This paper presents a new Python-based methodology that combines the advantages of two established techniques for analysing electrical power systems (i.e., (i) harmonic domain (HD) modelling and (ii) PSS/E software tool). By integrating these two approaches, the Python-based program enhances the accuracy of the solution while maintaining computational resources.The proposed methodology primarily employs the harmonic domain admittance matrix (HDAM) that describes the system's behaviour in terms of harmonics. PSS/E can then analyse the system's power flow while considering the effects of harmonics. The paper showcases the proposed algorithm when using two specific scenarios (i.e., (i) integrating high-voltage direct current (HVDC) systems into the grid and (ii) integrating photovoltaic (PV) solar power systems into the grid).The proposed model was validated on the IEEE 39-bus system incorporating HVDC and PV connections. Benchmarking against detailed PSCAD time-domain simulations demonstrated high accuracy, with maximum voltage magnitude errors maintained below 1.5 & times; 10-3 pu and power mismatches not exceeding 2.5 & times; 10-3 pu. Furthermore, the proposed method significantly improved computational efficiency, reducing simulation time by over 98% (e.g., from 1760 s in PSCAD to 5.94 s with the proposed method) while achieving strict convergence (10-6) in fewer than 18 iterations.

This paper presents a method for estimating moisture content (%MC) in power transformers. It primarily relies on the analysis of the statistical properties of relaxation times characterizing the dielectric frequency response (DFR), which is fitted as a sum of rational functions using the Vector Fitting (VF) tool. The DFR is modeled as a sum of Debye terms (accounting for materials exhibiting different relaxation times due to multiple polarization processes) characterized by poles and residues provided by VF. These parameters are then used to derive statistical factors that correlate with the shape of the dielectric response curve in the context of the Havriliak-Negami (HN) model, which is known for its effectiveness in characterizing materials with multiple relaxation times. By correlating the statistical factors with the HN model parameters, substantial insights into the insulation condition can be achieved. A moisture index (MI) is proposed from these parameters, which, when combined with conductivity, allows for accurate %MC estimation in the solid insulation system (cellulose). The combined MI and conductivity capture combined effects on moisture behavior, addressing both conductivity and polarization losses at different frequencies. The proposed method provides an efficient and straightforward non-invasive approach to insulation assessment without complex optimization algorithms. Experimental work on transformers at varying moisture levels provides validation of the proposed approach and demonstrates strong correlation with industry standards. The results confirm its reliability for moisture evaluation in transformer monitoring.

Frequency-dependent modeling of components and subsystems using rational functions allows fast and accurate representation of the component's terminal behaviors in electromagnetic transient simulation studies. One shortcoming with general state-space model formulations of the rational model is that it is not possible to directly simulate the component's active losses during transient excitations, for instance when the losses are to be used as input for thermal calculations. In this work we propose applying an add-on secondary model for calculation of the losses. The secondary model is based on a circuit equivalent formulation which takes the component terminal voltages as input. Precise calculation of instantaneous power dissipation (active losses) is achieved by using currents/voltages through/over resistors as state variables. The power absorption in dynamic elements is also calculated. The proposed approach is verified via theoretical examples and applied to a cable system within a motor drive.

The grouping of propagation modes in line/cable systems involving a large number of conductors has been central in recent research regarding the stability of simulations for electromagnetic transient (EMT) studies. This article presents three improvements to the existing wideband line/cable modeling techniques for EMT analysis. The first improvement consists of calculating optimum time delays such that the oscillations in the phase angle of the propagation function are reduced. The second is a novel strategy for grouping propagation modes. As a third improvement, the maximum fitting frequency of a rapidly decaying mode is limited when its magnitude is below a threshold value. It is demonstrated that these modifications improve the stability characteristics of time-domain simulation, compared to the existing implementation of line/cable models in an EMT-type software. The proposed modifications are applied to several cable systems, including a very challenging real system involving 96 cables and one double-circuit overhead line, all with a length of 250 m.

Accurate initialization of large-scale power systems involving inverter-based resources can significantly reduce the computation time needed for reaching steady-state conditions in electromagnetic transient simulations. This paper presents a novel method for initializing detailed Photovoltaic Park models for electromagnetic transient simulations. The proposed approach uses the load-flow simulation results of the system to initialize electrical components and control variables of the PV park. The validity of the proposed initialization is demonstrated through the EMT simulations of aggregated and detailed representations of the PV park and its integration into a large-scale power system via three case studies. The results indicate that the existing initialization method significantly reduces the computational time required to reach steady-state, compared to the existing technique.

Experience with the vector fitting algorithm for rational approximation of frequency domain responses has in some situations revealed problems with slow convergence. A typical example is the fitting of noisy responses. The problem is linked to a high frequency asymptotic condition of the nontriviality constraint used in the vector fitting formulation. Relaxation of the non-triviality constraint is shown to be an effective solution to this problem. Application to selected examples demonstrates a significant improvement of performance in terms of convergence speed and the accuracy of the final result

This paper pushes forward frequency domain (FD) modeling of switched networks aimed at transient simulation, with particular interest in photovoltaic (PV) systems. The PV system simulation is performed via the numerical Laplace transform (NLT) in a sequential (partitioned-time) fashion by using a set of time-windows. The proposed technique enhances existing FD PV models by a) averaging switching functions and b) using sample overlapping to alleviate numerical oscillations due to rise-time phenomenon at time-window interfaces. The proposed enhancements provide a more efficient dynamic simulation compared to both classical single-window full-sample NLT implementation and non-averaged FD PV models. Verification is performed via prevalent electromagnetic transient (EMT) software tools.

This paper presents an alternative vector fitting (VF)-based approach to generate a rational model of nonlinear system behaviors. The proposed approach, named here as nonlinear VF (NL-VF), has its fundament in the previously proposed time-domain VF (TD-VF). TD-VF generates rational models using TD input/output responses and numerical convolution. Unlike TD-VF, NL-VF utilizes input/output vectors in the frequency domain (FD) to produce a rational approximation of the corresponding transfer function/matrix. The input/output FD vectors are obtained in this paper by a numerical Laplace transform (NLT) algorithm. Alternatively, input/output TD vectors available from any solution algorithm can be transformed to FD via NLT and used in the NL-VF approach. Computational efficiency and accuracy of the NL-VF are compared with the TD-VF technique for the single-phase case. It is demonstrated that NL-VF gives more accurate results than TD-VF, in particular when the data is calculated from a time-window, which does not capture the system slow dynamics. Also, unlike TD-VF, NL-VF can handle in a natural way FD weighting, thus exhibiting error control. The proposed approach is applied in this paper to 1) a boost converter circuit, and 2) a network involving simultaneously a three-phase grid-tied photovoltaic (PV) system and a nonlinear reactor load.

Accurate initialization of large-scale power systems involving inverter-based resources can significantly reduce the computation time needed for reaching steady-state conditions usually utilized to start up an electromagnetic (EMT) simulation. This paper presents a novel method for initializing detailed Photovoltaic (PV) park models for electromagnetic transient (EMT) simulations, based on a previously proposed method. The proposed approach uses the load-flow (LF) simulation results of the system to initialize electrical components and control variables of the PV park. The validity of the proposed initialization is demonstrated through the EMT simulation of aggregated and detailed representations of the PV park and its integration into a large-scale system via three case studies. The results indicate that the existing initialization method significantly reduces the computational time required to reach steady-state, compared to the existing technique.

This paper presents the rational approximation of a three-phase photovoltaic (PV) system via the time-domain vector fitting (TD-VF) and nonlinear vector fitting (NL-VF) techniques. A significant feature of both methods is that the transfer function is unknown, and input/output variables are sufficient data for the transfer function's dynamics characterization. The accuracy and robustness of both techniques are compared via an example. It is shown that NL-VF is an effective alternative to TD-VF for rationally identifying a system with nonlinear components, such as a PV generator.

This paper presents frequency domain (FD) Norton-type equivalents of single- and three-phase photovoltaic (PV) generators, aimed at transient calculations. The proposed equivalents can be interfaced with a general frequency-dependent network and/or nonlinear loads and include voltage/current sources representing initial conditions aimed at partitioned-time transient simulation. The transient solution is performed via the numerical Laplace transform (NLT) in a sequential scheme by using time sub-windows with corresponding initial conditions and using reduced number of samples. The proposed approach overcomes the issue of computationally heavy simulations as in classical implementations of NLT. Two case studies involving PV generator and nonlinear load are presented to illustrate the performance of the proposed FD PV models.

The flexible extended harmonic domain (FEHD) is a HD-based technique which has been applied to the modeling of modern power systems for waveform distortion studies. The FEHD provides the dynamics of harmonics and/or interharmonics, not necessarily arranged in sequential order. However, those discrete frequencies are fixed for the complete simulation process. This hinders performing a more complete analysis involving a wider frequency spectrum in the system's dynamics. If a finer frequency discretization is chosen in FEHD, dimensions explosion occurs.To overcome the abovementioned issues of FEHD, this paper presents an improved technique, named here as dynamic FEHD (DFEHD). The DFEHD fundamentally relies on the analytic solution of a linear system applied to FEHD models. Such combination permits to model the system under study as a closed-form expression which involves a wider and finer frequency band. Two case studies are presented to validate the proposed methodology.

This paper proposes a rational approximation-based approach to find positive real parameters for the extended Debye model (EDM), aimed at condition assessment of insulation systems of power transformers. The EDM can model the slow and fast polarization phenomenon, including relaxation mechanisms with different relaxation times within a composite dielectric material. In the proposed approach, the complex permittivity of the transformer’s composite insulation is approximated via rational functions, as given by the vector fitting (VF) software tool, and the EDM parameters are identified from the obtained poles/residues. To guarantee positive real parameters, i.e., a physically realizable circuit, VF is internally modified to calculate the final residues of the rational approximation via a constrained linear least-squares problem without resorting to further post-processing algorithms, as in existing methods, hence without affecting fitting accuracy. The effectiveness of the parametrized EDM is demonstrated in two ways: (a) by reconstructing frequency domain spectroscopy (FDS) curves provided via measurements in new oil-immersed power transformers and (b) by the comparison of the calculated polarization current given by EDM versus real measurements in time domain. The achieved fitting accuracy in most of the cases is above 99 percent for the reconstructed FDS curves, while the polarization current waveform is reproduced with good agreement.

This paper proposes an approach to parameterize the extended Debye model (EDM) employing rational approximation of dielectric frequency response (DFR) curves via the vector fitting (VF) technique. The rational approximation, expressed as a set of partial fractions with corresponding poles and residues, provides straightforwardly the EDM RC parameters. In the proposed approach, the VF algorithm is internally modified to compute the residues as a constrained linear least-squares problem to guarantee positive RC elements, i.e., a physically realizable circuit, unlike existing methods that use post-processing algorithms.Validation of the proposed technique is performed by reconstructing the DFR curves of new transformers and by the comparison of calculated polarization current from DFR versus real-time domain measurements.

This paper presents a harmonic domain (HD) model of a high-voltage direct current (HVDC) system including controls for DC voltage and reactive power via the secant method. The proportional-integral (PI) control scheme is used to verify the results by the secant method. It is shown that the latter achieves computational savings compared to the PI control scheme. A case study is presented and verified with the PSCAD/EMTDC software tool in terms of harmonics and steady state waveforms.

In this paper, a photovoltaic (PV) generator is represented as a lumped-parameters network equivalent. This equivalent can be readily integrated into a power flow algorithm. The PV generator is initially modeled via its harmonic domain admittance matrix (HDAM). The HDAM is obtained by using the flexible harmonic domain (FHD) which permits to include frequency coupling. Reduction of the HDAM to fundamental power frequency quantities provides the lumped-parameters π-shaped circuit of the PV generator, amenable for power flow algorithm. It is demonstrated that multiple π-equivalents, emulating solar farms with distinct switching frequencies of the involved power electronics devices, can be integrated into the power flow solution algorithm.

This paper presents a novel approach, based on the theory of hyperplanes, for mode identification of linear systems. The proposed approach can operate on either a set of ordinary differential equations (converted to diagonal form, if needed) or a set of partial fractions derived from a synthesized transfer function of the system under analysis. For either format, the linear system is structured to have as unknown variable a vector containing the residues. Singular value decomposition is initially used to identify an initial sparsity of the residue vector where the number of nonzero values corresponds to the pre-defined order of the dominant poles (eigenvalues) under search. An algorithm based on geometrical search of hyperplanes is used to optimize the selection of the nonzero residue locations, minimizing the residual of the zero residue hyperplanes. Finally, a recalculation of the residues is carried out by using the obtained optimal sparsity.

This paper presents a frequency domain (FD) model of a three-phase photovoltaic (PV) system tied to a nonlinear load and aimed at electromagnetic transient (EMT) simulation. The simulation is performed by using the numerical Laplace transform (NLT) and by considering sequential time-windows until the total simulation time is reached. Compared to a previously proposed model, this paper presents a major enhancement, i.e., substantial decrease of rise-time oscillations when simulating with sequential time-windows. This is achieved by using interpolators to smooth the oscillatory behavior.

Electromagnetic transients (EMT) analysis heavily relies on computational simulation of power networks. Initialization of simulation at a given operating point, i.e., stationary state, requires either long simulation time-windows, especially for poorly damped networks or networks with very distinct time delays, or the use of alternative software tools. EMT-type programs are computationally very intensive as nonlinear components have to be brought to steady state and, in case of power electronic devices (PEDs)-based networks, a very small time-step has to be utilized. This article proposes a fast methodology to calculate steady state of networks involving both nonlinear loads and PEDs. The proposed approach relies on: (a) frequency-domain representation of PEDs via the chain-matrix and (b) using Fast Fourier Transform/Inverse Fast Fourier Transform (FFT/IFFT) operations for the nonlinear elements, thus avoiding problems with transients. The fact that the methodology uses frequency-domain operations permits to naturally obtain harmonics and/or interharmonics. A test system including a nonlinear reactor and a photovoltaic (PV) generator is used to illustrate the performance of the proposed methodology.

This paper presents a methodology, based on both Floquet theory and the participation matrix concepts, aimed at characterization of a time-periodic system in terms of its poles location in the complex plane. The participation matrix allows to find which poles are the most influenced by certain state-variables. The proposed methodology can be utilized as an auxiliary tool for parameter design, stability assessment, and for post-event analysis of time-periodic systems. The proposed methodology is readily applicable to distributed generation systems. A case study involving a stand-alone wind-photovoltaic-battery system is presented to support this statement.