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Microwave impedance microscopy and its application to quantum materials

We report materials in which quantum mechanical effects lead to novel properties on the macroscopic scale are of interest both fundamentally and for applications. For such quantum materials, the ability to measure local conductivity and permittivity with nanoscale spatial resolution, minimal sample preparation, in a contact-free fashion and under a wide range of external conditions (such as temperature and magnetic field) is highly desirable. To this end, microwave impedance microscopy, a scanning probe technique that measures tip–sample admittance (inverse of impedance) in a non-contact geometry at microwave frequencies, has matured over the past decade to become a tool that can do just that. This Technical Review describes its fundamental working principles and practical implementations, discusses its application to a wide range of quantum materials, including correlated, topological and 2D van der Waals materials, and outlines future opportunities in expanding the capabilities of microwave impedance microscopy.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Classifying photonic topology using the spectral localizer and numerical K -theory

Recently, the spectral localizer framework has emerged as an efficient approach for classifying topology in photonic systems featuring local nonlinearities and radiative environments. In nonlinear systems, this framework provides rigorous definitions for concepts such as topological solitons and topological dynamics, where a system’s occupation induces a local change in its topology due to nonlinearity. For systems embedded in radiative environments that do not possess a shared bulk spectral gap, this framework enables the identification of local topology and shows that local topological protection is preserved despite the lack of a common gap. However, as the spectral localizer framework is rooted in the mathematics of C*-algebras, and not vector bundles, understanding and using this framework requires developing intuition for a somewhat different set of underlying concepts than those that appear in traditional approaches for classifying material topology. In this tutorial, we introduce the spectral localizer framework from a ground-up perspective and provide physically motivated arguments for understanding its local topological markers and associated local measure of topological protection. In doing so, we provide numerous examples of the framework’s application to a variety of topological classes, including crystalline and higher-order topology. We then show how Maxwell’s equations can be reformulated to be compatible with the spectral localizer framework, including the possibility of radiative boundary conditions. To aid in this introduction, we also provide a physics-oriented introduction to multi-operator pseudospectral methods and numerical K-theory, two mathematical concepts that form the foundation for the spectral localizer framework. Finally, we provide some mathematically oriented comments on the C*-algebraic origins of this framework, including a discussion of real C*-algebras and graded C*-algebras that are necessary for incorporating physical symmetries. Looking forward, we hope that this tutorial will serve as an approachable starting point for learning the foundations of the spectral localizer framework.

97 MATHEMATICS AND COMPUTING↗

Graphical Optimization of Spectral Shift Reconstructions for Optical Backscatter Reflectometry

Optical backscatter reflectometry (OBR) is an interferometric technique that can be used to measure local changes in temperature and mechanical strain based on spectral analyses of backscattered light from a singlemode optical fiber. The technique uses Fourier analyses to resolve spectra resulting from reflections occurring over a discrete region along the fiber. These spectra are cross-correlated with reference spectra to calculate the relative spectral shifts between measurements. The maximum of the cross-correlated spectra—termed quality—is a metric that quantifies the degree of correlation between the two measurements. Recently, this quality metric was incorporated into an adaptive algorithm to (1) selectively vary the reference measurement until the quality exceeds a predefined threshold and (2) calculate incremental spectral shifts that can be summed to determine the spectral shift relative to the initial reference. Using a graphical (network) framework, this effort demonstrated the optimal reconstruction of distributed OBR measurements for all sensing locations using a maximum spanning tree (MST). By allowing the reference to vary as a function of both time and sensing location, the MST and other adaptive algorithms could resolve spectral shifts at some locations, even if others can no longer be resolved.

47 OTHER INSTRUMENTATION↗

Outer midplane neutral density measurements and H-mode fueling studies in NSTX-U

Neutral density profiles are measured on the outboard midplane of the National Spherical Torus Experiment Upgrade (NSTX-U) using a two dimensional camera (ENDD, edge neutral density diagnostic) filtered for deuterium Balmer α (D α ) emission interpreted via simulations using the Monte Carlo neutral transport code DEGAS 2. Deuterium atomic densities n D are calculated by inverting the line-integrated D α brightness and using local measurements of electron density ne and temperature T e to determine atomic rate coefficients. The assumptions used in the derivation of n D from D α emissivity are validated using DEGAS 2 to estimate contributions to emissivity due to electron impact excitation and molecular processes. In this work, experimental measurements and DEGAS 2 simulations are compared over a database of L- and H-mode discharges, showing good agreement in D α emissivity profiles. Residual disagreement between experiment and simulations is suggestive of the possible role of intermittent transport and uncertainties in the molecular data. DEGAS 2 simulations are further used to complement the ENDD diagnostic, extracting neutral (atomic and molecular) densities at locations where the ENDD measurement are not made or where the assumptions used in the ENDD analysis are not valid. Deuterium atomic densities and ionization profiles on the outer midplane are compared for L-mode and H-mode discharges in NSTX-U. One-way coupling of DEGAS 2 to UEDGE multi-fluid simulations constrained by experimental data is used to study edge fueling and neutral penetration in NSTX-U discharges.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

An artificial spiking quantum neuron

Abstract Artificial spiking neural networks have found applications in areas where the temporal nature of activation offers an advantage, such as time series prediction and signal processing. To improve their efficiency, spiking architectures often run on custom-designed neuromorphic hardware, but, despite their attractive properties, these implementations have been limited to digital systems. We describe an artificial quantum spiking neuron that relies on the dynamical evolution of two easy to implement Hamiltonians and subsequent local measurements. The architecture allows exploiting complex amplitudes and back-action from measurements to influence the input. This approach to learning protocols is advantageous in the case where the input and output of the system are both quantum states. We demonstrate this through the classification of Bell pairs which can be seen as a certification protocol. Stacking the introduced elementary building blocks into larger networks combines the spatiotemporal features of a spiking neural network with the non-local quantum correlations across the graph.

Kristensen, Lasse Bjørn (ORCID:0000000239398170)↗

Power-Law Entanglement and Hilbert Space Fragmentation in Nonreciprocal Quantum Circuits

Quantum circuits utilizing measurement to evolve a quantum wave function offer a new and rich playground to engineer unconventional entanglement dynamics. Here, in this work, we introduce a hybrid, nonreciprocal setup featuring a quantum circuit, whose updates are conditioned on the state of a classical dynamical agent. In our example the circuit is represented by a Majorana quantum chain controlled by a classical N-state Potts chain undergoing pair flips. The local orientation of the classical spins controls whether randomly drawn local measurements on the quantum chain are allowed or not. This imposes a dynamical kinetic constraint on the entanglement growth, described by the transfer matrix of an N-colored loop model. It yields an equivalent description of the circuit by an SU(N)-symmetric Temperley-Lieb Hamiltonian or by a kinetically constrained surface growth model for an N-component height field. For N = 2, we find a diffusive growth of the half-chain entanglement toward a stationary profile S(L) ~ L 1/2 for L sites. For N ≥ 3, the kinetic constraints impose Hilbert space fragmentation, yielding subdiffusive growth toward S(L) ~ L 0.57 . This showcases how the control by a classical dynamical agent can enrich the entanglement dynamics in quantum circuits, paving a route toward novel entanglement dynamics in nonreciprocal hybrid circuit architectures.

1-dimensional spin chains↗

Bi-Level Linear Programming Model for Automatic Load Shedding: A Distributed Wide-Area Measurement System-based Solution

Load shedding is currently implemented as a two-step based approach. In the first step, manual load shedding is taken place, were system operators, using estimates, inform distribution utilities of predicted stressful conditions. Information provided include the potential use of energy reserves, as well as load shedding amount. In a second step, automatic load shedding is done. The latter is realized using protection relays. While considering frequency variation, pre-defined values of load to be shed and correspondent number of stages for such to be realized are transformed into relay settings. Under-frequency protection relays use only local measurements towards decision making, thus operate in a decentralized architecture. Decision making is done in milliseconds plus breaker time. While this approach has provided much system reliability, considering the new smart grid paradigm, where system dynamics are much faster due to increasing renewable resources penetration, in some operating conditions it will generate sub-optimal solutions, such as islanding. Phasor measurement units provide a source of information which can be useful for this problem. Centralized architecture-based solutions for automatic load shedding, as present in the state-of-the-art, require though total processing times which are not acceptable for real-life implementation. In this work, considering the above, a bi-level linear programming model is presented. The model is implemented considering a distributed architecture while leveraging phasor measurement units data. The upper-level model estimates the current system state. Results of this model are embedded in a lower-level model, which decision variables are the location and load value to be shed. Easy-to-implement model, built-on the classic weighted least squares solution, highlight potential aspects towards real-life applications.

Bretas, Arturo Suman↗

SPT-3G D1: CMB temperature and polarization power spectra and cosmology from 2019 and 2020 observations of the SPT-3G main field

Here, we present measurements of the temperature and E-mode polarization angular power spectra of the cosmic microwave background (CMB) from observations of 4% of the sky with SPT-3G, the current camera on the South Pole Telescope (SPT). The maps used in this analysis are the deepest used in a CMB TT / TE / EE analysis to date. The maps and resulting power spectra have been validated through blind and unblind tests. The measurements of the lensed EE and TE spectra are the most precise to date at ℓ = 1800 – 4000 and ℓ = 2200 – 4000 , respectively. Combining our TT / TE / EE spectra with previously published SPT-3G CMB lensing results, we find parameters for the standard Λ cold dark matter ( Λ CDM ) model consistent with Planck and ACT DR6 with comparable constraining power. We report a Hubble constant of H 0 = 66.66 ± 0.60 km s - 1 Mpc - 1 from SPT-3G alone, 6.2 σ away from local measurements from SH0ES. For the first time, combined ground-based ( SPT + ACT ) CMB primary and lensing data have reached Planck’s constraining power on some parameters, a milestone for CMB cosmology. The combination of these three CMB experiments yields the tightest CMB constraints to date, with H 0 = 67.19 ± 0.38 km s - 1 Mpc - 1 , and the amplitude of clustering σ 8 = 0.8137 ± 0.0037 . CMB data alone show no evidence for physics beyond Λ CDM , however, we observe a 2.8 σ difference in Λ CDM between CMB and baryon acoustic oscillation (BAO) results from DESI-DR2, which is relaxed in extended models. The combination of CMB and BAO yields 2 − 3 σ shifts from Λ CDM in the curvature of the Universe, the amplitude of CMB lensing, or the dark energy equation of state. It also drives mild preferences for models that address the Hubble tension through modified recombination or variations in the electron mass in a nonflat Universe. This work highlights the growing power of ground-based CMB experiments and lays a foundation for further cosmological analyses with SPT-3G.

Camphuis, Etienne [Sorbonne Université; CNRS; Inst↗

Data-Driven Multi-agent Deep Reinforcement Learning for Distribution System Decentralized Voltage Control with High Penetration of PVs

This paper proposes a novel model-free/data-driven centralized training and decentralized execution multi-agent deep reinforcement learning (MADRL) framework for distribution system voltage control with high penetration of PVs. The proposed MADRL can coordinate both the real and reactive power control of PVs with existing static var compensators and battery storage systems. Unlike the existing DRL-based voltage control methods, our proposed method does not rely on a system model during both the training and execution stages. This is achieved by developing a new interaction scheme between the surrogate modeling of the original system and the multi-agent soft actor critic (MASAC) MADRL algorithm. In particular, the sparse pseudo-Gaussian process with a few-shots of measurements is utilized to construct the surrogate model of the original environment, i.e., power flow model. This is a data-driven process and no model parameters are needed. Furthermore, the MASAC enabled MADRL allows to achieve better scalability by dividing the original system into different voltage control regions with the aid of real and reactive power sensitivities to voltage, where each region is treated as an agent. This also serves as the foundation for the centralized training and decentralized execution, thus significantly reducing the communication requirements as only local measurements are required for control. Comparative results with other alternatives on the IEEE 123-nodes and 342-nodes systems demonstrate the superiority of the proposed method.

14 SOLAR ENERGY↗

Increasing Main Cooler Thermal Performance for sCO2 Power Cycles

Increasing sCO2 power cycle cooler performance will increase the efficiency and lower the cost of electricity for sCO2 power cycles. This work considered the heat transfer performance of two counter-flow, shell-and-tube heat exchangers using local measurements. The heat exchangers were constructed using commercial tubing. One heat exchanger was constructed with a conventional, smooth inner tube (7 mm nominal inner diameter). The other was constructed with an additively manufactured tube with a square cross section and angled rib turbulators. For mixed mean temperatures away from the pseudocritical temperature, the heat transfer coefficient matched the Dittus-Boelter correlation. As the mixed mean temperature approached the pseudocritical temperature, the heat transfer coefficient deviated from the Dittus-Boelter correlation due to transverse property variation and buoyancy. The local and average heat transfer coefficients for angled rib tubes were nominally 150% to 300% greater than the heat transfer coefficients for the smooth tube. Trends in heat exchanger effectiveness indicated that the ribs were most beneficial when the water flow rates were at the low end of the range considered: 0.016 – 0.126 kg/s. Using results from prior system modeling, it is anticipated that adding heat transfer enhancement features in a cycle cooler will increase the sCO2 cycle efficiency by 0.34 percentage points.

Searle, Matthew↗

Model-based real-time surface heat flux and temperature estimation for the DIII-D tokamak

A control-oriented model for monitoring of wall power flux densities on the DIII-D tokamak has been successfully implemented and validated experimentally. Future reactors will have to withstand severe steady state high heat flux loads on plasma-facing components (PFCs). Due to the difficulty of directly-measuring local heat fluxes on these components, monitoring and protection of PFCs during the plasma discharge can benefit from simplified physics-based real-time (RT) functional models to estimate and guide heat load control. As a first step into the development, a control-oriented model for monitoring of wall power flux densities and temperatures on DIII-D tokamak has been successfully implemented. The paper discusses the experimental demonstration and comparison of the 2-D model-based wall heat flux algorithm on the DIII-Dinner wall limiter (IWL) against infra-red camera heat flux measurements for limited plasma configurations. The paper also reports on the benchmarking of the field line tracing environment, SMITER, developed at ITER organization on DIII-D tokamak against experimental IR diagnostic data and the derivation of the component shaping weighting factors for the 2-D model-based approach. Here, the extension of the model-based approach for surface temperature estimation on the DIII-D IWL is also presented.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Thermal confinement and transport in spherical tokamaks: a review

Here, we review the thermal plasma confinement and transport properties observed and predicted in low aspect ratio tokamaks, or spherical tokamaks (STs), which can depart significantly from those observed at higher aspect ratio. In particular, thermal energy confinement scalings show a strong, near linear dependence of energy confinement time on toroidal magnetic field, while the dependence on plasma current is more modest, the opposite of what is seen at higher aspect ratio. STs have revealed a very strong improvement in normalized confinement with decreasing collisionality, much stronger than at higher aspect ratio, which bodes well for an ST-based fusion pilot plant should this trend continue at an even lower collisionality than has already been accessed. These differences arise because of fundamental differences in transport in STs due to the more extreme toroidicity (i.e. reduced region of bad curvature), and to the relatively larger $E_r \times B$ shearing rates, both of which can suppress electrostatic drift wave instabilities at both ion and electron gyroradius scales. In addition, electromagnetic effects are much stronger in STs because they operate at high β T . Gyrokinetic (GK) studies, coupled with low- and high-k turbulence measurements, have shed light on the underlying physics controlling transport. At lower β T , both ion- and electron-scale electrostatic drift turbulence may be responsible for transport. At higher β T , microtearing, kinetic ballooning, and hybrid trapped electron/kinetic ballooning modes increasingly play a role, and they have a much stronger impact in the core of ST plasmas than at higher aspect ratio. Flow shear affects the balance between ion- and electron-scale modes. Non-linear GK simulations find regimes where the electron heat flux decreases with decreasing collisionality, consistent with the experimental global normalized confinement scaling. The ST is unique in that the relatively low toroidal magnetic field allows for localized measurements of electron-scale turbulence, and this coupled with turbulence measurements at ion-scales has facilitated detailed comparisons with GK simulations. These data have provided compelling evidence for the presence of ion temperature gradient and electron temperature gradient turbulence in some plasmas, and direct experimental support for the impact of experimental actuators like rotation shear, density gradient and magnetic shear on turbulence and transport.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Warm Decaying Dark Matter and the Hubble Tension

If a fraction of the dark matter is unstable and decays into dark radiation at around the time of matter-radiation equality, it could impact the expansion history of the universe in a way that helps to ameliorate the long-standing tension between the locally measured value of the Hubble constant and the value inferred from measurements of the cosmic microwave background and baryon acoustic oscillations (assuming standard ΛCDM cosmology). If this component of decaying dark matter is cold, however, it will modify the evolution of the gravitational potentials, leading to inconsistencies with these same data sets. With this in mind, we consider here a component of decaying warm dark matter, with a free-streaming length that is long enough to remain consistent with existing data. We study the background and perturbation evolution of warm decaying dark matter, and use cosmological data to constrain the mass, abundance and decay rate of such a particle. We find that a component of warm decaying dark matter can significantly reduce the tension between local and cosmological determinations of the Hubble constant.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

High-throughput micro-scale bandgap mapping for perovskite-inspired materials with complex composition space

Abstract To realize the full promise of high-throughput experimental workflows, the rate of sample synthesis must be matched by that of characterization. Of growing interest are contactless optical techniques that can rapidly measure material homogeneity and properties. Here, we present a hyperspectral imaging method to measure local optical bandgap distributions within samples, utilizing spatially-resolved reflectance spectra coupled with automated data analysis. We collect approximately one million optical bandgap data across the compositional space of Cs 3 (Bi x Sb 1-x ) 2 (Br y I 1-y ) 9 perovskite-inspired materials. Our results show non-monotonic bandgap variations (i.e., bandgap bowing) along six composition gradient sequences, in addition to identifying samples with multiple bandgaps in statistics. High-throughput transient absorption spectroscopy reveals that within these compositions, the depletion of the ground state carriers to excited states occurred at discrete energy levels with independent carrier dynamics, consistent with the bandgap observation and indicative of phase separation. This work demonstrates the potential for rapid optical measurements to assess material quality and homogeneity in a high-throughput experimental setting, supporting screening and recipe optimization of optoelectronic material candidates with desired carrier dynamics and optical properties.

Science & Technology - Other Topics↗

Planck 2018 results. VI. Cosmological parameters

In this paper, we present cosmological parameter results from the final full-mission Planck measurements of the cosmic microwave background (CMB) anisotropies, combining information from the temperature and polarization maps and the lensing reconstruction. Compared to the 2015 results, improved measurements of large-scale polarization allow the reionization optical depth to be measured with higher precision, leading to significant gains in the precision of other correlated parameters. Improved modelling of the small-scale polarization leads to more robust constraints on many parameters, with residual modelling uncertainties estimated to affect them only at the 0.5 σ level. We find good consistency with the standard spatially-flat 6-parameter ΛCDM cosmology having a power-law spectrum of adiabatic scalar perturbations (denoted “base ΛCDM” in this paper), from polarization, temperature, and lensing, separately and in combination. A combined analysis gives dark matter density Ω c h 2 = 0.120 ± 0.001, baryon density Ω b h 2 = 0.0224 ± 0.0001, scalar spectral index n s = 0.965 ± 0.004, and optical depth τ = 0.054 ± 0.007 (in this abstract we quote 68% confidence regions on measured parameters and 95% on upper limits). The angular acoustic scale is measured to 0.03% precision, with 100 θ * = 1.0411 ± 0.0003. These results are only weakly dependent on the cosmological model and remain stable, with somewhat increased errors, in many commonly considered extensions. Assuming the base-ΛCDM cosmology, the inferred (model-dependent) late-Universe parameters are: Hubble constant H 0 = (67.4 ± 0.5) km s –1 Mpc –1 ; matter density parameter Ω m = 0.315 ± 0.007; and matter fluctuation amplitude σ 8 = 0.811 ± 0.006. We find no compelling evidence for extensions to the base-ΛCDM model. Combining with baryon acoustic oscillation (BAO) measurements (and considering single-parameter extensions) we constrain the effective extra relativistic degrees of freedom to be N eff = 2.99 ± 0.17, in agreement with the Standard Model prediction N eff = 3.046, and find that the neutrino mass is tightly constrained to Σ m ν < 0.12 eV. The CMB spectra continue to prefer higher lensing amplitudes than predicted in base ΛCDM at over 2 σ , which pulls some parameters that affect the lensing amplitude away from the ΛCDM model; however, this is not supported by the lensing reconstruction or (in models that also change the background geometry) BAO data. The joint constraint with BAO measurements on spatial curvature is consistent with a flat universe, Ω K = 0.001 ± 0.002. Also combining with Type Ia supernovae (SNe), the dark-energy equation of state parameter is measured to be w 0 = –1.03 ± 0.03, consistent with a cosmological constant. We find no evidence for deviations from a purely power-law primordial spectrum, and combining with data from BAO, BICEP2, and Keck Array data, we place a limit on the tensor-to-scalar ratio r 0.002 < 0.06. Standard big-bang nucleosynthesis predictions for the helium and deuterium abundances for the base-ΛCDM cosmology are in excellent agreement with observations. The Planck base-ΛCDM results are in good agreement with BAO, SNe, and some galaxy lensing observations, but in slight tension with the Dark Energy Survey’s combined-probe results including galaxy clustering (which prefers lower fluctuation amplitudes or matter density parameters), and in significant, 3.6 σ , tension with local measurements of the Hubble constant (which prefer a higher value). Simple model extensions that can partially resolve these tensions are not favoured by the Planck data.

79 ASTRONOMY AND ASTROPHYSICS↗

Speed of quantum information spreading in chaotic systems

In this work, we present a general theory of quantum information propagation in chaotic quantum many-body systems. The generic expectation in such systems is that quantum information does not propagate in localized form; instead, it tends to spread out and scramble into a form that is inaccessible to local measurements. To characterize this spreading, we define an information speed via a quench-type experiment and derive a general formula for it as a function of the entanglement density of the initial state. As the entanglement density varies from zero to one, the information speed varies from the entanglement speed to the butterfly speed. We verify that the formula holds both for a quantum chaotic spin chain and in field theories with an AdS/CFT gravity dual. For the second case, we study in detail the dynamics of entanglement in two-sided Vaidya-AdS-Reissner-Nordstrom black branes. We also show that, with an appropriate decoding process, quantum information can be construed as moving at the information speed, and, in the case of AdS/CFT, we show that a locally detectable signal propagates at the information speed in a spatially local variant of the traversable wormhole setup.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

The MOSAiC Distributed Network: Observing the coupled Arctic system with multidisciplinary, coordinated platforms

Central Arctic properties and processes are important to the regional and global coupled climate system. The Multidisciplinary drifting Observatory for the Study of Arctic Climate (MOSAiC) Distributed Network (DN) of autonomous ice-tethered systems aimed to bridge gaps in our understanding of temporal and spatial scales, in particular with respect to the resolution of Earth system models. By characterizing variability around local measurements made at a Central Observatory, the DN covers both the coupled system interactions involving the ocean-ice-atmosphere interfaces as well as three-dimensional processes in the ocean, sea ice, and atmosphere. The more than 200 autonomous instruments (“buoys”) were of varying complexity and set up at different sites mostly within 50 km of the Central Observatory. During an exemplary midwinter month, the DN observations captured the spatial variability of atmospheric processes on sub-monthly time scales, but less so for monthly means. They show significant variability in snow depth and ice thickness, and provide a temporally and spatially resolved characterization of ice motion and deformation, showing coherency at the DN scale but less at smaller spatial scales. Ocean data show the background gradient across the DN as well as spatially dependent time variability due to local mixed layer sub-mesoscale and mesoscale processes, influenced by a variable ice cover. The second case (May–June 2020) illustrates the utility of the DN during the absence of manually obtained data by providing continuity of physical and biological observations during this key transitional period. We show examples of synergies between the extensive MOSAiC remote sensing observations and numerical modeling, such as estimating the skill of ice drift forecasts and evaluating coupled system modeling. The MOSAiC DN has been proven to enable analysis of local to mesoscale processes in the coupled atmosphere-ice-ocean system and has the potential to improve model parameterizations of important, unresolved processes in the future.

54 ENVIRONMENTAL SCIENCES↗

Chapter 7: Learning Stable Local Volt/Var Controllers in Distribution Grids

This chapter describes a framework to synthesize provably stable local Volt/Var controllers for distributed energy resources (DERs) in power distribution grids (DGs). The goal is to control the reactive power injections of DERs to improve the system performance as quantified by a generic optimal reactive power flow (ORPF) problem. To achieve this, we jointly design for each DER the control function, which prescribes the reactive power update rule, and the equilibrium function, which approximates the ORPF solutions from local measurements of voltages and powers. We provide conditions on the equilibrium functions and the control parameters ensuring the stability of the closed-loop system. In particular, we discuss the trade-offs between each set of conditions accounting for practical considerations, like fully exploiting the DERs' generation capabilities and reducing the optimality gap. These conditions are then translated into learning constraints on the neural networks' parameters that are enforced in the training phase. We validate our framework with numerical simulations on the IEEE 37-bus network and through a comparison with an optimized version of standard piece wise linear control rules.

closed-loop asymptotic stability↗