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Highlights • An approach to estimate the probability distribution of electrified railway traction loads using a non-parametric method. • This method accounts for both optimal bandwidth selection and boundary bias. • This method obtains more accurate results compared with parametric methods and traditional Gaussian kernel density estimator. Abstract The probabilistic modeling for traction load is one of the most basic and challenging work in the field of electrified railway. The improved diffusion-based kernel density estimator (DKDE) is used for the first time to establish the probability distribution of traction loads. Based on the diffusion partial differential equation of finite domain, DKDE can be obtained by discrete and inverse discrete cosine transform. The DKDE effectively accounts for both the optimal bandwidth selection and boundary correction. Based on the measured data (feeder currents and re/active power), four goodness-of-fit tests are applied to test the estimated probability distribution of traction loads. Compared with the parametric estimation models and Gaussian kernel density estimator (GKDE) respectively, the results show that this probability distribution of traction loads by DKDE is more accurate and suitable. Moreover, this DKDE has strong applicability and versatility for the random variation of different traction loads. [ABSTRACT FROM AUTHOR]

• The reactor line-mode transient voltage fault component in dc grid is analyzed. • The mathematical morphology-based method to select the partition time for two parts is proposed. • The protection scheme using the ratio of the two-part voltage integral is proposed. • The high sensitivity to high impedance faults is validated in a four-terminal meshed DC grid. The fast, sensitive, and reliable primary protection method can ensure the continuous and reliable operation of the modular multilevel converter (MMC)–based HVDC grid during dc faults. This paper proposes a fault detection method based on the ratio of voltage integrals to improve the single-end method sensitivity to high-resistance faults. First, the reactor line-mode transient voltage caused by the initial traveling wave (TW) is analyzed in the fault component network. The exponential term of the voltage is independent of the fault resistance and can be reflected by the proposed voltage integral ratio. Next, the influence of subsequent TWs on the integral is analyzed. The partition time, selected by the mathematical morphology, is proposed to divide the voltage sampling data into two parts. Finally, the criterion using the ratio of the two-part voltage integral is proposed to detect internal faults. The ± 500 kV bipolar MMC-HVDC grid built on PSCAD/EMTDC validates that the proposed method can fast identify internal faults, and the protected zone covers the whole line length. Smulation cases analyze the influence of the reactor and sampling frequencies on the proposed method. Moreover, the method is sensitive to high-resistance faults and can work under white noise and different wiring modes. [ABSTRACT FROM AUTHOR]

• A quantitative calculation and analysis method of the correlation degree between traction load sequences based on the optimal delay is proposed. • The generalized EM algorithm combined with the normalized root mean square error (NRMSE) is proposed to solve the optimal delay model. • Considering the effects of no-load removal processing, inverse transformation, and different sampling intervals, the maximum correlation coefficient of Pearson, Spearman and Kendall and the corresponding optimal delay amount are obtained. • The proposed method provides the basic information of correlation for traction load modeling and its application. The correlation between traction load sequences is an important basic information for traction load modeling, but the correlation and time difference between traction load sequences restrict each other. The analysis method for the correlation between traction load sequences based on the optimal delay method is proposed in this paper. This method appropriately delays the sequences between different traction loads and taps the indirect correlation between traction load sequences. Firstly, the maximum correlation coefficient between the two traction load sequences after delaying is taken as the objective function to establish the optimal delay model. The delay time range and the monotonicity of sequence occurrence time constitute the constraints. Secondly, combined with the normalized root mean square error (NRMSE) index, the generalized expectation maximization (EM) algorithm is proposed to solve the optimal delay model. Finally, the traction load sequence with one and two trains at any period of time from the four traction substations of high-speed railway are taken as case examples. Considering the effects of no-load removal processing, inverse transformation, and different test intervals, the maximum correlation coefficients of Pearson, Spearman and Kendall, as well as the corresponding optimal delay amount are calculated. The analysis results show that the proposed method in this paper can dig out the potential delay correlation between traction load sequences, as well as provide basic reference information for traction load modeling and its application. [ABSTRACT FROM AUTHOR]

• A new pair of estimate points with only first three-order moments are added for additional calculation to improve the calculation accuracy of 3PEM. • It avoids the calculation of higher order center moments and the non-real number solutions that may occur with normalized center distances. • Taking into account the different coefficients of variation and the results of MCS method as a reference, the results have been compared with those obtained by the existing 2PEM and 3PEM in the presence of Normal distribution inputs, correlated inputs and non-normal inputs. • The performance and applicability to system size of the improved three point-estimate method scheme are discussed and compared. An improved three point-estimate method (I3PEM) scheme is proposed to estimate the probability moments of probabilistic power flow (PLF) in this paper. I3PEM is obtained by the existing three point-estimate method (3PEM), two point-estimate method (2PEM) and Chebyshev inequality. With only first three order moments (mean, standard deviation and skewness) of input random variables, the additional calculations in I3PEM are performed by a newly added pair of estimate points to improve the accuracy of estimated moments for output random variables. It is avoided the calculation of higher order center moments and the non-real number solutions that may occur with normalized center distances. The case studies of IEEE 14-bus and 118-bus system are employed to verify the performance of I3PEM in the presence of Normal distribution inputs, correlated inputs and non-normal inputs. Under the different coefficient of variation (CV) conditions, the results of MCS are used as benchmark. The accuracy of results obtained by I3PEM have been validated by comparing with those obtained from the basic 3PEM and 2PEM. Further, the advantages and applicability of I3PEM are compared. [ABSTRACT FROM AUTHOR]

• The time sequence probabilistic model of traction station load of high-speed railway considering the spatial correlation is proposed by applying the diffusion-based kernel density estimator (DKDE). • The model of time sequence three-phase probabilistic power flow (3PPF) calculation for power system connected with traction load is proposed through the port transformation of traction power supply system (TPSS). • The algorithm of time sequence 3PPF calculation model based on the three-point estimation method (3PEM) owned both the accuracy and the computational efficiency. • The time sequence probability density curves of voltage amplitude and three phase voltage unbalance degree of grid are obtained under the impact of the asymmetry and spatiotemporal of traction load of high-speed railway. The increasing traction station load of high-speed railway with single-phase, high-power, spatiotemporal mobility, and uncertainty has brought major impacts and challenges to the power system. The non-negligible characteristics are the spatial distribution of traction stations on high-speed railway lines, the time sequence of traction load moving with EMUs, as well as the nonparametric probabilistic features and correlation of traction load. A method of time sequence three-phase probabilistic power flow (3PPF) calculation for power system including traction station load of high-speed railway is proposed in this paper. Firstly, through the port transformation of traction power supply system (TPSS), the three-phase equivalent model of TPSS is established to construct an unbalanced three-phase power flow calculation model of grid with traction station load. Secondly, considering the spatial correlation, time sequence and non-parametric probabilistic feature of traction load, the time sequence probabilistic models of multiple traction station loads are built based on diffusion-based kernel density estimator (DKDE). Then, the algorithm of time sequence 3PPF calculation based on three-point estimation method (3PEM) is applied to deal with the computational efficiency and accuracy problems of 3PPF considering correlation and timing. Finally, the two modified three-phase power systems are employed for case studies in combination with the measured data of traction station of a high-speed railway. The validity and computing efficiency of the proposed method are verified. [ABSTRACT FROM AUTHOR]

An accurate awareness of the properties of the interarea electromechanical oscillation modes is essential for the secure operation of the power systems. Previously, the Ibrahim time domain (ITD) method has been applied to estimate the interarea oscillation modes using ring-down signals caused by major disturbances (such as three phase faults). In this paper, through the combination of the random decrement technique (RDT) and the ITD method, the RDT–ITD method is proposed to estimate the interarea modes during ambient operation of the power systems, using random responses caused by random changes of the loads. The estimated parameters include the frequencies, damping ratios and mode shapes. Considering the fact that both the ambient excitations and the measurement noise are stochastic in nature, Monte Carlo simulations were conducted to evaluate the performances of the RDT–ITD method in a statistical way. Simulation results in the 16 machine nonlinear power system model, as well as comparison results with the RDT-Prony method, the NExT-ERA method and the Subspace method realized by N4SID (Numerical algorithm for Subspace State Space System Identification) show that the RDT–ITD method is promising in estimating the interarea modes during ambient operation conditions. The RDT–ITD method was also validated using real PMU measurements from Sichuan power grid of China. [ABSTRACT FROM AUTHOR]