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DOE OSTI2023

A Sequential Quadratic Programming Algorithm for Nonsmooth Problems with Upper- ${\boldsymbol{\mathcal{C}^2}}$ Objective

An optimization algorithm for nonsmooth nonconvex constrained optimization problems with upper- ${\boldsymbol{\mathcal{C}^2}}$ objective functions is proposed and analyzed. Upper- ${\boldsymbol{\mathcal{C}^2}}$ is a weakly concave property that exists in difference of convex (DC) functions and arises naturally in many applications, particularly certain classes of solutions to parametric optimization problems e.g., recourse of stochastic programming and projection onto closed sets. The algorithm can be viewed as an extension of sequential quadratic programming (SQP) to nonsmooth problems with upper- ${\boldsymbol{\mathcal{C}^2}}$ objectives or a simplified bundle method. It is globally convergent with bounded algorithm parameters that are updated with a trust-region criterion. The algorithm handles general smooth constraints through linearization and uses a line search to ensure progress. The potential inconsistencies from the linearization of the constraints are addressed through a penalty method. In conclusion, the capabilities of the algorithm are demonstrated by solving both simple upper- ${\boldsymbol{\mathcal{C}^2}}$ problems and a real-world optimal power flow problem used in current power grid industry practices.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Flexible dynamic boundary microgrid operation considering network and load unbalances

Flexible microgrids with dynamic boundaries have recently been introduced in the literature. With the ability to reconfigure the topology of the microgrids dynamically through remotely controlled switches, flexible microgrids with dynamic boundaries can further improve the resiliency and energy efficiency of microgrids with distributed energy resources (DERs). This paper focuses on the optimal operation considering one of the predominant characteristics of microgrids and distribution systems – unbalanced networks and loads. In existing literature, balanced modeling of microgrids is more common due to its attractive simplicity. The three-phase power unbalance has not been considered as a constraint on the generation units in a microgrid. Further, negative sequence constraints have also been neglected. In this article, we propose a set of constraints that is specifically related to the capabilities of inverter interfaced resources to supply unbalanced current/power when the microgrid is islanded from the main distribution grid. We incorporate the new set of constraints into two optimization formulations leveraging two convex relaxations of the three-phase power flow equations: mixed-integer linear programming (MILP) and mixed-integer semidefinite programming (MISDP) that optimize the dispatch of controllable switches and DERs in the microgrid. The algorithms are then extended to networked microgrids with grid-forming sources. We test the algorithms on a realistic community microgrid model in Puerto Rico as well as standardized IEEE distribution test feeders. The testing results demonstrate the performance of the proposed algorithms. The MILP is fast and scalable, and the MISDP enforces the negative sequence voltage constraints.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2025

A bilevel multistage stochastic self-scheduling model with indivisibilities for trading in the continuous intraday electricity market

In this paper, we study the profit maximization problem of a virtual power plant trading in the continuous intraday electricity market. Our virtual power plant model is compatible with renewable, and thermal assets, covering a range of virtual power plants currently participating in energy markets. We model the trading problem as a bilevel multistage stochastic program. The upper level of the problem accounts for the profit maximization of the virtual power plant with explicit modeling of the technical constraints of the operational status of the thermal power plant including minimum start-up and shut-down times, ramp-up and ramp-down rates, and minimum generation level. The upper level also decides which continuous and indivisible (fill-or-kill) orders are submitted to the market. The lower-level problem accounts for the clearing of the continuous intraday market, i.e., matching of buy and sell orders. Because of the presence of fill-or-kill orders, the lower-level problem is mixed-integer, which prevents its direct conversion to a single-level problem using duality. In order to solve this challenging problem, we develop a convex-hull extended formulation for the lower-level problem, apply duality theory to obtain a single-level stochastic equivalent formulation, and employ McCormick envelopes to turn the problem into a multistage stochastic mixed-integer linear problem, which we solve using the stochastic dual dynamic integer programming algorithm. We conduct numerical experiments and analyze the optimal trading behavior of a virtual power plant trading in an ideal continuous market without arbitrage.

Bilevel multistage stochastic programming problem↗

DOE OSTI2024

Revealing Decision Conservativeness Through Inverse Distributionally Robust Optimization

This paper introduces Inverse Distributionally Robust Optimization (I-DRO) as a method to infer the conservativeness level of a decision-maker, represented by the size of a Wasserstein metric-based ambiguity set, from the optimal decisions made using Forward Distributionally Robust Optimization (F-DRO). By leveraging the Karush-Kuhn-Tucker (KKT) conditions of the convex F-DRO model, we formulate I-DRO as a bi-linear program, which can be solved using off-the-shelf optimization solvers. Additionally, this formulation exhibits several advantageous properties. We demonstrate that I-DRO not only guarantees the existence and uniqueness of an optimal solution but also establishes the necessary and sufficient conditions for this optimal solution to accurately match the actual conservativeness level in F-DRO. Furthermore, we identify three extreme scenarios that may impact I-DRO effectiveness. Our case study applies F-DRO for power system scheduling under uncertainty and employs I-DRO to recover the conservativeness level of system operators. Numerical experiments based on an IEEE 5-bus system and a realistic NYISO 11-zone system demonstrate I-DRO performance in both normal and extreme scenarios. An extended version of this paper with additional analyses is available at li2024revealing.

distributionally robust optimization↗

DOE OSTI2020

Voltage positioning using co-optimization of controllable grid assets in radial networks

With increasing penetration of solar PV, some distribution feeders are experiencing highly variable net-load flows and even reverse flows. To optimize distribution systems under such conditions, the scheduling of mechanical devices, such as OLTCs and capacitor banks, needs to take into account forecasted solar PV and actual grid conditions. However, these legacy switching assets are operated on a daily or hourly timescale, due to the wear and tear associated with mechanical switching, which makes them unsuitable for real-time control. Therefore, there is a natural timescale- separation between these slower mechanical assets and the responsive nature of inverter-based resources. In this paper, we present a network admissible convex formulation for holistically scheduling controllable grid assets to position voltage optimally against solar PV. An optimal hourly schedule is presented that utilizes mechanical resources to position the predicted voltages close to nominal values, while minimizing the use of inverter-based resources (i.e., DERs), making them available for control at a faster time-scale (after the uncertainty reveals itself). A convex, inner approximation of the OPF problem is adapted to a mixed-integer linear program that minimizes voltage deviations from nominal (i.e., maximizes voltage margins). Here, the resulting OPF solution respects all the network constraints and is, hence, robust against modeling simplifications. Simulation based analysis on IEEE distribution feeders validates the approach.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2021

Joint Topology Identification and State Estimation in Unobservable Distribution Grids

Many distribution system operations (e.g., state estimation, control, fault detection/localization) rely on the assumption that the underlying topology is accurately defined. In general, topology identification is a challenging problem in distribution systems as these systems are unobservable with a very limited number of available measurements. In this paper, we tackle this problem by designing a compressive sensing framework that jointly estimates the systems states and network topology via an integrated mixed integer nonlinear program (MINLP) formulation. Here, two reformulations of the original MINLP problems are investigated. Firstly, in order to remove the nonlinearity in the MINLP formulation, a mixed integer linear programming (MILP) problem that employs auxiliary variables is derived. Furthermore, to achieve a faster solution, convex relaxation of the original formulation is derived. Finally, using a Markovian model for topology changes, prior information about system topology is used to improve topology identification particularly when a limited amount of measurements is available. Simulation results on IEEE 37-bus test feeder and IEEE 123-bus test feeder illustrate the efficiency and scalability of the proposed approaches from both state estimation and topology identification point of view (even with 30% of available data).

42 ENGINEERING↗

DOE OSTI2023

Learning Distribution Grid Topologies: A Tutorial

Unveiling feeder topologies from data is of paramount importance to advance situational awareness and proper utilization of smart resources in power distribution grids. This tutorial summarizes, contrasts, and establishes useful links between recent works on topology identification and detection schemes that have been proposed for power distribution grids. The primary focus is to highlight methods that overcome the limited availability of measurement devices in distribution grids, while enhancing topology estimates using conservation laws of power-flow physics and structural properties of feeders. Grid data from phasor measurement units or smart meters can be collected either passively in the traditional way, or actively, upon actuating grid resources and measuring the feeder's voltage response. Analytical claims on feeder identifiability and detectability are reviewed under disparate meter placement scenarios. Such topology learning claims can be attained exactly or approximately so via algorithmic solutions with various levels of computational complexity, ranging from least-squares fits to convex optimization problems, and from polynomial-time searches over graphs to mixed-integer programs. Although the emphasis is on radial single-phase feeders, extensions to meshed and/or multiphase circuits are sometimes possible and discussed. Here this tutorial aspires to provide researchers and engineers with knowledge of the current state-of-the-art in tractable distribution grid learning and insights into future directions of work.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2024

Optimizing Desalination Operations for Energy Flexibility

Despite the value of energy optimization in desalination processes, modeling dynamic operations for monthly billing periods has remained a computational challenge. This work proposes a framework for energy flexibility optimization, which includes new modeling features for independent operation of parallel skids, start-up delays associated with chemical stabilization, the consideration of industrial energy tariff structures, and inclusion of hourly electrical carbon intensities. This is done using a modular and computationally efficient formulation that guarantees a globally optimal solution with standard optimization solvers. In this study, the approach is demonstrated in two distinct case studies: a seawater desalination plant in Santa Barbara, CA, and an indirect potable reuse facility in San Jose, CA. Trends predicted from the model are validated against operational facility measurements from a demand response shutdown event. Preliminary results show that optimizing energy flexibility can result in 18.51% monthly cost savings over energy efficiency-optimized operation. The value extracted from a facility-wide shutdown during peak electricity price hours is hampered by start-up delays in post-treatment chemical stabilization. In cases in which a facility does not have much excess capacity, using a flow equalization tank or operating over a wide recovery range may be cost-effective.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

Parallel hybrid quantum-classical machine learning for kernelized time-series classification

Supervised time-series classification garners widespread interest because of its applicability throughout a broad application domain including finance, astronomy, biosensors, and many others. Here, in this work, we tackle this problem with hybrid quantum-classical machine learning, deducing pairwise temporal relationships between time-series instances using a timeseries Hamiltonian kernel (TSHK). A TSHK is constructed with a sum of inner products generated by quantum states evolved using a parameterized time evolution operator. This sum is then optimally weighted using techniques derived from multiple kernel learning. Because we treat the kernel weighting step as a differentiable convex optimization problem, our method can be regarded as an end-to-end learnable hybrid quantum-classical-convex neural network, or QCC-net, whose output is a data set-generalized kernel function suitable for use in any kernelized machine learning technique such as the support vector machine (SVM). Using our TSHK as input to a SVM, we classify univariate and multivariate time-series using quantum circuit simulators and demonstrate the efficient parallel deployment of the algorithm to 127-qubit superconducting quantum processors using quantum multi-programming.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Resilience-Motivated Distribution System Restoration Considering Electricity-Water-Gas Interdependency

A major outage in the electricity distribution system may affect the operation of water and natural gas supply systems, leading to an interruption of multiple services to critical customers. Therefore, enhancing resilience of critical infrastructures requires joint efforts of multiple sectors. In this paper, a distribution system service restoration method considering the electricity-water-gas interdependency is proposed. The objective is maximizing the supply of electricity, water, and gas to critical customers after an extreme event. The operational constraints of electricity, water, and natural gas networks are considered. Additionally, the characteristics of electricity-driven coupling components, including water pumps and gas compressors, are also modeled. Relaxation techniques are applied to non convex constraints posed by physical laws of those networks. Consequently, the restoration problem is formulated as a mixed-integer second-order cone program, which can readily be solved by the off-the-shelf solvers. The proposed method is validated by numerical simulations on an electricity-water-gas integrated system, developed based on benchmark models of the subsystems. The results indicate that considering the interdependency refines the allocation of limited generation resources and demonstrate the exactness of the proposed convex relaxation

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2021

Optimal Power Flow in DC Networks with Robust Feasibility and Stability Guarantees

With high penetrations of renewable generation and variable loads, there is significant uncertainty associated with power flows in DC networks such that stability and operational constraint satisfaction are of concern. Most existing DC network optimal power flow (DN-OPF) formulations assume exact knowledge of loading conditions and do not provide stability guarantees. Here, in contrast, this paper studies a DN-OPF formulation which considers both stability and operational constraint satisfaction under uncertainty. The need to account for a range of uncertainty realizations in this paper's robust optimization formulation results in a challenging semi-infinite program (SIP). The proposed solution algorithm reformulates this SIP into a computationally tractable problem by constructing a tight convex inner approximation of the stability set using sufficient conditions for the existence of a feasible and stable power flow solution. Optimal generator set-points are obtained by optimizing over the proposed convex stability set. The validity and effectiveness of the propose algorithm is demonstrated through various DC networks adapted from IEEE test cases.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2024

Optimal Control of SOEC-Based Hydrogen Production Systems for Demand Response Using Deep Reinforcement Learning in Smart Grids

Solid oxide electrolysis cell (SOEC) hydrogen production technology can range in size from small, appliance-size equipment to large-scale, central production facilities that can be tied directly to renewable or non-greenhouse-gas-emitting forms of electricity production, making it an ideal resource for demand response (DR). The SOEC hydrogen production system is a complex integrated system that encompasses fluid dynamics, electrical dynamics, and electrochemical and thermal dynamics, all of which involve non-linearity and non-convexity. Proper control of the SOEC hydrogen production system is crucial to enable its participation in the DR program. Here, to overcome the difficulty of designing an explicit control law for such nonlinear systems with nonconvex optimization features in DR applications, deep reinforcement learning (DRL) is explored to achieve the optimal control of the SOEC system for DR participation. Specifically, a twin delayed deterministic policy gradient (TD3) control framework is applied to achieve optimal response performance during DR events by considering power tracking error and hydrogen production efficiency with a suitable reward function. Two case studies with grid connections for tracking different DR commands were investigated. The first case study involved operating conditions reaching the boundaries, while the second involved operating conditions within the boundaries. The results showed that the proposed DRL-based control for SOEC can track the DR signal in a timely manner while maintaining high energy efficiency.

08 HYDROGEN↗

DOE OSTI2025

Efficient proximal subproblem solvers for a nonsmooth trust-region method

In [R. J. Baraldi and D. P. Kouri, Mathematical Programming, (2022), pp. 1-40], we introduced an inexact trust-region algorithm for minimizing the sum of a smooth nonconvex and nonsmooth convex function. The principle expense of this method is in computing a trial iterate that satisfies the so-called fraction of Cauchy decrease condition—a bound that ensures the trial iterate produces sufficient decrease of the subproblem model. In this paper, we expound on various proximal trust-region subproblem solvers that generalize traditional trust-region methods for smooth unconstrained and convex-constrained problems. We introduce a simplified spectral proximal gradient solver, a truncated nonlinear conjugate gradient solver, and a dogleg method. Finally, we compare algorithm performance on examples from data science and PDE-constrained optimization.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

McCormick envelopes in mixed-integer PDE-constrained optimization

McCormick envelopes are a standard tool for deriving convex relaxations of optimization problems that involve polynomial terms. Such McCormick relaxations provide lower bounds, for example, in branch-and-bound procedures for mixed-integer nonlinear programs but have not gained much attention in PDE-constrained optimization so far. This lack of attention may be due to the distributed nature of such problems, which on the one hand leads to infinitely many linear constraints (generally state constraints that may be difficult to handle) in addition to the state equation for a pointwise formulation of the McCormick envelopes and renders bound-tightening procedures that successively improve the resulting convex relaxations computationally intractable. We analyze McCormick envelopes for a model problem class that is governed by a semilinear PDE involving a bilinearity and integrality constraints. We approximate the nonlinearity and in turn the McCormick envelopes by averaging the involved terms over the cells of a partition of the computational domain on which the PDE is defined. This yields convex relaxations that underestimate the original problem up to an a priori error estimate that depends on the mesh size of the discretization. These approximate McCormick relaxations can be improved by means of an optimization-based bound-tightening procedure. We show that their minimizers converge to minimizers to a limit problem with a pointwise formulation of the McCormick envelopes when driving the mesh size to zero. We provide a computational example, for which we certify all of our imposed assumptions. The results point to both the potential of the methodology and the gaps in the research that need to be closed. Our methodology provides a framework first for obtaining pointwise underestimators for nonconvexities and second for approximating them with finitely many linear inequalities in an infinite-dimensional setting.

Approximations and Expansions↗

DOE OSTI2026

Relaxations of the steady optimal gas flow problem for a non-Ideal gas

Natural gas ranks second in U.S. primary energy consumption. Because most production sites are remote, gas must be transported through pipeline networks equipped with compressors, valves, and other components. For both economic efficiency and system reliability, it is desirable to operate these networks optimally. The governing physics across pipeline components entails nonlinear, non-convex equality and inequality constraints, and the most general steady-flow operations problem is a Mixed-Integer Nonlinear Program (MINLP).This work focuses on one such steady-flow problem-the Optimal Gas Flow (OGF) for a natural gas pipeline network-which minimizes production cost subject to the steady-flow physics. For day-to-day operations, the ability to quickly compute a globally optimal solution and a strong lower bound for varying demand profiles is crucial. A promising strategy is to build tight relaxations of the OGF’s nonlinear constraints. However, many nonlinearities arising from non-ideal equations of state either lack relaxations or have relaxations that do not scale to realistic network sizes. We address this gap by combining recent advances in polyhedral relaxations for univariate functions to construct tight, computationally efficient relaxations of the OGF with a non-ideal equation of state. These relaxations solve within seconds on a standard laptop. In conclusion, we demonstrate their quality through extensive numerical experiments on very large-scale test networks from the literature and find that the proposed approach proves optimality in 92% of tested instances.

03 NATURAL GAS↗

DOE OSTI2026

Joint Optimization of Multimodal Transit Frequency and Shared Autonomous Vehicle Fleet Size with Hybrid Metaheuristic and Nonlinear Programming

Shared autonomous vehicles (SAVs) bring competition to traditional transit services but redesigning multimodal transit network can utilize SAVs as feeders to enhance service efficiency and coverage. This paper presents an optimization framework for the joint multimodal transit frequency and SAV fleet size problem, a variant of the transit network frequency setting problem. The objective is to maximize total transit ridership (including SAV-fed trips and subtracting boarding rejections) across multiple time periods under budget constraints, considering endogenous mode choice (transit, point-to-point SAVs, driving) and route selection, while allowing for strategic route removal by setting frequencies to zero. Due to the problem’s non-linear, non-convex nature and the computational challenges of large-scale networks, we develop a hybrid solution approach that combines a metaheuristic approach (particle swarm optimization) with nonlinear programming for local solution refinement. To ensure computational tractability, the framework integrates analytical approximation models for SAV waiting times based on fleet utilization, multimodal network assignment for route choice, and multinomial logit mode choice behavior, bypassing the need for computationally intensive simulations within the main optimization loop. Applied to the Chicago metropolitan area’s multimodal network, our method illustrates a 33.3% increase in transit ridership through optimized transit route frequencies and SAV integration, particularly enhancing off-peak service accessibility and strategically reallocating resources.

Ng, Max↗

DOE OSTI2025

Optimizing the design and operation of water networks: Two decomposition approaches

We consider the design and operation of water networks simultaneously. Water network problems can be divided into two categories: the design problem and the operation problem. The design problem involves determining the appropriate pipe sizing and placements of pump stations, while the operation problem involves scheduling pump stations over multiple time periods to account for changes in supply and demand. Our focus is on networks that involve water co-produced with oil and gas. While solving the optimization formulation for such networks, we found that obtaining a primal (feasible) solution is more challenging than obtaining dual bounds using off-the-shelf mixed-integer nonlinear programming solvers. Therefore, we propose two methods to obtain good primal solutions. One method involves a decomposition framework that utilizes a convex reformulation, while the other is based on time decomposition. To test our proposed methods, we conduct computational experiments on a network derived from the PARETO case study.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

The geometry of the modular bootstrap

Abstract We explore the geometry behind the modular bootstrap and its image in the space of Taylor coefficients of the torus partition function. In the first part, we identify the geometry as an intersection of planes with the convex hull of moment curves onR + ⊗ℤ, with boundaries characterized by the total positivity of generalized Hankel matrices. We phrase the Hankel constraints as a semi-definite program, which has several advantages, such as the validity of bounds irrespective of spin truncation. We derive bounds on the gap, twist-gap, and the space of Taylor coefficients themselves. We find that if the gap is above$$ {\Delta }_{\textrm{gap}}^{\ast } $$ ∆ gap ∗ , where$$ \frac{c-1}{12}<{\Delta}_{\textrm{gap}}^{\ast }<\frac{c}{12} $$ c − 1 12 < Δ gap ∗ < c 12 , all coefficients become bounded on both sides and kinks develop in the space. In the second part, we propose an analytic method of imposing the integrality condition for the degeneracy number in the spinless bootstrap, which leads to a non-convex geometry. We find that even at very low derivative order this condition rules out regions otherwise allowed by bootstraps at high derivative order.

Physics↗