Sparse Tableau Research Articles

Complete Response of Linear Circuits to Periodic Nonsinusoidal Input in MATLAB

The paper deals with the proposal for finding<br />the complete response of dynamic linear circuits to a<br />periodic nonsinusoidal input in the MATLAB environment.<br />A very powerful tool for solving the given problem is to<br />transform the circuits directly into the complex frequency<br />domain using the Laplace transform and then apply the<br />sparse tableau analysis technique to solve them. Applying<br />above-mentioned methods in the MATLAB environment, it<br />is not difficult to find the complete response of dynamic<br />linear circuits to the periodic input.

Benefits of Sparse Tableau Over Nodal Admittance Formulation for Power-Flow Studies

Assembly of power-flow equations has traditionally begun from a nodal analysis formulation of the underlying transmission circuit's behavior. Most power-flow formulations encapsulate network constraints in the bus admittance matrix, $Y_{\text{bus}}$ . From a circuit perspective, this admittance representation restricts the network elements to be voltage controlled; resulting drawbacks treating zero-impedance branches in such applications as state estimation have long been recognized. This paper explores the advantages of the alternatives to $Y_{\text{bus}}$ -based formulations in power flow. It proposes a sparse tableau formulation (STF), and it demonstrates its computational efficiency, robustness, and generality in detailed comparison to traditional $Y_{\text{bus}}$ -based solution algorithms. In the examples of power networks ranging from 1888 to 82 000 buses, computational case studies indicate that STF provides comparable computational speed, while allowing simple treatment of zero-impedance branches and more reliably converging to solutions in many cases for which $Y_{\text{bus}}$ -based Newton algorithms diverge.

Advanced modeling of DERs and UPFC devices with Sparse Tableau Formulation for ACOPF

Distributed energy resources (DERs) are projected to be significant contributors in electric power grid modernization efforts. Yet many distribution and transmission system operators are concerned about potential reliability and security impacts of high penetration of DERs, and cite the need of greater visibility and controllability of decision variables associated with these DERs. Tractable modeling of DERs in AC optimal power flow (ACOPF) is one of the barriers to their integration into the existing operation, planning and market paradigm. Similarly, as more intermittent renewable resources are deployed, power flow control becomes more valuable, and accurate representation of Flexible AC Transmission System (FACTS) devices in ACOPF will likely grow in importance. This paper focuses on the detailed representation of substations with the high penetration of DERs, and on the Unified Power Flow Controller (UPFC) FACTS devices, in OPF. To this end, a novel Sparse Tableau Formulation (STF) for ACOPF is used to accurately model such features as substation reconfiguration to accommodate DERs, and to represent decision variables appropriate to the voltage source converters within UPFCs. Numerical studies are provided to demonstrate the impacts of detailed DER and UPFC representations on optimal operation, and the computational efficiency.

Convex relaxation of Sparse Tableau Formulation for the AC optimal power flow

Optimal power flow (OPF) approaches employing such methods as semi-definite programming (SDP) have garnered considerable interest in the literature of the last decade. The OPF formulations for these approaches have almost universally relied on Ybus admittance matrix representations, which derive from nodal analysis, and restrict allowable network elements to be voltage-controlled only. Limitations of nodal analysis long been recognized, and to overcome these, commercial power system software often employs modified nodal analysis (MNA). Here, we consider a novel general power system modeling approach based on multi-port representation of individual components with Sparse Tableau Formulation (STF) of network constraints, which is more versatile than MNA for OPF requiring many monitored links with constrained flows. In STF, one is better able to exploit the fact that the vast majority of power grid network elements have voltage-current behavior that is well-modeled as linear. This opens the door to simple, engineering-based convex relaxations. We discuss two relaxations, admittance-based and current-based. The tightness of the relaxation is shown to improve when angle constraints and narrow bounds for active power generation are provided. Sequential bound tightening and reduced spatial branch-and-bound are discussed to obtain stronger relaxation solution. We conduct case studies to show the effectiveness of our relaxations with standard test cases.

A Sparse Tableau Formulation for Node-Breaker Representations in Security-Constrained Optimal Power Flow

Realistic representations of contingencies in AC optimal power flow (ACOPF) often challenge traditional bus-branch network models. Derived from nodal analysis, such approaches represent network constraints in terms of the familiar bus admittance matrix, $Y_{\text{bus}}$ . A fixed $Y_{\text{bus}}$ is unable to represent common circuit breaker actions such as bus splitting. Work-arounds for $Y_{\text{bus}}$ -based analysis typically rely on topology processing, switching between different $Y_{\text{bus}}$ matrices depending on breaker settings. In this paper, we propose a very general node-breaker approach, employing multi-port element models and using a sparse tableau formulation (STF) for network constraints. Instead of treating breaker action as altering network topology, and hence changing the structure of Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL) equations, this approach represents a breaker's position as impacting only constraints associated with a single component, thereby maintaining fixed structure in the KVL and KCL constraints. While larger numbers of variables are required, STF proves sparser than $Y_{\text{bus}}$ formulations. Numerical case studies herein demonstrate that STF provides computational efficiency comparable to a $Y_{\text{bus}}$ -based ACOPF at the scale of several hundred buses, and lower computational cost in an example of over one thousand buses.

Real-Time Monitoring of Points of Common Coupling in Distribution Systems Through State Estimation and Geometric Tests

This paper presents an extended state estimation method which allows the identification of unexpected power injections at multiple points of common coupling (PCCs) of power distribution grids. The developed state estimator makes use of a three-phase network modeling where active and reactive power injection levels at PCCs are estimated and checked in real time with the values expected by the system operator. The estimation algorithm applies a sparse tableau formulation in which equality constraints are employed to model power injection assumptions at PCCs. The identification of unexpected power injections is then performed using a methodology based on normalized Lagrange multipliers and geometric tests. Simulations conducted on a modified version of the IEEE 34-node test feeder illustrate the accuracy and applicability of the proposed approach, fostering discussions regarding the future of real-time monitoring on power distribution systems.

Sybolical Index Calculation For Linear Circuits

The condition for the DAE index to be higher than one is calculated symbolically. Using Analog Insydes a function for the computer algera system Mathmatica is written. It calculates the index condition if the sparse tableau analysis method, or the modified nodal analysis method is applied.

Symbolically calculated higher index conditions for linear circuits

For linear dynamic circuits, the condition of the index to be higher than one is calculated symbolically. Using Analog Insydes a function for the computer algebra system Mathematica is written. It calculates the index condition if the sparse tableau analysis method is applied to the circuit. Examples illustrate the performance of the method.

A study of the effects of the lower index methods on ADAMS sparse tableau formulation for the computational dynamics of multi-body mechanical systems

This paper studies the effects of reducing the index of the original ADAMS (automatic dynamic analysis of mechanical systems) formulation. For this, a pilot computer program MECH-SIM (MECHanical SIMulations) was developed on the basis of the original ADAMS attributes such as sparse tableau formulation (STF) and backward difference formula (BDF) [1]. The STF implemented in the original ADAMS computer program uses differential and algebraic equations (DAE). In the new formulation it eliminates the three relations connecting the body angular velocities and the angular momentum. It also reformulates the implicit force equations by substituting their functions directly into the corresponding differential equations of motion. The above reduction maintains a reasonable number of equations when the geometric constraints stabilize the velocity and/or the acceleration constraints. A method is found to measure ‘the drift’ allowed by the velocity or acceleration constraints. This permits the use of constraints stabilization only ‘when needed’. Consequently, the BDF error control improves maintenance of the efficiency of the method. The pilot computer program MECH-SIM incorporates these improvements. The program was used to test the numerical accuracy and efficiency on mechanical systems used in agricultural mechanization, industrial equipment and auto suspensions. Hence, such results can be extended to the modern ADAMS computer program.

Stable Numerical Algorithms for Equilibrium Systems

An equilibrium system (also known as a Karush–Kuhn–Tucker (KKT) system, a saddlepoint system, or a sparse tableau) is a square linear system with a certain structure. Strang [SIAM Rev., 30 (1988), pp. 283–297] has observed that equilibrium systems arise in optimization, finite elements, structural analysis, and electrical networks. Recently, Stewart [Linear Algebra Appl., 112 (1989), pp. 189–193] established a norm bound for a type of equilibrium system in the case when the “stiffness” portion of the system is very ill-conditioned. This paper investigates the algorithmic implications of Stewart’s result. It is shown that several algorithms for equilibrium systems appearing in applications textbooks are unstable. A certain hybrid method is then proposed, and it is proved that the new method has the right stability property.

A user oriented active network simulator

A digital computer simulator for the frequency response and tolerance network comprising <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RLCM</i> elements, ideal operational amplifiers and controlled sources is presented. The simulator is based on `tableau approach'. Reordering of the sparse tableau matrix is done using Markowitz Criterion and the diagonal pivots are chosen for simplicity. The simulator also employs dynamic allocation for maximum utilization of memory and faster turn around time. Three network are simulated and their results are presented. A network in which the operational amplifiers are assumed to have single pole behaviour is also analyzed.

The bordered triangular matrix and minimum essential sets of a digraph

A partitioning strategy of sparse matrices is dealt with. In particular, the problem of transforming a nonsingular matrix by symmetric permutation to an optimal bordered triangular form (BTF) is solved. It is shown that the problem is equivalent to the determination of a minimum essential set of a directed graph. An efficient algorithm is given for finding minimum essential sets of a digraph. The method depends on, as a preliminary step, graph simplication using local information at a vertex. A circuit-generation technique based on vertex elimination is then introduced. The algorithm is illustrated with a complete example. A simple electrical network is used to illustrate the use of the BTF in the sparse tableau approach of network analysis.