Results for “mixed integer programming” — Page 19

DOE OSTI2021

Machine-learning-aided cognitive reconfiguration for flexible-bandwidth HPC and data center networks [Invited]

This paper proposes a machine-learning (ML)-aided cognitive approach for effective bandwidth reconfiguration in optically interconnected datacenter/high-performance computing (HPC) systems. The proposed approach relies on a Hyper-X-like architecture augmented with flexible-bandwidth photonic interconnections at large scales using a hierarchical intra/inter-POD photonic switching layout. We first formulate the problem of the connectivity graph and routing scheme optimization as a mixed-integer linear programming model. A two-phase heuristic algorithm and a joint optimization approach are devised to solve the problem with low time complexity. Then, we propose an ML-based end-to-end performance estimator design to assist the network control plane with intelligent decision making for bandwidth reconfiguration. Numerical simulations using traffic distribution profiles extracted from HPC applications traces as well as random traffic matrices verify the accuracy performance of the ML design estimator ( < <#comment/> 9 % <#comment/> error) and demonstrate up to 5 × <#comment/> throughput gain from the proposed approach compared with the baseline Hyper-X network using fixed all-to-all intra/inter-portable data center interconnects.

Chen, Xiaoliang (ORCID:0000000278056237)↗

DOE OSTI2020

Dakota A Multilevel Parallel Object-Oriented Framework for Design Optimization Parameter Estimation Uncertainty Quantification and Sensitivity Analysis: Version 6.12 Theory Manual

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a theoretical manual for selected algorithms implemented within the Dakota software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of Dakota-related research publications in the areas of surrogate-based optimization, uncertainty quantification, and optimization under uncertainty that provide the foundation for many of Dakota's iterative analysis capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization Parameter Estimation Uncertainty Quantification and Sensitivity Analysis: Version 6.12 User's Manual

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a user's manual for the Dakota software and provides capability overviews and procedures for software execution, as well as a variety of example studies.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Dakota A Multilevel Parallel Object-Oriented Framework for Design Optimization Parameter Estimation Uncertainty Quantification and Sensitivity Analysis (V.6.14) (Theory Manual)

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a theoretical manual for selected algorithms implemented within the Dakota software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of Dakota-related research publications in the areas of surrogate-based optimization, uncertainty quantification, and optimization under uncertainty that provide the foundation for many of Dakota's iterative analysis capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Dakota A Multilevel Parallel Object-Oriented Framework for Design Optimization Parameter Estimation Uncertainty Quantification and Sensitivity Analysis (V.6.14) (User's Manual)

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a users manual for the Dakota software and provides capability overviews and procedures for software execution, as well as a variety of example studies.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 6.13 Theory Manual

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a theoretical manual for selected algorithms implemented within the Dakota software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of Dakota-related research publications in the areas of surrogate-based optimization, uncertainty quantification, and optimization under uncertainty that provide the foundation for many of Dakota's iterative analysis capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 6.13 User's Manual

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a user’s manual for the Dakota software and provides capability overviews and procedures for software execution, as well as a variety of example studies.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

AGGREGATE: dAta-driven modelinG preservinG contRollable dEr for outaGe mAnagemenT and rEsiliency (Final Report)

The AGGREGATE project team successfully developed and validated various modules for outage management. Brief summaries of each module are provided to showcase their strength for outage management and restoration for a distribution system with a high penetration of connected distribution energy resources (DERs). In recent years, inverter-based DERs have been widely deployed in distribution system. A most of behind-the-meter (BTM) solar power generation is not visible to the utility. The data-driven DER and load estimation modules are using machine learning (ML) and artificial intelligence (AI) to manage this issue, which provides an opportunity for distribution system operators (DSOs) to operate systems and make decisions in real-time for a distribution system with a high penetration of DERs deployed. Also, the estimated DER and true load can be further leveraged in network aggregation and cold-load pick up estimation for reducing the computing complexity and providing for fast restoration. After load demand and DER power generations have been estimated, the information will support topology and state estimation (SE). The topology estimation module demonstrated the viability of mixed integer linear programming (MILP) formulation to estimate the most likely operational radial topology and outage sections using power flow measurements, historical/estimated load and DERs data and smart meter ping measurements. Formulation includes continuous (power flow, load and DERs data) and binary measurements (smart meter ping measurements) in a single formulation. Errors in continuous data and binary data are modeled as normal distribution and Bernoulli distribution, respectively. In the future distribution grid, the power injection from controllable DERs will be essential for efficient and resilient grid operation. However, determining the optimal DER injections and restoration actions is dependent on knowledge of the system states. State estimation (SE), already the cornerstone of transmission energy management systems, will become commonplace in distribution management systems as more measurements become available from deployment of automated metering infrastructure (AMI). Observability analysis is the first step in SE, as it determines the sufficiency of the available measurements for accurately estimating the current system states. A new type of pseudo-measurement called a Correlational Measurement (CM) is introduced in this module, to enhance the observability of the system to enable more accurate SE. CMs encapsulate knowledge of correlation between demand patterns for similar classes of loads as well as injection patterns for same-technology renewable DERs. During grid contingency scenarios, DERs have been traditionally disconnected, without any fault ride-through capabilities. However, with new regulations and better technology, it is feasible for these resources to contribute to the grid’s restoration after an adverse event and hence enhance resilience. The controllability module proposes a two-step restoration scheme for the power system restoration process by leveraging additional degrees of freedom in power electronics interfaced DERs for mitigating voltage problems. In a resilience mode without the utility system, the distribution grid relies on DERs to serve critical load. In such a severe event with multiple faults on the distribution feeders, actuation of various protective devices (PDs) divides the distribution system into electrical islands. The undetected actuated PDs due to fault current contributions from DERs can delay the restoration process, thereby reducing the system resilience. The Advanced Outage Management (AOM) and the Advanced Feeder Restoration (AFR) modules developed in this project provide improved system resilience with multiple DERs. AOM identifies the faulted sections and actuated PDs in a distribution system with DERs by incorporating smart meter data. The most credible outage scenario including fault locations, PD actuations, and fault indicator (FI) failures is identified by a set of binary integer linear programming incorporating hypotheses. The AFR module serves to restore a distribution system with available energy resources taking into consideration the availability of utility sources and DERs. By partitioning the system into islands, critical load will be served with the available generation resources within islands based on the solution of a MILP. When the utility systems become available, the optimal path will be determined by a spanning tree search algorithm that reconnects these islands back to substations and restores the remaining load. The transmission and distribution (T&D) co-simulation module was used to validate the effect of a control action performed on the distribution side assets as it propagates to the transmission side. This ensures that the control action performed results in a feasible operating point on both the transmission and the distribution system. In addition to validation, the team used the T&D co-simulation module to demonstrate how distribution system assets can be used to mitigate issues on the transmission system. Specifically, the team demonstrated that appropriate switching operations on the distribution side can alleviate the line overload condition on the transmission side without causing new operational constraint violations.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2021

Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 6.15 User's Manual

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Theory Manual (V.6.15)

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a theoretical manual for selected algorithms implemented within the Dakota software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of Dakota-related research publications in the areas of surrogate-based optimization, uncertainty quantification, and optimization under uncertainty that provide the foundation for many of Dakota's iterative analysis capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis (V.6.16 User's Manual)

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a user's manual for the Dakota software and provides capability overviews and procedures for software execution, as well as a variety of example studies.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 6.16 Theory Manual

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a theoretical manual for selected algorithms implemented within the Dakota software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of Dakota-related research publications in the areas of surrogate-based optimization, uncertainty quantification, and optimization under uncertainty that provide the foundation for many of Dakota’s iterative analysis capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Models and Strategies for Optimal Demand Side Management in the Chemical Industries

Deregulation and the increase of renewable electricity generation from wind and solar photovoltaics have transformed the U.S. electricity market. Economic and environmental benefits notwithstanding, the presence of renewables has increased variability and uncertainty on the supply side of the grid. Managing demand, rather than generation – a strategy referred to as “demand response (DR)” – is an attractive approach for mitigating this imbalance. DR efforts aim to reduce electricity usage during peak demand times, lessening stress on the grid. Industrial users are particularly attractive entities for DR participation since they present large, localized loads that can provide significant relief on grid demand and –unlike other large loads, such as buildings – are minimally dependent on human needs and preferences. In this project, we accomplished three main objectives. (1) We developed data-driven low-order DR scheduling-relevant dynamic models of chemical processes. Concurrently, we studied the formulation and solution of the associated optimal DR production scheduling problems. (a) A prototype air separation unit (ASU) model was used to generate simulated operating data for initial modeling efforts, which enabled the later use of industrial data for data-driven modeling. (b) We utilized Hammerstein-Wiener (HW) and Finite Step Response (FSR) models to represent nonlinear plant dynamics. (c) The HW models were linearized using exact linearization so they could potentially be embedded in power system models, which are formulated as mixed integer linear programs (MILPs). (d) We solved DR optimization problems under uncertainty and found that even naïve predictions of electricity price and product demand led to significant cost savings benefits. (2) Our DR scheduling optimization problem formulations are amenable to real-time solution. (a) We utilized Lagrangian Relaxation (LR) to efficiently solve the optimization problem by decoupling subproblems linked by complicating constraints. (b) We have achieved computation times for the 3-day DR scheduling problem of an ASU as low as 1.88 minutes. (3) Our representations of the DR behavior of chemical process as grid-level batteries were embedded in power system models. (a) For a small-scale grid, we found that incorporating the dynamics of the chemical plant in the optimal power flow calculations resulted in better resource management leading to up to 15% and 46% cost reduction for the grid and chemical plant operations, respectively, during periods of power line congestion. We have published several works dedicated to modeling and solving DR optimization problems from the user side. These were published in top peer-reviewed journals and are summarized in this report. The most recent work (and papers in preparation) considers DR scheduling from the grid side. Future efforts will consider networked plants (e.g., air separation units operating on a common pipeline) for DR participation, which is expected to amplify the capabilities of industrial DR participants to perform load-shifting. Our consideration of uncertainty in DR has inspired future directions in this area as well: we plan to develop multistage methods to fully account for the effects of uncertainty in DR scheduling.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2022

Integrating Concentrating Solar Power Technologies into the Hybrid Optimization and Performance Platform (HOPP)

As the world increases renewable energy deployment, there is an increasing interest in hybridizing various generation and storage technologies to maximize net benefit to the developer and/or off-taker. A particularly interesting combination of renewable technologies is concentrating solar power (CSP) with thermal energy storage (TES), photovoltaics (PV), and electrochemical battery energy storage (BESS). Due to the system complexity of CSP technology, it is difficult to evaluate the technological and financial performance of a CSP-PV hybrid system without detailed modeling of annual operations. To address this challenge, we have developed a modeling framework for evaluating the performance and financial viability of CSP systems hybridized with PV and battery technologies. This modeling effort incorporates CSP tower and trough systems into an existing modeling tool recently developed by NREL referred to as the Hybrid Optimization and Performance Platform (HOPP). This report outlines the modeling methodology as well as preliminary results from example case studies conducted using the model. The methodology describes: (i) the integration of CSP tower and troughs into HOPP using python interfaces to access System Advisor Model (SAM) underlining technology models, (ii) the mathematical formulation of the mixed integer linear program dispatch optimization model which optimizes operations of storage asset to either maximize system revenue or minimize operating cost while load following, (iii) the design analysis methods implemented within HOPP, and (iv) simulation clustering for the purposes of reducing computational expense. We exercise the model using a case study of a future scenario where we assume (i) CSP and PV technologies achieve the 2030 cost targets provided by the Solar Energy Technologies Office (SETO), (ii) battery costs reduce to the 2030 mid cost projection presented by NREL. Lastly, (iii) electricity prices for southern California in 2030 are provided by NREL's Cambium database, and (iv) a capacity payment of $150/kW-yr based on the system capacity factor during the to 100 net-load hours.

14 SOLAR ENERGY↗

DOE OSTI2025

New Results on Communication- and Memory-Aware Load Balancing Model and Algorithms

While load balancing in distributed-memory computing has been well-studied, we present an innovative approach to this problem: a unified, reduced-order model that combines three key components to describe “work” in a distributed system: computation, communication, and memory. Our model enables an optimizer to explore complex tradeoffs in task placement, such as augmented parallelism, at the expense of data replication increasing memory usage. We propose a fully distributed, heuristic-based load balancing optimization algorithm, and demonstrate that it quickly finds close-to-optimal solutions. We formalize the complex optimization problem as a mixed-integer linear program, and compare it to our strategy. Finally, we show that when applied to an electromagnetics code, our approach obtains up to 2.3x speedups for the imbalanced execution.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Substation-Level Grid Topology Optimization Using Bus Splitting

Operations of substation circuit breakers are important for maintenance needs and topology reconfiguration in power systems. Bus splitting is one type of topology change where the two bus bars at a substation can become electrically disconnected under certain actions of circuit breakers. Because these events involve detailed substation modeling, they are typically not considered in routine power system operation and control. In this paper, an improved substation-level topology optimization framework is developed by expanding traditional line switching decisions by breaker-level bus splitting, which can further reduce grid congestion and generation costs. A tight McCormick relaxation is proposed to reformulate the bilinear terms in the resultant optimization problem to linear inequality constraints. Thus, a tractable mixed-integer linear program reformulation is attained that allows for efficient solutions in real-time operations. Numerical studies on the IEEE 14-bus and 118-bus systems demonstrate the computational performance and economic benefits of the proposed topology optimization approach.

bus split↗

DOE OSTI2019

Co-Optimized Analysis and Design of Electric and Natural Gas Infrastructures

This paper proposes and implements a long-term deterministic capacity expansion model for the co-optimization of electric and natural gas infrastructures. It determines the required investments in generation units, transmission lines and pipelines for meeting future demands, while representing electricity and natural gas flows using DC Power Flow and Weymouth equations, respectively. A Mixed Integer Nonlinear Programming (MINLP) problem is developed, from which a linearized version is derived. A 26 node integrated gas-electric system for the Eastern Region of the United States is used to demonstrate the model’s capabilities. Results show that the model provides an accurate operational representation of the integrated system, and, therefore, enhances the expansion planning process.

47 OTHER INSTRUMENTATION↗

DOE OSTI2021

Per-Phase and 3-Phase Optimal Coordination of Directional Overcurrent Relays Using Genetic Algorithm

Penetration of the power grid by renewable energy sources, distributed storage, and distributed generators is becoming increasingly common. Increased utilization of these distributed energy resources (DERs) has given rise to additional protection coordination concerns, particularly where they are utilized in an unbalanced manner or where loading among phases is unbalanced. Digital relays such as the SEL-751 (produced by Schweitzer Engineering Laboratories, Pullman, WA, USA) series have the capability of being set on a per-phase basis. This capability is underutilized in common practice. Additionally, in optimization algorithms for determining relay settings, the time-overcurrent characteristics (TOCs) of relays are generally not treated as variables and are assigned before running the optimization algorithm. In this paper, TOC options themselves are treated as discrete variables to be considered in the optimization algorithm. A mixed integer nonlinear programming problem (MINLP) is set up where the goal is to minimize relay operating times. A genetic algorithm (GA) approach is implemented in MATLAB where two cases are considered. In the first case, the TOC and Time dial setting (TDS) of each relay is set on a three-phase basis. In the second case, per-phase settings are considered. Relay TDSs and TOCs are both considered as simultaneous discrete control variables. Despite the stochastic nature of using per-phase settings for unbalanced systems is found to generally allow for shorter operating times. However, for relatively balanced systems, it is best to use three-phase settings if computation time is of importance.

Matthews, Ronald C.↗