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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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16 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Enhancing ACPF Analysis: Integrating Newton-Raphson Method with Gradient Descent and Computational Graphs

This paper presents a new method for enhancing Alternating Current Power Flow (ACPF) analysis. The method integrates the Newton-Raphson (NR) method with Enhanced-Gradient Descent (GD) and computational graphs. The integration of renewable energy sources in power systems introduces variability and unpredictability, and this method addresses these challenges. It leverages the robustness of NR for accurate approximations and the flexibility of GD for handling variable conditions, all without requiring Jacobian matrix inversion. Furthermore, computational graphs provide a structured and visual framework that simplifies and systematizes the application of these methods. The goal of this fusion is to overcome the limitations of traditional ACPF methods and improve the resilience, adaptability, and efficiency of modern power grid analyses. We validate the effectiveness of our advanced algorithm through comprehensive testing on established IEEE benchmark systems. Furthermore, our findings demonstrate that our approach not only speeds up the convergence process but also ensures consistent performance across diverse system states, representing a significant advancement in power flow computation.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2023

Solution and sensitivity analysis of nonlinear equations using a hypercomplex-variable Newton-Raphson method

Here, the classical Newton-Raphson (NR) method for solving nonlinear equations is enhanced in two ways through the use of hypercomplex variables and algebra. In particular, i) the Jacobian is computed in a highly accurate and automated way, and ii) the derivative of the solution to the nonlinear equations is computed with respect to any parameter contained within the system of equations. These advances provide two significant enhancements in that it is straightforward to provide an accurate Jacobian and to construct a reduced order model (ROM) of arbitrary order with respect to any parameter of the system. The ROM can then be used to approximate the solution for other parameter values without requiring additional solutions of the nonlinear equations. Several case studies are presented including 1D and 2D academic examples with fully functioning Python code provided. Additionally, a case of study of the catenary of an elastic cable subject to its own weight and a vertical point load. Derivatives up to 10th order were computed with respect to material, loading, and geometrical parameters. The derivatives were used to generate reduced order models of the cable deformation and reaction forces at its ends with respect to multiple input parameters. Results show that from a single hypercomplex evaluation of the cable under a single vertical point load, it is possible to generate an accurate reduced order model capable of predicting the cable deformation with 1.5 times the load in the opposite direction and with 3.5 times the load in the same direction without resolving the system of equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

RESOLVING THE ELECTROCHEMICAL EQUATIONS OF A SOLID OXIDE FUEL CELL FOR USE IN TRANSIENT SIMULATION AND INTEGRATION INTO CYBER-PHYSICAL SYSTEMS

A major challenge with complex cyber-physical systems stems from long model computational time that creates a mismatch between the model system and the physical system. The numerical modeling of solid oxide fuel cells (SOFCs) presents particular challenges due to the highly coupled nature of the underlying equations and the multiphysics needed to fully resolve their behavior during a transient event. To this end current approaches revolve around splitting the computational efforts into resolving temperature effects and resolving electrochemical effects. Current methods employed for the transient simulation of an SOFC for implementation in the Hybrid Performance (HyPer) facility cyberphysical plant at the National Energy Technology Laboratory reveal a distinct need for accelerated results with a high degree of stability. To this aim, an investigation into the computational time for the code reveals that the underlying electrochemical algorithm takes an order of magnitude more time than its thermal counterpart and has a tendency to vary in terms of iteration time and as such a rework of the underlying system is proposed. The primary method for accelerated electrochemical algorithm solutions is to employ higher order root finding recipes for the resolution of the highly coupled electrochemical equations. This is done with the intention to reduce the overall number of subiterations necessary for resolving voltage, current density, and species concentration, properties of the fuel cell that are all directly coupled and require nested iterative approaches. The overall objective of this approach is an order of magnitude reduction in calculation time without sacrificing stability and increasing accuracy. Specific approaches involve using both bounded and unbounded techniques, such as the False Position method and the Secant method (or if applicable Newton-Raphson) respectively, the drawbacks being slower convergence for False Position and instability for the Secant or Newton-Raphson methods. Current preliminary results on simplified versions of the parent functions involved for electrochemical calculations indicate a reduction in computational steps by a factor of two for the secant method and a factor of three for Newton-Raphson. When implemented into new modified electrochemical algorithms, the results indicate a possible order of magnitude reduction in calculation time.

Arias, Jesus↗
DOE OSTI2022

Numerical Solution of the Steady-State Network Flow Equations for a Non-Ideal Gas

Herein we formulate a steady-state network flow problem for non-ideal gas that relates injection rates and nodal pressures in the network to flows in pipes. For this problem, we present and prove a theorem on uniqueness of generalized solution for a broad class of non-ideal pressure-density relations that satisfy a monotonicity property. Further, we develop a Newton-Raphson algorithm for numerical solution of the steady-state problem, which is made possible by a systematic non-dimensionalization of the equations. The developed algorithm has been extensively tested on benchmark instances and shown to converge robustly to a generalized solution. Previous results [1]-[4], indicate that the steady-state network flow equations for an ideal gas are difficult to solve by the Newton-Raphson method because of its extreme sensitivity to the initial guess. In contrast, we find that non-dimensionalization of the steady-state problem is key to robust convergence of the Newton-Raphson method. We identify criteria based on the uniqueness of solutions under which the existence of a non-physical generalized solution found by a non-linear solver implies non-existence of a physical solution, i.e., infeasibility of the problem. Finally, we compare pressure and flow solutions based on ideal and non-ideal equations of state to demonstrate the need to apply the latter in practice. The solver developed in this article is open-source and is made available for both the academic and research communities as well as the industry.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Efficient and Robust Dynamic Simulation of Power Systems With Holomorphic Embedding

Dynamic simulation is vitally important in power system analysis, but traditional approaches based on numerical integration over small time steps are time-consuming. Also, the Newton-Raphson method suffers from difficulty in convergence when solving nonlinear algebraic equations. In this paper, we propose a novel dynamic simulation approach based on holomorphic embedding. By obtaining a high-order approximation of system dynamics, it achieves a much larger time step and thus enhances the computational efficiency significantly. In addition, the new approach avoids non-convergence issues in solving algebraic equations, which improves robustness. The approach includes flexible modeling of synchronous generators and controllers, and we propose a method for modeling generator coordinate transformations. The approach is tested on the IEEE 39-bus, 10-generator system and a Polish 2383-bus, 327-generator system. The results demonstrate promising computational efficiency and satisfactory numerical robustness for the analysis of large-scale power systems.

42 ENGINEERING↗
DOE OSTI2021

Contingency Analysis Based on Partitioned and Parallel Holomorphic Embedding

In the steady-state contingency analysis, the traditional Newton-Raphson method suffers from non-convergence issues when solving post-outage power flow problems, which hinders the integrity and accuracy of security assessment. In this paper, we propose a novel robust contingency analysis approach based on holomorphic embedding (HE). Here, the HE-based simulator provides theoretical convergence guarantee, which is desirable because it avoids the influence of numerical issues and provides a credible security assessment conclusion. In addition, based on the multi-area characteristics of real-world power systems, a partitioned HE (PHE) method is proposed with an interfacebased partitioning of HE formulation. The PHE method does not undermine the numerical robustness of HE and significantly reduces the computation burden in large-scale contingency analysis. The PHE method is further enhanced by parallel or distributed computation to become parallel PHE (P2HE). Tests on a 458-bus system, a synthetic 419-bus system and a large-scale 21447-bus system demonstrate the advantages of the proposed methods in robustness and efficiency.

42 ENGINEERING↗
DOE OSTI2019

H2OI95: A Stand-Alone Fortran Code for Evaluating the IAPWS-95 Equation-of-State Model for Water

H2OI95 is a stand-alone Fortran code for evaluating the IAPWS-95 equation-of-state model (Wagner and Pruss, 2002) for the thermodynamic properties of water. It further evaluates the corresponding thermochemical properties of water consistent with the CODATA recommendations (Cox et al., 1989). The IAPWS-95 model is based on a model equation for the dimensionless Helmholtz energy for which the primary variables are the inverse reduced temperature and reduced density. Here is the absolute temperature (K), is density (kg/m3), and the subscript refers to the critical point of water (647.096 K and 22.064 MPa pressure in this model, for which is 322 kg/m3). The code solves four basic types of problems, distinguished by the specified inputs: 1, Temperature (K) and density (kg/m3) or reduced density 2. Temperature (K) and pressure (MPa). 3. Temperature (K) on the saturation (liquid-vapor equilibrium) curve 4. Pressure (MPa) on the saturation curve Each type of problem is run using a corresponding input (text) file. All but the first type of problem require iteration. For example, to solve for desired temperature and pressure, the reduced density must be adjusted to give the desired pressure. Iteration is accomplished using the Newton-Raphson method, though the secant method is also used in solving the fourth type of problem. H2OI95 has been used to conduct numerical studies of convergence and the problem of multiple numerical solutions, only some of which correspond to valid results. Obtaining valid results depends mainly on appropriate choice of starting values. The default values in H2OI95 appear to consistently lead to generally desired results. With modification (not addressed here), H2OI95 can be used to support SUPCRT92 (Johnson et al., 1992) and similar codes that compute chemical thermodynamic properties of species and reactions over a wide range of temperature and pressure (273.16-1273K and 0-1000 MPa).

Wolery, ThomasJ.↗
DOE OSTI2018

H2OI95: A Stand-Alone Fortran Code for Evaluating the IAPWS-95 Equation-of-State Model for Water

H2OI95 is a stand-alone Fortran code for evaluating the IAPWS-95 equation-of-state model (Wagner and Pruss, 2002) for the thermodynamic properties of water. It further evaluates the corresponding thermochemical properties of water consistent with the CODATA recommendations (Cox et al., 1989). The IAPWS-95 model is based on a model equation for the dimensionless Helmholtz energy for which the primary variables are the inverse reduced temperature (τ = T cr /T) and reduced density ( δ= ρ/ρ cr ). Here T is the absolute temperature (K), ρ is density (kg/m 3 ), and the subscript “cr” refers to the critical point of water (647.096 K and 22.064 MPa pressure in this model, for which ρ cr is 322 kg/m 3 ). The code solves four basic types of problems, distinguished by the specified inputs: 1, Temperature (K) and density (ρ kg/m 3 ) or reduced density (δ) 2. Temperature (K) and pressure (MPa). 3. Temperature (K) on the saturation (liquid-vapor equilibrium) curve 4. Pressure (MPa) on the saturation curve Each type of problem is run using a corresponding input (text) file. All but the first type of problem require iteration. For example, to solve for desired temperature and pressure, the reduced density must be adjusted to give the desired pressure. Iteration is accomplished using the Newton-Raphson method, though the secant method is also used in solving the fourth type of problem. H2OI95 has been used to conduct numerical studies of convergence and the problem of multiple numerical solutions, only some of which correspond to valid results. Obtaining valid results depends mainly on appropriate choice of starting values. The default values in H2OI95 appear to consistently lead to generally desired results. With modification (not addressed here), H2OI95 can be used to support SUPCRT92 (Johnson et al., 1992) and similar codes that compute chemical thermodynamic properties of species and reactions over a wide range of temperature and pressure (273.16-1273K and 0-1000 MPa).

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

H2OI95: A Stand-Alone Fortran Code for Evaluating the IAPWS-95 Equation-of-State Model for Water (Rev. 1)

H2OI95 is a stand-alone Fortran code for evaluating the IAPWS-95 equation-of-state model for the thermodynamic properties of water. It also evaluates the corresponding thermochemical properties of water consistent with the CODATA recommendations. The IAPWS-95 model is based on a model equation for the dimensionless Helmholtz energy for which the primary variables are the inverse reduced temperature (τ = T cr /T ) and reduced density (δ = ρ/ρ cr ). Here T is the absolute temperature (K), ρ is density (kg/m³), and the subscript “ cr ” refers to the critical point of water (647.096 K and 22.064 MPa pressure in this model, for which ρ cr is 322 kg/m³). The code solves four basic types of problems, distinguished by the specified inputs: 1. Temperature (K) and density (ρ kg/m³) or reduced density (δ) 2. Temperature (K) and pressure (MPa). 3. Temperature (K) on the saturation (liquid-vapor equilibrium) curve 4. Pressure (MPa) on the saturation curve. Each type of problem is run using a corresponding input (text) file. All but the first type of problem require iteration. For example, to solve for desired temperature and pressure, the reduced density must be adjusted to give the desired pressure. Iteration is accomplished using the Newton-Raphson method, though the secant method is also used in solving the fourth type of problem. For the last three types of problems, H2OI95 has been used to conduct numerical studies of convergence and the problem of multiple numerical solutions, some of which are not physically valid. Obtaining valid results depends mainly on appropriate choice of starting values for the density. The default values used in H2OI95 appear to consistently lead to generally desired results. With modification (not addressed here), H2OI95 can be used to support SUPCRT92 and similar codes that compute chemical thermodynamic properties of species and reactions over a wide range of temperature and pressure (273.16-1273K and 0-1000 MPa). Version 1.1 of this software calculates an extended set of water properties, including some derivable solely from IAPWS-95, such as the isobaric thermal expansivity and the isothermal compressibility, and still other properties based on supplementary models sanctioned by IAPWS, including the dynamic viscosity, thermal conductivity, surface tension, and static dielectric constant, and various other properties (notably Debye-Hückel parameters) derivable from the dielectric constant and its partial derivatives with respect to temperature and pressure. A coding error in Version 1.0 was identified while testing the implementation of the viscosity model. This error is described in Appendix C. It mainly affects calculation of “higher order” properties such as compressibility and heat capacity near the critical point.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

The W4 method: A new multi-dimensional root-finding scheme for nonlinear systems of equations

We propose a new class of method for solving nonlinear systems of equations, which, among other things, has four nice features: (i) it is inspired by the mathematical property of damped oscillators, (ii) it can be regarded as a simple extension to the Newton-Raphson (NR) method, (iii) it has the same local convergence as the NR method does, (iv) it has a significantly wider convergence region or the global convergence than that of the NR method. In this article, we present the evidence of these properties, applying our new method to some examples and comparing it with the NR method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Mitigating Voltage Instability in the Saudi Grid for a Decarbonized, Fully Solar Power System

Lately, solar photovoltaic (PV) has received significant interest due to its economic and environmental benefits. As the integration of renewable energy sources (RES) increases into existing power grids, challenges such as the decrease in short circuit ratio (SCR) are introduced. Here, this paper investigates a case study of the Saudi power grid, examining the voltage stability as the grid transitions from traditional power generation to 100% penetration of solar PV gradually. Moreover, the paper explores the relationship between the increase in solar penetration and the potential effects on the short circuit MVA (SCMVA), which could significantly impact the overall SCR of the system. To meet the North American Electric Reliability Corporation (NERC) recommendation of maintaining SCR at a particular level, synchronous condensers (SC)s were integrated into the grid. The effectiveness of utilizing SCs to maintain the system voltage at optimal levels in a fully solar grid is also considered. In addition, this paper covers the weak grid analysis via utilizing the Newton-Raphson load flow method, along with PV and QV curve analyses. The purpose behind that is to determine weak bus locations in need of voltage improvements, and to meet the amount of reactive power to be injected. Finally, the amount of power being delivered by SCs will be added progressively in three scenarios to show the enhancements on SCR more precisely.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2025

Hierarchical Network Partitioning for Solution of Potential-Driven, Steady-State Nonlinear Network Flow Equations

The solution of potential-driven steady-state flow in large networks is a task which manifests in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology, and in general, there is no numerical algorithm that offers guaranteed convergence to the solution (assuming a solution exists). Some methods offer guarantees in cases where the network topology satisfies certain assumptions, but these methods fail for larger networks. On the other hand, the Newton-Raphson algorithm offers a convergence guarantee if the starting point lies close to the (unknown) solution. It would be advantageous to compute the solution of the large nonlinear system through the solution of smaller nonlinear sub-systems wherein the solution algorithms (Newton-Raphson or otherwise) are more likely to succeed. Here, this letter proposes and describes such a procedure, a hierarchical network partitioning algorithm that enables the solution of large nonlinear systems corresponding to potential-driven steady-state network flow equations.

42 ENGINEERING↗
DOE OSTI2024

A Machine Learning Initializer for Newton-Raphson AC Power Flow Convergence

Power flow computations are fundamental to many power system studies. Obtaining a converged power flow case is not a trivial task especially in large power grids due to the non-linear nature of the power flow equations. One key challenge is that the widely used Newton based power flow methods are sensitive to the initial voltage magnitude and angle estimates, and a bad initial estimate would lead to non-convergence. This paper addresses this challenge by developing a random-forest (RF) machine learning model to provide better initial voltage magnitude and angle estimates towards achieving power flow convergence. This method was implemented on a real ERCOT 6102 bus system under various operating conditions. By providing better Newton-Raphson initialization, the RF model precipitated the solution of 2,106 cases out of 3,899 non-converging dispatches. These cases could not be solved from flat start or by initialization with the voltage solution of a reference case. Finally, results obtained from the RF initializer performed better when compared with DC power flow initialization, Linear regression, and Decision Trees.

random forest↗
DOE OSTI2024

Newton-Raphson AC Power Flow Convergence Based on Deep Learning Initialization and Homotopy Continuation

Power flow forms the basis of many power system studies. With the increased penetration of renewable energy, grid planners tend to perform multiple power flow simulations under various operating conditions and not just selected snapshots at peak or light load conditions. Getting a converged AC power flow (ACPF) case remains a significant challenge for grid planners especially in large power grid networks. This paper proposes a two-stage approach to improve Newton-Raphson ACPF convergence and was applied to a 6102 bus Electric Reliability Council of Texas (ERCOT) system. The first stage utilizes a deep learning-based initializer with data re-training. Here a deep neural network (DNN) initializer is developed to provide better initial voltage magnitude and angle guesses to aid in power flow convergence. This is because Newton-Raphson ACPF is quite sensitive to the initial conditions and bad initialization could lead to divergence. The DNN initializer includes a data re-training framework that improves the initializer's performance when faced with limited training data. The DNN initializer successfully solved 3,285 cases out of 3,899 non-converging dispatch and performed better than random forest and DC power flow initialization methods. ACPF cases not solved in this first stage are then passed through a hot-starting algorithm based on homotopy continuation with switched shunt control. The hot-starting algorithm successfully converged 416 cases out of the remaining 614 non-converging ACPF dispatch. In conclusion, the combined two-stage approach achieved a 94.9% success rate, by converging a total of 3,701 cases out of the initial 3,899 unsolved cases.

Deep learning↗
DOE OSTI2020

Development of the finite slab geometry in the NORA code

In this work we develop the complete linearization formalism for the radiation transfer equation in slab geometry. This development was motivated by an observed convergence instability arising from the NORA code’s inner boundary while modeling a suite of simple test problems in slab geometry presented at the Line Transfer Workshop held in LLNL in May of 2019. The convergence issue was seen to emerge early on in Newton-Raphson iterations. These test problems required parameters in the former NORA inner boundary to achieve the asymptotic limit of zero incoming intensity. In this work we derive the limiting case directly to gain insight into the former algorithm’s pathology.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Linear and Nonlinear Solvers for Simulating Multiphase Flow within Large-Scale Engineered Subsurface Systems

Simulation of multiphase flow in the subsurface is well-known to be computationally challenging. While there have been many studies that have explored approaches to overcoming these challenges, they often utilize relatively simple case studies. In this paper, we focus on the unique numerical challenges posed by modeling large-scale engineered subsurface systems, characterized by discrete features embedded in a heterogeneous natural subsurface setting. The man-made features such as shafts, tunnels, and barriers often cause multiple challenges in modeling the domain for multiphase porous media flow. This flow scenario can have a wide range of applications such as nuclear waste repositories, enhanced recovery of a petroleum reservoir, geothermal engineering, and carbon sequestration. An example of these severe numerical challenges is the case of performance assessment (PA) for Waste Isolation Pilot Plant (WIPP), the only operating deep geological repository in the US, which simulates extreme material properties of bedded salt rock formation and extreme contrast due to open excavation next to the formation. The models have extremes not only of permeability and porosity but also of the constitutive models needed for multiphase flow; additionally, they have process models like salt creep closure reducing porosity over time, fracturing in clay and anhydrite interbeds of the bedded salt, gas generation from the waste materials, and unintentional human borehole intrusions in some scenarios. Numerical simulations require the solution of coupled systems of nonlinear PDEs; in our work, we use the open-source simulator PFLOTRAN which is based on Finite Volume discretization. The solution of the nonlinear equations requires use of the Newton-Raphson iteration at each time step, which entails the solution of the linearized Jacobian system at each iteration. The effects of all the processes (i.e., large number of unknowns, highly nonlinear constitutive relations, large contrasts in material properties in short distances) lead to an ill-conditioned Jacobian matrix that severely challenges traditional linear solver, i.e., stabilized biconjugate gradient with block Jacobi incomplete LU preconditioner (BCGS-ILU) leading to non-convergence for traditional Newton-Raphson nonlinear solver causing unacceptably long computation time for each model. This paper presents linear solvers such as constrained pressure residual (CPR) two-stage preconditioner with alternate-block-factorization (ABF) and quasi- implicit pressure and explicit saturation (QIMPES) decouplers and flexible generalized residual solver (FGMRES). The new general-purpose nonlinear solver, Newton trust-region dogleg Cauchy (NTRDC), is also introduced to resolve extreme nonlinearities in the models. We demonstrate the effectiveness of each method relative to the default BCGS-Newton solver. The two best cases had nearly 50 times speed-up and achieved completion of a simulation in 14 hours that never completed due to non-convergence with the default solver. We also investigate the strong scalability of each method and discuss some of the deficiencies found for Block Jacobi preconditioner using parallel domain decomposition, and node packing effects of modern processor architecture.

Preconditioner, Nonlinear, Porous media, Multiphas↗