Stochastic Differential Algebraic Equations Research Articles

Stochastic Differential-Algebraic Equations in Power Systems–Semi Implicit Approach

Stochastic Differential-Algebraic Equations in Power Systems–Semi Implicit Approach

Central exponents of linear stochastic differential-algebraic equations of index 1

Our aim in this paper is to establish a concept of central exponents of a stochastic differential-algebraic equation (SDAE) of index 1. For this purpose, we introduce the inherent regular stochastic differential equation (SDE) associated with a SDAE of index 1. Then, the central exponents are defined samplewise via the induced two-parameter stochastic flow generated by the inherent regular SDE. We prove that the central exponents are nonrandom and greater than or equal to the Lyapunov exponents.

On the Calculation of the Variance of Algebraic Variables in Power System Dynamic Models With Stochastic Processes

This letter presents a technique to calculate the variance of algebraic variables of power system models represented as a set of stochastic differential-algebraic equations. The technique utilizes the solution of a Lyapunov equation and requires the calculation of the state matrix of the system. The IEEE 14-bus system serves to demonstrate the accuracy of the proposed technique over a wide range of variances of stochastic processes. The accuracy is evaluated by comparing the results with those obtained with Monte Carlo time domain simulations. Finally, a case study based on a 1479-bus dynamic model of the all-island Irish transmission system shows the computational efficiency of the proposed approach compared to the Monte Carlo method.

Upper central exponent of linear stochastic differential algebraic equations of index 1

The current study was carried out to assess the antioxidant expression of the seed’s black-testa peanut cultivar namely CNC1 in the artifical drought condition. The oxidative stress with a burst of O2 .- and H2O2 in peanut leaves resulted the obviously cellular membrane damage that was illustrated by an increase of lipid peroxidation. A high induced activity of enzymatic antioxidants such as SOD, CAT and POX would detoxify the excess of O2 .- and H2O2. An increase in content of amino acid proline protects the stressed cells by adjusting intercellular osmotic potential. The antioxidative expression of CNC1 plants was similar to that in L20 cutivar - a drought tolerant cutivar of peanut.

Harvesting electricity from random vibrations via a nonlinear energy sink

Integration of a vibration absorber and an energy harvester is a promising approach to achieve simultaneously vibration reductions and vibratory energy harvesting. The present work investigates random responses of a nonlinear energy sink (NES) combined with a giant magnetostrictive energy harvester (GMEH). When a structure is modeled as a one-degree-of-freedom oscillator subjected to a random excitation, the structure with the integrated NES-GMEH is governed by two simultaneous nonlinear stochastic differential equations coupled with a nonlinear algebraic equation. The Ferrari formula is applied to decouple the algebraic equations from the stochastic differential-algebraic equations. The resulting stochastic algebraic differential equations are transformed into the steady Fokker-Planck-Kolmogorov equations. A fourth finite difference scheme is used to solve numerically the FPK equations. The FPK equations based solutions are qualitatively verified via Monte Carlo simulations. The parametric effects on vibration suppression and electricity production are examined, where the electricity is indicated by the square expectations of voltage and power. The responses of the structure, the structure with the NES only and the structure with both NES-GMEH are compared under the same Gaussian white noise and system parameters. It is found that the integrated NES-GMEH system achieves the best vibration suppression. It is demonstrated that a properly designed NES-GMEH device can simultaneously suppress vibration and produce electricity under Gaussian white noise excitations.

Stochastic Runge–Kutta methods for multi-dimensional Itô stochastic differential algebraic equations

In this paper, we discuss the numerical solutions to index 1 stochastic differential algebraic equations. We introduce a new class of weak second-order stochastic Runge–Kutta methods for finding the numerical approximate solutions to multi-dimensional stochastic differential algebraic equations. A four-stage stiffly accurate stochastic Runge–Kutta methods for approximating analytical solutions to index 1 stochastic differential algebraic equations are derived. By colored rooted tree analysis, the order conditions for the stochastic Runge–Kutta methods of order two satisfying the weak convergence is obtained. The scalar test equations are considered to obtain the mean-square stability and the T-stability of weak second-order stochastic Runge–Kutta methods. Finally, some numerical illustrations are provided to prove the theoretical findings.

Modeling of Correlated Stochastic Processes for the Transient Stability Analysis of Power Systems

This paper proposes a systematic and general approach to model correlated stochastic processes in power systems by means of stochastic differential-algebraic equations. The paper discusses the theoretical background of stochastic differential-algebraic equations and provides a variety of examples of correlated stochastic models for power system applications. With this aim, stochastic processes with Normal and Weibull distributions are considered. The case study utilizes the well-known two-area system to demonstrate that the presence of correlation between the stochastic processes can cause instability, despite the fact that the same system is stable if the same processes are uncorrelated. The case study also considers a 1479-bus dynamic model of the all-island Irish transmission system to show the scalability of the proposed technique, and to compare scenarios with different levels of correlation among stochastic processes. Results indicate that correlation has a non-negligible impact on short-term dynamics. A high level of correlation among the processes, in fact, can give raise to instability.

Numerical Study of Stochastic Leontief-Type Model with Impulses

ABSTRACT In this article, a stochastic Leontief model with impulses has been studied, which is represented by a system of stochastic differential-algebraic equations, in both sides of the rectangular constant numerical matrices that form a singular pencil. The system has been considered in terms of the current velocity of the solution, which is a direct analog of the physical velocity of deterministic processes. The proposed approach in this work does not impose restrictions on the size and the form of the matrices included in the Leontief-type system. The Kroenke-Weierstrass transformation of the pencil was conducted by the coefficient matrices to the canonical form has been used to simplify the study of equations. This study also involves two methods: Firstly, using a stochastic differential equation, this was followed by using the so-called mean derivatives of Nelson random processes to describe the solutions of this equation. The distinguishing feature of the work proposed an approach based on the convergence of the theoretical results to the exact one. The findings show that explicit formulas for solutions and solvability conditions are obtained, and for a subsystem resolved with respect to the symmetric derivative. The theorem of existence of solutions for the system under consideration has been proved under certain conditions on the coefficients of the system. Conducting computational experiments on the model confirming the effectiveness of the proposed approach. Error, maximum and minimum of the singular values of matrices.

Strong convergence of a half-explicit Euler scheme for constrained stochastic mechanical systems

AbstractThis paper is concerned with the numerical approximation of stochastic mechanical systems with nonlinear holonomic constraints. The considered systems are described by second-order stochastic differential-algebraic equations involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation, the drift coefficient of which is typically not globally one-sided Lipschitz continuous. We investigate a half-explicit, drift-truncated Euler scheme that fulfills the constraint exactly. Pathwise uniform $L_p$-convergence is established. The proof is based on a suitable decomposition of the discrete Lagrange multipliers and on norm estimates for the single components, enabling the verification of consistency, semistability and moment growth properties of the scheme.

Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems

In this paper, we discuss stochastic differential-algebraic equations (SDAEs) and the asymptotic stability assessment for such systems via Lyapunov exponents (LEs). We focus on index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Via ergodic theory, it is then feasible to analyze the LEs via the random dynamical system generated by the underlying SDEs. Once the existence of well-defined LEs is guaranteed, we proceed to the use of numerical simulation techniques to determine the LEs numerically. Discrete and continuous QR decomposition-based numerical methods are implemented to compute the fundamental solution matrix and use it in the computation of the LEs. Important computational features of both methods are illustrated via numerical tests. Finally, the methods are applied to two applications from power systems engineering, including the single-machine infinite-bus (SMIB) power system model.

Analytical Study of the Impacts of Stochastic Load Fluctuation on the Dynamic Voltage Stability Margin Using Bifurcation Theory

This paper studies the impacts of stochastic load fluctuations, namely the fluctuation intensity and the load power variation speed, on power system dynamic voltage stability. Additionally, the trade-off relationship between the two parameters is revealed, which provides important insights regarding the potential of using energy storage to maintain voltage stability under high uncertainty. To this end, Stochastic Differential-Algebraic Equations (SDAEs) are used to model the stochastic load variation; bifurcation analysis is carried out to explain the influence of stochasticity. Numerical study and Monte Carlo simulations on the IEEE 14-bus system demonstrate that a larger fluctuation intensity or a slower load power variation speed may lead to a smaller voltage stability margin. To the best of authors' knowledge, this work uncovers the impacts of the time evolution property of the driving parameters, i.e., the load power variation speed and its trade off effect with the fluctuation intensity on the size of the dynamic voltage stability margin.

Stability analysis of high order Runge–Kutta methods for index 1 stochastic differential-algebraic equations with scalar noise

In this paper, a new class of implicit stochastic Runge-Kutta (SRK) methods is constructed for numerically solving systems of index 1 stochastic differential-algebraic equations (SDAEs) with scalar multiplicative noise. By applying rooted tree theory analysis, the family of coefficients of the proposed methods of order 1.5 are calculated in the mean-square sense. In particular, we derive some four-stage stiffly accurate semi-implicit SRK methods for approximating index 1 SDAEs with scalar noise. For these methods, first, MS-stability functions, applied to a scalar linear test equation with multiplicative noise, are calculated. Then, their regions of MS-stability are compared with the corresponding MS-stability region of the original SDE. Accordingly, we illustrate that the proposed schemes seems to have good stability properties. Numerical results will be presented to check the convergence order and computational efficiency of the new methods.

Analytic Estimation Method of Forced Oscillation Amplitude Under Stochastic Continuous Disturbances

This paper proposes an analytic method for the amplitude estimation of the forced oscillation under stochastic continuous disturbances (SCDs). There is considerable interest in the integration of intermittent renewable energy resources into a power system, which leads to the power system in an environment with SCDs. Thus, the forced oscillation issues under SCDs are of concern. In this paper, the SCDs are described as continuous-time stochastic processes, and then the systems under SCDs are modeled by stochastic differential algebraic equations. Furthermore, by employing stochastic theory, an efficient analytic method for calculating the statistics of system states during the forced oscillation is proposed. Moreover, a novel forced oscillation phenomenon, which presents system states oscillate with stationary distributions under SCDs, is clearly characterized and theoretically explained. Based on the proposed analytic method, a scheme is put forth to estimate amplitudes of system states during forced oscillation under SCDs. Finally, compared to Monte Carlo simulation, the proposed scheme has two significant advantages. First, the proposed scheme offers a fast and effective solution for engineering applications. Second, the impact mechanism of SCDs on forced oscillation in power system is clearer, by the proposed scheme.

Nonlinear Model Predictive Control for Disturbance Rejection in Isoenergetic-isochoric Flash Processes

Abstract We present a nonlinear model predictive control (NMPC) algorithm for semi-explicit index-1 stochastic differential-algebraic equations. It is natural to model isoenergetic-isochoric (constant energy-constant volume) flash processes with such equations. The algorithm uses the continuous-discrete extended Kalman filter (EKF) for state estimation, and it uses a single-shooting method to solve the involved optimal control problems. It computes the gradients with an adjoint method. The isoenergetic-isochoric flash is also called the UV flash, and it is important to rigorous models of phase equilibrium processes because it is a mathematical statement of the second law of thermodynamics. NMPC algorithms for UV flash processes are therefore relevant to both safe and economical operation of phase equilibrium processes such as flash separation, distillation, two-phase flow in pipes, and oil production. We design the NMPC algorithm for disturbance rejection, and we therefore augment the state vector with the unknown disturbance variables in the continuous-discrete EKF. We present a numerical example of economical NMPC of a UV flash separation process. It involves output constraints and the estimation of an unknown and unmeasured disturbance. The computation time of the NMPC algorithm does not exceed 3.14 s in any of the 5 min control intervals. This indicates that real-time NMPC of UV flash separation processes is computationally feasible.

Geometric Methods in Analysis and Control of Implicit Differential Systems

In this article, we discuss the application of differential geometric methods in analyzing the structure and designing control laws for implicit differential systems or differential algebraic equation (DAE) systems. While there have been several efforts toward numerical and quantitative analysis of DAE problems, the theoretical contributions especially in the case of nonlinear systems are scarce. We discuss two popular techniques from differential geometric control theory and bring out their merits in addressing implicit differential systems. In the first section, we review the theory of noninteracting control via input–output decoupling and its application in analyzing the intrinsic structure of DAE control problems. In particular, we focus on addressing the problem of well-posedness of DAE systems as well as feedback control design through a regularization process which allows one to solve the DAE by expressing the constraint variable as a dynamically dependent endogenous function of the states, inputs, and their derivatives. Further, extensions of these techniques to stochastic differential algebraic equations have been presented. In the second section, we review the theory of differential flatness and its applicability to feedback control design for a class of DAE systems. Here, the DAE system is expressed as a Cartan field on a manifold of jets of infinite order, and necessary and sufficient conditions for its equivalence to a linear, controllable system have been derived in order to design globally stabilizing nonlinear feedback laws. Examples from constrained mechanics have been presented in order to demonstrate the practical applicability of these methods.

Operator differential-algebraic equations with noise arising in fluid dynamics

We study linear semi-explicit stochastic operator differential algebraic equations (DAEs) for which the constraint equation is given in an explicit form. In particular, this includes the Stokes equations arising in fluid dynamics. We combine a white noise polynomial chaos expansion approach to include stochastic perturbations with deterministic regularization techniques. With this, we are able to include Gaussian noise and stochastic convolution terms as perturbations in the differential as well as in the constraint equation. By the application of the polynomial chaos expansion method, we reduce the stochastic operator DAE to an infinite system of deterministic operator DAEs for the stochastic coefficients. Since the obtained system is very sensitive to perturbations in the constraint equation, we analyze a regularized version of the system. This then allows to prove the existence and uniqueness of the solution of the initial stochastic operator DAE in a certain weighted space of stochastic processes.

Stability analysis and classification of Runge–Kutta methods for index 1 stochastic differential-algebraic equations with scalar noise

The problem of solving stochastic differential-algebraic equations (SDAEs) of index 1 with a scalar driving Wiener process is considered. Recently, the authors have proposed a class of stiffly accurate stochastic Runge–Kutta (SRK) methods that do not involve any pseudo-inverses or projectors for the numerical solution of the problem. Based on this class of approximation methods, classifications for the coefficients of stiffly accurate SRK methods attaining strong order 0.5 as well as strong order 1.0 are calculated. Further, the mean-square stability of the considered class of SRK methods is analyzed. As the main result, families of A-stable efficient order 0.5 and 1.0 stiffly accurate SRK methods with a minimal number of stages for SDEs as well as for SDAEs are presented.

Stochastic Nash Games for Markov Jump Linear Systems with State- and Control-Dependent Noise

This paper investigates Nash games for a class of linear stochastic systems governed by Ito’s differential equation with Markovian jump parameters both in finite-time horizon and infinite-time horizon. First, stochastic Nash games are formulated by applying the results of indefinite stochastic linear quadratic (LQ) control problems. Second, in order to obtain Nash equilibrium strategies, cross-coupled stochastic Riccati differential (algebraic) equations (CSRDEs and CSRAEs) are derived. Moreover, in order to demonstrate the validity of the obtained results, stochastic H 2/H ∞ control with state- and control-dependent noise is discussed as an immediate application. Finally, a numerical example is provided.

Understanding Early Indicators of Critical Transitions in Power Systems From Autocorrelation Functions

Many dynamical systems, including power systems, recover from perturbations more slowly as they approach critical transitions---a phenomenon known as critical slowing down. If the system is stochastically forced, autocorrelation and variance in time-series data from the system often increase before the transition, potentially providing an early warning of coming danger. In some cases, these statistical patterns are sufficiently strong, and occur sufficiently far from the transition, that they can be used to predict the distance between the current operating state and the critical point. In other cases CSD comes too late to be a good indicator. In order to better understand the extent to which CSD can be used as an indicator of proximity to bifurcation in power systems, this paper derives autocorrelation functions for three small power system models, using the stochastic differential algebraic equations (SDAE) associated with each. The analytical results, along with numerical results from a larger system, show that, although CSD does occur in power systems, its signs sometimes appear only when the system is very close to transition. On the other hand, the variance in voltage magnitudes consistently shows up as a good early warning of voltage collapse. Finally, analytical results illustrate the importance of nonlinearity to the occurrence of CSD.

Application of Stochastic Differential-Algebraic Equations in Hybrid MTL Systems Analysis

The paper deals with the application of stochastic differential-algebraic equations (SDAE) in the field of the time-domain simulation of hybrid (lumped/distributed) systems with randomly varying parameters. A core of the method lies on the theory of stochastic differential equations (SDE) considering the system responses as stochastic processes. However, due to a hybrid nature of the system, namely its lumped parameter part, non-differential (algebraic) parts arise generally in the solution. Herein, multiconductor transmission lines (MTL) play a role of the distributed-parameter parts of the hybrid system. The MTL model is designed as a cascade connection of generalized RLCG T-networks, while the state-variable method is applied for its description. The MTL boundary conditions are incorporated through a modified nodal analysis (MNA) to cover arbitrarily complex circuits. System responses are formed by the sets of stochastic trajectories completed by corresponding sample means and respective confidence intervals. To get the results a weak stochastic backward Euler scheme is used, consistent with the Itô stochastic calculus. All the computer simulations have been performed in the Matlab language environment. DOI: http://dx.doi.org/10.5755/j01.eee.20.5.7098