A series of digital-logic design laboratory experiments have been created for a first course in digital logic design.These laboratory experiments are aimed primarily at first and second year electrical engineering and computer science/engineering students.The laboratory exercises include a set of six hardware laboratory experiments, and eight digital-logic simulation experiments.To receive a copy of the digital-design experiments discussed in this paper, send a request to Dr.Dans@ieee.org I. Hardware Laboratory ExperimentsThe objective of the hardware laboratory design is to start students with basic experiments that emphasize common laboratory measuring and debugging techniques.Later more sophisticated experiments emphasize the design skills students have acquired in the lecture portion of the class as shown in Table 1.

We present a novel method for calculating Padé approximants that is capable of eliminating spurious poles placed at the point of development and of identifying and eliminating spurious poles created by precision limitations and/or noisy coefficients. Information contained in in the eliminated poles is assimilated producing a reduced order Padé approximant (PA). While the [m + k/m] conformation produced by the algorithm is flexible, the m value of the rational approximant produced by the algorithm reported here is determined by the number of spurious poles eliminated. Spurious poles due to coefficient noise/precision limitations are identified using an evidence-based filter parameter applied to the singular values of a matrix comprised of the series coefficients. The rational function poles are found directly by solving a generalized eigenvalue problem defined by a matrix pencil. Spurious poles place at the point of development, responsible in some algorithms for degeneracy, are identified by their magnitudes. Residues are found by solving an overdetermined linear matrix equation. The method is compared with the so-called Robust Padé Approximation (RPA) method [6] and shown to be competitive on the problems studied. By eliminating spurious poles, particularly in functions with branch points, such as those encountered solving the power-flow problem, solution of these complex-valued problems is made more reliable.

What has become known as Stahl's Theorem in power engineering circles has been used to justify a convergence guarantee of the Holormorphic Embedding Method (HEM) as it applies to the power flow (PF) problem. In this two-part paper, we examine in more detail the implications of Stahl's theorems to both theoretcial and numerical convergence for a wider range of problems to which these theorems are now being applied. In Pt. 1, we introduce the theorem using the necessary mathematical parlance and then translate the language to show its implications to convergence of nonlinear problems in general and the PF problem specifically. We show that among other possibilities the existence of the Chebotarev points, which are embedding specific, are a possible theoretical impediment to convergence. Numerical impediments to convergences are discussed in the companion paper.

What has become known as Stahl's Theorem in power-engineering circles has been used to justify a convergence guarantee of the Holomorphic Embedding Method (HEM) as it applies to the power-flow problem. In this, the second part of a two-part paper, we examine implications to numerical convergence of HEM and the numerical properties of a Pad\'e approximant algorithm. We show that even if the convergence domain is identical to the function's domain, numerical convergence of the sequence of Pad\'e approximants computed with finite precision is not guaranteed. We also show that the study of convergence properties of the Pad\'e approximant is the study of the location of branch-points of the function, which dictate branch-cut topology and capacity and, therefore, convergence rate. We show how poorly chosen embeddings can prevent numerical convergence.

The requirement for solving nonlinear algebraic equations is ubiquitous in the field of electric power system simulations. While Newton-based methods have been used to advantage, they sometimes do not converge, leaving the user wondering whether a solution exists. In addition to improved robustness, one advantage of holomorphic embedding methods (HEM) is that, even when they do not converge, roots plots of the Padé approximants (PAs) to the functions in the inverse-$α$ plane can be used to determine whether a solution exist. The convergence factor (CF) of the near-diagonal PAs applied to functions expanded about the origin is determined by the logarithmic capacity of the associated branch cut (BC) and the distance of the evaluation point from the origin. However the underlying mechanism governing this rate has been obscure. We prove that the ultimate distribution of the PA roots on the BC in the complex plane converges weakly to the equilibrium distribution of electrostatic charges on a 2-D conductor system with the same topology in a physical setting. This, along with properties of the Maclaurin series can be used to explain the structure of the CF equation We demonstrate the theoretical convergence behavior, with numerical experiments.

Engineering approximations of physical systems sometimes produce models in which real-valued model-based physical-distances are added to complex-valued distances and/or (for electrical systems), real-valued current/charge image intensities are replaced with complex-valued quantities. These models are arrived at often using ad hoc approximations that allow infinite integrals or series to be approximated in closed form. Arriving at accurate ad hoc approximations in a compatible analytic form is often the difficult step in the derivation of these approximations. In this paper, we show that this difficult ad hoc step can be replaced for many classes of functions with the use of analytic continuation via Fade approximants, along with some reasonable engineering judgement. We apply our approach to several existing approximations in the electrical engineering field (overhead transmission line impedance, underground cable impedance and Green's functions used in ground potential rise calculations) and show that these approximations can be derived elegantly, without the need for grand leaps of insight, and provide a basis for both distance parameters and current/charge intensities that are complex-valued.

This paper presents the modeling and simulation of power loads due to plug-in electric vehicles' (EVs) charging events at a dc fast charging station. Two algorithms for modeling the loads are introduced and compared, one based on sampling and one based on statistical distribution built from the sample database. Simulation of load versus time was performed using a horizon of 7 days using both techniques. The cause for the difference in the result of these two approaches is explored. Regardless of the method used, the results show that a dc fast charging station with 6 fast chargers potentially serving 700 plug-in EVs generally gets 105 charging events per day with a peak load of 375 kW.

Network reduction is an effective tool for reducing the complexity of many analysis, design, and optimization problems. However, many of the conventional reduction methods, such as Ward and Radial, Equivalent, Independent are only accurate at the base case. When the operating condition changes, the reduced model does not match the full model performance because linearization is used somewhere in the process. In this paper, a new reduction method that preserves the model's nonlinear structure using the holomorphic embedding (HE) technique is proposed, to generate network reductions which are accurate over a broader range of operating conditions. When applied to the power-flow problem, simulation results showthat the proposed method can significantly improve bus-voltage and branch-flow accuracy, matching the full-model power-flow solution exactly when moving along the so-called a line. In addition, the HE reduction is more efficient than traditional methods when calculating nonlinear network solutions under many operating conditions.

The true value of a battery energy storage system (BESS) can only be established when multiple technically and operationally compatible services rendered by the BESS are 'stacked' and valued. This paper makes an attempt towards estimating the stacked value of a BESS providing multiple services such as peak shaving, frequency regulation, and reserve support etc. in an Arizona-based test system. This has been achieved by developing a mixed-integer programming (MIP) based optimization model that captures the operation of the BESS in the test system. The obtained results suggest that BESS with stacked benefits yield significantly more value than BESS delivering a single primary service in isolation.

A holomorphic embedding method (HEM)-based algorithm for finding Type-1 power-flow solutions is introduced whose complexity is the same as that of the HEM power-flow algorithm for calculating the high voltage (operable) power-flow solution. The algorithm is tailored to finding Type-1 solutions by using a modified embedded system and a numerical mapping from a set of integer-based boundary conditions to a floating-point-number-based reference state. The modified system can also be viewed as a homotopy whose initial point (no-load reference state) is consistent with the Type-1 solution homotopy path of the modified system, with an embedding parameter that functions simultaneously as the homotopy parameter. By using analytic continuation, starting from the initial point/reference state, the solution obtained for the modified system matches the one obtained from the original system model at the load of interest. Numerical results for three-/five-/seven-/14- and 118-bus systems are presented to demonstrate the numerical robustness.

The holomorphic embedding method (HEM) applied to the power flow problem has more robust performance than the Newton-Raphson method. But as the saddle-node bifurcation point (SNBP) is approached, more terms must be included in the Maclaurin series representation of the voltage to achieve a converged solution, leading to matrices that are ill-conditioned. This ill-conditioning may prevent HEM from converging or may interfere with the ability to predict the SNBP using the so-called roots method. In this paper, we look at the effect of the robust Padé approximation (RPA) method on both of these issues.

A class of nonlinear equation solvers known as the holomorphically embedding method, when applied to the power-flow problem is theoretically guaranteed to find an operable solution to the power-flow problem, if one exists, provided rather mild conditions are satisfied. To date, all of the published approaches use a Gauss-Seidel-based fixed-point form as the starting point for the embedding. We show that a fixed-point form based on a Newton-Raphson scheme may also be use and has some advantageous properties when attempting to find the saddle-node bifurcation point (SNBP).

A new method of solving the power-flow problem, the holomorphically embedded load-flow method (HELM) is theoretically guaranteed to find the high-voltage solution, if one exists, up to the saddle-node bifurcation point (SNBP), provided sufficient precision is used and the conditions of Stahl's theorem are satisfied. Sigma (σ) indices, have been proposed as estimators of the distance from the present operating point to the SNBP, and indicators of the weak buses in a system. This paper investigates the theoretical foundation of the σ method and shows that the σ condition proposed in [10] will not produce reliable results and that a modified requirement can be used to produce a tight upper bound on the SNBP. A new HEM-based method is then proposed that can be used to estimate the weak buses in a system (from a steady-state voltage stability perspective) at all operating points through to the SNBP, using a single power-flow solution. Numerical results for the proposed approach are compared to traditional modal analysis for the IEEE 14-bus and 118-bus systems.

The holomorphic embedding load flow method (HELM) is an application for solving the power-flow problem based on a novel method developed by Dr. Trias. The advantage of the method is that it comes with a theoretical guarantee of convergence to the high-voltage (operable) solution, if it exists, provided the equations are suitably framed. While theoretical convergence is guaranteed by Stahl's theorem, numerical convergence is not; it depends on the analytic continuation algorithm chosen. Since the holomorphic embedding method (HEM) has begun to find a broader range of applications (it has been applied to non-linear structure-preserving network reduction, weak node identification and saddle-node bifurcation point determination), examining which algorithms provide the best numerical convergence properties, which do not, why some work and not others, and what can be done to improve these methods, has become important. The numerical Achilles heel of HEM is the calculation of the Fade approximant, which is needed to provide both the theoretical convergence guarantee and accelerated numerical convergence. In the past, only two ways of obtaining Fade approximants applied to the power series resulting from power-system-type problems have been discussed in detail: the matrix method and the Viskovatov method. This paper explores several methods of accelerating the convergence of these power series and/or providing analytic continuation and distinguishes between those that are backed by the theoretical convergence guarantee of Stahl's theorem (i.e., those computing Pade approximants), and those that are not. For methods that are consistent with Stahl's theoretical convergence guarantee, we identify which methods are computationally less expensive, which have better numerical performance and what remedies exist when these methods fail to converge numerically. (C) 2017 Elsevier Ltd. All rights reserved.

•Four HEM-based methods that estimate the SNBP with a single PF problem solution.•Formulation that allows all loads and real power generation to be scaled by α.•Using roots of Padé approximants, sigma indices to get the SNBP, detect weak buses.•Formulation that allows loads at different buses to be scaled by different amounts.•A new HEM model for polynomial ZIP loads.

A new method of solving the power-flow problem, the holomorphically embedded load-flow method (HELM) is theoretically guaranteed to find the high-voltage solution, if one exists, up to the saddle-node bifurcation point (SNBP), provided sufficient precision is used and the conditions of Stahls theorem are satisfied. Sigma indices, have been proposed as estimators of the distance from the present operating point to the SNBP, and indicators of the weak buses in a system. In this paper, it is shown that the sigma condition proposed in [2] will not produce reliable results and that a modified requirement can be used to produce a tight upper bound on the SNBP. Introduced is an approach to estimate the weak buses in the system using the HEM power series with numerical results compared to traditional modal analysis for a 14-bus system.

Reduced order models of dc networks using Ward, REI, and bus-aggregation approaches have been used to advantage for some time. Proposed in this paper is an optimization-based framework that can easily be modified to generate a Ward reduction, a bus-aggregation-based reduction, or a range of hybrid reductions between these two extremes, generating reduced-order models that are tailored to the purpose of the study. The so-called optimization-based Ward reduction method is demonstrated to be numerically stable on practical examples and is specifically applied to the problem of large-reactance branch elimination. We show that it offers accuracy advantages over the traditional approach of removing high-reactance branches using a threshold criterion.

Iterative methods for solving the power flow problem, including the Newton Raphson method and fast decoupled methods require good starting points, otherwise they may not converge or may converge to the wrong (low voltage) solution. The holomorphic embedding method (HEM) is a recursive, not iterative, method which uses the no-load condition as its starting points and is theoretically guaranteed to converge to the operable, high-voltage (HV) solution if one exists. The HEM method uses Padé approximant as a means of analytic continuation and is capable of finding the HV solution up to the saddle node bifurcation point (SNBP). The univariate HEM has been proven to be an efficient tool in experiments. However, a key drawback of the univariate HEM is that it lacks flexibility: the method can calculate the solutions only when the load/generation profile is scaled as a whole. A straightforward improvement is to use a multi-variate HEM combined with multi-variate Padé approximants. This paper presents a bivariate HEM formulation, which uses a corresponding bivariate Padé (Chisholm) approximant. Simulations on a three-bus and a modified IEEE 14-bus system show that the method can yield accurate voltage solutions and accurate values of the SNBPs.

This paper presents three different true nonlinear reduction methods to obtain network equivalents for radial (distribution-type) networks (using the holomorphically embedded power flow algorithm), which are exact, given computational precision limitations, even when the loads and the real-power generations are scaled. The proposed reduction methods are applied in this paper to reduce a radial distribution system and provide a two-bus-model equivalent which accurately models the real and reactive power load seen at the transmission network due to random changes in the distribution system load. Numerical results are provided for a radial 14-bus system to show the accuracy of the proposed methods in preserving voltages and slack bus power. The approach is shown to have better performance than Ward reduction even when the loads are increased in a random manner.