Differential-algebraic Equations Research Articles

Power System Transient Stability Preventive Control via Aptenodytes Forsteri Optimization with an Improved Transient Stability Assessment Model

Transient stability preventive control (TSPC), a method to efficiently withstand the severe contingencies in a power system, is mathematically a transient stability constrained optimal power flow (TSC-OPF) issue, attempting to maintain the economical and secure dispatch of a power system via generation rescheduling. The traditional TSC-OPF issue incorporated with differential-algebraic equations (DAE) is time consumption and difficult to solve. Therefore, this paper proposes a new TSPC method driven by a naturally inspired optimization algorithm integrated with transient stability assessment. To avoid solving complex DAE, the stacking ensemble multilayer perceptron (SEMLP) is used in this research as a transient stability assessment (TSA) model and integrated into the optimization algorithm to replace transient stability constraints. Therefore, less time is spent on challenging calculations. Simultaneously, sensitivity analysis (SA) based on this TSA model determines the adjustment direction of the controllable generators set. The results of this SA can be utilized as prior knowledge for subsequent optimization algorithms, thus further reducing the time consumption process. In addition, a naturally inspired algorithm, Aptenodytes Forsteri Optimization (AFO), is introduced to find the best operating point with a near-optimal operational cost while ensuring power system stability. The accuracy and effectiveness of the method are verified on the IEEE 39-bus system and the IEEE 300-bus system. After the implementation of the proposed TSPC method, both systems can ensure transient stability under a given contingency. The test experiment using AFO driven by SEMLP and SA on the IEEE 39-bus system is completed in about 35 s, which is one-tenth of the time required by the time domain simulation method.

Research and analysis of tower crane load behavior when the rope breaks

The object of research is the behavior of the load of the tower crane during the break of the sling. One of the most problematic areas is the safety of work and the prevention of emergency situations. Despite the presence of mandatory safety measures, during cargo transportation, one of the sling branches may be destroyed due to the presence of a dynamic component during the operation of the crane, or errors of the slinger when securing the cargo. Also, the presence of hidden internal or unnoticed defects in the sling construction itself cannot be ruled out. Also, one of the most problematic places is the chaotic fluctuations of the load, which negatively affect the stability of the crane and safety. The paper describes the case of the destruction of one of the branches of a two-rope sling during the transportation of a long product by a tower crane. The proposed method of cargo behavior analysis is based on the use of a dynamic description of cable system failure modes within the framework of setting and solving differential-algebraic equations. This makes it possible to more accurately describe the behavior of the cargo when the sling breaks. The obtained results show that the application of the proposed method makes it possible to bring the mathematical model of the two-link mathematical pendulum significantly closer to the actual mutual oscillations of the load during the sling break. This is due to the fact that the proposed method has a number of features, in particular, high sensitivity to changes in the behavior of the cargo and a quick reaction to a rope break. These results can be used in practice in the design and operation of tower cranes. Thanks to the application of the proposed method, it is possible to obtain accurate values of cargo behavior indicators and timely detection of a rope break. Compared to similar known methods, this method has such advantages as high efficiency, reliability and safety of operation.

Improving the robustness of distributed secondary control in autonomous microgrids to mitigate the effects of communication delays

Distributed control has been employed in autonomous microgrids (MGs) to attain secondary control goals. However, the reliance of MG’s distributed secondary control on communication makes it vulnerable to degraded performance and the risk of instability due to communication delays. This paper enhances the flexibility of the MG control framework, with each distributed generator incorporating supplementary local feedback signals. This added flexibility strengthens the MG’s ability to counteract disturbances caused by delays and enhances its dynamic performance. The introduction of extra feedback signals has a direct impact on the dynamic response of the MG, leaving its steady-state condition unaffected. In comparison to the methods found in existing literature, the increased flexibility relaxes the inherent trade-off between the objectives of improved transient response and the system’s ability to mitigate disturbances caused by delays. To model the MG, a set of delay differential–algebraic equations (DDAEs) is used. The small-signal linearized model is derived and employed to evaluate the stability of the MG as well as fine-tune the control parameters. The effectiveness of this proposed control structure is demonstrated using the MATLAB/Simulink environment and a controller in the loop (CIL) setup, tested against both fixed universal and asynchronous delays, as well as time-varying delays.

Index‐aware learning of circuits

AbstractElectrical circuits are present in a variety of technologies, making their design an important part of computer aided engineering. The growing number of parameters that affect the final design leads to a need for new approaches to quantify their impact. Machine learning may play a key role in this regard; however, current approaches often make suboptimal use of existing knowledge about the system at hand. In terms of circuits, their description via modified nodal analysis is well‐understood. This particular formulation leads to systems of differential‐algebraic equations (DAEs), which bring with them a number of peculiarities, for example, hidden constraints that the solution needs to fulfill. We use the recently introduced dissection index that can decouple a given system of DAEs into ordinary differential equations, only depending on differential variables, and purely algebraic equations, that describe the relations between differential and algebraic variables. The idea is to then only learn the differential variables and reconstruct the algebraic ones using the relations from the decoupling. This approach guarantees that the algebraic constraints are fulfilled up to the accuracy of the nonlinear system solver, and it may also reduce the learning effort as only the differential variables need to be learned.

Transforming Radical Differential Equations to Algebraic Differential Equations

In this paper, we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial relations among them by means of a rational change of variables. The solutions of the given equation and its transformation correspond one-to-one. This work can be seen as a generalization of previous work on reparametrization of ODEs and PDEs with radical coefficients.

Physics-Informed Neural Networks for Solving High-Index Differential-Algebraic Equation Systems Based on Radau Methods

As is well known, differential algebraic equations (DAEs), which are able to describe dynamic changes and underlying constraints, have been widely applied in engineering fields such as fluid dynamics, multi-body dynamics, mechanical systems, and control theory. In practical physical modeling within these domains, the systems often generate high-index DAEs. Classical implicit numerical methods typically result in varying order reduction of numerical accuracy when solving high-index systems. Recently, the physics-informed neural networks (PINNs) have gained attention for solving DAE systems. However, it faces challenges like the inability to directly solve high-index systems, lower predictive accuracy, and weaker generalization capabilities. In this paper, we propose a PINN computational framework, combined Radau IIA numerical method with an improved fully connected neural network structure, to directly solve high-index DAEs. Furthermore, we employ a domain decomposition strategy to enhance solution accuracy. We conduct numerical experiments with two classical high-index systems as illustrative examples, investigating how different orders and time-step sizes of the Radau IIA method affect the accuracy of neural network solutions. For different time-step sizes, the experimental results indicate that utilizing a 5th-order Radau IIA method in the PINN achieves a high level of system accuracy and stability. Specifically, the absolute errors for all differential variables remain as low as 10−6, and the absolute errors for algebraic variables are maintained at 10−5. Therefore, our method exhibits excellent computational accuracy and strong generalization capabilities, providing a feasible approach for the high-precision solution of larger-scale DAEs with higher indices or challenging high-dimensional partial differential algebraic equation systems.

Problem for differential-algebraic equations with a significant loads

In this article, the problem for a differential-algebraic equation with a significant loads is studied. Unlike previously studied problems for differential equations with a significant loads, in the considered equation, there is a matrix in the left part with a derivative that is not invertible. Therefore, the system of equations includes both differential and algebraic equations. To solve the problem, we propose a modification of the Dzhumabaev’s parametrization method. The considered problem is reduced to a parametric problem for the differential-algebraic equation with significant loads. We apply the Weierstrass canonical form to this problem. We obtain parametric initial value problem for a differential equations and an algebraic equations with a significant loads. The solvability conditions for the considered problem are established.

On the update of algebraic states during state estimation of differential–algebraic equation (DAE) systems

This manuscript presents a discussion on the algebraic state update step performed during recursive filtering of the differential–algebraic equation (DAE) systems. Existing DAE state estimation approaches follow a two-step state update procedure at each sampling instant. In particular, they first estimate the differential states using the Kalman update, and then update algebraic states by explicitly solving the algebraic equations. Specifically, for the case of DAE systems involving linear algebraic equations though the differential equations are nonlinear, we show that when appropriately initialized, this two-step state update procedure is not needed. It can instead be replaced with a one-step state update procedure that computes the differential and algebraic state estimates simultaneously through the Kalman update. The satisfaction of algebraic equations is guaranteed by this one-step update without it being explicitly enforced. Towards this end, we show that the error covariance matrix of augmented states, when properly initialized, satisfies a null-space property after prediction and update step at each sampling instant. This property ensures that the state estimates obtained using the proposed one-step update approach, satisfy the algebraic equations. This holds for both analytical linearization based extended Kalman filtering and statistical linearization based sigma-point filtering approaches. We also propose a heuristic-based update procedure for state estimation of DAE systems that involve nonlinear algebraic equations. This procedure draws out inferences from the case of DAE systems involving linear algebraic equations and is based on the analysis of algebraic equations residuals obtained from the updated differential and algebraic state estimates with a one-step state update. The efficacy of the proposed state update procedures is demonstrated by performing simulation studies on a benchmark drum boiler system case study. Results demonstrate that the proposed update procedures satisfactorily estimate the differential and algebraic states of a DAE system when compared to the traditional two-step update procedure.

A Posteriori Error Estimation for Parabolic Problems with Dynamic Boundary Conditions

This paper is concerned with adaptive mesh refinement strategies for the spatial discretization of parabolic problems with dynamic boundary conditions. This includes the characterization of inf–sup stable discretization schemes for a stationary model problem as a preliminary step. Based on an alternative formulation of the system as a partial differential–algebraic equation, we introduce a posteriori error estimators which allow local refinements as well as a special treatment of the boundary. We prove reliability and efficiency of the estimators and illustrate their performance in several numerical experiments.

Load Margin Assessment of Power Systems Using Physics-Informed Neural Network with Optimized Parameters

Challenges in the operation of power systems arise from several factors such as the interconnection of large power systems, integration of new energy sources and the increase in electrical energy demand. These challenges have required the development of fast and reliable tools for evaluating the operation of power systems. The load margin (LM) is an important index in evaluating the stability of power systems, but traditional methods for determining the LM consist of solving a set of differential-algebraic equations whose information may not always be available. Data-Driven techniques such as Artificial Neural Networks were developed to calculate and monitor LM, but may present unsatisfactory performance due to difficulty in generalization. Therefore, this article proposes a design method for Physics-Informed Neural Networks whose parameters will be tuned by bio-inspired algorithms in an optimization model. Physical knowledge regarding the operation of power systems is incorporated into the PINN training process. Case studies were carried out and discussed in the IEEE 68-bus system considering the N-1 criterion for disconnection of transmission lines. The PINN load margin results obtained by the proposed method showed lower error values for the Root Mean Square Error (RMSE), Mean Square Error (MSE) and Mean Absolute Percentage Error (MAPE) indices than the traditional training Levenberg-Marquard method.

A Lie group variational integrator in a closed-loop vector space without a multiplier

Abstract. As a non-tree multi-body system, the dynamics model of four-bar mechanism is a differential algebraic equation. The constraints breach problem leads to many problems for computation accuracy and efficiency. With the traditional method, constructing an ODE-type dynamics equation for it is difficult or impossible. In this exploration, the dynamics model is built with geometry mechanic theory. The kinematic constraint variation relation of a closed-loop system is built in matrix and vector space with Lie group and Lie algebra theory respectively. The results indicate that the attitude variation between the driven body and the follower body has a linear recursion relation, which is the basis for dynamics modelling. With the Lie group variational integrator method, the closed-loop system Lagrangian dynamics model is built in vector space, with Legendre transformation. The dynamics model is reduced to be the Hamilton type. The kinematic model and dynamics model are solved using Newton iteration and the Runge–Kutta method respectively. As a special case of a crank and rocker mechanism, the dynamics character of a parallelogram mechanism is presented to verify the good structure conservation character of the closed-loop geometry dynamics model.

Simplified Reactor Model for Microreactor Coupled with Helium Closed Brayton Cycle

Microreactors comprise a new actively developing class of very small advanced reactors that have the potential to be an alternative to carbon-intensive energy technologies. A microreactor based on high-temperature gas reactor (HTGR) technology is a very promising advanced reactor with inherent safety, and it can couple with a closed Brayton cycle for higher efficiency. Since dynamics characteristics are fundamental to analyzing a power generation system and a reactor is the main source of the dynamics characteristics of a system, it is necessary to study a microreactor model suitable for system analysis. The main goal is to simulate the performance of the previously mentioned integrated system, focusing on the details of the power conversion unit while still ensuring acceptable calculation times. Hence, a simplified reactor model is needed that could supply sufficiently accurate values of pressure drop and heat transfer across the core. In this paper, by simplifying the physical processes in a microreactor, a dynamic model described by differential algebraic equations is obtained based on the lumped parameter modeling methodology and the basic conservation of fluid mass, momentum, and energy. Coupling thermal hydraulics with neutron kinetics, the temperature coefficient of reactivity and xenon poisoning are considered. Finally, the model is programmed and calculated using Modelica language. The transient responses of the main parameters under typical perturbations are analyzed, and the results show that the responses are correct. Because of the effect of reactivity feedback, fluctuations of the main parameters caused by microperturbations eventually tend to stabilize. In addition, the effects of negative reactivity introduced by xenon poisoning under two typical dynamic processes are analyzed. In power regulation, excess reactivity is required to compensate for the negative reactivity introduced by 135Xe. The model and results can properly predict the systematic parameters and serve as a basis for system analysis of microreactor coupling with the helium closed Brayton cycle.

Inertial Parameter Identification for Closed-Loop Mechanisms: Adaptation of Linear Regression for Coordinate Partitioning

Abstract This study investigates the use of linear-regression-based identification in rigid multibody system applications. A multibody system model, originally described with differential-algebraic equations (DAE), is transformed into a set of ordinary differential equations using coordinate partitioning. This allows the identification framework (where the system is described with ordinary differential equations) to be applied to rigid multibody systems described with nonminimal coordinates. The methodology is demonstrated via numerical and experimental validation on a slider–crank mechanism. The results show that the presented methodology is capable of accurately identifying the system's inertial parameters even with a short motion trajectory used for training. The presented linear-regression-based identification approach opens new opportunities to develop more accurate multibody models. The resulting updated multibody models can be considered especially useful for state-estimation and the control of multibody systems.

Plasticity in nonlinear elastic metamaterials: Low-order dynamic modeling

Nonlinear elastic metamaterials have been shown to admit a variety of rich, dynamical features that can be leveraged to tailor the propagation of mechanical waves. Since these materials derive their properties from intricate, subwavelength geometries, direct numerical simulations are often prohibitively expensive at scales of interest. To overcome this limitation, reduced-order models, typically in the form of effective continua or discrete lattices that capture the essential features of the material at sufficiently long wavelengths, have been developed. While many prior studies have implemented these models successfully, the vast majority have considered only recoverable elastic deformations with linear damping and neglected history-dependent effects, such as plasticity and friction. In this presentation, we introduce an effective lattice modeling framework for nonlinear elastic metamaterials undergoing plastic deformation. Due to the history-dependent nature of plasticity, this framework generally yields a system of differential-algebraic equations whose computational cost is significantly greater than a purely elastic system of similar size. We apply the method to several examples of interest and explore means to obtain phenomenological elastic-plastic models for general material architectures.

Differential Fault and Algebraic Equation Combined Analysis on PICO

In modern information technology, research on block cipher security is imperative. Concerning the ultra lightweight block cipher PICO, there has been only one study focused on recovering its complete master key, with a large search space of 264, and no fault analysis yet. This paper proposes a new fault analysis approach, combining differential fault and algebraic equation techniques. It achieved the recovery of PICO’s entire master key with 40 faults in an average time of 0.57 h. S-box decomposition was utilized to optimize our approach, reducing the time by a remarkable 75.83% under the identical 40-fault condition. Furthermore, PICO’s complete master key could be recovered with 28 faults in an average time of 0.78 h, indicating a significant 237 reduction in its search space compared to the previous study. This marks the first fault analysis on PICO. Compared to conventional fault analysis methods DFA (differential fault analysis) and AFA (algebraic fault analysis), our approach outperforms in recovering PICO’s entire master key, highlighting the cruciality of key expansion complexity in block cipher security. Therefore, our approach could serve to recover master keys of block ciphers with comparably complicated key expansions, and production of more secure block ciphers could result.

Effective numerical simulations of synchronous generator system

Synchronous generator system is a complicated dynamic system for energy transmission, which plays an important role in modern industrial production. In this article, we propose some predictor-corrector methods and structure-preserving methods for a generator system based on the first benchmark model of subsynchronous resonance, among which the structure-preserving methods preserve a Dirac structure associated with the so-called port-Hamiltonian descriptor systems. To illustrate this, the simplified generator system in the form of index-1 differential-algebraic equations has been derived. Our analyses provide the global error estimates for a special class of structure-preserving methods called Gauss methods, which guarantee their superior performance over the PSCAD/EMTDC and the predictor-corrector methods in terms of computational stability. Numerical simulations are implemented to verify the effectiveness and advantages of our methods.

A robust Gated-PINN to resolve local minima issues in solving differential algebraic equations

Physics-Informed Neural Network (PINN) has emerged as a promising tool for solving various physical problems with differential equations. However in practice, PINN often suffers from the local minima issue while solving problems with minimal initial conditions. In this paper, we propose a robust Gated-PINN to address this issue, which blends numerical methods and PINN. The Gated-PINN overcomes the local minima issue by controlling the flow of information of the neural network similar to a numerical method. We first investigate the use of conventional PINN to the Cartesian-coordinate simple pendulum problem with minimal initial conditions, as one of basic differential algebraic equation (DAE) problem. Our results show the advantage of PINN over existing numerical methods in that it does not require sophisticated mathematical techniques such as index reduction. But we also observe that conventional PINN can lead to inaccurate solutions; such solutions partially satisfy the differential equation requirement, but does not meet the given initial condition and fail to further improve. We demonstrate the effectiveness of the proposed Gated-PINN by showing that it yields accurate solutions, such that for a pendulum of length 1, the mean Euclidean error between the Gated-PINN model and the traditional numerical method model is less than 0.01 and pointwise maximum Euclidean error is less than 0.04. Moreover, Gated-PINN can operate without any complicated index reduction, and unlike conventional PINN, accurate solution can be obtained consistently without falling into a local minima. Overall, our study presents the potential of the Gated-PINN for solving DAE problems and provides a valuable insight into the challenges and limitations of using PINN for solving physical problems.

Solutions to the output regulation problem of time-varying descriptor systems

This paper studies the output regulation problem of time-varying descriptor systems and the problem of designing state feedback and dynamic measurement output feedback control laws which asymptotically achieves output regulation and disturbance rejection is considered. New regulator equations are proposed for time-varying descriptor systems in the form of differential-algebraic matrix equations. The unique solution of the proposed regulator equations is given as well. We prove that the output regulation problem of time-varying descriptor systems is solvable if and only if the given regulator equations are solvable. Based on the solution of the regulator equations, the state feedback and dynamic measurement output feedback control laws are designed to solve the output regulation problem. The work extends the existing results of output regulation problem for time-varying linear systems to the time-varying descriptor systems. Numerical examples are given to show the effectiveness of our methodology.

Solving differential‐algebraic equations in power system dynamic analysis with quantum computing

AbstractPower system dynamics are generally modeled by high dimensional nonlinear differential‐algebraic equations (DAEs) given a large number of components forming the network. These DAEs' complexity can grow exponentially due to the increasing penetration of distributed energy resources, whereas their computation time becomes sensitive due to the increasing interconnection of the power grid with other energy systems. This paper demonstrates the use of quantum computing algorithms to solve DAEs for power system dynamic analysis. We leverage a symbolic programming framework to equivalently convert the power system's DAEs into ordinary differential equations (ODEs) using index reduction methods and then encode their data into qubits using amplitude encoding. The system nonlinearity is captured by Hamiltonian simulation with truncated Taylor expansion so that state variables can be updated by a quantum linear equation solver. Our results show that quantum computing can solve the power system's DAEs accurately with a computational complexity polynomial in the logarithm of the system dimension. We also illustrate the use of recent advanced tools in scientific machine learning for implementing complex computing concepts, that is, Taylor expansion, DAEs/ODEs transformation, and quantum computing solver with abstract representation for power engineering applications.

Алгоритмы формирования последовательности псевдослучайных чисел в натурном эксперименте при имитации процессов функционирования сложных информационных систем

In this work is considered information systems: its structure and influencing factors of the external environment that allows to describe its simulation means. Elements for the description of similar systems are algebraic differential equations, Markov chains or similar of the Research objective consists in forming of conditions of carrying out a natural experiment not on the studied system, and on some other (with the known distribution parameters of the studied random variable, for example, of the distribution law), rather “similar” to given and such which experimental functioning will not lead to a little notable costs. Research methods. At simulation of processes of Markov the difficulties connected with need of input in memory of the machine of stochastic matrixes of infinite dimension very often meet. It is actually very difficult to present an infinite stochastic matrix in which all elements are selected “purely accidentally”, i.e. without any patterns. In practice there are always only such matrixes in which, since some line, a certain pattern in formation of elements is shown. The solution of the specified problem is based on this property. Results. The algorithm of finding of a random variable consists in the choice of that from expressions which at this value is true. Then the following implementation is on the formula corresponding to the selected expression. If at the same time the selected formula contains a random variable, then by means of already known receptions there is implementation of this random variable and is substituted in it. To be convinced of correctness of the offered algorithm and to understand reasons on the basis of which expressions are constructed it is enough to substitute all possible values of random variables and to compare result to received by means of a matrix. The considered logical reception is applicable also at simulation of behavior of the objects, more difficult, than a Markov chain. Its use in more difficult situations – a component of the idea of automatic modeling of information systems. Conclusion. The algorithms of forming of pseudorandom numbers received in work for simulation of components of an information system are rather adequate, however their efficiency is defined by the relation of a part of the square of a rectangle located under a curve to all its area. If this relation is not enough, then a considerable part of machine time is spent for unproductive receiving the accidental points lying under a curve and discarded in the course of calculations. From here follows, in particular, that for exponential distribution the specified method least, and for uniform is most effective. In practice it is reasonable to use such method when the specified relation exceeds value 0.3. In certain cases it can be applied and when this relation more than 0.1.