Differential Algebraic Equation Model Research Articles

An integrated approach for sensor placement in linear dynamic systems

This paper investigates the sensor placement in linear dynamic systems for fault detectability and isolability. A linear system is modelled by a bond graph (BG) that can be associated in a natural way with a set of linear differential-algebraic equations (DAEs). Simultaneously, possible sensor locations are modelled as junctions on the BG. Causal paths capture cause–effect relationships of a linear system and provide a means to analyze what subset of junctions contributes to fault detectability and isolability. Furthermore, this paper exploits DAEs associated with a BG and proves a necessary and sufficient condition of sensor placement to fulfill fault detectability. Based on the fault detectability condition, a necessary condition of sensor placement to achieve two-fault distinguishability is developed in the DAEs model and serves as the basis of formulating the sensor placement problem with regard to a fault set F. For efficiency, a dedicated dynamic programming (DP) algorithm is devised to attain the optimal set of junctions for fault isolability. The two-tank system is employed to illustrate the sensor placement steps in a linear system. The optimal set(s), computed by the proposed sensor placement methodology, is/are validated by deriving primary analytical redundancy relations (ARRs) of the two-tank system.

Nonlinear programming strategies for dynamic chemical process optimization

Problem formulations and algorithms are considered for optimization problems with differential-algebraic equation (DAE) models. In particular, we provide an overview of direct methods, based on nonlinear programming (NLP), and indirect, or variational, methods. We further classify each method and tailor it to the appropriate applications. For direct methods, we briefly describe current approaches including the sequential approach (or single shooting), multiple shooting method, and the simultaneous collocation (or direct transcription) approach. In parallel to these strategies we discuss NLP algorithms for these methods and discuss optimality conditions and convergence properties. In particular, we present the simultaneous collocation approach, where both the state and control variable profiles are discretized. This approach allows a transparent handling of inequality constraints and unstable systems. Here, large scale nonlinear programming strategies are essential and a novel barrier method, called IPOPT, is described. This NLP algorithm incorporates a number of features for handling large-scale systems and improving global convergence. The overall approach is Newton-based with analytic second derivatives and this leads to fast convergence rates. Moreover, it allows us to consider the extension of these optimization formulations to deal with nonlinear model predictive control and real-time optimization. To illustrate these topics we consider a case study of a low density polyethylene (LDPE) reactor. This large-scale optimization problem allows us to apply off-line parameter estimation and on-line strategies that include state estimation, nonlinear model predictive control and dynamic real-time optimization.

Robust Kinetic Modelling of Heterogeneously Catalyzed Free Fatty Acids Esterification in (Monophasic Liquid)/Solid Packed-Bed Reactor: Rival Model Discrimination

Biofuels are key products for the sustainability of the world energy consumption in the next years. Biodiesel in particular is a non-toxic, biodegradable, environmentally friendly alternative diesel fuel. Nowadays, the main problem for the commercialization of biodiesel is its final cost, that is strongly dependent (about 85 % of the total) by the feedstock used. A possible way to lower the biodiesel production costs is using raw oils, which contain a higher amount of Free Fatty Acids (FFA) that should be eliminated before the transesterification reaction to avoid soaps formation. In this work the regression of kinetic parameters of heterogeneously catalyzed esterification with a packed-bed reactor was made. Robust techniques for kinetic parameters estimation and simultaneous discrimination of rival models were adopted and combined with a dedicated differential-algebraic equation (DAE) model that characterizes the system. The main kinetic parameters were regressed using kinetic models other than literature models in considering the activity (UNIQUAC model was used to calculate the activity coefficients) of the components instead of the concentration because the oil/methanol/water/FAME system is highly non ideal. The kinetic parameters were obtained using equilibrated resins, i.e. using the catalyst after having let it to adsorb reactants and products at the operative conditions. The regressed parameters allow to represent the system in a wide range of operating conditions with a little residual error.

A brand new nonlinear robust control design of SSSC for transient stability and damping improvement of multi-machine power systems via pseudo-generalized Hamiltonian theory

Nonlinear robust control of static synchronous series compensator (SSSC) is investigated in multi-machine multi-load power systems by using the pseudo-generalized Hamiltonian method. First, the uncertain nonlinear differential algebraic equation model is constructed for the power system. Then, the dissipative pseudo-generalized Hamiltonian realization of the system is established by means of variable transformation and prefeedback control. Finally, based on the obtained dissipative pseudo-generalized Hamiltonian realization, a brand new nonlinear robust controller is put forward. The proposed controller can effectively use the internal structure and the energy balance property of the power system. Simulation results demonstrate the effectiveness and robustness of the control scheme.

Converting DAE models to ODE models: application to reactive Rayleigh distillation

This paper illustrates the application of an index reduction method to some differential algebraic equations (DAE) modelling the reactive Rayleigh distillation. After two deflation steps, this DAE is converted to an equivalent first-order explicit ordinary differential equation (ODE). This ODE involves a reduced number of dependent variables, and some evaluations of implicit functions defined, either from the original algebraic constraints, or from the hidden ones. Consistent initial conditions are no longer to be computed; at the opposite of some other index reduction methods, which generate a drift-off effect, the algebraic constraints remain satisfied at any time; and, finally, the computational effort to solve the ODE may be less than the one associated to the original DAE.

Analysis of Power System Dynamics Subject to Stochastic Power Injections

We propose a framework to study the impact of stochastic active/reactive power injections on power system dynamics with a focus on time scales involving electromechanical phenomena. In this framework, the active/reactive power injections evolve according to a continuous-time Markov chain (CTMC), while the power system dynamics are described by the standard differential algebraic equation (DAE) model. The DAE model is linearized around a nominal set of active/reactive power injections, and the combination of the linearized DAE model and the CTMC forms a stochastic process known as a stochastic hybrid system (SHS). The extended generator of the SHS yields a system of ordinary differential equations that governs the evolution of the power system dynamic and algebraic state moments. We illustrate the application of the framework to the computation of long-term power system state statistics, and to short-term probabilistic dynamic performance/reliability assessment.

STRUCTURAL ANALYSIS OF HIGH-INDEX DAE FOR PROCESS SIMULATION

This paper deals with the structural analysis problem of dynamic lumped process high-index differential algebraic equations (DAE) models. The existing graph theoretical method depends on the change in the relative position of underspecified and overspecified subgraphs and has an effect to the value of the differential index for complex models. In this paper, we consider two methods for index reduction of such models by differentiation: Pryce's method and the symbolic differential elimination algorithm rifsimp. They can remedy the above drawbacks. Discussion and comparison of these methods are given via a class of fundamental process simulation examples. In particular, the efficiency of Pryce's method is illustrated as a function of the number of tanks in process design. Moreover, a range of nontrivial problems are demonstrated by the symbolic differential elimination algorithm and fast prolongation.

Nonlinear Coordinated Control of STATCOM and Generator Excitation Based on Improved Dynamic Surface Control Method

For the single machine connected to infinite bus power system with uncertainties, one nonlinear coordinated control scheme for Static Synchronous Compensator (STATCOM) and excitation is proposed in this paper. Firstly, in order to avoid solving the differential algebraic equations (DAEs) model of system, we simplify the DAEs into the classical differential equations, and then a new nonlinear parameter strict feedback model is given. Secondly, in order to make the system achieve the desired results, the controller is designed in two parts based on improved dynamic surface control method (IDSC) and passive control techniques. The theoretical analysis shows that the derived controller can not only attenuate the influences of external disturbances, but also has strong robustness for system parameters variety. The control law obtained is more effective and the system globally and uniformly ultimately bounded can be achieved using full nature of nonlinear dynamic. Lastly, the further simulation results indicate that the proposed controller can ensure transient stability of the power system under large sudden fault.

Decentralized robust control of multiple static var compensators via Hamiltonian function method

Using the energy-based Hamiltonian function method, this paper investigates the decentralized robust nonlinear control of multiple static var compensators (SVCs) in multimachine multiload power systems. First, the uncertain nonlinear differential algebraic equation model is constructed for the power system. Then, the dissipative Hamiltonian realization of the system is completed by means of variable transformation and prefeedback control. Finally, based on the obtained dissipative Hamiltonian realization, a decentralized robust nonlinear controller is put forward. The proposed controller can effectively utilize the internal structure and the energy balance property of the power system. Simulation results verify the effectiveness of the control scheme.

CTDAE & CTODE models and their applications to power system stability analysis with time delays

In this paper, two new models suitable for power system stability analysis with considering time delays are derived. They are constraint time-delayed ordinary differential equation model (CTODE model) and constraint time-delayed differential algebraic equation model (CTDAE model). In these two models, dimension of the time-delayed equation is significantly reduced so as to speed up the calculation efficiency. Detail derivations to their nonlinear and incremental linear forms are given. And, some of their good features are discussed. Finally, a typical time-delayed system and single-machine-infinite-bus system are selected to validate the new models through Lyapunov-Krasovskii criterion and eigenvalue locus tracing procedure. It reveals that some unnecessary calculations in the algorithm can be avoided to achieve higher efficiency based on new models. The new models and new criterion given in this paper are helpful to design wide area protectors for bulk power systems.

Model-Based Dynamic Optimization with OpenModelica and CasADi

This paper demonstrates model-based dynamic optimization through the coupling of two open source tools: OpenModelica, which is a Modelica-based modeling and simulation platform, and CasADi, a framework for numerical optimization. The coupling uses a standardized XML format for exchange of differential-algebraic equations (DAE) models. OpenModelica supports export of models written in Modelica and the optimization language extension using this XML format, while CasADi supports import of models represented in this format. This allows users to define optimal control problems (OCP) using Modelica and optimization language specification, and solve the underlying model formulation using a range of optimization methods, including direct collocation and direct multiple shooting. The proposed solution has been tested on several industrially relevant optimal control problems, including a diesel-electric power train.

Adaptive observer based fault diagnosis applied to differential-algebraic systems

Some engineering systems are naturally described by differential-algebraic equations (DAE), whereas it may be difficult or impossible to model them with ordinary differential equations (ODE). This paper proposes an approach to fault diagnosis for systems described by DAEs. Through a particular discretization method and under realistic assumptions, the considered continuous time DAE model is transformed to an explicit state space model in discrete time. An adaptive observer is then applied to the discretized system for monitoring faults possibly affecting the system and represented by changes in model parameters. As an illustrative example, the diagnosis of faults in a heat exchanger modeled by nonlinear DAEs is studied by numerical simulations.

Order-reduction Strategy of Non-linear Differential-algebraic Equation Models with Application on Power Systems

The time evolution of many physical processes in power systems is described by non-linear differential-algebraic equations. In this article, a new reduced-order modeling technique for constructing explicit state-space approximations of non-linear power system models described by differential-algebraic equation models is developed. The proposed technique takes into account the interconnection structure and provides information on the effect of control actions on system behavior. A general second-order theory is presented for treating non-linear behavior in large non-linear power system models. In this approach, the algebraic equations are first reduced by projecting the original system model onto a truncated basis of the subspace generated by the right and left eigenvectors of the linear system model. A technique for constructing an explicit state-space approximation of the non-linear differential-algebraic equation model is then developed. The method is highly efficient and capable of developing state-space approximations of interest for non-linear simulation and control design while maintaining acceptable accuracy. Examples of the application of the developed procedures on a 50-machine test system are presented to estimate the validity of the reduction technique.

Determination of reduced order model for torsional interaction investigations using selective modal analysis – Part II: Application to practical power systems

A new strategy is proposed based on selective modal analysis (SMA) for the determination of reduced order model in torsional interaction investigations of large electric power system. SMA computes eigenvalues associated with selected modes, then constructs a reduced order model that involves only the relevant variables, incorporating the effect of the less relevant dynamics by low order model. The reduction process is carried out based on linear differential algebraic equation (LDAE) model and illustrated on IEEE benchmark model. The model does not require preprocessing of data for removal of network state redundancy. A case study is conducted on a very large electric power system for validation of the results. The reduced order model finds potential application for real-time simulation and also for testing of SSR damping controllers.

Pareto Approach in Designing Optimal Semicontinuous Water Networks

Pareto optimization approach was used in searching for the particular successions of topologies and operating conditions for a semicontinuous water network (batch with respect to the raw materials) that minimize both the freshwater consumption and the investment and operating costs. The semicontinuous wastewater network (SWN) formed of N water-using units (WUs), each handling at most K contaminants, and one storage tank (ST) is optimized on time intervals by increasing the wastewater reuse opportunities. A single freshwater source is available and later a regeneration unit (RU) is added to the network as a fine-tuning tool in the Pareto optimization, and further its influence is discussed. The WUs are ordered simultaneously by schedule and by maximum allowable outlet concentrations of contaminants. Thus, the rules concerning the internal reuse between the overlapping WUs correspond to an oriented graph: no recycle is allowed against the ordered sequence. The differential-algebraic equations model consisting in a set of differential equations for the ST and a nonlinear system mass balance equations with restrictions for the overlapping WUs is optimized using the Matlab built in functions for both the genetic algorithms (GA) and the RK-type integrator. The topology of the SWN changes for each time interval, its optimal structure depending upon the minimum freshwater consumption and the costs. A complex synthetic case study was subjected to optimization, and several scenarios are analyzed.

Optimal location and signal selection of UPFC device for damping oscillation

Unified power flow controller (UPFC) is used for controlling the real and reactive power in transmission line and bus voltage simultaneously and independently. An additional task of UPFC is to increase transmission capacity as result of power oscillation damping. The effectiveness of this controller depends on its optimal location and proper signal selection in the power system network. A residue factor has been proposed to find the optimal location of the UPFC controllers and eigenvalue analyses are used to assess the most appropriate input signals (stabilizing signal) for supplementary damping control of UPFC to damp out the inter-area mode of oscillations. The proposed residue factor is based on the relative participation of the parameters of UPFC controller to the critical mode. A simple approach of computing the residue factor has been proposed, which combines the linearized differential algebraic equation model of the power system and the UPFC output equations. While for signal selection a right-half plane zeros (RHP zeros) and Hankel singular value (HSV) is used as tools to select the most receptive signal to a mode of the inter-area oscillation. The placements of UPFC controllers have been obtained for the base case and for the dynamic critical contingences. The effectiveness of the proposed method of placement and selection of signals are demonstrated on practical network of TNB 25 bus system of south Malaysian network and New England 39 bus system.

A perturbation approach for consistent initialization of index-1 explicit differential–algebraic equations arising from battery model simulations

Estimation of consistent initial conditions is very crucial for the successful solution of differential–algebraic equation (DAE) systems that arise in many fields of science and engineering. In this paper, an efficient perturbation approach for initialization of DAE systems of index-1 is proposed and implemented for DAE models governing batteries. In addition, different existing solvers are compared for consistent initialization of DAE systems. The proposed approach does not necessarily require a nonlinear solver for initialization and builds on the applicability and usability of robust and efficient explicit, linearly implicit and semi-implicit integrators in time. Three different problems are presented wherein the proposed approach is observed to work for a wider range of inconsistent initial conditions compared to other existing generally used routines. It is also observed that the present approach is computationally efficient compared to the other existing approaches in a given environment.

Power system voltage small-disturbance stability studies based on the power flow equation

This study first studies power system small-disturbance stability at the operating point where the power flow (PF) equation encounters a saddle-node bifurcation. The authors demonstrate that the linearised model of the differential-algebraic equation (DAE) that describes the power system dynamics will have a zero eigenvalue at the equilibrium precisely when the PF Jacobian is singular. Note that the PF equation and DAE models are general ones. This clarifies a point in previous contributions on this relationship. Numerical results for two power system examples are used to demonstrate the theory, and finally the extension of the theory is discussed for the limit-induced bifurcation associated with the PF equation when some generators reach their reactive power limits.

Regularized numerical integration of multibody dynamics with the generalized [formula omitted] method

This paper discusses the consistent regularization property of the generalized α method when applied as an integrator to an initial value high index and singular differential-algebraic equation model of a multibody system. The regularization comes from within the discretization itself and the discretization remains consistent over the range of values the regularization parameter may take. The regularization involves increase of the smallest singular values of the ill-conditioned Jacobian of the discretization and is different from Baumgarte and similar techniques which tend to be inconsistent for poor choice of regularization parameter. This regularization also helps where pre-conditioning the Jacobian by scaling is of limited effect, for example, when the scleronomic constraints contain multiple closed loops or singular configuration or when high index path constraints are present. The feed-forward control in Kane’s equation models is additionally considered in the numerical examples to illustrate the effect of regularization. The discretization presented in this work is adopted to the first order DAE system (unlike the original method which is intended for second order systems) for its A-stability and same order of accuracy for positions and velocities.

Optimal Location of Static Var Compensator Device for Damping Oscillations

Problem statement: Static Var Compensators (SVC) devices are used to improve voltage and reactive power conditions in AC systems. An additional task of SVC is to increase transmission capacity as result of power oscillation damping. The effectiveness of this controller depends on its optimal location and proper signal selection in the power system network. A residue factor had been proposed to find the optimal location of the SVC controllers to damp out the inter-area mode of oscillations. Approach: The proposed residue factor was based on the relative participation of the parameters of SVC controller to the critical mode. A simple approach of computing the residue factor had been proposed, which combined the linearized differential algebraic equation model of the power system and the SVC output equations. Input-output controllability analyses were used to assess the most appropriate input signals (stabilizing signal) for SVC device. Results: The placements of SVC controller had been obtained for the base case as well as for the critical contingency cases. Conclusion: The effectiveness of the proposed method was demonstrated on 25 bus of south Malaysian power system.