Deep learning identifies leak in water pipeline system using transient frequency response

The integration of a high proportion of renewable energy has significantly reduced the grid inertia level and markedly increased the risk of transient frequency instability in power systems. Meanwhile, the large-scale integration of diverse heterogeneous resources-such as wind power, photovoltaics, energy storage, and high voltage direct current (HVDC) transmission systems-has considerably enriched the portfolio of frequency regulation assets in modern power grids. However, the marked disparities in the dynamic response characteristics and actuation speeds among these resources introduce significant nonlinearity and high-dimensional complexity into the system's transient frequency behavior. As a result, conventional methods face considerable challenges in achieving accurate and timely prediction of such responses. However, the substantial differences in the frequency regulation characteristics and response speeds of these resources have led to a highly nonlinear and high-dimensional complex transient frequency response process, which is difficult to accurately and rapidly predict using traditional methods. To address this challenge, this paper proposes an online prediction method for transient frequency response that deeply integrates physical principles with data-driven approaches. First, a frequency dynamic response analysis model incorporating the frequency regulation characteristics of multiple resource types is constructed based on the Single-Machine Equivalent (SME) method, which is used to extract key features of the post-fault transient frequency response. Subsequently, information entropy theory is introduced to quantify the informational contribution of each physical feature, enabling the adaptive weighted fusion of physical frequency response features and Wide-Area Measurement System (WAMS) data. Finally, a physics-guided machine learning framework is proposed, in which the weighted physical features and the complete frequency curve predicted by the physical model are jointly embedded into the prediction process. An MLP-GRU-Attention model is designed as the data-driven predictor for frequency response. A physical consistency constraint is incorporated into the loss function to ensure that predictions strictly adhere to physical laws, thereby enhancing the accuracy and reliability of the transient frequency prediction model. Case studies based on the modified IEEE 39-bus system demonstrate that the proposed method significantly outperforms traditional data-driven approaches in terms of prediction accuracy, generalization capability under small-sample conditions, and noise immunity. This provides a new avenue for online frequency security awareness in renewable-integrated power systems with multiple heterogeneous frequency regulation resources.

The fault location problem has been tackled mainly through impedance-based techniques, the travelling wave principle and more recently machine learning algorithms. These techniques require both current and voltage measurement. In the case of impedance-based methods they can provide multiples solutions. In the case of the travelling wave approach it usually requires high sampling frequency measurements together with sophisticated identification algorithms. Machine learning techniques require training data and re-tuning for different grid topologies. This paper proposes a new fault location method based on the fault’s transient intermediate frequency response of the system immediately after a fault occurs. The transient response is characterized by the travelling wave phenomenon together with intermediate frequencies of oscillation, which are dependent on the faulted section and the fault location. In the proposed fault location solution, an offline methodology identifies these intermediate frequencies and their dependency on the fault location is fitted using a polynomial regression. The online fault location is performed using those polynomial regressions together with voltage measurements from the system and simple signal processing techniques. The full method is tested with an EMT simulation in PSCAD, using the exact frequency dependent model for underground cables.

The nonlinear frequency response of a chromatographic column for the adsorption of two dissolved components is analyzed using the concept of higher order frequency response functions (FRFs) which is based on the Volterra series and generalized Fourier transform. By applying this concept a nonlinear model of a system is replaced by an infinite series of the FRFs of the first, second, etc. order. The FRFs up to the third order are derived theoretically starting from the equilibrium-dispersive model, which is used for description of a chromatographic column, and applying the harmonic probing method. The functions that relate outlet concentration changes of each component to the corresponding inlet concentration changes are derived. At the inlet of a chromatographic column, it is considered: (a) the concentration change of one of the components keeping the concentration of the other component constant and (b) the concentration change of both components keeping their ratio constant. The FRFs are calculated numerically for different steady-state concentrations and relative mixture compositions. It has been found that, despite certain differences in initial conditions, the FRFs exhibit similar behavior. For higher frequencies, the amplitudes of the FRFs tend to zero and phases to −∞. In the low frequency range, which is of interest for investigation of equilibrium parameters, these functions have similar behavior, but tend to different asymptotic values. Correlations between coefficients of competitive adsorption isotherms, i.e. partial isotherm derivatives, and the derived FRFs are established. This theoretical result offers the potential to use the analysis of the nonlinear frequency response of a chromatographic column for estimation of competitive adsorption isotherms.

Estimating frequency response functions by using transient responses of systems to simple excitations

This paper investigates the measurement errors involved in estimating frequency response functions (FRFs) - and related quantities such as the coherence function - from weighted overlapped segment averaging, a technique that has become a standard in modern data analyzers due to its computational advantages. Particular attention is paid to leakage errors, for which this technique has frequently been criticized. Our main result is that a half-sine or diff window with about 2/3 overlap achieves the best compromise to reduce leakage errors in the case of stationary random excitations, and this is independently of the system FRF. This conclusion is to be contrasted with the customary habit of using a Hanning window with 1/2 overlap. The same reasoning confirms that a rectangular window without overlap minimizes measurement errors in the special case of multisine excitations. Moreover, practical formulas are provided to the reader for computing the bias and variance of the frequency response and coherence functions in the general case.

Frequency response functions (FRFs) can be calculated from a composite function of other FRFs composition and decomposition problems. Considering that an experimentally measured FRF has an uncertainty in its magnitude and phase, what would be the uncertainty of the magnitude and phase of an FRF calculated from a composite function of experimentally measured FRFs? This work derives analytical expressions for the uncertainty of magnitude and phase of FRFs calculated from functions composed of the basic arithmetic operations involving measured FRFs and their inherent uncertainties. The derivation of the uncertainty values comes from the application of multivariable error analysis to the basic arithmetic operations considering the real and the imaginary parts of the FRFs involved. The resultant uncertainty of the composite FRF is obtained by applying the methodology to every operation in the composite function. Hence, an analytical method is proposed for applying to both composition and decomposition problems. An example of a decomposition problem is presented, and the calculated uncertainty of the magnitude and the phase of the composite FRF showed good agreement with results obtained from Monte Carlo simulation. In addition, it is shown that the blind application of the root-mean-square rule to the composite function written in terms of magnitude is not an appropriate procedure. The proposed methodology is accurate in predicting the uncertainty of composite FRFs.

Inverse substructuring method has been recently proposed and applied for inverse analysis of the dynamical response of product transport system. The component-level frequency response functions (FRFs) and the coupling dynamic stiffness for facilitating the cushioning packaging design are all predicted from only the system-level FRFs. However, the system-level FRFs from coupling degree of freedoms may not be measured accurately because of the difficulties of vibration excitation and response measurement for the coupled interface between packaged product and vehicle within the limited accessible space. The aim of this paper is to develop a new FRF-based indirect inverse substructuring method for the analysis of the dynamic characteristics of a three-substructure coupled product transport system without measuring system-level FRFs at the coupling degree of freedoms. By enforcing the dynamic equilibrium conditions at the coupling coordinates and the displacement compatibility conditions, a closed-form analytical solution to inverse sub-structuring analysis of multi-substructure coupled product transport system is derived based on the relationship of easy-to-monitor component-level FRFs and the system-level FRFs at the coupling coordinates.. The proposed method is validated by a lumped mass-spring-damper model, and the predicted coupling dynamic stiffness is compared with the direct computation, showing exact agreement. Then, the FRF tests of a physical prototype of multi-substructure coupled product transport system are performed to further check the accuracy of the suggested method. The method developed offers an approach to predict the unknown coupling dynamic stiffness from measured FRFs purely. The proposed method may help to obtain the main controlling factors and contributions from the various structure-borne paths for product transport system. Copyright © 2014 John Wiley & Sons, Ltd.

In this study, the influence of inerters on the dynamic characteristics of mass-spring chain systems is systematically investigated through both theoretical analysis and numerical simulations. Specifically, the effects of the inerter coefficient on the natural frequencies, mode shape vectors, and frequency response functions (FRFs) are examined. A novel concept, termed the critical inerter coefficient, is introduced to identify a threshold beyond which the dynamic behavior of a vibration system undergoes fundamental change. At this critical value, both the mode shapes and FRFs exhibit significant deviations from those observed in conventional mass-spring chain systems. Moreover, by introducing a new pole-cancellation mechanism, resonance peaks in the FRFs can be effectively eliminated, leading to essential modifications in the dynamic response. The simulation results further reveal that the combined action of dampers and inerters gives rise to frequency response behaviors that are qualitatively different from those in systems employing traditional damping mechanisms. In particular, not only is the resonance peak suppressed, but distinctive dips or bulges emerge in the FRFs near the natural frequency. Building on these findings, the analysis is extended to an n -degree-of-freedom ( n -DOF) mass-spring chain system incorporating inerters. A theoretical criterion for determining the critical inerter coefficients is derived, providing analytical insights into the influence of inerters on the natural frequencies, mode shape vectors, and FRFs of n -DOF systems. The theoretical predictions are corroborated through numerical simulations, which also uncover the intricate interaction mechanisms between damping and inerters in these systems. This work advances the understanding of how inerters modify the dynamic behavior of vibration systems and offers a theoretical framework for their utilization in structural vibration control and mitigation strategies.

Due to lack of kinetic energy, the power electronic inverters based renewable energy microgrids have no inherent inertia capability to address the power swings. This degrades the transient response in island or weak-grid connected operational modes during disturbances. To address this issue, an important evolution that has been cited in the literature is, design of droop control techniques with the capability of virtual inertia injection to strengthen the frequency response during transients. But, adaptivity and voltage response aspects have not been covered. So, to address these issues, this paper proposes modified droop control technique using fuzzy logic based injection of adaptive virtual moment of inertia. The philosophy involved in the design is, varying the p-w and q-v droop coefficients adaptively with respect to frequency and voltage deviations occurred during disturbances. So, the droop controller can efficiently cope-up with disturbances and provide better frequency/voltage response during transients. The efficacy of the proposed technique is analyzed using MATLAB/Simulink®. From results, it is seen that the proposed technique improved voltage and frequency transient response over the conventional technique by providing adaptive control action during disturbances in microgrids.

The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy. The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy.

In everyday life, especially in the electrical circuit, there are many usage of matrix. One use of a matrix is found in the system of linear equations. In the field of electrical circuits there are also problems involving systems of linear equations in matrix form. To solve the system of linear equations in matrix form, in addition to using elementary row operations, also used matlab.

This paper presents frequency response function (FRF) expansion techniques that minimize the difference between the analytical and predicted FRF matrices to satisfy the FRF constraints. The measured FRF relationships at a small number of locations were used as constraints. The expansion method is useful for estimating the rotational FRFs that are difficult to measure or apply using frequency-based substructuring (FBS) techniques. The validity of the proposed method, including the effects of external noise, was confirmed using numerical examples. An FBS algorithm was also derived by incorporating the FRFs of each substructure, and the compatibility conditions were transformed into FRFs with pseudomasses at the joint nodes. A discrepancy between the FRFs of the synthesized system estimated using the proposed method and the analysis results of the entire system was observed in the numerical example, and the reasons for the discrepancy were investigated and discussed. It is estimated that the proposed FBS method can be enhanced by combining it with other dynamic substructuring techniques and supplementing additional information, such as measured FRFs.

The stability lobe diagrams predicted using the tool frequency response function (FRF) at the idle state usually have discrepancies compared with the actual stability cutting boundary. These discrepancies can be attributed to the effect of spindle rotating on the tool FRFs which are difficult to measure at the rotating state. This paper proposes a new tool FRF identification method without using noncontact sensor for the rotating state of the spindle. In this method, the FRFs with impact applied on smooth rotating tool and vibration response tested on spindle head are measured for two tools of different lengths clamped in spindle–holder assembly. Based on those FRFs, an inverse receptance coupling substructure analysis (RCSA) algorithm is developed to identify the FRFs of spindle–holder–partial tool assembly. A finite-element modeling (FEM) simulation is performed to verify the validity of inverse RCSA algorithm. The tool point FRFs at the spindle rotating state are obtained by coupling the FRFs of the spindle–holder–partial tool and the other partial tool. The effects of spindle rotational speed on tool point FRFs are investigated. The cutting experiment demonstrates that this method can accurately identify the tool point FRFs and predict cutting stability region under spindle rotating state.

Measurement of Frequency Response Functions (FRFs) on the Millennium Bridge in London was a key part in the validation of the retrofit solution for excessive sway vibration caused by walking pedestrians during the bridge opening in June 2000. The measured FRFs were employed to verify the analytical response analysis and to estimate and confirm analytical predictions of natural frequencies, modal damping ratios, mode shapes and modal masses corresponding to various configurations of prototype viscous and tuned mass dampers. These results were used to experimentally assess the effectiveness of the proposed retrofit solution. Although FRF measurement technology is fairly standard in many engineering disciplines, its application on the Millennium Bridge was quite problematic and required some unique solutions which are described in this paper. Modal testing of the Millennium Bridge clearly demonstrated that it was possible to measure good quality FRFs around a frequency as low as 0·5 Hz, which was the natural frequency of the first lateral mode of vibration. This was achieved by designing, constructing and installing a bespoke hydraulic shaker featuring a translatory moving mass of 1000 kg. As the shaker was not able to produce perfectly sinusoidal excitation for stepped-sine testing, FRFs were successfully estimated using the so-called Resolver method in conjunction with an Hc estimator. The standard H1 FRF estimator, which is used in the vast majority of modern digital spectrum analysers nowadays, was demonstrably unable to produce meaningful FRF estimates when using the Resolver approach. The FRF estimation and curve-fitting performed in situ proved to be crucial for the success of the modal testing of the Millennium Bridge. By evaluating the estimates of the experimental FRFs and the corresponding modal properties during the testing programme on site, a number of test errors were spotted. This, in turn, enabled immediate repetition of measurements and/or evaluation of the structural behaviour which would have been impossible if the FRF data analysis had been performed after the testing had been completed.

A metric on the similarity between two frequency response functions