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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Average Incremental Cost Pricing for the AC Unit Commitment Problem [Rev. 1]

Unit Commitment (UC) problems that consider the Alternating Current (AC) model of the transmission network have long been considered intractable to solve at scale by the power system community. Recently, the Grid-Optimization (GO) Competition held by the Advanced Research Project Agency-Energy (ARPA-E) has facilitated the development of the first algorithms to solve large-scale ACUC problems. This new capability opens a path towards the explicit consideration of the AC transmission network model in UC problems used to clear day-ahead electricity markets. This calls for the analysis of electricity market structures that accommodate both the continuous non-linearity of the AC transmission network and the discrete non-linearity of the UC problem simultaneously. This paper serves as an initial effort to do so by proposing an Average Incremental Cost (AIC) pricing structure that is designed around the ACUC problem. In particular, an AIC one-pass pricing problem is proposed that represents a continuously constrained variant of the ACUC problem and allows for the computation of Locational Incremental Prices (LIPs) for both real and reactive power as the local optimal Lagrange multipliers of the power balance constraints. To avoid degeneracy, the pricing problem includes a small parameter ϵ > 0. Under certain assumptions market participants are shown to realize profit that converges to a non-negative value as ϵ approaches zero, practically ensuring profitability for small values of ϵ. We additionally provide many simple and important examples that provide intuition and insights into the proposed prices. Examples illustrate the basic concept of AIC pricing, the derived profitability results, the existence of multiple LIPs, the importance of including reactive power in the dispatch and pricing problems, the need for reactive power prices, and the improved incentives exhibited by LIPs as compared to traditional Locational Marginal Prices (LMPs). We additionally indicate many directions for future work including analysis of larger test cases.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2022

Ab-Initio Investigation of Finite Size Effects in Rutile Titania Nanoparticles with Semilocal and Nonlocal Density Functionals

In this work, we employ hybrid and generalized gradient approximation (GGA) level density functional theory (DFT) calculations to investigate the convergence of surface properties and electronic gap of rutile titania nanoparticles with particle size. The surface energies and electronic gaps are calculated for cuboidal particles with minimum dimension ranging from 3.7 Angstrom (24 atoms) to 10.3 Angstrom (384 atoms) using a highly-parallel real-space DFT code to enable hybrid level DFT calculations of larger nanoparticles than are typically practical. We deconvolute the geometric and electronic finite size effects in surface energy, and evaluate the influence of defects on electronic gap and density of states (DOS). The electronic finite size effects in surface energy vanish when the minimum length scale of the nanoparticles becomes greater than 10 Angstrom. We show that this length scale is consistent with a computationally efficient numerical analysis of the characteristic length scale of electronic interactions. The surface energy of nanoparticles having minimum dimension beyond this characteristic length can be approximated using slab calculations that account for the geometric defects. In contrast, the finite size effects on the electronic gap and DOS is highly dependent on the shape and size of these particles. Furthermore, the DOS for cuboidal particles and more realistic particles constructed using the Wulff algorithm reveal that defect states within the electronic gap play a key role in determining the eigen value distribution of nanoparticles and the electronic gap does not converge to the bulk limit for the particle sizes investigated.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Fracture mechanics approach to TRISO fuel particle failure analysis

Weibull stress-based methods for failure probability assessment have been developed and analyzed to assess the integrity of tristructural isotropic (TRISO) fuel particles during fuel life cycles and accident operating conditions. While simple, these methods entail a number of drawbacks when stress concentrates near crack tips, including finite element mesh size dependency when the Weibull stress is averaged over the finite element domain. Fracture mechanics approaches involving the use of interaction integrals eliminate this lack of mesh convergence and produce consistent fracture predictions. In this work, we use an interaction integral approach to computing stress intensity factors for a crack in the inner pyrolytic carbon layer perpendicular to the silicon carbide layer, which is simplified representation of a failure mode in TRISO particles. Further, the interface between these two TRISO layers has been shown to become porous, which we simulate by considering a transition of mechanical properties over such porous length, i.e. the layers are modeled as a functionally graded material. Aspects such as porosity and thermal and irradiation eigenstrains are considered in computing the stress intensity factor from a fracture mechanics approach and compared with the known Weibull stress failure approach. The methodology introduced in this paper enables a more general fracture probability assessment in TRISO particles and eliminates the need to identify best suited parameters when using local or averaged stress-based failure criterion. Finally, the numerical sensitivity studies show how parameters such as the porous transition zone length, the material stiffness, and creep affect the probability of TRISO fuel particle failure.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2023

On unifying randomized methods for inverse problems

This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems with Gaussian priors by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a sample average approximation (SAA). More importantly, we are able to prove a single theoretical result that guarantees the asymptotic convergence for a variety of randomized methods. Additionally, viewing randomized methods as an SAA enables us to prove, for the first time, a single non-asymptotic error result that holds for randomized methods under consideration. Another important consequence of our unified framework is that it allows us to discover new randomization methods. Here, we present various numerical results for linear, nonlinear, algebraic, and PDE-constrained inverse problems that verify the theoretical convergence results and provide a discussion on the apparently different convergence rates and the behavior for various randomized methods.

42 ENGINEERING↗
DOE OSTI2024

Uncertainties in greenhouse gas emission factors: A comprehensive analysis of switchgrass‐based biofuel production

Abstract This study investigates uncertainties in greenhouse gas (GHG) emission factors related to switchgrass‐based biofuel production in Michigan. Using three life cycle assessment (LCA) databases—US lifecycle inventory (USLCI) database, GREET, and Ecoinvent—each with multiple versions, we recalculated the global warming intensity (GWI) and GHG mitigation potential in a static calculation. Employing Monte Carlo simulations along with local and global sensitivity analyses, we assess uncertainties and pinpoint key parameters influencing GWI. The convergence of results across our previous study, static calculations, and Monte Carlo simulations enhances the credibility of estimated GWI values. Static calculations, validated by Monte Carlo simulations, offer reasonable central tendencies, providing a robust foundation for policy considerations. However, the wider range observed in Monte Carlo simulations underscores the importance of potential variations and uncertainties in real‐world applications. Sensitivity analyses identify biofuel yield, GHG emissions of electricity, and soil organic carbon (SOC) change as pivotal parameters influencing GWI. Decreasing uncertainties in GWI may be achieved by making greater efforts to acquire more precise data on these parameters. Our study emphasizes the significance of considering diverse GHG factors and databases in GWI assessments and stresses the need for accurate electricity fuel mixes, crucial information for refining GWI assessments and informing strategies for sustainable biofuel production.

Kim, Seungdo↗
DOE OSTI2022

Projection-Based Model Reduction for Coupled Conduction—Enclosure Radiation Systems

We report a projection-based reduced order model (pROM) methodology has been developed for transient heat transfer problems involving coupled conduction and enclosure radiation. The approach was demonstrated on two test problems of varying complexity. The reduced order models demonstrated substantial speedups (up to 185×) relative to the full order model with good accuracy (less than 3% L ∞ error). An attractive feature of pROMs is that there is a natural error indicator for the ROM solution: the final residual norm at each time-step of the converged ROM solution. Using example test cases, we discuss how to interpret this error indicator to assess the accuracy of the ROM solution. The approach shows promise for many-query applications, such as uncertainty quantification and optimization. The reduced computational cost of the ROM relative to the full-order model (FOM) can enable the analysis of larger and more complex systems as well as the exploration of larger parameter spaces.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

The Heavy Rainfall Mechanism Revealed by a Terrain-Resolving 4DVar Data Assimilation System—A Case Study

In this study, a newly developed terrain-resolving four-dimensional variational (4DVar)-based data assimilation system, Immersed Boundary Method_Variational Doppler Radar Analysis System (IBM_VDRAS), is applied to investigate the mechanisms leading to a heavy precipitation event that occurred in Taiwan during the Southwesterly Monsoon Experiment (SoWMEX) conducted in 2008. The multivariate analyses using IBM_VDRAS and surface observations reveal that the warm and moist southwesterly flow from the ocean decelerates after making landfall, forming a surface convergence zone along the coast, which is further strengthened during the passage of a prefrontal rainband. The flow ascends as it advances inland until reaching the mountains, producing persistent precipitation and the enhancement of evaporative cooling as well as a widespread high pressure zone. A very shallow (<0.4 km) layer of offshore flow can be identified over the southwestern plain, which helps to generate a quasi-stationary convergence zone near the coast. Furthermore, sensitivity studies are carried out to quantify the relative importance of the contributions made by topographic blockage, evaporative cooling, and their nonlinear interaction, to the evolution of this type of convective system. The influence of the topography is identified as the dominant factor in modulating the flow structure of the rainfall system. However, it is the nonlinear interaction between terrain and evaporation that determines the distribution of the temperature and pressure fields.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

TDCOSMO XXIII. Measurement of the Hubble constant from the doubly lensed quasar HE 1104−1805

Time-delay cosmography leverages strongly lensed quasars to measure the Universe’s current expansion rate, H 0 , independently from other methods. The latest TDCOSMO milestone measurement primarily used quadruply lensed quasars for their mass profile constraints. However, doubly lensed quasars, being more abundant and offering precise time delays, could expand the sample by a factor of 5, significantly advancing towards a 1% precision measurement of H 0 . We present the first TDCOSMO analysis of a doubly imaged source, HE 1104−1805, including the measurement of the four necessary ingredients. First, by combining 17 years of data from the SMARTS, Euler, and WFI telescopes, we measured a time delay of 176.3 +11.4 −10.3 days. Second, using MUSE data, we extracted stellar velocity dispersion measurements in three radial bins with 5% to 13% precision. Third, employing F160W HST imaging for lens modelling and marginalising over various modelling choices, we measured the Fermat potential difference between the images. Fourth, using wide-field imaging, we measured the convergence added by objects not included in the lens modelling. By combining these four ingredients, we measured the time delay distance and the angular diameter distance to the deflector, favouring a power-law mass model over a baryonic and dark matter composite model. The measurement was performed blindly to prevent experimenter bias and resulted in a Hubble constant of H 0 = +5.8 −5.0 × λ int km s −1 Mpc −1 , where λ int is the internal mass sheet degeneracy parameter. This is in agreement with the TDCOSMO-2025 milestone and its precision for λ int = 1 is comparable to that obtained with the best-observed quadruply lensed quasars (4–6%). This work is a stepping stone towards a precise measurement of H 0 using a large sample of doubly lensed quasars, supplementing the current sample. The next TDCOSMO milestone paper will include this system in its hierarchical analysis, constraining λ int and H 0 jointly with multiple lenses.

cosmological parameters↗
DOE OSTI2026

Dark Energy Survey year 6 results: Magnification modeling and its impact on galaxy clustering and galaxy-galaxy lensing cosmology

Gravitational lensing magnification alters the observed spatial distribution of galaxies and must be accounted for to prevent biases in cosmological probes of the large-scale structure. We investigate its effects on the Dark Energy Survey Year 6 galaxy clustering and galaxy-galaxy lensing analyses using the fiducial lens (position tracer) sample M ag L im++. Magnification bias is parameterized by a coefficient that describes the response of the number of selected objects per unlensed area element to a change in the lensing convergence. We quantify this coefficient using the BALROG synthetic source injection catalog to account for the complexity of the selection function, and compare these results with simplified estimates. The resulting values of the magnification coefficients for each redshift bin are [3.16 ± 0.08, 2.76 ± 0.21, 4.09 ± 0.15, 4.42 ± 0.16, 4.90 ± 0.29, 4.83 ± 0.25]. Relative to Year 3, this analysis provides more precise and accurate magnification bias estimates through a larger BALROG area and reweighting to better match the data properties. Here, the cosmological results are robust when tested against various magnification parameter prior choices and also when adding cross-clustering between lens redshift bins. Neglecting magnification, however, introduces significant systematic shifts: relative to the fiducial analysis with Gaussian priors centered on the BALROG -derived estimates, we observe shifts of 1.37σ in S 8 and -0.84σ in Ω m (with cosmic shear included: -0.61σ in S 8 and -0.71σ in Ω m ), in agreement with findings from simulated data, demonstrating that magnification must be modeled to avoid biases. Freeing the magnification bias in lens bin 2 leads to unphysical negative values, further justifying its exclusion from the fiducial Year 6 analysis.

Cosmological parameters↗
DOE OSTI2024

Integration of 5G and Time Sensitive Networks in Fossil Energy Generation Systems: A Case Study

Precise timing and data transmission within stringent time constraints are critical for numerous applications, such as robotics, virtual and augmented reality, industrial automation, energy and medical business, and various other sectors. Time-sensitive networks (TSN) and fifth-generation wireless communications (5G) are crucial for industrial communications, enabling convergent communication for various services using a common network core. Applications that are time-sensitive and require deterministic communications with low latency fall into this category, such as generation plant systems. This paper presents a simulated model of 5G-TSN for a fossil power plant based on the wired network parameters implemented in situ. Metrics analysis, comparison and future work are presented. © 2024 IEEE.

01 COAL, LIGNITE, AND PEAT↗
DOE OSTI2022

System identity and transformation in petroleum jurisdictions: A multi-method approach for the North Slope Borough, Alaska

Capturing the multidimensionality of a bounded social-environmental system (SES) presents a range of challenges to interdisciplinary researchers due to the need to integrate divergent scientific paradigms, scalar data, and social theories. Contemporary Arctic circumpolar SESs studied under conditions of rapid and unprecedented climatic, ecological, economic, and sociopolitical change, defy any singular established methodological approach that aims to schematize and interpret the system for decision-making purposes. As a small interdisciplinary team working within a large Arctic SES modeling effort, we have found that developing systems models to support resilience in the Arctic requires an understanding of system dynamics that is attentive to holistic indicators of change, measured both quantitatively and qualitatively. Using the Alaska North Slope Borough as a case study, we apply three convergent frameworks to capture significant dimensions of the system for improved problem definition in confronting the challenges of Arctic climate change. We describe contemporary “oil and gas” social-ecological system components and dynamics, the historical processes and transformations that fundamentally altered the system, and the scientific projections for the most likely catalysts of future change. This analysis results in a typology for defining subnational Arctic hydrocarbon SESs. We conclude that the future of oil and gas development as a policy pathway in different locations experiencing rapid climate change can be evaluated when difficult-to-quantify variables are included.

Lovecraft, Amy Lauren↗
DOE OSTI2020

Atmospheric teleconnection processes linking winter air stagnation and haze extremes in China with regional Arctic sea ice decline

Abstract. Recent studies suggested significant impacts of boreal cryosphere changes on wintertime air stagnation and haze pollution extremes in China. However, the underlying mechanisms of such a teleconnection relationship remains unclear. Here we use the Whole Atmosphere Community Climate Model (WACCM) to investigate dynamic processes leading to atmospheric circulation and air stagnation responses to Arctic sea ice changes. We conduct four climate sensitivity experiments by perturbing sea ice concentrations (SIC) and corresponding sea surface temperature (SST) in autumn and early winter over the whole Arctic and three subregions in the climate model. The results indicate distinct responses in circulation patterns and regional ventilation to the region-specific Arctic changes, with the largest increase of both the probability (by 132 %) and the intensity (by 30 %) of monthly air stagnation extremes being found in the experiment driven by SIC and SST changes over the Pacific sector of the Arctic (the East Siberian and Chukchi seas). The increased air stagnation extremes are mainly driven by an amplified planetary-scale atmospheric teleconnection pattern that resembles the negative phase of the Eurasian (EU) pattern. Dynamical diagnostics suggest that convergence of transient eddy forcing in the vicinity of Scandinavia in winter is largely responsible for the amplification of the teleconnection pattern. Transient eddy vorticity fluxes dominate the transient eddy forcing and produce a barotropic anticyclonic anomaly near Scandinavia and wave train propagation across Eurasia to the downstream regions in East Asia. The piecewise potential vorticity inversion analysis reveals that this long-range atmospheric teleconnection of Arctic origin takes place primarily via the middle and upper troposphere. The anomalous ridge over East Asia in the middle and upper troposphere worsens regional ventilation conditions by weakening monsoon northwesterlies and enhancing temperature inversions near the surface, leading to more and stronger air stagnation and pollution extremes over eastern China in winter. Ensemble projections based on state-of-the-art climate models in the Coupled Model Intercomparison Project Phase 6 (CMIP6) corroborate this teleconnection relationship between high-latitude environmental changes and midlatitude weather extremes, though the tendency and magnitude vary considerably among each participating model.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Atmospheric river representation in the Energy Exascale Earth System Model (E3SM) version 1.0

Abstract. The Energy Exascale Earth System Model (E3SM) project is an ongoing, state-of-the-science Earth system modeling, simulation, and prediction project developed by the US Department of Energy (DOE). With an emphasis on supporting the DOE's energy mission, understanding and quantifying how well the model simulates water cycle processes is of particular importance. Here, we evaluate E3SM version 1.0 (v1.0) for its ability to represent atmospheric rivers (ARs), which play significant roles in water vapor transport and precipitation. The characteristics and precipitation associated with global ARs in E3SM at standard resolution (1∘ × 1∘) are compared to the Modern-Era Retrospective analysis for Research and Applications, version 2 (MERRA2). Global patterns of AR frequencies in E3SM show high degrees of correlation (≥0.97) with MERRA2 and low mean absolute errors (MAEs; <1 %) annually, seasonally, and across different ensemble members. However, some large-scale condition biases exist, leading to AR biases – most significant of which are the double intertropical convergence zone (ITCZ), a stronger and/or equatorward-shifted subtropical jet during boreal and austral winters, and enhanced Northern Hemisphere westerlies during summer. By comparing atmosphere-only and fully coupled simulations, we attribute the sources of the biases to the atmospheric component or to a coupling response. Using relationships revealed in Dong et al. (2021), we provide evidence showing the stronger North Pacific jet in winter and the enhanced Northern Hemisphere westerlies during summer, associated with E3SM's double ITCZ and related weaker Atlantic meridional overturning circulation (AMOC), respectively, which are significant sources of the AR biases found in the coupled simulations.

58 GEOSCIENCES↗
DOE OSTI2025

The HTR-Proteus Benchmark: Analysis and Use as a Verification and Validation Case

This presentation outlines the evaluation and application of the HTR-Proteus benchmark as a verification and validation (V&V) case for advanced reactor modeling tools. The work supports the U.S. Department of Energy’s HALEU Availability Program (HAP) and the joint DOE/NRC DNCSH project, which aims to reduce criticality safety uncertainties in commercial-scale HALEU fuel cycle and transportation systems. The HTR-Proteus experiments, conducted at the Paul Scherrer Institute, provide high-fidelity data for TRISO-fueled, graphite-moderated pebble bed reactors with high neutron leakage—conditions relevant to HALEU transport scenarios. This study focuses on Cores 4.2 and 4.3 of the HTR-Proteus benchmark, analyzing key sources of uncertainty including pebble packing, TRISO particle positioning, and core height. Using Project Chrono for realistic pebble geometries and Serpent, SHIFT, and MCNP for neutronics simulations, the study quantifies the impact of these uncertainties on the effective multiplication factor (keff). Results show that a sample size of 110 pebble configurations is sufficient to converge keff, with ±30 pcm uncertainty due to packing randomness. TRISO positioning and core height variations also significantly influence keff, highlighting the importance of detailed modeling in V&V efforts. The benchmark serves as a valuable test case for validating the Griffin reactor physics code and improving confidence in HALEU system simulations.

73 - NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2023

Flash x-ray radiography analysis of detonation wave propagation in additive-manufactured high explosives

Recent research has demonstrated that additive manufacturing (AM) can be used to produce directionally sensitive high explosives (HEs), but detonation wave propagation in AM HEs with variable internal structure has not been studied. In this work, samples were printed using a 73 wt. % octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine-based HE ink. Flash x-ray radiography imaging was used to observe density changes in two imaging planes of each sample during detonation, and high-speed imaging was used to calculate the detonation velocity at the HE surface. The detonation front initially appears to fail in internal channel regions of the HE, but late reactions occurred in two samples, which increased the material bulk density by 2.8%, possibly due to shock convergence phenomena. The calculated detonation wave pressure similarly increased, but the accuracy of the results is uncertain because of the deviation of the low bulk density printed samples from the cast charge principal isentrope. Here, this work demonstrates that structure can be effectively utilized to guide detonation wave propagation through HE charges.

3D printing↗
DOE OSTI2023

Behavior and properties of helium cavities in ductile-phase-toughened tungsten irradiated with He + ions at an elevated temperature

Ductile-phase toughened tungsten (DPT W) composites have demonstrated a great potential for plasma-facing components in fusion power systems. These materials can retain the outstanding thermomechanical properties of W while significantly enhancing the fracture toughness. The DPT W composite in this study consists of W particles (88 wt.%) embedded in a ductile NiFeW (12 wt.%) solution matrix. Irradiation of the composite was performed with 90 keV He + ions to 1.0 × 10 17 He + /cm 2 at 973 K. The He cavities formed in both W and NiFeW phases are examined using convergent-beam scanning transmission electron microscopy (CB-STEM) at an atomic-level resolution. Full-range depth profiles of the He cavity diameters and number densities are analyzed. The results show that the average cavity diameter in NiFeW is ~2.7 times that in W, while the number density is an order of magnitude lower. A quantitative analysis of the He density and pressure in the cavities is also achieved using STEM electron energy loss spectroscopy (STEM-EELS) mapping. The He atoms in the cavities in NiFeW are found to be under-pressurized, while those in similar-sized cavities in W are not detectable. Further, the He energy-loss shift exhibits a linear relationship with the He density in cavities in NiFeW, with the slope comparable to that for martensitic steel.

36 MATERIALS SCIENCE↗
DOE OSTI2025

On the discretization error of the discrete generalized quantum master equation

The transfer tensor method (TTM) [Cerrillo and Cao, Phys. Rev. Lett. 112 , 110401 (2014)] can be considered a discrete-time formulation of the Nakajima–Zwanzig quantum master equation (NZ-QME) for modeling non-Markovian quantum dynamics. A recent paper [Makri, J. Chem. Theory Comput. 21 , 5037 (2025)] raised concerns regarding the consistency of the TTM discretization, particularly a spurious term at the initial time t = 0. Here, this work presents a detailed analysis of the discretization structure of the TTM, clarifying the origin of the initial-time correction and establishing a consistent relationship between the TTM discrete-time memory kernel K N and the continuous-time NZ-QME kernel $\mathscr{K}$( N Δ t ). This relationship is validated numerically using the spin-boson model, demonstrating convergence of reconstructed memory kernels and accurate dynamical evolution as Δ t → 0. While the TTM provides a consistent discretization, we note that alternative schemes are also viable, such as the midpoint derivative/midpoint integral scheme proposed in Makri’s work. The relative performance of various schemes for either computing accurate $\mathscr{K}$( N Δ t ) from exact dynamics or obtaining accurate dynamics from exact $\mathscr{K}$( N Δ t ) warrants further investigation.

Density-matrix↗
DOE OSTI2021

A Novel Feature of Valence Quark Distributions in Hadrons

Examining the evolution of the maximum of valence quark distribution, qV, weighted by Bjorken x, h(x,t)≡xqV(x,t), it is observed that h(x,t) at the peak becomes a one-parameter function; h(xp,t)=Φ(xp(t)), where xp is the position of the peak, t=logQ2, and Q2 is the resolution scale. This observation is used to derive a new model-independent relation which connects the partial derivative of the valence parton distribution functions (PDFs) in xp to the quantum chromodynamics (QCD) evolution equation through the xp derivative of the logarithm of the function Φ(xp(t)). A numerical analysis of this relation using empirical PDFs results in an observation of the exponential form of the Φ(xp(t))=h(xp,t)=CeDxp(t) for leading to next-to-next leading order approximations of PDFs for the range of Q2, covering four orders in magnitude. The exponent, D, of the observed “height-position” correlation function converges with the increase in the order of approximation. This result holds for all the PDF sets considered. A similar relation is observed also for the pion valence quark distribution, indicating that the obtained relation may be universal for any non-singlet partonic distribution. The observed “height-position” correlation is used also to indicate that no finite number of exchanges can describe the analytic behavior of the valence quark distribution at the position of the peak at fixed Q2.

Leon, Christopher↗