Instantaneous Values Research Articles

Improved Instantaneous Current Value-Based Protection Methods for Faulty Synchronizations of Synchronous Generators

Faulty synchronizations of synchronous generators can cause significant detrimental effects, primarily due to a large current and high electromagnetic torque. These effects not only impact the generator but they can also extend to the prime mover and the step-up transformer. Furthermore, such events can trigger disturbances in the power system, potentially leading to system collapse if not promptly cleared. Although the autosynchronizers and synchro-check technologies are well established in the industry, faulty synchronizations, such as those caused by incorrect wiring during maintenance or commissioning operations, can go undetected by these systems. Existing protections do not allow for the detection of faulty synchronizations in a timely manner. This paper presents novel protection methods specifically designed for this issue: one based on instantaneous current value and the other on the instantaneous current-derivative value. These schemes are activated exclusively during the synchronizations process, allowing for faster fault detection compared to existing methods, thereby reducing the duration of harmful electrical and mechanical stresses after a faulty synchronization. The effectiveness of the proposed schemes has been validated through computer simulations of a 362 MVA turbo-generator from a thermal power plant and also through experimental tests on a 5 kVA synchronous generator using a specialized laboratory synchronization test bench, yielding promising results.

A new adaptive peridynamic framework for modeling large deformation and fracture behavior of hyperelastic materials

Accurate prediction of deformation and fracture behaviors of hyperelastic materials is crucial in engineering applications and industry. The paper presents a new three-dimensional computational model to simulate large deformations and fractures in hyperelastic materials using the peridynamic method. The main innovation of the proposed three-dimensional hyperelastic peridynamic (3D-HPD) model lies in its incorporation of the stress–strain relationship of hyperelastic materials into the peridynamic framework. Therefore, the peridynamic parameters, such as bond stiffness and strength, are dynamically updated based on the instantaneous value of Young's modulus derived from the stress–strain relation of hyperelastic materials. The 3D-HPD model highlights the following novelties: 1) The 3D-HPD model directly employs the stress–strain relation for each peridynamic bond. Thus, the large deformations and fractures in hyperelastic materials are accurately captured in three dimensions by integrating the responses in all peridynamic bonds; 2) Unlike other models that may require specific hyperelastic formulations, the 3D-HPD model offers a more straightforward application, enhancing usability and reducing complexity; 3) The 3D-HPD model is capable of simulating the response of hyperelastic materials under various loading conditions, including tensile, torsional, and lateral loadings. We confirm the validity of the 3D-HPD model by evaluating it against experimental measurements and observations. For the first time, the model examines hyperelastic materials' fracture and deformation behaviors under combined loading conditions. The new findings of the mechanical performance of hyperelastic materials under multi-axial stresses provide new insights to enhance their design and usage.

Application of the Beating Method to Define the Electroencephalogram α−Rhythm Frequency Content

Among plenty of the brain oscillation electrical patterns, the spindle oscillations in the alpharhythm band has main functional significance, in particular in emotional, memory, motivation and attention processes. At the same time, signal processing techniques well developed in radio engineering are not used in electroencephalogram (EEG) processing. For the first time the beating method was applied to electroencephalograms. It is shown that the beating method allows the determination of α−rhythm frequencies, which are not always resoluted using Furrier transform. EEG is non-stationary process that’s why instantaneous frequency is a very important characteristics of EEG. The calculation of instantaneous frequency values was carried out based on the Hilbert transform and intersection detector method. A comparative analysis of electroencephalograms frequency content obtained by Furrier transform, Hilbert transform and Zero-level intersection detector method was carried out.

Weather at the core: Defining and categorizing geomagnetic excursions and reversals

Summary Paleomagnetic records provide us with information about the extreme geomagnetic events known as excursions and reversals, but the sparsity of available data limits detailed knowledge of the process and timing. To date there are no agreed on criteria for categorizing such events in terms of severity or longevity. In an analogy to categorizing storms in weather systems, we invoke the magnitude of the global (modified) paleosecular variation index $P_{i_D}$ to define the severity of the magnetic field state, ranging in level from 0 to 3, and defined by instantaneous values of $P_{i_D}$ with level 0 being normal ($P_{i_D}<0.5$) to extreme ($P_{i_D}\ge 15$). We denote the time of entry to an excursional (or reversal) event by when $P_{i_D}$ first exceeds 0.5, and evaluate its duration by the time at which $P_{i_D}$ first returns below its median value, termed the end of event threshold. We categorize each excursional event according to the peak level of $P_{i_D}$ during the entire event, with a range from Category-1 (Cat-1) to Cat-3. We explore an extended numerical dynamo simulation containing more than 1200 events and find that Cat-1 events are the most frequent (72%), but only rarely lead to actual field reversals where the axial dipole, $g_1^0$, has reversed sign at the end of the event. Cat-2 account for about 20% of events, with 34% of those leading to actual reversals, while Cat-3 events arise about 8% of the time but are more likely to produce reversals (43%). Higher category events take as much as 10 times longer than Cat-1 events. Two paleomagnetic field models separately cover the Laschamp excursion and Matuyama-Brunhes (M-B) reversal which are Cat-2 events with respective durations of 3.6 and 27.4 ky. It seems likely that Cat-2 may be an underestimate for M-B due to limitations in the paleomagnetic records. Our overall results suggest no distinction between excursions and reversals other than a reversal having the ending polarity state opposite to that at the start.

Experimental Evaluation of Gas-Dynamic Conditions of Heat Exchange of Stationary Air Flows in Vertical Conical Diffuser

Conical diffusers are widely used in technical devices (gasifiers, turbines, combustion chambers) and technological processes (ejectors, mixers, renewable energy). The perfection of flow gas dynamics in a conical diffuser affects the intensity of heat and mass transfer processes, the quality of mixing/separation of working media and the flow characteristics of technical devices. These parameters largely determine the efficiency and productivity of the final product. This article presents an analysis of experimental data on the gas-dynamic characteristics of stationary air flows in a vertical, conical, flat diffuser under different initial boundary conditions. An experimental setup was created, measuring instruments were selected, and an automated data collection system was developed. Basic data on the gas dynamics of air flows were obtained using the thermal anemometry method. Experimental data on instantaneous values of air flow velocity in a diffuser for initial velocities from 0.4 m/s to 2.22 m/s are presented. These data were the basis for calculating and obtaining velocity fields and turbulence intensity fields of the air flow in a vertical diffuser. It is shown that the value of the initial flow velocity at the diffuser inlet has a significant effect on the gas-dynamic characteristics. In addition, a spectral analysis of the change in air flow velocity both by height and along the diffuser axis was performed. The obtained data may be useful for refining engineering calculations, verifying mathematical models, searching for technical solutions and deepening knowledge about the features of gas dynamics of air flows in vertical diffusers.

Comparison of Short Fast Fourier Transform and Continuous Wavelet Transform in Study of Stride Interval

Neurodegenerative diseases (NDD) are a heterogeneous group of complex diseases characterized by neuronal loss and progressive degeneration of different areas of the nervous system. Gait analysis presents an early recognition system for NDD which is important to increase the patient’s awareness of their health conditions. However, it is very difficult to identify and formulate suitable digital biomarkers from the data collected from gait experiments such as stride interval and swing. The objective of this paper is to compare the result of Short - Time Fourier Transform (STFT) and Continuous Wavelet Transform (CWT) on the collected stride interval of healthy young people and healthy old people. In this paper, STFT and CWT are performed on the collected stride interval and from the result of the STFT and CWT, further features are extracted like instantaneous RMS and maximum RMS value. STFT is performed on the collected stride interval from a window length of 64 to 512 while CWT is performed on the collected stride interval from the scale of 128 to 2048. The processing time of the STFT and CWT with varied window lengths and scales respectively are collected. Besides, the actual maximum time from the time - frequency plot derived from STFT and CWT is also collected. Both STFT and CWT show that the young group has a higher maximum RMS, an indication of higher stride interval than the old group and higher variance, an indication of higher gait complexity. The suitable window lengths for STFT in analyzing the stride interval are 64 and 128 while the scale for CWT should be set to the lowest scale. Overall, STFT with a window length of 64 and 128 is better in analyzing the stride interval due to low processing time at the expense of slightly less accurate time - frequency representation.

Variable frequency interharmonic domain methodology for transient analysis and simulation of distributed generation systems

As the integration of renewable energy resources into the power grid increases, accurate modeling of variable frequency distributed generation systems becomes essential for ensuring grid stability, reliability, and optimal operation. Existing models often fail in effectively capturing the frequency dynamics of these systems, particularly when dealing with variable frequency sources. This gap hinders comprehensive analysis and the development of effective planning, control, and mitigation strategies.To address this challenge, this paper introduces a novel modeling methodology named here as the variable frequency interharmonic domain (VFID), which is based on the flexible extended harmonic domain (FEHD) approach. VFID dynamically adjusts the set of pre-selected frequencies within the state-space system to reflect changes in variable frequency sources. This method eliminates the need for post-processing routines, accurately representing both instantaneous values and frequency evolution of time-varying harmonics and interharmonics.Key results demonstrate that VFID offers significant improvements in simulation accuracy over conventional FEHD and PSCAD methods, making it a valuable tool for analyzing and simulating modern distributed generation systems. This advancement not only fills a critical research gap but also contributes to the development of more resilient and sustainable energy infrastructures.

Application of the FDTD Method to Analyze the Influence of Brick Complexity on Electromagnetic Wave Propagation

This article presents a numerical analysis of the effects related to the propagation of electromagnetic waves in an area containing a non-ideal, non-uniform, and absorbing dielectric. The analysis concerns the influence of electrical parameters, the structure of the building material, and the layering of the wall on the values of the electric field intensity. A multivariate analysis was carried out with different conductivity values. Homogeneous materials (e.g., solid brick) can be analyzed using the analytical method. In the case of complex materials containing, e.g., hollows (brick with hollows, hollow block), it is necessary to use the numerical method. The FDTD (finite difference time domain) method was used to assess the dependence of the electric field intensity on the layering, the length of hollows in bricks, and the material loss. In order to check the correctness of the adopted numerical assumptions, a series of tests related to the discretization of the model was carried out. The article also presents the influence of changing the length of hollows in bricks on the values of the electric field intensity at a frequency of 2.4 GHz. The instantaneous field distributions and maximum values of the electric field intensity are presented. In the model with a two-layer wall, regardless of the conductivity, the field values were the same for the two models, where the difference in the percentage of ceramic mass in the brick was 8%. A 12% decrease in the percentage of ceramic mass in the brick resulted in a 15% increase in the value of the area between a single-layer and a double-layer wall made of clinker bricks. At a conductivity of 0.04 S/m for a single-layer wall, the field values were similar for all brick variants.

OUT–OF–STEP protection in rotating generation networks for hydrogen fuel production. Modeling conditions to test its algorithms

The testing of relay protection devices is an extremely important task on which the stable and safe operation of the power systems depends. One of the most complex and responsible kinds of protection in networks with rotating generation, which includes hydrogen fuel generation, is protection against loss-of-synchronism. Integration of heterogeneous generating electrical devices based on renewable energy sources and based on a hydrogen fuel cell is associated with the need for reliable relay protection. It disconnects tie-lines between the systems in out-of-step conditions. And it is a challenging task to simulate input signals corresponding to the exact settings of the out-of-step protection to make sure it operates correctly. This is because one has to simulate exact parameters of impedance locus (its radius, coordinates of the center in the impedance plane, spin direction, speed, etc.) while available tools embedded in testing devices often don’t support such parameters directly. To deal with the problem, the authors present the approach to simulate instantaneous values of line currents and phase-to-ground voltages during the out-of-step operation. These electrical signals form the locus of the impedance vector with the desired parameters to test the operation of the out-of-step protection devices. The software based on the proposed algorithm also provides a graphical representation of the simulation results. All the proposed solutions proved to reduce the time required for commissioning of out-of-step protection and thereby increase tests quality and speed.

A high temporal resolution global gridded dataset of human thermal stress metrics

The human thermal stress indices and datasets are vital for promoting public health and reducing negative environmental impacts as global climate change and extreme meteorological events increase. The current thermal indices generally use an instantaneous or average value to describe thermal stress which cannot reflect the distribution of thermal comfort conditions over time, and there are no global-scale thermal stress datasets with both 0.1° or higher spatial resolution and hourly temporal resolution available yet. A novel human thermal metric, Thermal Stress Duration (TSD), is proposed to represent the accumulative time of different thermal stress levels within a certain period. A high temporal resolution global gridded dataset of human thermal stress metrics (HiGTS) is presented, which consists of hourly gridded maps of Universal Thermal Climate Index (UTCI), Universal Thermal Stress (UTS), and daily TSD at 0.1° × 0.1° spatial resolution over the global land surface, spanning from January 1, 2000, to December 31, 2023.

Faulty section location method for high impedance grounding fault in resonant grounding system based on discrete Fréchet distance

When a High Impedance Fault (HIF) occurs in a resonant grounding system, the differentiation of the current flowing upstream and downstream of the fault point is inconspicuous due to the weak electrical signal. This paper proposes a faulty section location approach based on Discrete Fréchet Distance (DFD) to address this issue. Initially, taking the zero-sequence current as the characteristic quantity, the instantaneous value of the zero-sequence current at each sampling point and its previous sampling moments are accumulated to amplify the fault signal, and furthermore to obtain the cumulative waveform of the zero-sequence current at each monitoring point. The DFD values between the cumulative waveforms of zero-sequence currents flowing through each section are also calculated. For a branchless section, if the Coefficient of Variation (CV) of the DFD in each section does not exceed the threshold value, it is identified as an end-of-line fault, otherwise, the section with the maximum DFD is identified as a faulty section. For a section with branches, the branching coefficient is calculated to determine whether it is a faulty section or not. Finally, MATLAB/Simulink simulation results and field-recorded data demonstrate the validity and robustness of the approach under various fault conditions, despite the connection of Distributed Generators (DGs).

A critical assessment of the onset strain of densification in the evaluation of energy absorption for additively manufactured cellular materials

Densification strain is an essential parameter in the characterization of energy absorption of additively manufactured cellular structures. In addition to its own merits as a metric that indicates usable stroke length for energy absorbers, it is central to the computation of energy absorbed by the structure. However, at least four different approaches have been used in the literature, each with its own limitations. In this work, a critical review of these approaches is first presented. While the maximum efficiency approach has been demonstrated to be optimal for cellular foams, this work shows how, for some additively manufactured cellular materials, it can fail to estimate densification strain accurately due to its sensitivity to instantaneous stress values in the plateau region. An alternative method is proposed in this work that leverages peak stress instead to determine the onset strain of densification and is shown to be consistently accurate across a range of cellular materials. The method is validated with the results from an experimental study of energy absorption in six different types of cellular structures across three relative densities, with identical geometries fabricated in two different base materials and processes: AlSi10Mg with Laser Powder Bed Fusion, and Nylon-12 with Selective Laser Sintering.

The influence of longitudinal duct profiling on unsteady gas dynamics and the heat transfer of pulsating gas flows in the outlet system of reciprocating-engine

Heat machines based on reciprocating-engines remain in demand in various fields of engineering and technology. Therefore, further research is needed to improve the efficiency, reliability, and environmental friendliness of engines. The thermomechanical improvement of non-stationary processes in outlet systems is an appropriate way to improve engine performance. The purpose of this research is to obtain and analyse the gas-dynamic and heat transfer characteristics of pulsating gas flows in an outlet system with ducts of various designs, using a laboratory model to simulate the outlet process in an engine. Thermal anemometry is used to receive data on the instant velocity values and local heat transfer coefficient of unsteady flows in ducts. The article examines two designs of outlet ducts, namely cylindrical (basic) and conical (with a taper of 0.0225) ducts. Spectral analysis of velocity, pressure and heat transfer coefficient pulsations, assessment of turbulence intensity, and calculation of flow characteristics of pulsating flows were performed to obtain detailed information on gas dynamics in the outlet system. The use of a conical duct (in comparison with a cylindrical one) leads to a slight increase in the turbulence intensity by up to 12 %, a decrease in the heat transfer coefficient by 15–20 %, and a change in volumetric gas flow within ± 7.5 %. Thus, the use of a conical duct will lead to the stabilisation of the flow in the outlet system, improved cleaning of the cylinder from outlet gases, a reduction in thermal stress, and a slight growth in the specific power of engines.

Opracowanie konstrukcji świdra wiertniczego do zwiercania skał o różnych właściwościach mechanicznych

This paper presents a description of a bit design with a combined cutter layout, aimed at increasing the efficiency of breaking rocks with different mechanical properties. The results from industrial testing of the manufactured bits are analyzed, revealing the need to improve their hydraulic flushing system. Computer modeling was conducted using the Flow Simulation CAD/CAM package of the Solid Works system, where various models of the design options for the bit, the borehole bottom, and a segment of the drill string were created. For each variant, numerical calculations of the fluid motion were carried out using the finite element method. Based on the simulation results, optimal angles of inclination for the jet nozzles relative to the bit axis and their distance from this axis were determined. Key factors influencing the stability of the bit cutters, which destroy the rock by cutting with axial vibrations, were identified, and analytical dependencies for their predictive stability were derived. The wear rate of the bit cutters was determined by instantaneous values of such physical parameters as: the degree of wear, rock cutting speed, temperature, axial loading force on the cutter, and the friction coefficient at the contact between the cutter and rock. Temperature is especially impactful, reaching levels exceeding 1000°C in the cutting zone; it is often regarded as the primary factor influencing bit cutter stability. To maximize the efficiency of rock-breaking in the process of drilling wells, it is necessary to set the following vibration parameters: frequency, amplitude, and vibration mode, which can be determined from the results of mathematical modeling.

Inverse problem for 2d Laplace equation in polar coordinate system

The article presents the results of study on finding the thermophysical characteristics of a heterogeneous material based on the heat conductivity equation in polar coordinates, carried out within the framework of the grant project of the Ministry of Science and Higher Education of the Republic of Kazakhstan AP19677594 «Development of methods for finding all nonlinear moisture-heat-conducting characteristics of nonhomogeneous medium, realization of methods by creating machine learning program». The temperature state of a body of a multilayer system of bodies can be characterized using a temperature field, which is understood as a set of instantaneous temperature values at all points of the studied space. It occurs when two bodies with different temperatures come into contact or within one body with areas with different temperatures.

ЗАДАЧА ОПТИМАЛЬНОГО СТОХАСТИЧЕСКОГО УПРАВЛЕНИЯ И ОЦЕНКИ СТОИМОСТИ КОРПОРАТИВНЫХ ОБЛИГАЦИЙ, ЗАВИСЯЩИХ ОТ ВРЕМЕНИ И СЛУЧАЙНЫХ СОСТОЯНИЙ ПАРАМЕТРОВ

In the optimal stochastic control problem, one estimates, using the utility indifference method, the bond price and the premium of a default swap contract (Credit default swap) when the model parameters (market interest rate, drift coef-ficient, and volatility of risky underlying assets) are random functions of time and state. Namely, the interest rate of the risk-free asset depends on time, and the prices of risky assets are described by linear homogeneous stochastic dif-ferential equations (SDEs) with multiplicative noise. To do this, the authors consider a portfolio with a risk-free asset and a risky asset with no default risk. For each such portfolio, the authors determine the amount of risky assets that maximizes the expected utility of its final wealth. This quantity allows one to solve the parabolic partial differential equations arising from the Hamilton-Jacobi-Bellman (HJB) equation by transforming them into ordinary differential equations using the method of variable separation to obtain the instantaneous value function of each portfolio. The authors derive the bond price and the default swap-contract premium (CDS) rate, which are the amounts that provide the same level of the expected utility by investing all of one’s wealth in a portfolio that does not contain these credit instruments, or by investing these amounts in credit instruments and the remainder of one’s wealth in the portfolio.

Study of patterns of PM2.5 and PM10 changes in the atmospheric air of Ivano-Frankivsk region

The purpose of this research is to analyse the temporal patterns of changes in concentrations of dust particles – aerosols, regression modeling of the interdependence of PM2.5 and PM10 at the level of instantaneous, average hourly, average daily and average weekly concentrations, and assessment of the impact of the anthropogenic component of dust pollution of atmospheric air in the Ivano-Frankivsk region on regularities of the temporal distribution of these shares.The database for the study included measurements of the concentration of PM10, PM2.5 every hour at four Ecocity public monitoring stations: in the central part of the city of Ivano-Frankivsk, in the village of Bovshiv within the influence of the Burshtyn TPP, in the village of Broshniv-Osada within the influence of the woodworking enterprise "SVISS KRONO" and within the recreational territory of the village of Mykulychyn, Nadvirnyan district. The general regularity of the temporal distribution of PM10, PM2.5 was established for all posts for daily fluctuations – the highest concentrations are observed at night, the lowest – during the period of maximum daily temperatures. It has been proven by actual data that the greater the level of atmospheric air pollution (the greater the anthropogenic component of PM), the higher the daily concentrations of PM10, PM2.5 and the more often the one-time standards of solid suspended particles are exceeded.According to the data of monitoring stations from the territories with different anthropogenic influence, functional dependences of PM2.5 content on PM10 content were obtained for instantaneous values, hourly average values, daily average values, and weekly average values. The scientific novelty of the conducted research consists in the establishment of identical functional interdependence of temporal changes in concentrations of PM2.5 and PM10 dust particles within a conditionally clean area and within the limits of the influence of stationary sources of pollution with a high level of direct regression dependence and a coefficient of determination greater than 0.9 in all cases. This allows us to conclude that the temporal patterns of changes in both PM2.5 and PM10 will not differ in clean and polluted air conditions for the Ivano-Frankivsk region. In conditions of incomplete data on the temporal distribution of the concentration of PM10, the obtained equations can be used to forecast the temporal distribution of the concentration of PM2.5.

Annular two-phase flow in a small diameter tube: OpenFOAM simulations with turbulence damping vs optical measurements

In this work, numerical simulations are performed to predict two-phase annular flow of refrigerant R245fa inside a 3.4 mm diameter vertical channel. The VOF (Volume of Fluid) method implemented in an OpenFOAM solver is used to accurately track the vapor-liquid interface. A 2D axisymmetric domain is considered and the Adaptive Mesh Refinement (AMR) method is applied to the cells near the liquid/vapor interface. The Reynolds-Averaged Navier Stokes (RANS) equations are solved and the k-ω SST model is adopted for turbulence modelling in both the liquid and vapor phase. Simulations are used to calculate instantaneous and mean values of the liquid film thickness at mass flux G = 100 kg m-2 s-1 and vapor quality ranging between 0.2 and 0.85. Numerical results are compared against measurements of the liquid film thickness taken during vertical annular downflow. Previous works from the literature and the deviations observed between present numerical and experimental results suggest the need for turbulence damping at the vapor-liquid interface by adding a source term in the ω equation. The simulations show that a low value of the turbulence damping parameter (e.g. 1) causes the average liquid film thickness to increase by 25 %–52 % compared to the non-damped scenario. The interface presents large amplitude disturbance waves in the non-damped case, whereas small ripple waves are predicted when turbulence damping is introduced. Furthermore, the difference between the application of a symmetric and asymmetric treatment for the source term is analysed. From the comparison between experimental data and numerical simulations, it emerges that the value of the correct damping source term to be applied is strictly dependent on the vapor quality.

Flow induced form: Shape optimization of bridge piers using CFD analyses and adjoint method

This study investigates the optimization of cylindrical bridge pier geometry using a relatively new technique involving Computational Fluid Dynamics (CFD) and adjoint shape optimization methods. Initially, Reynolds Averaged Navier-Stokes (RANS) equations were used in CFD models, integrated with an adjoint algorithm to reshape the pier and reduce bed shear stress. This approach is hereinafter called the “Flow induced Form”, FiF, aiming to minimize shear stress on the bed while maintaining the pier's cross-sectional and projected areas in the flow direction. Subsequently, Large Eddy Simulation (LES) transient analyses assessed the performance of the optimized geometry. Results show a reduction of approximately 5% in the time average of maximum shear stresses on the bed and a reduction of around 50% in instantaneous maximum values compared to the original pier design. These improvements were achieved with minimal changes to the cross-sectional area and projected area of the original geometry (∼2%), considering that the load-bearing capacity of the bridge pier.

Dopamine neurons encode trial-by-trial subjective reward value in an auction-like task

The dopamine reward prediction error signal is known to be subjective but has so far only been assessed in aggregate choices. However, personal choices fluctuate across trials and thus reflect the instantaneous subjective reward value. In the well-established Becker-DeGroot-Marschak (BDM) auction-like mechanism, participants are encouraged to place bids that accurately reveal their instantaneous subjective reward value; inaccurate bidding results in suboptimal reward (“incentive compatibility”). In our experiment, male rhesus monkeys became experienced over several years to place accurate BDM bids for juice rewards without specific external constraints. Their bids for physically identical rewards varied trial by trial and increased overall for larger rewards. In these highly experienced animals, responses of midbrain dopamine neurons followed the trial-by-trial variations of bids despite constant, explicitly predicted reward amounts. Inversely, dopamine responses were similar with similar bids for different physical reward amounts. Support Vector Regression demonstrated accurate prediction of the animals’ bids by as few as twenty dopamine neurons. Thus, the phasic dopamine reward signal reflects instantaneous subjective reward value.