Nonlinear Partial Differential Algebraic Equations Research Articles

Non-Iterative Calculation of Quasi-Dynamic Energy Flow in the Heat and Electricity Integrated Energy Systems

Quasi-dynamic energy flow calculation is an indispensable tool for the heat and electricity integrated energy system (HE-IES) analysis. One solves the nonlinear partial differential algebraic equations to obtain thermal, hydraulic and electric variations. However, mainstream iteration solvers face the challenges of inefficiency and bad robustness. For one thing, the frequent update and factorization of Jacobian matrices utilize high CPU time. For another, the per-step iteration numbers grow exponentially as the system loading level creeps up. This paper presents a novel non-iterative algorithm for the quasi-dynamic energy flow calculation. The kernel of the proposed algorithm is to transform these nonlinear equations into linear recursive ones, by solving which, we obtain explicit closed-form solutions of unknown variables. In each step, the proposed algorithm requires only one matrix factorization and fixed times of arithmetic operations regardless of the loading levels, so that it achieves small and consistent per-step time costs. A semi-discrete scheme is used in PDE solution to avoid dissipative and dispersive errors that are often overlooked in previous literature. To ensure convergence, we also propose to control the temporal step sizes adaptively by estimating the simulation errors. Case studies showed that the proposed method manifested efficient and robust time performance compared with the iterative algorithms, and meanwhile preserved high accuracy.

Simulation of Reactive Transport in Fractured Porous Media

Numerical simulations of reactive transport in fractured porous media require the solution of coupled physical and chemical processes that depend on the fractures. Such coupled processes are described by a system of nonlinear partial differential-algebraic equations, while strong heterogeneities characterise fractures. This paper presents an approach to simulate single-phase flow and non-isothermal reactive transport with mineral dissolution and precipitation in fractured porous media. Our numerical solution strategy is based on two ingredients. First, the model equations consist of coupled partial differential equations for the fluid flow, heat transfer and solute transport and nonlinear algebraic equations representing the chemical reactions. Second, fractures are explicitly represented and treated as lower-dimensional objects. The partial differential equations are discretised using finite-volume methods, and at each time step, we solve a nonlinear system of equations using Newton’s method. With numerical simulations, we illustrate our model’s ability to accurately describe the two-way interaction between coupled multi-physical processes and two- and three-dimensional porous media with intersecting fractures.

Nonlinear Dynamic Energy Flow Calculation for Integrated Gas and Power System Based on Method of Lines

The rapid development of gas turbine (GT) and power-to-gas (P2G) technology promotes the integration of power systems (PS) and natural gas systems (NGS). To accurately describe the state distribution, this paper first introduces the model of integrated gas and power system (IGPS) governed by nonlinear partial differential-algebraic equations (PDAEs). A third-order method of lines (MoL) is proposed to solve the nonlinear partial differential equations (PDEs) in the natural gas system. The distribution of mathematical equations and variables in the energy flow model is then analyzed. The solution strategy for nonlinear dynamic energy flow calculation (EFC) in integrated gas and power system is then developed, including the equation set formulation, solvability analysis, and initial guess selection method. Finally, the accuracy and efficiency of the proposed method are verified through simulation in two different cases.

Nonlinear Identifiability Analysis of the Porous Electrode Theory Model of Lithium-Ion Batteries

Porous electrode theory (PET) is widely used to model battery cycling behavior by describing electrochemical kinetics and transport in solid particles and electrolyte, and modeling thermodynamics by fitting an open-circuit potential. The PET model consists of tightly coupled nonlinear partial differential-algebraic equations in which effective kinetic and transport parameters are fit to battery cycling data, and then the model is used to analyze the effects of variations in design parameters or operating conditions such as charging protocols. In a detailed nonlinear identifiability analysis, we show that most of the effective model parameters in porous electrode theory are not practically identifiable from cycling data for a lithium-ion battery. The only identifiable parameter that can be identified from C/10 discharge data is the effective solid diffusion coefficient, indicating that this battery is in the diffusion-limited regime at this discharge rate. A resistance in series correlation was shown for the practically unidentifiable parameters by mapping out the confidence region. Alternative experiments in addition to discharge cycles are required in order to uniquely determine the full set of parameters.

Automatic control of simulated moving bed process with deep Q-network

Optimal control of a simulated moving bed (SMB) process is challenging because the system dynamics is represented as nonlinear partial differential-algebraic equations combined with discrete events. In addition, product purity constraints are active at the optimal operating condition, which implies that these constraints can be easily violated by disturbance. Recently, artificial intelligence techniques have received significant attention for their ability to address complex problems, involving a large number of state variables. In this study, a data-based deep Q-network, which is a model-free reinforcement learning method, is applied to the SMB process to train a near-optimal control policy. Using a deep Q-network, the control policy of a complex dynamic system can be trained off-line as long as a sufficient number of data is provided. These data can be efficiently generated by performing numerical simulations in parallel on multiple machines. The on-line computation of the control input using a trained Q-network is fast enough to satisfy the computational time limit for the SMB process. However, because the Q-network does not predict the future state, it is not possible to explicitly impose state constraints. Instead, the state constraints are indirectly imposed by providing a relatively large penalty (negative reward) when the constraints are violate. Furthermore, logic-based switching control is utilized to limit the ranges of the extract and raffinate purities, which helps to satisfy the state constraints and reduce the regions in the state space for reinforcement learning to explore. The simulation results demonstrate the advantages of applying deep reinforcement learning to control the SMB process.

A Reduced-order Model for Real-time NMPC of Ethanol Steam Reformers

One of the approaches to green energy of interest in recent years is through the use of hydrogen technologies in which the waste product of combustion is water rather than carbon dioxide, nitrous oxides, and other pollutants generated by the burning of hydrocarbon fuels. Ethanol steam reformers is one of the primary technologies with the potential to generate hydrogen efficiently and reliably. This article investigates the design of nonlinear optimal feedback control systems for ethanol steam reformers based on mechanistic models described by a system of nonlinear partial differential-algebraic equations, aka singular/descriptor systems. The mechanistic models are too complicated to simulate in real-time using the standard numerical discretization methods, much less to directly incorporate such models as nonlinear constraints into mechanistic model-based nonlinear model predictive control formulations. This article employs a rather sophisticated multivariate form of the method of characteristics to reformulate the nonlinear constraints into a much simpler form amenable to real-time control implementations.

Time Stepping Methods and Solvers for Battery Models

There are a wide range of battery models at different scales, from empirical models to molecular dynamics models, that can describe the battery behavior. The Pseudo 2-Dimensional (P2D) model considers the porous electrode theory, concentrated solution theory, Ohm’s law, kinetic relationships, as well as charge and material balances.1 These physic-based behaviors are described by a set of stiff nonlinear partial differential algebraic equations (DAEs) which can be only solved numerically. In this talk, we discuss and review different methods to simulate the battery models, specifically about integration in the time domain. As the P2D model discretized in spatial coordinates by using any suitable method such as finite difference2-3, finite volume4 and spectral methods5-8, it results in a system of nonlinear DAEs. Typically, nonlinear DAEs can be solved based on Runge-Kutta (RK) methods (explicit or implicit) or multistep methods. The different orders of accuracy will affect the accuracy of the numerical solutions for each time step and it will further affect the computational efficiency. However, most of the time the stability region becomes smaller when the order of accuracy of the method increases. For the multistep method, Backward Differentiation Formula (BDF) method is commonly used, because we can get more than second order of accuracy without increasing the number of variables.9 Both RK and BDF methods will be reviewed for simulating battery models. Subtle differences among different methods and efficiency improvement and robustness of all these methods will be analyzed and discussed. In particular, the compromise between, stability, accuracy, and ease of programming will be discussed. Implementation in different programming languages and platforms will also be compared. Acknowledgements The authors are thankful for the financial support of this work by the Clean Energy Institute (CEI) at the University of Washington.

Modeling boyciana–fish–human interaction with partial differential algebraic equations

Under the influence of human population distribution, the boyciana–fish ecological system is considered. First, the system can be described as a nonlinear partial differential algebraic equations system (PDAEs) with Neumann boundary conditions and ratio-dependent functional response. Second, we examine the system’s persistence properties: the loacl stabilities of positive steady states, the absorbtion region and the global stability. And the proposed approach is illustrated by numerical simulation. Finally, by using the realistic data collected in the past fourteen years, the PDAEs parameter optimization model is built to predict the boyciana population.

Low-order mathematical modelling of electric double layer supercapacitors using spectral methods

This work investigates two physics-based models that simulate the non-linear partial differential algebraic equations describing an electric double layer supercapacitor. In one model the linear dependence between electrolyte concentration and conductivity is accounted for, while in the other model it is not. A spectral element method is used to discretise the model equations and it is found that the error convergence rate with respect to the number of elements is faster compared to a finite difference method. The increased accuracy of the spectral element approach means that, for a similar level of solution accuracy, the model simulation computing time is approximately 50% of that of the finite difference method. This suggests that the spectral element model could be used for control and state estimation purposes. For a typical supercapacitor charging profile, the numerical solutions from both models closely match experimental voltage and current data. However, when the electrolyte is dilute or where there is a long charging time, a noticeable difference between the numerical solutions of the two models is observed. Electrical impedance spectroscopy simulations show that the capacitance of the two models rapidly decreases when the frequency of the perturbation current exceeds an upper threshold.

과산화수소 농축을 위한 투과증발공정 모델링

본 연구의 목적은 과산화수소 농축을 위한 투과증발공정의 수학적 모델을 제시하는 것이다. 대상공정은 NASA에서 개발한 것으로 쉘과 멤브레인 튜브들로 구성되어 있다. 쉘과 튜브에는 각각 과산화수소용액과 sweep gas가 향류로 흐른다. 이때 막을 통해 투과증발되는 기체에는 과산화수소보다 물분자가 더 많기 때문에 과산화수소를 농축할 수 있다. 이 공정의 수학적 모델은 투과물의 흡수-확산 메커니즘에 기초한 비선형 편미분 및 대수 방정식, 투과물 플럭스의 온도 의존성에 대한 아레니우스 관계, 그리고 막 모듈 내의 액상 농도와 흐름에 대한 질량 및 운동량 수지식의 형태로 개발하였다. 과산화수소 농축물 농도의 동적 거동은 제안된 모델의 단순화된 형태를 풀어 모사하였고 그 결과값을 NASA 특허에서 보고된 실험자료와 비교하였다. The objective of this study is to propose a mathematical model for a pervaporation process for concentrating hydrogen peroxide. The process was developed by NASA, which consists of a shell and membrane tubes, where a liquid hydrogen peroxide solution flows in the shell, and a sweep gas flows in the tubes countercurrent to each other. The liquid retentate is concentrated as more water molecules permeate and evaporate through the membrane than hydrogen peroxide. For this process, a mathematical model has been developed in the form of a system of nonlinear partial differential algebraic equations based on a sorption-diffusion mechanism for permeation, an Arrhenius relationship for the temperature dependency of the permeate flux, and mass and momentum balances for the liquid concentrations and flows in the membrane module. The dynamic behavior of the concentration of hydrogen peroxide in the retentate side has been simulated by solving a simplified version of the proposed model, and the result is compared with the experimental data reported in the NASA patent.

Solution of the nonlinear PDAEs by variational iteration method and its applications in nanoelectronics

In this paper, the system of nonlinear partial differential-algebraic equations is solved by the wellknown variational iteration method and the results with high accuracy are obtained by only one iteration. Furthermore, some nanoelectronics models are expressed by partial differential-algebraic equations and one of them is successfully solved by the proposed method. Although solving nonlinear PDAEs is difficult but it is shown that the variational iteration method using Taylor expansion is an efficient method to solve these nonlinear problems.

Self-heating in a coupled thermo-electric circuit-device model

The self-heating of a coupled thermo-electric circuit-semiconductor system is modeled and numerically simulated. The system consists of semiconductor devices, an electric network with resistors, capacitors, inductors, and voltage sources, and a thermal network. The flow of the charge carriers is described by the energy-transport equations coupled to a heat equation for the lattice temperature. The electric circuit is modeled by the network equations from modified nodal analysis coupled to a thermal network describing the evolution of the temperatures in the lumped and distributed circuit elements. The three subsystems are coupled through thermo-electric, electric circuit-device, and thermal network-device interface conditions. The resulting system of nonlinear partial differential-algebraic equations is discretized in time by the 2-stage backward difference formula and in space by a mixed finite-element method. Numerical simulations of a one-dimensional ballistic diode and a frequency multiplier circuit containing a pn-junction diode illustrate the heating of the semiconductor device and circuit resistors.

Combination of an adaptive multilevel SQP method and a space-time adaptive PDAE solver for optimal control problems

We present an adaptive multilevel generalized SQP method to solve PDAE-constrained optimization problems. It explicitly allows the use of independent integration schemes such that all involved systems can be solved highly efficient and as accurate as desired. We will couple the optimization with the state-of-the-art PDAE solver KARDOS and apply the new tool to a radiative heat transfer problem described by space-time dependent non-linear partial differential algebraic equations. Results are presented, compared and discussed.

A global strategy for solving reactive transport equations

Reactive transport models are complex nonlinear Partial Differential Algebraic Equations (PDAE), coupling the transport engine with the geochemical operator. We propose an efficient and robust global numerical method, based on a method of lines and Differential Algebraic Equations (DAE) solvers, combined with a Newton method using a powerful sparse linear solver. Numerical experiments show the performances of the method. We also propose a unified framework to describe classical methods such as Sequential Non-Iterative Approach, Sequential Iterative Approach, Direct Substitution Approach and compare them with our global method.

SIMULATION OF THERMAL EFFECTS IN OPTOELECTRONIC DEVICES USING COUPLED ENERGY-TRANSPORT AND CIRCUIT MODELS

A coupled model with optoelectronic semiconductor devices in electric circuits is proposed. The circuit is modeled by differential-algebraic equations derived from modified nodal analysis. The transport of charge carriers in the semiconductor devices (laser diode and photo diode) is described by the energy-transport equations for the electron density and temperature, the drift-diffusion equations for the hole density, and the Poisson equation for the electric potential. The generation of photons in the laser diode is modeled by spontaneous and stimulated recombination terms appearing in the transport equations. The devices are coupled to the circuit by the semiconductor current entering the circuit and by the applied voltage at the device contacts, coming from the circuit. The resulting time-dependent model is a system of nonlinear partial differential-algebraic equations. The one-dimensional transient transport equations are numerically discretized in time by the backward Euler method and in space by a hybridized mixed finite-element method. Numerical results for a circuit consisting of a single-mode heterostructure laser diode, a silicon photo diode, and a high-pass filter are presented.

Index analysis and numerical solution of a large scale nonlinear PDAE system describing the dynamical behaviour of Molten Carbonate Fuel Cells

This paper deals with the efficient simulation of the dynamical behaviour of Molten Carbonate Fuel Cells (MCFCs). MCFCs allow an efficient and environmentally friendly energy production via electrochemical reactions. Their dynamics can be described by large scale systems of up to currently 22 nonlinear partial differential algebraic equations (PDAE). The paper also serves as a basis for later parameter identification and optimal control purposes. Therefore, the numerical simulations are particularly based on hierarchically embedded systems of PDAE, first of all in one space dimension. The PDAE are of mixed parabolic-hyperbolic type and are completed by nonlinear initial and boundary conditions of mixed type. For a series of embedded models in one space dimension, the vertical method of lines (MOL) is used throughout this paper. For the semi-discretization in space appropriate difference schemes are applied depending on the type of equations. The resulting system of ordinary differential algebraic equations (DAE) in time is then solved by a standard RADAU5 method. In order to justify the numerical procedure, a detailed index analysis of the PDAE systems with respect to time index, spatial index and MOL index is carried through. Because of the nonlinearity of the PDAE system, the existing theory has to be generalized. Moreover, MOL is especially suited for near optimal real time control on the basis of a sensitivity analysis of the semi-discretized DAE system, since a theoretically safeguarded sensitivity analysis does not exist so far for PDAE constrained optimal control problems of the above type. Numerical results complete the paper and show their correspondence with the expected dynamical behaviour of MCFCs.

A Differentiation Index for Partial Differential-Algebraic Equations

A differentiation index for nonlinear partial differential-algebraic equations is presented. Determination of the differentiation index with respect to a direction in the space of independent variables uncovers all equations that must be satisfied by Cauchy data on the hyperplane orthogonal to that direction. This index is thus a generalization of the differentiation index of differential-algebraic equations. The motivation for this work is consistent initialization of partial differential-algebraic systems. The analysis is demonstrated through several examples, including some taken from fluid dynamics and chemical engineering.