Search NASA⌕ Search

SEARCH · Search NASA

Results for “network optimization”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 325 records · Page 18
All sourcesNASA NTRSDOE OSTI
NASA NTRS2006

Single- and Multiple-Objective Optimization with Differential Evolution and Neural Networks

Genetic and evolutionary algorithms have been applied to solve numerous problems in engineering design where they have been used primarily as optimization procedures. These methods have an advantage over conventional gradient-based search procedures became they are capable of finding global optima of multi-modal functions and searching design spaces with disjoint feasible regions. They are also robust in the presence of noisy data. Another desirable feature of these methods is that they can efficiently use distributed and parallel computing resources since multiple function evaluations (flow simulations in aerodynamics design) can be performed simultaneously and independently on ultiple processors. For these reasons genetic and evolutionary algorithms are being used more frequently in design optimization. Examples include airfoil and wing design and compressor and turbine airfoil design. They are also finding increasing use in multiple-objective and multidisciplinary optimization. This lecture will focus on an evolutionary method that is a relatively new member to the general class of evolutionary methods called differential evolution (DE). This method is easy to use and program and it requires relatively few user-specified constants. These constants are easily determined for a wide class of problems. Fine-tuning the constants will off course yield the solution to the optimization problem at hand more rapidly. DE can be efficiently implemented on parallel computers and can be used for continuous, discrete and mixed discrete/continuous optimization problems. It does not require the objective function to be continuous and is noise tolerant. DE and applications to single and multiple-objective optimization will be included in the presentation and lecture notes. A method for aerodynamic design optimization that is based on neural networks will also be included as a part of this lecture. The method offers advantages over traditional optimization methods. It is more flexible than other methods in dealing with design in the context of both steady and unsteady flows, partial and complete data sets, combined experimental and numerical data, inclusion of various constraints and rules of thumb, and other issues that characterize the aerodynamic design process. Neural networks provide a natural framework within which a succession of numerical solutions of increasing fidelity, incorporating more realistic flow physics, can be represented and utilized for optimization. Neural networks also offer an excellent framework for multiple-objective and multi-disciplinary design optimization. Simulation tools from various disciplines can be integrated within this framework and rapid trade-off studies involving one or many disciplines can be performed. The prospect of combining neural network based optimization methods and evolutionary algorithms to obtain a hybrid method with the best properties of both methods will be included in this presentation. Achieving solution diversity and accurate convergence to the exact Pareto front in multiple objective optimization usually requires a significant computational effort with evolutionary algorithms. In this lecture we will also explore the possibility of using neural networks to obtain estimates of the Pareto optimal front using non-dominated solutions generated by DE as training data. Neural network estimators have the potential advantage of reducing the number of function evaluations required to obtain solution accuracy and diversity, thus reducing cost to design.

Rai, Man Mohan↗
DOE OSTI2026

A Hierarchical Optimization Method for Electric Vertical Takeoff and Landing Aircraft Network Design

Electric vertical takeoff and landing aircraft (eVTOLs) are expected to serve urban air mobility in a station-to-station configuration, which makes the optimal network design of eVTOL stations a critical question to explore. Existing approaches often face limitations, such as the inability to interact station locations with demand or difficulty in finding the optimal solution for large study regions. Here, this paper first proposes a mathematical model to generate optimal eVTOL station locations while considering associated potential eVTOL demand, and then proposes a heuristic algorithm, Hierarchical Optimization MEthod (HOME), to efficiently solve the model. With a case study of Southern California, HOME was compared to 1) directly solving the original integer linear programming-based network design problem, and 2) employing the widely used genetic algorithm. Results suggest that HOME can find optimal solutions with limited computational resources. The proposed framework powered by HOME provides a computationally efficient way to support urban air mobility planning.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Risk-based Post Injection Site Care and Monitoring for Commercial-Scale Carbon Storage: Reevaluation of the FutureGen 2.0 Site using NRAP-Open-IAM and DREAM

Modeling was conducted to assess the effectiveness of the NRAP toolset, specifically NRAP-IAM-CS v2 and DREAM, to determine a risk-based PISC period and optimized monitoring network for a commercial-scale CO2 storage project. Realizations from NRAP-IAM-CS v2 revealed that leakage rates of both CO2 and brine were small, on the order of 10-5 kg/s, and occurred primarily during the injection phase in the Ironton-Galesville, the thief zone immediately overlying the injection reservoir. Using this information to design an optimized monitoring well network eliminated one of the two originally planned above zone monitoring wells, resulting a substantial cost reduction for the project. Perhaps the most significant finding from this effort is that NRAP-IAM-CS v2 can be used to define a risk-based, and substantially shorter, PISC period for the site. NRAP-IAM-CS v2 realizations indicate that the majority of risk of endagerment to USDWs decreases within the first 5 years after CO2 injection ends. Doubling this timeframe would still lead to a net PISC period reduction of 40-years and an operational cost reduction in excess of $50M for the project. This study serves as a first-of-its-kind effort to apply NRAP tools to evaluate components of a commercial-scale Class VI UIC permit and provides a foundation for broader application and adoption by the CCS community.

system model, integrated assessment, reservoir, aq↗
NASA NTRS2019

High-Performance Computing Optimization for Aladyn – Adaptive Neural Network Molecular Dynamics Mini-Application

This report provides a description and performance evaluation of the optimization techniques for high performance computing (HPC) implementation of the open source Computational Materials mini-application Aladyn (https://github.com/nasa/aladyn). Aladyn is a basic molecular dynamics code written in FORTRAN 2003, which is designed to demonstrate the use of adaptive neural networks (ANNs) in atomistic simulations. The role of ANNs is to efficiently reproduce the very complex energy landscape resulting from the atomic interactions in materials with the accuracy of the more expensive quantum mechanics-based calculations. The ANN is trained on a large set of atomic structures calculated using the density functional theory (DFT) method. While achieving orders of magnitude faster computational performance than DFT, the ANN-based approach was still very computationally demanding compared to the conventional approach of using empirically fitted energy functions. After its initial development, Aladyn was evaluated and optimized by experts at the NASA Advanced Supercomputing (NAS) division to exploit modern supercomputer architectures. The code has been optimized for execution on multicore central processing units (CPUs), including Intel® Skylake microarchitecture, and on graphic accelerators, such as Nvidia® V100 graphic processing units (GPUs), using Open Multi-Processing (OpenMP) and Open Accelerators (OpenACC) programming interfaces. The optimization achieved a speedup of 4.7 times the baseline version on CPU performance and an additional 2.4 times on CPU+GPU performance. Atomistic computer simulations are a fundamental tool in materials research to model material properties form physics-based first principles. Atomic interaction, governed by Quantum Mechanics (QM) require sophisticated and highly computationally demanding mathematical models to calculate [1]. Classical methods use approximate functional forms, empirically fitted through a set of variable parameters to emulate atomic energies as direct functions of atomic coordinates [2]. While empirical potentials are computationally much simpler, allowing simulations of large-scale systems of up to a trillion (1012) atoms [3], they are substantially less accurate compared to quantum calculations and applicable only to very specific atomic configurations or predefined crystallographic phases. A recently suggested approach is to use heuristic machine learning methods [4], such as those based on Adaptive Neural Networks (ANNs) to predict atomic energies, after being trained on a sufficiently large database of QM-calculated structures [5,6]. This approach reduces significantly the computational complexity, allowing for simulations of orders of magnitude larger systems compared to QM-based methods without compromising accuracy. Still, compared to classical methods using empirical energy functions, ANN methods remain two- to three orders of magnitude more computationally demanding. Hence, the computational cost of simulations, together with the need for extensive training of ANNs, still makes the practical implementation of ANN-based methods quite challenging. The purpose of the Aladyn mini-application software [7], available as open source at https://github.com/nasa/aladyn, is to be a testbed for exploring possible optimization strategies to develop highly scalable parallel algorithms for ANN-based atomistic simulations. Aladyn is aimed at utilizing the architecture of the high-end modern highperformance computing (HPC) hardware based on multicore central processing units (CPUs) equipped with graphic processing unit (GPU) accelerators. Specifically, the goal is to optimize the performance on a single HPC compute node, before implementing scaling to multi-node parallelization using message passing interface (MPI). At the same time, the open source code of Aladyn can serve as a training model for students and professors in academia.

Yamakov, Vesselin I.↗
DOE OSTI2025

Hierarchical Resilience Planning for Networked Microgrids: A Case Study of Puerto Rico

Microgrids can be designed to enhance the energy resilience of communities and critical infrastructures, such as hospitals, data centers, and communication networks, which are vulnerable to frequent weather-related disruption. Coordinating multiple microgrids in a network can leverage the geographical diversity of load and generation resources while enabling resilient and cost-effective planning of the distribution system. Designing a networked microgrid is complex, involving intricate technical assessment, cost-benefit analysis, site-specific requirements, and the evaluation of existing resources. Therefore, this paper proposes a hierarchical resilience planning framework and performs an extensive techno-economic analysis for the design of a networked microgrid. Hierarchical resilience planning involves technology sizing at an individual community level to meet the critical load and satisfy resilience criteria, and resource optimization at networked microgrid level to provide a higher level of resilience and energy adequacy. A real-world case of Puerto Rico's cooperative microgrid “Microrred de la Montaña” is investigated considering localized electricity tariffs, site-specific demand profiles, solar generation, and existing hydro resources. Multiple optimization scenarios are developed based on the resiliency requirement to estimate the capacity of solar photovoltaic and battery energy storage (BES) to be installed at each substation. The results provide the optimal sizing for individual community and networked microgrid to withstand 1day and 3-day outages along with the criteria for critical load.

13 - HYDRO ENERGY↗
NASA NTRS1993

Synthesis of Optimal Nonlinear Feedback Laws for Dynamic Systems Using Neural Networks

Open-loop solutions of dynamical optimization problems can be numerically computed usingexisting software packages. The computed time histories of the state and control variables, formultiple sets of end conditions can then be used to train a neural network to 'recognize' the optimal,nonlinear feedback relation between the states and controls of the system. The 'learned' network canthen be used to output an approximate optimal control given a full set (or a partial set) of measuredsystem states. With simple neural networks, we have successfully demonstrated the efficacy of theproposed approach using a minimum-time orbit injection problem. The usefulness and limitations ofthis novel approach on real-life optimal guidance and control problems, with many state and control variables as well as path inequality constraints, remain to be seen.

Orbit injection problem↗
NASA NTRS1992

Optimal mapping of neural-network learning on message-passing multicomputers

A minimization of learning-algorithm completion time is sought in the present optimal-mapping study of the learning process in multilayer feed-forward artificial neural networks (ANNs) for message-passing multicomputers. A novel approximation algorithm for mappings of this kind is derived from observations of the dominance of a parallel ANN algorithm over its communication time. Attention is given to both static and dynamic mapping schemes for systems with static and dynamic background workloads, as well as to experimental results obtained for simulated mappings on multicomputers with dynamic background workloads.

Chu, Lon-Chan↗
DOE OSTI2022

Co-optimization of repairs and dynamic network reconfiguration for improved distribution system resilience

In this work, a post-disaster distribution system repair and restoration (DSRR) strategy is proposed to improve distribution system resilience. The DSRR strategy is formulated as a two-stage optimization. The first stage is a comprehensive co-optimization of repair crew scheduling, dynamic network reconfiguration, and distributed energy resource (DER) dispatch based on the forecast load profile. The goal is to minimize the accumulative operating cost caused by the load reduction payment as well as DER operating cost. In particular, since the number of available repair crews is usually smaller than the number of faulted lines after a disaster event, the DSRR strategy determines the optimal scheduling for repairing faulted lines. The second stage is a re-dispatch of the DER power output and load shedding based on the real-time load demand of each bus. The proposed algorithm is validated by case studies of the IEEE 33-bus and 123-bus test systems. We consider those scenarios in which faults occur in multiple heavy-loaded feeders. The simulation results demonstrate that the DSRR strategy effectively coordinate the repair scheduling, network reconfiguration and load shedding to minimize the operating cost.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

GAINN: The Galaxy Assembly and Interaction Neural Networks for High-redshift JWST Observations

We present the Galaxy Assembly and Interaction Neural Networks (Gainn), a series of artificial neural networks for predicting the redshift, stellar mass, halo mass, and mass-weighted age of simulated galaxies based on James Webb Space Telescope (JWST) photometry. Our goal is to determine the best neural network for predicting these variables at 11 < z < 15. The parameters of the optimal neural network can then be used to estimate these variables for real, observed galaxies. The inputs of the neural networks are JWST filter magnitudes of a subset of five broadband filters (F150W, F200W, F277W, F356W, and F444W) and two medium-band filters (F162M and F182M). We compare the performance of the neural networks using different combinations of these filters, as well as different activation functions and numbers of layers. The best neural network predicted redshift with a normalized rms error of $0.010^{+0.003}_{-0.001}$, stellar mass with rms = $0.089^{+0.044}_{-0.022}$, halo mass with a mean-squared error of $0.022^{+0.014}_{-0.008}$, and mass-weighted age with rms = $12.466^{+5.065}_{-2.408}$. We also test the performance of Gainn on real data from MACS0647JD, an object observed by JWST. Predictions from Gainn for the first projection of the object (JD1) have normalized bias $\langle$Δz$\rangle$ < 0.00228, which is significantly smaller than found with template-fitting methods. We find that the optimal filter combination is F277W, F356W, F162M, and F200W when considering both theoretical accuracy and observational resources from JWST.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

PANNA: Properties from Artificial Neural Network Architectures

We report prediction of material properties from first principles is often a computationally expensive task. Recently, artificial neural networks and other machine learning approaches have been successfully employed to obtain accurate models at a low computational cost by leveraging existing example data. Here, we present a software package “Properties from Artificial Neural Network Architectures” (PANNA) that provides a comprehensive toolkit for creating neural network models for atomistic systems following the Behler–Parrinello topology. Besides the core routines for neural network training, it includes data parser, descriptor builder for Behler–Parrinello class of symmetry functions and force-field generator suitable for integration within molecular dynamics packages. PANNA offers a variety of activation and cost functions, regularization methods, as well as the possibility of using fully-connected networks with custom size for each atomic species. PANNA benefits from the optimization and hardware-flexibility of the underlying TensorFlow engine which allows it to be used on multiple CPU/GPU/TPU systems, making it possible to develop and optimize neural network models based on large datasets.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

LOCOMOTIVES - Comprehensive Impact and Cost Assessment Framework of Carbon Lowering Approaches for the US Rail Freight System

The goal of this project is to develop a tool to aid railroads and other stakeholders assess and approach the decarbonization of freight rail operations by identifying new, viable low-carbon energy storage and conversion systems for future locomotive systems and how they should be deployed on the existing US freight rail network. In the first quarter, the project focused on collecting data, establishing a simulation workflow, and engaging industry through the creation of the Industry Advisory Board (IAB). In the second quarter, the project focused on selecting fuel pathways and powertrain technologies, setting performance targets, conducting a techno-economic analyses, and developing the simulation framework that would serve as the backbone of the future toolhead. The third quarter involved developing an industry-oriented interactive dashboard powered by a five-step sequential framework, as well as holding industry advisory board meetings as per the initial technology-to-market plan. In the remaining project quarters, the NUFRIEND dashboard were fine-tuned with the help of IAB member feedback and in-depth scenario analyses were conducted to support the techno-economic analysis of energy sources. Additionally, dashboard documentation, project insights, and open-source code on GitHub were prepared and released. Throughout the project, the team completed testing and analysis of all model components, integrated all initial test scenarios, and conducted stakeholder engagement. Lower-carbon drop-in fuels can be deployed as admixtures and are considered uniform across the network at a desired penetration rate, while hydrogen and battery-electric technology deployment poses a more complex problem as they require significant investments to be made in the siting of refueling/charging facilities and the replacement of locomotive fleets. Thus, strategies for locating and sizing refueling/charging facilities on a railroad’s network to meet their energy demands were developed to inform deployment decisions. To address this challenge, the Northwestern University Freight Rail Infrastructure & Energy Network Decarbonization (NUFRIEND) framework presents a five-step sequential framework to select O-D paths, locate facilities, reroute flows, size facilities, and evaluate the deployment for alternative energy sources that require locomotive powertrains to be converted and new refueling infrastructure to be deployed. The NUFRIEND Framework is an industry-oriented tool for simulating the deployment of new energy technologies across the US freight rail network. The framework provides a comprehensive network-level optimization and scenario simulation tool for decarbonizing the freight rail sector, addressing the uncertainties surrounding technological developments by supporting sensitivity analyses for different operational and technological parameters through a transparent and flexible input module. It offers practical alternatives to diesel locomotives and can be applied for any railroad considering the specific network structure and freight demand, outputting evaluation metrics for the associated emissions and costs relative to diesel operations. A number of relevant simulation scenarios were run and analyzed for key insights on the value of different alternative technologies for freight rail decarbonization. The project developments and findings have been presented at numerous conferences and events.

08 HYDROGEN↗
DOE OSTI2023

Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions

Here, in this paper, we propose and study neural network-based methods for solutions of high-dimensional quadratic porous medium equation (QPME). Three variational formulations of this nonlinear PDE are presented: a strong formulation and two weak formulations. For the strong formulation, the solution is directly parameterized with a neural network and optimized by minimizing the PDE residual. It can be proved that the convergence of the optimization problem guarantees the convergence of the approximate solution in the $L^1$ sense. The weak formulations are derived following (Brenier in Examples of hidden convexity in nonlinear PDEs, 2020) which characterizes the very weak solutions of QPME. Specifically speaking, the solutions are represented with intermediate functions who are parameterized with neural networks and are trained to optimize the weak formulations. Extensive numerical tests are further carried out to investigate the pros and cons of each formulation in low and high dimensions. This is an initial exploration made along the line of solving high-dimensional nonlinear PDEs with neural network-based methods, which we hope can provide some useful experience for future investigations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A tri-level optimization model for interdependent infrastructure network resilience against compound hazard events

Resilient operation of interdependent infrastructures against compound hazard events is essential for maintaining societal well-being. To address consequence assessment challenges in this problem space, we propose a novel policy-guided tri-level optimization model applied to a proof-of-concept case study with fuel distribution and transportation networks – encompassing one realistic network; one fictitious, yet realistic network; as well as networks drawn from three synthetic distributions. Mathematically, our approach takes the form of a defender-attacker-defender (DAD) model—a multi-agent tri-level optimization, comprised of a defender, attacker, and an operator acting in sequence. Here, in this study, our notional operator may choose proxy actions to operate an interdependent system comprised of fuel terminals and gas stations (functioning as supplies) and a transportation network with traffic flow (functioning as demand) to minimize unmet demand at gas stations. A notional attacker aims to hypothetically disrupt normal operations by reducing supply at the supply terminals, and the notional defender aims to identify best proxy defense policy options which include hardening supply terminals or allowing alternative distribution methods such as trucking reserve supplies. We solve our DAD formulation at a metropolitan scale and present practical defense policy insights against hypothetical compound hazards. We demonstrate the generalizability of our framework by presenting results for a realistic network; a fictitious, yet realistic network; as well as for three networks drawn from synthetic distributions. Additionally, we demonstrate the scalability of the framework by investigating runtime performance as a function of the network size. Steps for future research are also discussed.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
NASA NTRS1990

A technique for optimal temperature estimation for modeling sunrise/sunset thermal snap disturbance torque

A predictive temperature estimation technique which can be used to drive a model of the Sunrise/Sunset thermal 'snap' disturbance torque experienced by low Earth orbiting spacecraft is described. The twice per orbit impulsive disturbance torque is attributed to vehicle passage in and out of the Earth's shadow cone (umbra), during which large flexible appendages undergo rapidly changing thermal conditions. Flexible members, in particular solar arrays, experience rapid cooling during umbra entrance (Sunset) and rapid heating during exit (Sunrise). The thermal 'snap' phenomena has been observed during normal on-orbit operations of both the LANDSAT-4 satellite and the Communications Technology Satellite (CTS). Thermal 'snap' has also been predicted to be a dominant source of error for the TOPEX satellite. The fundamental equations used to model the Sunrise/Sunset thermal 'snap' disturbance torque for a typical solar array like structure will be described. For this derivation the array is assumed to be a thin, cantilevered beam. The time varying thermal gradient is shown to be the driving force behind predicting the thermal 'snap' disturbance torque and therefore motivates the need for accurate estimates of temperature. The development of a technique to optimally estimate appendage surface temperature is highlighted. The objective analysis method used is structured on the Gauss-Markov Theorem and provides an optimal temperature estimate at a prescribed location given data from a distributed thermal sensor network. The optimally estimated surface temperatures could then be used to compute the thermal gradient across the body. The estimation technique is demonstrated using a typical satellite solar array.

Zimbelman, D. F.↗
DOE OSTI2019

Systems and methods for convex relaxations and linear approximations for optimal power flow in multiphase radial networks

Centralized node controllers in accordance with embodiments of the invention enable linear approximation of optimal power flow. One embodiment includes a centralized node controller including: a network interface, a processor, and a memory containing: a centralized power control application a network topology, where the network is multiphase unbalanced and comprises a plurality of connected nodes; wherein the processor is configured by the centralized controller application to: request node operating parameters from the plurality of connected nodes; calculate network operating parameters using a linear approximation of optimal power flow and the node operating parameters from the plurality of connected nodes; send network operating parameters to the plurality of connected nodes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Switch Opening and Exchange Method for Stochastic Distribution Network Reconfiguration

The purpose of distribution network reconfiguration (DNR) is to determine the optimal topology of an electricity distribution network, which is an efficient measure to reduce network power losses. Electricity load demand and photovoltaic (PV) output are uncertain and vary with time of day, and will affect the optimal network topology. Single-hour deterministic DNR is incapable of handling this uncertainty and variability. Therefore, this paper proposes to solve a multi-hour stochastic DNR (SDNR). Existing solution methods for DNR are either inaccurate or excessively time-consuming, and are therefore incapable of solving multi-hour SDNRs for large distribution networks. In this regard, a switch opening and exchange (SOE) method is proposed. Starting from a looped network with all switches closed, the SOE consists of three steps. The first step is to sequentially open the switches until all of the loops are opened. The second and third steps modify the status of branches obtained in the first step to obtain better radial topologies. Five test systems are used to validate the accuracy and fast solution speed of the SOE and the superiority of multi-hour SDNR over single-hour deterministic DNR.

distribution network reconfiguration↗
NASA NTRS1992

Application of a neural network to simulate analysis in an optimization process

A new experimental software package called NETS/PROSSS aimed at reducing the computing time required to solve a complex design problem is described. The software combines a neural network for simulating the analysis program with an optimization program. The neural network is applied to approximate results of a finite element analysis program to quickly obtain a near-optimal solution. Results of the NETS/PROSSS optimization process can also be used as an initial design in a normal optimization process and make it possible to converge to an optimum solution with significantly fewer iterations.

Rogers, James L.↗
NASA NTRS2003

Optimal Feedback Control of Thermal Networks

An improved approach to the mathematical modeling of feedback control of thermal networks has been devised. Heretofore software for feedback control of thermal networks has been developed by time-consuming trial-and-error methods that depend on engineers expertise. In contrast, the present approach is a systematic means of developing algorithms for feedback control that is optimal in the sense that it combines performance with low cost of implementation. An additional advantage of the present approach is that a thermal engineer need not be expert in control theory. Thermal networks are lumped-parameter approximations used to represent complex thermal systems. Thermal networks are closely related to electrical networks commonly represented by lumped-parameter circuit diagrams. Like such electrical circuits, thermal networks are mathematically modeled by systems of differential-algebraic equations (DAEs) that is, ordinary differential equations subject to a set of algebraic constraints. In the present approach, emphasis is placed on applications in which thermal networks are subject to constant disturbances and, therefore, integral control action is necessary to obtain steady-state responses. The mathematical development of the present approach begins with the derivation of optimal integral-control laws via minimization of an appropriate cost functional that involves augmented state vectors. Subsequently, classical variational arguments provide optimality conditions in the form of the Hamiltonian equations for the standard linear-quadratic-regulator (LQR) problem. These equations are reduced to an algebraic Riccati equation (ARE) with respect to the augmented state vector. The solution of the ARE leads to the direct computation of the optimal proportional- and integral-feedback control gains. In cases of very complex networks, large numbers of state variables make it difficult to implement optimal controllers in the manner described in the preceding paragraph.

Papalexandris, Miltiadis↗