Application of return difference matrix to stability analysis for multivariable feedback networks

Due to the internal dynamics of the modular multilevel converter (MMC), the coupling between the positive and negative sequences in impedance, which is defined as frequency coupling, inherently exists in MMC. Ignoring the frequency coupling of the MMC impedance model may lead to inaccurate stability assessment, and thus the multi-input multi-output (MIMO) impedance model has been developed to consider the frequency coupling effect. However, the generalized Nyquist criterion (GNC), which is used for the stability analysis of an MIMO model, is more complicated than the stability analysis method applied on single-input-single-output (SISO) models. Meanwhile, it is not always the case that the SISO model fails in the stability assessment. Therefore, the conditions when the MIMO impedance model needs to be considered in the stability analysis of an MMC system should be analyzed. This paper quantitatively analyzes the effect of frequency coupling on the stability analysis of grid-connected MMC, and clarifies the frequency range and grid conditions that the coupling effect required to be considered in the stability analysis. Based on the quantitative relations between the frequency coupling and the stability analysis of the grid-connected MMC system, a simple and accurate stability analysis method for the grid-connected MMC system is proposed, where the MIMO impedance model is applied when the frequency coupling has a significant effect and the SISO impedance model is used if the frequency coupling is insignificant.

The dq impedance stability analysis for a grid-connected current-control inverter is based on the impedance ratio matrix. However, the coupled matrix brings difficulties in deriving its eigenvalues for the analysis based on the general Nyquist criterion. If the couplings are ignored for simplification, unacceptable errors will be present in the analysis. In this paper, the influence of the couplings on the dq impedance stability analysis is studied. To take the couplings into account simply, the determinant-based impedance stability analysis is used. The mechanism between the determinant of the impedance-ratio matrix and the inverter stability is unveiled. Compared to the eigenvalues-based analysis, only one determinant rather than two eigenvalue s-function is required for the stability analysis. One Nyquist plot or pole map can be applied to the determinant to check the right-half-plane poles. The accuracy of the determinant-based stability analysis is also checked by comparing with the state-space stability analysis method. For the stability analysis, the coupling influence on the current control, the phase-locked loop, and the grid impedance are studied. The errors can be 10% in the stability analysis if the couplings are ignored.

When the renewable energy with a voltage source converter (VSC) as the interface circuit is connected to a weak grid, the power system is more likely to become unstable. The common method to analyze the interaction stability is the impedance-based stability analysis method. However, the traditional impedance model only considers the same frequency response as the perturbation frequency, but ignores the frequency coupling characteristic, so the stability analyses based on the traditional impedance model is inaccurate in some frequency ranges. Hence, considering multiple frequency coupling factors, the frequency coupling impedance model for a VSC based on the dual second order generalized integrator phase-locked loop (DSOGI-PLL) is established which can be more accurate, and the proposed impedance model is verified by simulation in MATLAB/Simulink. Furthermore, the influences of frequency coupling characteristic, short-circuit ratio (SCR) and outer-loop bandwidth on the stability of the grid-converter interconnected system has been analyzed by the generalized Nyquist criterion (GNC). Finally, the time-domain simulation results proved that the GNC can accurately forecast the resonance frequencies of the grid-converter interconnected system.

This study presents a dq impedance‐decoupled network modelling method of offshore wind power plant (OWPP) for stability analysis. DQ impedance frequency responses of grid‐connected inverters (GCIs) and long transmission cables (LTCs) are first measured by frequency scanning method, which are fitted as transfer function matrices using matrix fitting algorithm. Then, the GCIs are modelled as Norton equivalent circuits. In addition, per‐unit‐length electrical parameters of the LTCs are extracted from the measured dq impedance frequency responses of a specific LTC, based on which dq impedance‐decoupled two port network models of the LTCs are established. DQ impedance‐decoupled network model of the whole OWPP is then established based on connection relationships of these GCIs and LTCs. Finally, generalised Nyquist criterion (GNC) is performed in all the dq impedance‐decoupled subsystems, and the subsystems where the GNC is not satisfied are identified as instability sources. Compared with conventional impedance‐based stability analysis methods of OWPP, the proposed dq impedance‐decoupled modelling method is able to facilitate the application of GNC and further instability source identification based on partitioning the whole OWPP into several decoupled subsystems. The effectiveness of the network modelling method is validated based on time‐domain simulation results in Matlab/Simulink and real‐time verification results in OPAL‐RT.

A survey of some recent results in the stability and instability analyses of time-varying systems

A novel general stability criterion of time-delay fractional-order nonlinear systems based on WILL Deduction Method

The high penetration of power electronic converters with complex control systems is changing the power system dynamics, introducing new challenges such as converter-driven stability incidents. Traditional stability analysis methods, suitable for classical problems like voltage, frequency, and rotor angle stability in large systems, are insufficient for addressing the fast control dynamics of converters, which involve electromagnetic phenomena. These phenomena require detailed converter and network modeling, which can be performed in both the frequency and time domains, enabling the respective stability analyses to be carried out. However, frequency domain methods, based on small-signal impedances linearized at a single operating point, inherently ignore time domain phenomena like switching events and nonlinear behaviors. In contrast, time domain electromagnetic transient (EMT) simulations are effective for analyzing converter-driven stability but are computationally intensive when applied to large transmission systems with numerous use cases. Therefore, to reduce the simulation complexity in EMT tools, a complexity reduction procedure is proposed in this paper. Leveraging the advantages of the frequency domain, such as faster simulation times and information on wideband frequency characteristics of the system, this procedure utilizes the small-signal impedances and introduces a method for network reduction. The procedure also uses the frequency domain stability analysis method to screen for critical network use cases. Primarily, this procedure is a frequency domain toolchain encompassing frequency domain stability analysis and frequency domain network reduction. The result of the toolchain is a reduced network size and reduced network use cases that can be used for EMT simulations. The procedure is applied to an IEEE 39 bus system, where converter-driven stability is evaluated for two use cases. Furthermore, the network reduction method is tested on a critical use case, demonstrating reductions in network size and computation times without compromising the quality of stability analysis results.

In this paper an algebraic method is presented whereby one can obtain the relevant values of the gain, K, of a multivariable (multi-input-multi-output) feedback system to be stable or relatively stable. The basis of the method involves the algebraic solution of the two equations obtained from the real and imaginary parts of the characteristic equation of the system. The proposed method offers an alternative procedure for determining stability (relative stability) to the graphical methods based on the ‘ generalized Nyquist criterion ’ and the ‘ multivariable root locus method ’. This method is analogous to the analytical form of the root locus combined with the Routh-Hurwitz criterion, as compared to the Nyquist or graphical root locus methods when applied to single-input-single-output feedback systems. Several examples are presented illustrating the applications of the method and compared with the graphical methods.

According to the interaction between the inverter and the grid in the large-scale photovoltaic grid-connected system, an auxiliary system based on grid connection of a single inverter is introduced. Research shows that the stability of the inverter output current in large-scale photovoltaic grid-connected systems depends on the stability of the auxiliary system and the stability of the inverter itself. But the stability of the total grid-connected current depends only on the stability of the auxiliary system. Therefore, a method for stability analysis of large-scale photovoltaic grid-connected systems is proposed. For a large-scale photovoltaic grid-connected system that is connected to the ideal grid through the grid impedance Zg after the inverters of the same specification and model are connected in parallel, the stable determination conditions are: (1) The inverter itself is stable, that is, a single inverter is stable when it is directly connected to the ideal power grid; (2) A single inverter in a large-scale PV grid-connected system is stable through the grid impedance nZg to the ideal grid. The stability analysis method can transform the research on the stability of large-scale photovoltaic grid-connected systems into the stability of the grid-connected system of a single inverter. For the multi-inverter parallel grid-connected system with uniform specifications and models, the stability analysis method is applicable regardless of whether all inverter current loop input reference currents are identical. The simulation results verify the correctness of the theoretical analysis.

This paper extends small-signal stability analysis methods developed for balanced three-phase systems to the more general unbalanced three-phase system. The impedance-based stability analysis method in the phase-domain is generalized to unbalanced systems characterized by coupled positive-sequence and negative-sequence impedance networks. Compared to balanced three-phase systems, modeling of unbalanced converter systems is more complicated due to the negative-sequence trajectory that is impressed on the ac system by the unbalanced condition. Small-signal models are developed to determine whether or not a small unbalance voltage condition introduces any significant change to the inverter impedance models of the perfectly balanced case. Simulation and experimental measurements are used for verification of the small-signal models and the proposed stability analysis method.

Stability analysis programs are a primary tool used by power system planning and operating engineers to predict the response of the system to various disturbances.Important conclusions and decisions are made based on the results of stability studies.The conventional method of analyzing stability is to calculate the transient behaviour of generators due to a given disturbance.Direct methods of stability analysis identify whether or not the system will remain stable once the disturbance is removed by comparing it with a calculated threshold value.Direct methods not only avoid the time consuming solutions required in the conventional method, but also provide a quantitative measure of the degree of system stability.This additional information makes direct methods very attractive when the relative stability of different plans must be compared or when stability limits must be calculated quickly.Direct methods of transient stability analysis of a multi machine power system, using a function describing the system's transient energy, are discussed.By examining the trajectory of the disturbed system, the following fundamental questions are dealt with: the concept of a controlling unstable equilibrium point (U.E.P), the manner in which some generators tend to lose synchronism, and identifying the energy directly responsible for system separation.Resolving this issue will substantially improve transient stability analysis by direct method.

Many existing stability analysis methods for fuzzy logic control systems like TS fuzzy model based methods usually tackle plants that are linear with respect to control. However, these methods may be difficult to apply to plants with dead zones. This paper presents an improved stability analysis method to resolve this difficulty. Our proposed method employs a Lyapunov function to prove the stability of nonlinear fuzzy logic control systems with asymmetric dead zone and saturation. An illustrative example will be given to demonstrate the ability of the method.

This paper is mainly focused on the stability-related issues of a grid-tied interlinking converter system. First, hybrid AC/DC admittance characteristics of an interlinking converter operating under unity and non-unity power factors are analyzed separately. Then, by using the hybrid AC/DC admittance and determinant-based General Nyquist Criterion (GNC), the stability assessment, instability root cause identification, and instability mitigation of a grid-tied interlinking converter system are studied. In addition, decoupled mitigation of the AC- and DC-side instabilities are discussed based on the hybrid AC/DC admittance characteristics. The study results show that when an interlinking converter operates under unity power factor, the stabilities of the AC and DC sides are decoupled and can be regulated independently. To be more specific, the DC-side stability is subjected to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d$</tex-math></inline-formula> -axis admittance, DC admittance, and their coupling admittances while the AC-side stability is dominated by the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -axis admittance. According to the hybrid AC/DC admittance characteristics, this paper further reveals that the DC-side stability is determined by the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d$</tex-math></inline-formula> -axis controllers. In contrast, the AC-side stability is governed by the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -axis controllers and the phase-locked loop (PLL). Finally, the analysis presented in this paper is verified based on frequency- and time-domain simulations in the Matlab/Simulink environment.

Uncertainty of parameters in control systems was described by interval variables, and some idea of the non-probabilistic reliability method that we proposed in previous papers was used for stability analysis and synthesis of uncertain systems. A new method, called robust reliability method, was presented for quadratic stability analysis and robust stabilization of dynamic systems with interval parameters. In the method, the performance function for robust reliability calculation was established by criterion of quadratic stability, and a solving method in the form of optimization for robust reliability was presented. The proposed stabilization method can take both the required energy supply for control and the control robustness into account. This makes the reasonable strategy of compromising between the control cost and system reliability become possible. By the method, robust reliability measure of the stability degree of uncertain control systems can be provided, and the maximum allowable ranges of the basic parameters can be obtained. The presented method is based on the linear matrix inequality (LMI) approach, and is convenient to implement Three numerical examples were provided to illustrate the validity and efficiency of the presented method.

The performance of the stability chart (SC), limit equilibrium (LE), and finite element (FE) methods for stability analysis of weathered volcanic rock slopes under static and earthquake loads is not fully understood. This research aimed to evaluate the performance of the SC, LE, and FE methods for slope stability analyses by studying a case of a weathered volcanic rock slope at the Leuwikeris Dam diversion tunnel in Indonesia. Site characterisations were carried out to obtain the input parameters for the static and pseudostatic slope stability analyses. The results showed that using various search methods of failure surface and the suggested optimum number of finite elements, multiple non-circular failure surfaces were predicted from the LE and FE methods that were not anticipated in the initial slope design based on the SC method. The location and shape of the failure surfaces from the LE method agreed with those from the FE method. The difference between the Fs values from the static LE analysis and those from the static FE analyses was less than 14%, but the discrepancy increased up to 78% in the pseudostatic analyses. This study highlights the importance of validating the results from one method with the other methods to prevent slope failures.