swarm optimization (QPSO) algorithm, which ensures that the optimized DBN architecture can extract distinguishing and deep features from compressed samples. Furthermore, the choice of compression ratio (CR) is realized in this process with the least-squares support vector machine (LSSVM) classifier. Third, testing samples are input into the DBN structure. A regression layer added on the last hidden layer, which stores extracted features, achieves the prediction of fault. In addition, LSSVM is exploited to distinguish fault types of features. The experimental results show that the modified wireless sensor can collect and transmit signals stably and achieve better performance in transmission interval and energy management. More importantly, the experimental results of prediction and diagnosis show that the proposed approach achieves excellent performance in terms of condition monitoring for the WT planetary gearbox.

Condition Monitoring of Wind Turbine Planetary Gearboxes Under Different Operating Conditions

Abstract Digitally enhanced services for wind power could reduce operations and maintenance costs as well as the levelized cost of energy. Therefore, there is a continuous need for advanced monitoring techniques, which can exploit the opportunities of internet of things and big data analytics, revolutionizing the future of the energy sector. The heart of wind turbines is a rather complex epicyclic gearbox. Among others, extremely critical gearbox components, which are often responsible for machinery stops, are the rolling element bearings. The vibration signatures of bearings are rather weak compared to other components, such as gears, and as a result, an extended number of signal processing techniques and tools have been proposed during the last decades, focusing toward accurate, early, and on time bearing fault detection with limited false alarms and missed detections. Envelope analysis is one of the most important methodologies, where an envelope of the vibration signal is estimated usually after filtering around a frequency band excited by impacts due to the bearing faults. Different tools, such as Kurtogram, have been proposed in order to accurately select the optimum filter parameters (center frequency and bandwidth). Cyclic spectral correlation (CSC) and cyclic spectral coherence (CSCoh), based on cyclostationary analysis, have been proved as very powerful tools for condition monitoring. The monitoring techniques seem to have reached a mature level in case a machinery operates under steady speed and load. On the other hand, in case the operating conditions change, it is still unclear whether the change of the monitoring indicators is due to the change of the health of the machinery or due to the change of the operating parameters. Recently, the authors have proposed a new tool called improved envelope spectrum via feature optimization-gram (IESFOgram), which is based on CSCoh and can automatically select the filtering band. Furthermore, the CSCoh is integrated along the selected frequency band leading to an improved envelope spectrum (IES). In this paper, the performance of the tool is evaluated and further extended on cases operating under different speeds and different loads. The effectiveness of the methodology is tested and validated on the National Renewable Energy Laboratory (NREL) wind turbine gearbox vibration condition monitoring benchmarking dataset, which includes various faults with different levels of diagnostic complexity as well as various speed and load operating conditions.

Adaptive data-driven nonlinear synchro squeezed transform with single class radial basis function kernel support vector machine applied to wind turbine planetary gearbox fault diagnostics

Planetary stage gears operated at low rotational speed and varying wind speed result variation in load. Variable speed and variable load induce nonstationary operating conditions. Vibration signal measured from Wind power gear transmission systems are embedded with multiple sources of vibration and attenuated considerably as it travels from source of vibration to measuring point. Efficacious multi-component decomposition without mode mixing ensures the accurate fault signature recognition. Synchro squeezing transform is the promising tool that represents the ridges with high resolution in time as well as in frequency axis. An efficient vibration analysis technique, short windowed Fourier synchro squeezing transform with nonlinear radial basis function kernel support vector machine is proposed to detect the mechanical faults in low speed planetary stage of wind turbines. Raw vibration is modeled in time–frequency plane to extract fault pattern signatures effectively with high resolution by adapting an empirical nonlinear tool synchro squeezing transforms. Amplitude modulation and frequency modulation parameters are sculpted from instantaneous amplitude and instantaneous phase, frequency. Hybrid feature space with signal attributes, statistical moments, and randomness measures are extricated from amplitude modulation-frequency modulation components. Single class radial basis function support vector machine is trained with hybrid features. The fault detection accuracy of the proposed method is compared with the standard variants of empirical mode decomposition. The proposed short windowed Fourier synchro squeezing transform-radial basis function kernel support vector machine shows 98.2% accuracy, 98% sensitivity, and 98% specificity.

Vibration modulation sidebands mechanisms of equally-spaced planetary gear train with a floating sun gear

Abstract The sun gear of planetary gear train is generally designed to float in the radial direction to achieve the uniform load distribution among different planet gears, but brings periodic gear mesh errors to generate more complicated vibration signals. There lack effective feature indicators for fault diagnosis at present. On the basis of researches on signal transfer path functions and measuring-direction projection functions of the gear mesh force, a set of vibration signal models are built to represent the modulation sidebands induced by a single planet gear mesh. Taking phase differences among equally-spaced planet gears into consideration, vibration modulation sidebands generating from meshes of multiple planet gears are further studied. The mechanisms of sidebands are obtained, especially the influences of floating sun gear on some specific frequency components. Vibration signals of a single-stage planetary gear train and a wind turbine planetary gearbox show good frequency consistencies with the model-based signal. The comparison analyses effectively verify the correctness of the proposed model. An important feature indicator is obtained to distinguish whether the planetary gear train is assembled with a floating sun gear, or whether the planetary gear train without a floating sun gear exists some distributed defects.

Time-Frequency demodulation analysis via Vold-Kalman filter for wind turbine planetary gearbox fault diagnosis under nonstationary speeds

Abstract Wind turbine planetary gearbox fault diagnosis under nonstationary speeds is a challenging topic, because of the high complexity and strong time variability of vibration signals. In order to resolve time-varying gear fault features, a quality time-frequency analysis method is in demand. Time-frequency representations based on Hilbert transform and analytic signal approach have fine time-frequency resolution, and are free from both outer (cross-term) and inner (auto-term) interferences, thus providing an effective approach to nonstationary signal analysis. However, they rely on accurate instantaneous frequency estimation, and thereby are subject to mono-component constraint. To address this issue, Vold-Kalman filter is exploited to construct time-frequency representation, by virtue of its capability to decompose the multi-component vibration signal of rotating machinery into constituent mono-component harmonic waves. Even so, intricate time-varying sidebands inherent with raw planetary gearbox vibration signals in joint time-frequency domain are still a hurdle, because they do not link to gear fault frequency directly. To solve this problem, the proposed time-frequency analysis method is further extended to generate time-varying amplitude and frequency demodulated spectra, inspired by the fact that gear fault frequency is manifested straight by the amplitude and frequency modulating frequencies. The proposed method is illustrated by numerical simulation, and is further validated using lab experimental signals of a wind turbine planetary gearbox. Both the localized and distributed faults on gears are successfully diagnosed under nonstationary speeds.

Vibration based condition monitoring and fault diagnosis of wind turbine planetary gearbox: A review

Abstract As one of the most immensely growing renewable energies, the wind power industry also experiences a high failure rate and operation & maintenance cost. Therefore, the condition monitoring and fault diagnosis of a wind turbine (WT) generator set are highly needed. Among different components of a WT generator set, WT planetary gearbox plays a crucial role in transmission and leads to relatively higher failure rate and longer downtime. Towards this, a number of studies have been reported in both the academic journals and conference proceedings. This paper provides a systemic and pertinent state-of-art review on WT planetary gearbox condition monitoring techniques on the topics of fundamental analysis, signal processing, feature extraction, and fault detection. Moreover, a few valuable open issues are pointed out and potential research directions are suggested.

Dynamics modeling and vibration modulation signal analysis of wind turbine planetary gearbox with a floating sun gear

In wind turbine planetary gearboxes, the sun gears are generally designed to float in the radial direction to achieve the uniform load distribution among different planet gears, but will lead to gear mesh errors and cause complicated vibration signals on the contrary. The floating sun gear is misdiagnosed as a distributed defect easily. Additionally, the floating sun gear's rotating motion and excitation force are too disorderly to be characterized by mathematical or empirical formulas. In the paper, a set of rigid-flexible coupling dynamics models of planetary gear train are built to research vibration characteristics of the floating sun gear under different conditions. Depend on the model-based vibration analysis, the rotating motion of floating sun gear is revealed well, which can be characterized by its frequency features, meanwhile mechanisms of modulation sidebands indicating different kinds of defects are deduced clearly. The experimental data keep good consistency with the simulation signals, which verifies the validity of the proposed well. Vibration feature indicators are obtained to distinguish fault signals and diagnose the distributed or localized defect of floating sun gear effectively, which is helpful to avoid the misdiagnosis of floating sun gear.

A tacholess order tracking method for wind turbine planetary gearbox fault detection

Wind turbine (WT) usually works under a poor condition, its operating speed varies almost all the time, therefore, the transmission mechanism of planetary gearbox tends to occur fault to a large extent, and an effective fault detection method for WT planetary gearbox is urgently needed. However, the traditional fault detection methods which are based on constant operating speed assumption will be invalid in such a complex situation, this paper provides a novel tacholess order tracking method based on generalized demodulation (GD) for WT fault detection. Firstly, the conventional GD is modified with dual path optimization ridge estimation (DPORE) strategy, and the main innovative idea of the proposed method is that the phase reference information is obtained from the generator shaft vibration signal through GD and Hilbert transform rather than the gearbox vibration signal. Then the raw gearbox vibration signal could be resampled with uniform angle. Finally, the envelope order spectrum is obtained and the fault characteristic order (FCO) related to the WT planetary gearbox fault can be identified without auxiliary sensors. The effectiveness of the proposed method is demonstrated by real-world WT vibration signals, compound-faults on planetary gearbox can be effectively uncovered and a better performance is obtained when compared with the conventional method.

Acoustic Emission Source Localization in Ring Gears from Wind Turbine Planetary Gearboxes

Condition monitoring and fault diagnosis are methods to achieve a higher reliability for complex mechanical systems. Due to the ability of acoustic emission (AE) technology to detect damages during at an early stage, AE has high potential to be considered in novel machine condition monitoring systems. Since AE waves are generated due to rapid releases of strain energy, such as crack propagation, roughness contact and mechanical impacts, it is possible to use multiple AE sensors to track down the AE source location. This paper presents an accurate and easy-to-use methodology to locate AE sources in ring gears from planetary gearboxes. The methodology uses AE wave’s time of arrival calculated from three signal differential envelopes. The time of arrival is subsequently used to analytically determine the angular position of the AE source. The proposed methodology was tested in two different real-sized wind turbine gearboxes using Hsu-Nielsen sources. The results showed that, with the correct selection of filter cut-off frequencies, the localization could be carried out with an error equivalent to only two teeth from the correct position, indicating the good performance of the approach.

Self-Powered Wireless Sensor for Fault Diagnosis of Wind Turbine Planetary Gearbox

This paper proposes a wind turbine planetary gearbox (PGB) fault diagnosis method based on a self-powered wireless sensor. The proposed wireless sensor, which consists of a piezoelectric energy harvester, a power management circuit, a microcontroller unit (MCU), a radio-frequency (RF) module, and an accelerometer, can acquire the vibration signals of wind turbine PGB by the accelerometer. The piezoelectric energy harvester utilizing vibration environment is optimized as a power supply for the proposed wireless sensor, including the MCU, RF module, and accelerometer. An ac–dc converter combined with a low-dropout voltage regulator is developed to provide stable dc voltage for the proposed wireless sensor. Stacked denoising autoencoder (SDAE) shows excellent performance in learning robust features from the noised signal. Thus, in this paper, the SDAE method is adopted to learn robust and distinguishable features from measured signals. Then, the least squares support vector machine (LSSVM) is employed to classify features extracted by the SDAE. Both the SDAE and LSSVM are optimized by quantum particle swarm optimization (QPSO). The experimental results show that the presented power supply can generate 3.3-V dc voltage, which ensures regular operation of the rest of the wireless sensor. The proposed wireless sensor can achieve a reliable communication distance of 40.8 m in the test environment. Furthermore, the SDAE approach and LSSVM show excellent performance in feature extraction and fault diagnosis, respectively. The experimental results indicate that the proposed method is effective in terms of fault diagnosis for the wind turbine PGB.

Wind Turbine Planetary Gearbox Fault Diagnosis Based on Self-Powered Wireless Sensor and Deep Learning Approach

In this paper, a novel and more effective fault diagnosis approach for wind turbine planetary gearbox (PGB) is proposed. In order to better detect the faults in the early stage of the faults of the wind turbine PGB, the corresponding maintenance measures can be carried out to prevent the faults from becoming more serious, so as to seriously affect the normal operation of the fan gearbox. The gear with lighter fault degree is used to simulate the early fault signal. Compared with the fault which has seriously affected the normal working condition, the fault characteristics of the early fault signal are more difficult to detect. So in this design, deep belief network (DBN) optimized by quantum particle swarm optimization (QPSO) algorithm is used to extract deeper and more identifiable features of slight fault signal. After optimization by QPSO algorithm, DBN can get a most suitable structure according to the actual working signal of fan gearbox. Then these extracted features are input into the least squares support vector machine (LSSVM) optimized by QPSO for fault diagnosis test. At the same time, the wireless sensor nodes using self-energy in vibration state are optimized. By using microcontroller unit (MCU) MSP430F149 and nRF24L01 radio frequency (RF) chip with lower energy consumption, the normal dormant state can be maintained, the power requirement of transmission mode can be met, the stability of the whole node can be improved, and the phenomenon of energy shortage caused by short-term fluctuation can be prevented. The comparative experiments in this paper show that this method has good effect on the fault diagnosis of wind turbine PGB.

Automated Fault Detection of Wind Turbine Gearbox using Data- Driven Approach

Wind turbine manufacturers have adopted condition monitoring systems to monitor and report a turbine is health and operating parameters to ensure that the system operates within its design specifications. While the present systems use specialized condition monitoring hardware to detectabnormal acoustic or vibration signals, it is not capable of pinpointing the exact location of the fault apart from isolating the system from which the signal originated. This drawback can be attributed to the requirement of powerful signal processors in order to decode the signal and efforts to train a system to identify the signal emitted by a faulty component. In the light of recent advancement of datadriven approaches and signal processing, these drawbacks can be overcome with increased computation power and sophisticated algorithms that foray into every integratedsystem. This paper reports such an investigation conducted on a miniature wind turbine planetary gearbox subjected to multi-component failures. The vibration signals were acquired using two accelerometers placed inside the gearbox. The speed of the gearbox was varied according to asimulated wind flow pattern. The primary goal of the study was to investigate the practicality of implementing datadriven approaches to categorise multi-component faults from a composite non-stationary signal. Short time Fourier transforms ( STFT ) coefficients were used as attributes by aset of data-driven algorithms to build machine learning models. Each model built was tested with a randomised set of instances which was reserved from the main dataset and tested multiple times by means of cross validation. The novelty in the paper entails a methodology which has been devised to classify faults using a randomised vibration dataset with little human intervention by means of machine learning algorithms. The authors propose that this methodology can also be used for real-time fault detection and classification for various machinery and components.

Meshing frequency modulation assisted empirical wavelet transform for fault diagnosis of wind turbine planetary ring gear

Abstract Condition monitoring and fault diagnosis for wind turbine gearbox is significant to save operation and maintenance costs. However, strong interferences from high-speed parallel gears and background noises make fault detection of wind turbine planetary gearbox challenging. This paper addresses the fault diagnosis for wind turbine planetary ring gear, which is intractable for traditional spectral analysis techniques, since the fault characteristic frequency of planetary ring gear can be resulted from the revolving planet gears inducing modulations even in healthy conditions. The main contribution is to establish an adaptive empirical wavelet transform framework for fault-related mode extraction, which incorporates a novel meshing frequency modulation phenomenon to enhance the planetary gear related vibration components in wind turbine gearbox. Moreover, an adaptive Fourier spectrum segmentation scheme using iterative backward-forward search algorithm is developed to achieve adaptive empirical wavelet transform for fault-related mode extraction. Finally, fault features are identified from envelope spectrums of the extracted modes. The simulation and experimental results show the effectiveness of the proposed framework for fault diagnosis of wind turbine planetary ring gear. Comparative studies prove its superiority to reveal evident fault features and avoid the ambiguity from the planet carrier rotational frequency over ensemble empirical mode decomposition and spectral kurtosis.

Wind turbine planetary gearbox feature extraction and fault diagnosis using a deep-learning-based approach

Planetary gearboxes are widely used in the drivetrain of wind turbines. Planetary gearbox fault diagnosis is very important for reducing the downtime and maintenance cost and improving the safety, reliability, and life span of the wind turbines. The wind energy industry is currently using condition monitoring systems to collect massive real-time data and conventional vibratory analysis as a standard method for planetary gearbox condition monitoring. As an attractive option to process big data for fault diagnosis, deep learning can automatically learn features that otherwise require much skill, time, and experience. This article presents a new deep-learning-based method for wind turbine planetary gearbox fault diagnosis developed by a large memory storage and retrieval neural network with dictionary learning. The developed approach can automatically extract self-learned fault features from raw vibration monitoring data and perform planetary gearbox fault diagnosis without supervised fine-tuning process. From the raw vibration monitoring data, a dictionary is first learned by a large memory storage and retrieval with dictionary learning network. Based on the learned dictionary, a sparse representation of the raw vibration signals is generated by shift-invariant sparse coding and input to a large memory storage and retrieval network classifier to obtain fault diagnosis results. The structure of the large memory storage and retrieval with dictionary learning is determined by optimal selection of the sliding box size to generate sub-patterns from the vibration data. The effectiveness of the presented method is tested and validated with a set of seeded fault vibration data collected at a planetary gearbox test rig in laboratory. The validation results have shown a promising planetary gearbox fault diagnosis performance with the presented method.

Joint High-Order Synchrosqueezing Transform and Multi-Taper Empirical Wavelet Transform for Fault Diagnosis of Wind Turbine Planetary Gearbox under Nonstationary Conditions.

Wind turbines usually operate under nonstationary conditions, such as wide-range speed fluctuation and time-varying load. Its critical component, the planetary gearbox, is prone to malfunction or failure, which leads to downtime and repair costs. Therefore, fault diagnosis and condition monitoring for the planetary gearbox in wind turbines is a vital research topic. Meanwhile, the signals measured by the vibration sensors mounted in the gearbox exhibit time-varying and nonstationary features. In this study, a novel time-frequency method based on high-order synchrosqueezing transform (SST) and multi-taper empirical wavelet transform (MTEWT) is proposed for the wind turbine planetary gearbox under nonstationary conditions. The high-order SST uses accurate instantaneous frequency approximations to obtain a sharper time-frequency representation (TFR). As the acquired signal consists of many components, like the meshing and rotating components of the gear and bearing, the fault component may be masked by other unrelated components. The MTEWT is used to separate the fault feature from the masking components. A variety of experimental signals of the wind turbine planetary gearbox under nonstationary conditions have been analyzed to demonstrate the effectiveness and robustness of the proposed method. Results show that the proposed method is effective in diagnosing both gear and bearing faults.

Adaptively Detecting the Transient Feature of Faulty Wind Turbine Planetary Gearboxes by the Improved Kurtosis and Iterative Thresholding Algorithm

It is significant for diagnosing the early faults of wind turbine planetary gearboxes. However, the weak transient feature generated by the early gear failure is often submerged in other vibration signals and noise, thus this paper discusses a new automatic sparse representation method for detecting weak transients. An improved Morlet wavelet which strictly satisfies the admissibility condition is constructed, and it is highly beneficial to sparsely separating signal component. By the use of the improved Morlet wavelet dictionary and Fourier dictionary, the transient component can be extracted by solving a sparse problem. Aiming at the drawback of the kurtosis and squared envelope spectrum (SES) negentropy in evaluating the similarity between the extracted transient and the original transient, an improved kurtosis index is designed, which can be used to search the optimal sparse parameters. Then, by the use of the improved kurtosis index and iterative thresholding method, an adaptively iterative thresholding shrinkage algorithm is proposed to detect the repetitive transients from the planetary gearbox signal. The comparative results in simulation and experiments show that the proposed method can more effectively and accurately perform the fault diagnosis of wind turbine planetary gearboxes than the infogram method and the resonance-based signal decomposition method.

Fault detection enhancement in wind turbine planetary gearbox via stationary vibration waveform data

Planetary gearbox fault diagnosis is very important for reducing the downtime, maintenance cost, and for improving the safety, reliability, and lifespan of wind turbines. The present work reports the results concluded by long-term experiments to a defected planetary gearbox system, with a transverse cut with a depth of 1.0 mm and thickness of 0.2 mm to simulate the planetary gearbox component crack. For each defect, recordings every 60.0 min were acquired and a total of 7 recordings (∼ 6.0 h of test duration) were resulted until the termination of the test. Fault is assured by increasing the test period to the point of where the remaining metal in the tooth area has enough stress to be in the plastic deformation region. An experimental procedure is developed to assess the severity of the gearbox component fault. Gearbox components faults of cracked planet gear tooth, cracked planet gears carrier, and cracked main bearing inner race were tested under accelerated fault conditions, where a comparative analysis of condition monitoring indicators for various crack detection has been done. The experimental localized fault signals (vibration acceleration signals) were subjected to the same diagnostic techniques such as spectrum comparisons, spectral kurtosis analysis, skewness analysis, and crest factor analysis. The method is validated on a set of seeded localized faults on all gears and components: sun gear, ring gear, etc. The results look promising, where the root mean square value analysis could be a good indicator when compared with the other indicators in terms of early detection and characterization of faults.

Fault diagnosis of wind turbine planetary gearbox based on order analysis and divergence index

Fault diagnosis of wind turbine planetary gearbox based on order analysis and divergence index

Frequency response model and mechanism for wind turbine planetary gear train vibration analysis

Frequency components of complicated asymmetric modulation sidebands, existing in the vibration of the healthy planetary gear train, are prone to be erratically diagnosed as fault characteristics, which leads to difficulties in fault diagnosis of wind turbine planetary gearbox. The factors affecting the modulation sideband, i.e. the periodical time-varying transmission path and meshing force direction, are analysed. Considering both the meshing vibrations of the planet–ring and planet–sun gear pairs, a mathematical model was developed to analyse the planetary gear train's vibration response. Simulation and experiments were conducted, and the mechanism of vibration modulation sidebands was revealed. The modulation sideband is not caused by the meshing vibration itself, but by the testing method that sensors are fixed on the ring gear or gearbox casing. The frequency components and amplitudes of the sidebands are determined by the tooth number of the ring gear and sun gear, the number of planet gears and their initial assembling phases. The asymmetric modulation sideband is mainly caused by the phase difference of the initial planets' assembling phase.