Nonlinear Differential-algebraic Equations Research Articles

Faster-Than-Real-Time Dynamic Simulation of AC/DC Grids on Reconfigurable Hardware

Dynamic simulation of integrated AC and DC grids is paramount to address real-time operation challenges in energy control centers, such as available transfer capacities, relieving grid congestion, and taking effective control actions for improving the integrated grid system stability and reliability. This paper proposes a faster-than-real-time (FTRT) dynamic simulation of integrated AC/DC grids on the reconfigurable parallel hardware architecture of the field programmable gate array (FPGA). A fine-grained relaxation algorithm (FGRA) is proposed for a more efficient solution of the nonlinear differential algebraic equations of the integrated system model, including the detailed nonlinear models of the synchronous generators in the AC system which can be solved in parallel without matrix on the FPGA. The system solution is massively parallelized and pipelined in hardware to realize the lowest latencies and minimum utilization of hardware resource. Two case studies are used to illustrate the efficacy of the proposed algorithm and demonstrated a closed-loop prediction scenario for improving grid stability. Computational acceleration of up to 134 times faster than real-time are reported for the two case studies, and the accuracy of the dynamic interaction is validated using the off-line transient stability simulation tool TSAT of the DSATools package.

A parallel-in-time algorithm for variable step multistep methods

This paper presents a multigrid reduction in time (MGRIT) algorithm for achieving time parallelism using multistep backward difference formula (BDF) methods on variably-spaced temporal grids. This MGRIT approach transforms the linear multistep methods into single step methods applied to groups of time steps. Stability considerations are addressed through lowering of the order on coarse grids. The methods are presented for fixed and variable time step formulations. Numerical results show moderate speedups for the heat equation as well as two IEEE power grid test problems characterized by nonlinear differential algebraic equations (DAE) of index 1. The performance of MGRIT with BDF is compared to MGRIT with Runge-Kutta for both the fixed and variable step methods of the same order.

Spectral time-dependent solutions for Darcy model of natural convection in a porous enclosure

According to his high accuracy in the simply connected domains, the Galerkin spectral method having Fourier series as trial functions is chosen to develop a space-time dependent solution for the Darcy model coupled with the convection-diffusion heat transfer equation in a square porous cavity. Applying the Fourier-Galerkin (FG) procedure for the stream function formulation of the governing equations, we obtain a system of nonlinear algebraic differential equations that we simplify in ordinary differential equations and integrate in time using the variable order Runge-Kutta method. Two configurations dealing with transient and unsteady regimes are considered where the solution is derived for a wide range of Rayleigh numbers. The developed FG solution shows high accuracy with a reduced computational cost and excellent agreement with a finite element solution. The FG solution is used to provide physical insight into the effect of time variation on the mechanisms of natural convection (NC). The results provide a set of high-accurate data that can be used for testing numerical codes of heat transfer in time-dependent configurations.

Robust Control for Renewable-Integrated Power Networks Considering Input Bound Constraints and Worst Case Uncertainty Measure

Uncertainty from renewable energy and loads is one of the major challenges for stable grid operation. Various approaches have been explored to remedy these uncertainties. In this paper, we design centralized or decentralized state-feedback controllers for generators while considering worst-case uncertainty. Specifically, this paper introduces the notion of $\mathcal{L}_{\infty}$ robust control and stability for uncertain power networks. Uncertain and nonlinear differential algebraic equation model of the network is presented. The model includes unknown disturbances from renewables and loads. Given an operating point, the linearized state-space presentation is given. Then, the notion of $\mathcal{L}_{\infty}$ robust control and stability is discussed, resulting in a nonconvex optimization routine that yields a state feedback gain mitigating the impact of disturbances. The developed routine includes explicit input-bound constraints on generators' inputs and a measure of the worst-case disturbance. The feedback control architecture can be centralized, distributed, or decentralized. Algorithms based on successive convex approximations are then given to address the nonconvexity. Case studies are presented showcasing the performance of the $\mathcal{L}_{\infty}$ controllers in comparison with automatic generation control and $\mathcal{H}_{\infty}$ control methods.

Index Reduction for Differential-algebraic Equations with Mixed Matrices

Differential-algebraic equations (DAEs) are widely used for the modeling of dynamical systems. The difficulty in numerically solving a DAE is measured by its differentiation index. For highly accurate simulation of dynamical systems, it is important to convert high-index DAEs into low-index DAEs. Most of the existing simulation software packages for dynamical systems are equipped with an index-reduction algorithm given by Mattsson and Söderlind. Unfortunately, this algorithm fails if there are numerical cancellations. These numerical cancellations are often caused by accurate constants in structural equations. Distinguishing those accurate constants from generic parameters that represent physical quantities, Murota and Iri introduced the notion of a mixed matrix as a mathematical tool for faithful model description in a structural approach to systems analysis. For DAEs described with the use of mixed matrices, efficient algorithms to compute the index have been developed by exploiting matroid theory. This article presents an index-reduction algorithm for linear DAEs whose coefficient matrices are mixed matrices, i.e., linear DAEs containing physical quantities as parameters. Our algorithm detects numerical cancellations between accurate constants and transforms a DAE into an equivalent DAE to which Mattsson–Söderlind’s index-reduction algorithm is applicable. Our algorithm is based on the combinatorial relaxation approach, which is a framework to solve a linear algebraic problem by iteratively relaxing it into an efficiently solvable combinatorial optimization problem. The algorithm does not rely on symbolic manipulations but on fast combinatorial algorithms on graphs and matroids. Our algorithm is proved to work for any linear DAEs whose coefficient matrices are mixed matrices. Furthermore, we provide an improved algorithm under an assumption based on dimensional analysis of dynamical systems. Through numerical experiments, it is confirmed that our algorithms run sufficiently fast for large-scale DAEs and output DAEs such that physical meanings of coefficients are easy to interpret. Our algorithms can also be applied to nonlinear DAEs by regarding nonlinear terms as parameters.

Optimal power flow considering transient thermal behavior of overhead transmission lines

Electrothermal coordination (ETC) is a concept which exploits the potential transfer capability of transmission lines by coordinating the transient thermal behavior of the conductor and the system control actions. For the realization of ETC decision, the model of optimal power flow considering transient thermal behavior (OPFCTTB) of overhead lines and the solution methods are proposed in this paper. The OPFCTTB model is established by introducing the heat balance equations (HBEs) of overhead lines as equality constraints and the temperature of lines as the decision variable into the ac optimal power flow model. To solve the OPFCTTB model whose constraints consist of nonlinear differential-algebraic equations, the model is first discretized into a dynamic optimization problem whose constraints are algebraic equations, and then the primal-dual interior point method (PDIPM) is utilized to solve the discretized model. To improve the solution efficiency, the large-scale correction equations of the Karush-Kuhn-Tucker conditions in PDIPM are decoupled into several small-scale groups which can be solved in parallel. Numerical simulations on benchmark systems verify the validity of the proposed OPFCTTB model and corresponding solution methods.

Theoretical and Experimental Investigation of Thermal Dynamics of Steinhart–Hart Negative Temperature Coefficient Thermistors

The temperature-dependent dynamics of a negative temperature coefficient (NTC) thermistor conducting variable electric current is modeled using the differential approach. The thermistor is assumed to follow the Steinhart–Hart resistance-temperature equation. The developed mathematical model consists of a nonlinear differential-algebraic equations system, and it was analyzed by the Adomian decomposition method (ADM) and its time-marching version known as the multistage Adomian decomposition method (MADM) as well as the Dormand–Prince (DP) numerical method. Five sets of experiments were conducted on five different NTC thermistors and the laboratory measurements were compared with the model predictions. It is demonstrated that the proposed model, when combined with the MADM, can accurately simulate the thermal behavior of the NTC thermistors. The MADM reproduces the experimental temperature dynamics of the five NTC thermistors with an average absolute relative error of about 2.601% while the corresponding errors for the DP method and the classic ADM are 8.122% and 51.255%, respectively. Also, it is shown that the MADM is highly efficient in terms of computational efficiency and it is approximately 6.5 times faster than the classic DP method, when tuned appropriately.

Coupling Load-Following Control With OPF

In this paper, the optimal power flow (OPF) problem is augmented to account for the costs associated with the load-following control of a power network. Load-following control costs are expressed through the linear quadratic regulator (LQR). The power network is described by a set of nonlinear differential algebraic equations (DAEs). By linearizing the DAEs around a known equilibrium, a linearized OPF that accounts for steady-state operational constraints is formulated first. This linearized OPF is then augmented by a set of linear matrix inequalities that are algebraically equivalent to the implementation of an LQR controller. The resulting formulation, termed LQR-OPF, is a semidefinite program which furnishes optimal steady-state setpoints and an optimal feedback law to steer the system to the new steady state with minimum load-following control costs. Numerical tests demonstrate that the setpoints computed by LQR-OPF result in lower overall costs and frequency deviations compared to the setpoints of a scheme where OPF and load-following control are considered separately.

Implementation of a Mathematical Modelling of a Rotary Cement Kilns

Rotary cement kiln is the main equipment in the cement industry that has complex dynamic behavior, where any changes will affect the quality of the product and the consumed energy. A one-dimensional model of rotary kiln is needed to understand kiln’s behavior and improve kiln operating and design to achieve the optimum condition of product quality and energy required. In this study, the onedimensional mathematical model of a dry rotary cement kiln with pulverized coal combustion is developed. This model consists of a set of nonlinear ordinary differential equations and nonlinear algebraic equations that describe material and energy balance equations. The model has been solved numerically by using Matlab R2015a, and it has been validated by comparing the result with published experimental data. Based on the result, the steady-state simulation shows that the behavior of the model developed is appropriate with the results presented in the literature. It can be concluded that the model is accurate (error < 6%) to describe the profile of temperature and bed composition along with the kiln. It can be used to obtain a better understanding of kiln’s behavior and improve the kiln operating and design to achieve the optimum condition.

Painlevé Analysis and Abundant Meromorphic Solutions of a Class of Nonlinear Algebraic Differential Equations

In this paper, a class of nonlinear algebraic differential equations (NADEs) is studied. The Painlevé analysis of the NADEs is considered. Abundant meromorphic solutions of the NADEs are obtained by means of the complex method. Then, meromorphic exact solutions of the Schamel‐Korteweg‐de Vries (S‐KdV) equation and (2 + 1)‐dimensional sine‐Gordon equation are derived via the applications of the NADEs.

State Estimation in Direct Contact Membrane Distillation based Desalination Using Nonlinear Observer

Abstract In desalination plants, both measurable and unmeasurable process states play important roles in maintaining closed-loop stability and process operational performance. To effectively control the closed-loop system, an observer design that can estimate the values of the unmeasurable states is required. Motivated by that, in this paper we propose an observer design for an advanced water desalination technique called Direct Contact Membrane Distillation (DCMD). The DCMD process is modeled by a set of nonlinear Differential Algebraic Equations (DAEs). In addition, the effect of the estimation gain matrix on the differentiation index of the DAE system is investigated. Numerical simulations are presented to illustrate the effectiveness of the proposed observer design.

Internal and External Linearization of Semi-Explicit Differential Algebraic Equations

In this paper, we study two kinds of linearization (internal and external) of nonlinear differential-algebraic equations DAEs of semi-explicit SE form. The difference of external and internal linearization is illustrated by an example of a mechanical system. Moreover, we define different levels of external equivalence for two SE DAEs. The proposed explicitation procedure allows us to treat a given SE DAE as a control system defined up to feedback transformation (a class of control systems). Then sufficient and necessary conditions, expressed via explicitation procedure, are given to describe when a given SE DAE is level-3 externally equivalent to a linear SE DAE of some specific forms. At last, we show by an example that level-2 external linearization of a DAE can be achieved if its explicitation is level-2 input-output linearizable.

A direct method for the computation of the oscillatory voltage stability boundary

This paper proposes a two-stage approach for directly computing Hopf bifurcation (HB) points and the associated oscillatory voltage stability boundary in electric power systems. The mathematical formulation of the proposed approach is developed in a unified framework based on both a set of nonlinear differential algebraic equations representing the power system and a set of algebraic equations related to the conditions that must be satisfied for the occurrence of HBs. Both sets of equations are simultaneously solved in a two-stage approach for overcoming the difficulty faced by traditional Newton-based techniques regarding the definition of proper initial conditions for improving convergence towards the problem’s solution. In this context, the original set of equations is first reformulated and solved by using a homotopy method, such that proper initial conditions for the set of state variables to be solved are computed. The original set of equations is then solved in a second stage by the Newton method using the computed initial conditions. Because the oscillatory voltage stability boundary is only composed of HBs points, a practical procedure for directly computing this boundary based on the proposed approach is also presented. Lastly, the effectiveness of the proposed methodology is presented through numerical examples using the IEEE 3-generator 9-bus test system.

Computationally efficient modeling strategy for evaporator performance under frost conditions

Growth of a frost layer on an evaporator surface due to low evaporator temperature as well as moisture contained in surrounding air deteriorates performance of a refrigeration system significantly and requires significant energy for defrost. Many studies have been performed to model the heat and mass transfer phenomena in an attempt to have insight and accurate prediction. However, many models form nonlinear algebraic differential equations which require iterative numerical solvers. Computationally efficient but accurate models are needed in order to evaluate overall system performance. The objective of this paper is to introduce a modeling approach to overcome the problem. A solution strategy based on an enthalpy-based reformulation and linearization method will be presented. Comparisons of the proposed and detailed model results for both flat plate and finned tube heat exchangers are provided. The proposed modeling approach is around 10 times faster than reference models while maintaining comparable accuracy.

The effect of material density on load rate sensitivity in nonlinear viscoelastic material models

This work presents the results of a numerical sensitivity analysis of material density on crack propagation in nonlinear viscoelastic softening material models. Two basic material models are analyzed: Maxwell and Kelvin material models, under the loading with changing rate (slow or fast impact loading). The material is discretized as a lattice model, and the analysis is performed on simple examples consisting of several lattice bars. The mathematical description is based on the theory of dynamical systems, i.e., it is a system of nonlinear differential equations for Kelvin, and a system of nonlinear differential-algebraic equations for Maxwell material model. Sensitivity of displacements on mass and load rate accompanies the analysis.

OPTIMAL CONTROL OF NONLINEAR DUFFING OSCILLATORS: LIE-GROUP APPROACHES

In the optimal control theory, the Hamiltonian formulation is a famous one convenient to find an optimal designed control force. However, when the performance index is a complicated function of control force, the Hamiltonian method is not easy to find the optimal solution, because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations (DAEs). In this paper we address this issue via a quite novel and effective approach, of which the optimally controlled vibration problem of Duffing oscillator is recast into a two-point nonlinear DAEs by identifying the unknown control force. We develop the corresponding SL(n, R) and GL(n, R) shooting methods, as well as a Lie-group differential algebraic equations (LGDAE) method to numerically solve the optimal control forces. Eight examples of a single Duffing oscillator and one coupled Duffing oscillators are used to test the performance of the present method.

OPTIMAL CONTROL OF NONLINEAR DUFFING OSCILLATORS: LIE-GROUP APPROACHES

In the optimal control theory, the Hamiltonian formulation is a famous one convenient to find an optimal designed control force. However, when the performance index is a complicated function of control force, the Hamiltonian method is not easy to find the optimal solution, because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations (DAEs). In this paper we address this issue via a quite novel and effective approach, of which the optimally controlled vibration problem of Duffing oscillator is recast into a two-point nonlinear DAEs by identifying the unknown control force. We develop the corresponding SL(n, R) and GL(n, R) shooting methods, as well as a Lie-group differential algebraic equations (LGDAE) method to numerically solve the optimal control forces. Eight examples of a single Duffing oscillator and one coupled Duffing oscillators are used to test the performance of the present method.

Nonlinear coordinated control of STATCOM and generator excitation for multi‐machine power systems based on improved multi‐objective holographic feedback

Aiming at the coordination control problem of the static synchronous compensator (STATCOM) and generator excitation in a multi‐machine system, an improved multi‐objective holographic feedback (IMOHF) control method is proposed to ensure good dynamic and static state performance of each state in the system. First, the essence of IMOHF is deduced by the Hartman–Grobman theorem. The nonlinear differential algebraic equation (NDAE) system is used to arrange the poles in the vicinity of the equilibrium point by nonlinear feedback; the feedback coefficients are adjusted to constrain the objective variables, so that the variables converge rapidly in their respective references. Then this design concept is applied to the decentralized coordination control strategy of STATCOM and generator excitation. Two kinds of multi‐objective holographic feedback nonlinear control laws with active power, synchronous speed, generator terminal voltage, and STATCOM terminal voltage as control objectives are obtained. Simulation result of the three‐machine nine‐bus system shows that these two kinds of control laws can effectively restrain the power system oscillation and improve the transient stability of the system, and consequently the system voltage will not generate any static shift under different disturbances. © 2018 Institute of Electrical Engineers of Japan. Published by John Wiley &amp; Sons, Inc.

Generalized Jacobi–Galerkin method for nonlinear fractional differential algebraic equations

In this paper, we provide an approximate approach based on the Galerkin method to solve a class of nonlinear fractional differential algebraic equations. The fractional derivative operator in the Caputo sense is utilized and the generalized Jacobi functions are employed as trial functions. The existence and uniqueness theorem as well as the asymptotic behavior of the exact solution are provided. It is shown that some derivatives of the solutions typically have singularity at origin dependence on the order of the fractional derivative. The influence of the perturbed data on the exact solutions along with the convergence analysis of the proposed scheme is also established. Some illustrative examples provided to demonstrate that this novel scheme is computationally efficient and accurate.

Index reduction of differential algebraic equations by differential Dixon resultant

High index differential algebraic equations (DAEs) are ordinary differential equations (ODEs) with constraints and arise frequently from many mathematical models of physical phenomenons and engineering fields. In this paper, we generalize the idea of differential elimination with Dixon resultant to polynomially nonlinear DAEs. We propose a new algorithm for index reduction of DAEs and establish the notion of differential Dixon resultant, which can provide the differential resultant of the enlarged system of original equations. To make use of structure of DAEs, variable pencil technique is given to determine the termination of differentiation. Moreover, we also provide a heuristic method for removing the extraneous factors from differential resultant. The experimentation shows that the proposed algorithm outperforms existing ones for many examples taken from the literature.