Michael Chertkov

Daniel Bienstock, Michael Chertkov, Sean Harnett

When uncontrollable resources fluctuate, Optimum Power Flow (OPF), routinely used by the electric power industry to re-dispatch hourly controllable generation (coal, gas and hydro plants) over control areas of transmission networks, can result in grid instability, and, potentially, cascading outages. This risk arises because OPF dispatch is computed without awareness of major uncertainty, in particular fluctuations in renewable output. As a result, grid operation under OPF with renewable variability can lead to frequent conditions where power line flow ratings are significantly exceeded. Such a condition, which is borne by simulations of real grids, would likely resulting in automatic line tripping to protect lines from thermal stress, a risky and undesirable outcome which compromises stability. Smart grid goals include a commitment to large penetration of highly fluctuating renewables, thus calling to reconsider current practices, in particular the use of standard OPF. Our Chance Constrained (CC) OPF corrects the problem and mitigates dangerous renewable fluctuations with minimal changes in the current operational procedure. Assuming availability of a reliable wind forecast parameterizing the distribution function of the uncertain generation, our CC-OPF satisfies all the constraints with high probability while simultaneously minimizing the cost of economic re-dispatch. CC-OPF allows efficient implementation, e.g. solving a typical instance over the 2746-bus Polish network in 20 seconds on a standard laptop.

Petr Šulc, Scott Backhaus, Michael Chertkov

We formulate the control of reactive power generation by photovoltaic inverters in a power distribution circuit as a constrained optimization that aims to minimize reactive power losses subject to finite inverter capacity and upper and lower voltage limits at all nodes in the circuit. When voltage variations along the circuit are small and losses of both real and reactive powers are small compared to the respective flows, the resulting optimization problem is convex. Moreover, the cost function is separable enabling a distributed, on-line implementation with node-local computations using only local measurements augmented with limited information from the neighboring nodes communicated over cyber channels. Such an approach lies between the fully centralized and local policy approaches previously considered. We explore protocols based on the dual ascent method and on the Alternating Direction Method of Multipliers (ADMM) and find that the ADMM protocol performs significantly better.

Deepjyoti Deka, Scott Backhaus, Michael Chertkov

Distribution grids represent the final tier in electric networks consisting of medium and low voltage lines that connect the distribution substations to the end-users. Traditionally, distribution networks have been operated in a radial topology that may be changed from time to time. Due to absence of a significant number of real-time line monitoring devices in the distribution grid, estimation of the topology is a problem critical for its observability and control. This paper develops a novel graphical learning based approach to estimate the radial operational grid structure using voltage measurements collected from the grid loads. The learning algorithm is based on conditional independence tests for continuous variables over chordal graphs and has wide applicability. It is proven that the scheme can be used for several power flow laws (DC or AC approximations) and more importantly is independent of the specific probability distribution controlling individual bus power usage. The complexity of the algorithm is discussed and its performance is demonstrated by simulations on distribution test cases.

Sidhant Misra, Michael W. Fisher, Scott Backhaus et al.

Natural gas transmission pipelines are complex systems whose flow characteristics are governed by challenging non-linear physical behavior. These pipelines extend over hundreds and even thousands of miles. Gas is typically injected into the system at a constant rate, and a series of compressors are distributed along the pipeline to boost the gas pressure to maintain system pressure and throughput. These compressors consume a portion of the gas, and one goal of the operator is to control the compressor operation to minimize this consumption while satisfying pressure constraints at the gas load points. The optimization of these operations is computationally challenging. Many pipelines simply rely on the intuition and prior experience of operators to make these decisions. Here, we present a new geometric programming approach for optimizing compressor operation in natural gas pipelines. Using models of real natural gas pipelines, we show that the geometric programming algorithm consistently outperforms approaches that mimic existing state of practice.

Sejun Park, Deepjyoti Deka, Michael Chertkov

Limited presence of nodal and line meters in distribution grids hinders their optimal operation and participation in real-time markets. In particular lack of real-time information on the grid topology and infrequently calibrated line parameters (impedances) adversely affect the accuracy of any operational power flow control. This paper suggests a novel algorithm for learning the topology of distribution grid and estimating impedances of the operational lines with minimal observational requirements - it provably reconstructs topology and impedances using voltage and injection measured only at the terminal (end-user) nodes of the distribution grid. All other (intermediate) nodes in the network may be unobserved/hidden. Furthermore no additional input (e.g., number of grid nodes, historical information on injections at hidden nodes) is needed for the learning to succeed. Performance of the algorithm is illustrated in numerical experiments on the IEEE and custom power distribution models.

Michael Chertkov, Nikolai N. Novitsky

Heat fluxes in a district heating pipeline systems need to be controlled on the scale from minutes to an hour to adjust to evolving demand. There are two principal ways to control the heat flux - keep temperature fixed but adjust velocity of the carrier (typically water) or keep the velocity flow steady but then adjust temperature at the heat producing source (heat plant). We study the latter scenario, commonly used for operations in Russia and Nordic countries, and analyze dynamics of the heat front as it propagates through the system. Steady velocity flows in the district heating pipelines are typically turbulent and incompressible. Changes in the heat, on either consumption or production sides, lead to slow transients which last from tens of minutes to hours. We classify relevant physical phenomena in a single pipe, e.g. turbulent spread of the turbulent front. We then explain how to describe dynamics of temperature and heat flux evolution over a network efficiently and illustrate the network solution on a simple example involving one producer and one consumer of heat connected by "hot" and "cold" pipes. We conclude the manuscript motivating future research directions.

Deepjyoti Deka, Saurav Talukdar, Michael Chertkov et al.

Graphical models are a succinct way to represent the structure in probability distributions. This article analyzes the graphical model of nodal voltages in non-radial power distribution grids. Using algebraic and structural properties of graphical models, algorithms exactly determining topology and detecting line changes for distribution grids are presented along with their theoretical limitations. We show that if distribution grids have cycles/loops of size greater than three, then nodal voltages are sufficient for efficient topology estimation without additional assumptions on system parameters. In contrast, line failure or change detection using nodal voltages does not require any structural assumption. Under noisy measurements, we provide the first non-trivial bounds on the maximum noise that the system can tolerate for asymptotically correct topology recovery. The performance of the designed algorithms is validated with nonlinear AC power flow samples generated by Matpower on test grids, including scenarios with injection correlations and system noise.

Ali Hassan, Robert Mieth, Michael Chertkov et al.

Distribution system operators (DSO) world-wide foresee a rapid roll-out of distributed energy resources. From the system perspective, their reliable and cost effective integration requires accounting for their physical properties in operating tools used by the DSO. This paper describes an approach to leverage the dispatch flexibility of thermostatically controlled loads (TCLs) for operating distribution systems with a high penetration level of photovoltaic resources. Each TCL ensemble is modeled using the Markov Decision Process (MDP). The MDP model is then integrated with the chance-constrained optimal power flow that accounts for the uncertainty of PV resources. Since the integrated optimization model cannot be solved efficiently by existing dynamic programming methods or off-the-shelf solvers, this paper proposes an iterative Spatio-Temporal Dual Decomposition algorithm (ST-D2). We demonstrate the usefulness of the proposed integrated optimization and ST-D2 algorithm on the IEEE 33-bus test system.

Michael Chertkov, Mikhail Stepanov, Feng Pan et al.

In this manuscript we continue the thread of [M. Chertkov, F. Pan, M. Stepanov, Predicting Failures in Power Grids: The Case of Static Overloads, IEEE Smart Grid 2011] and suggest a new algorithm discovering most probable extreme stochastic events in static power grids associated with intermittent generation of wind turbines. The algorithm becomes EXACT and EFFICIENT (polynomial) in the case of the proportional (or other low parametric) control of standard generation, and log-concave probability distribution of the renewable generation, assumed known from the wind forecast. We illustrate the algorithm's ability to discover problematic extreme events on the example of the IEEE RTS-96 model of transmission with additions of 10%, 20% and 30% of renewable generation. We observe that the probability of failure may grow but it may also decrease with increase in renewable penetration, if the latter is sufficiently diversified and distributed.

Sejun Park, Deepjyoti Deka, Scott Backhaus et al.

Efficient operation of distribution grids in the smart-grid era is hindered by the limited presence of real-time nodal and line meters. In particular, this prevents the easy estimation of grid topology and associated line parameters that are necessary for control and optimization efforts in the grid. This paper studies the problems of topology and parameter estimation in radial balanced distribution grids where measurements are restricted to only the leaf nodes and all intermediate nodes are unobserved/hidden. To this end, we propose two exact learning algorithms that use balanced voltage and injection measured only at the end-users. The first algorithm requires time-stamped voltage samples, statistics of nodal power injections and permissible line impedances to recover the true topology. The second and improved algorithm requires only time-stamped voltage and complex power samples to recover both the true topology and impedances without any additional input (e.g., number of grid nodes, statistics of injections at hidden nodes, permissible line impedances). We prove the correctness of both learning algorithms for grids where unobserved buses/nodes have a degree greater than three and discuss extensions to regimes where that assumption doesn't hold. Further, we present computational and, more importantly, the sample complexity of our proposed algorithm for joint topology and impedance estimation. We illustrate the performance of the designed algorithms through numerical experiments on the IEEE and custom power distribution models.

Deepjyoti Deka, Scott Backhaus, Michael Chertkov

Traditionally power distribution networks are either not observable or only partially observable. This complicates development and implementation of new smart grid technologies, such as those related to demand response, outage detection and management, and improved load-monitoring. In this two part paper, inspired by proliferation of metering technology, we discuss estimation problems in structurally loopy but operationally radial distribution grids from measurements, e.g. voltage data, which are either already available or can be made available with a relatively minor investment. In Part I, the objective is to learn the operational layout of the grid. Part II of this paper presents algorithms that estimate load statistics or line parameters in addition to learning the grid structure. Further, Part II discusses the problem of structure estimation for systems with incomplete measurement sets. Our newly suggested algorithms apply to a wide range of realistic scenarios. The algorithms are also computationally efficient -- polynomial in time -- which is proven theoretically and illustrated computationally on a number of test cases. The technique developed can be applied to detect line failures in real time as well as to understand the scope of possible adversarial attacks on the grid.

Michael Chertkov, Vladimir Y. Chernyak, Deepjyoti Deka

A Markov decision process (MDP) framework is adopted to represent ensemble control of devices with cyclic energy consumption patterns, e.g., thermostatically controlled loads. Specifically we utilize and develop the class of MDP models previously coined linearly solvable MDPs, that describe optimal dynamics of the probability distribution of an ensemble of many cycling devices. Two principally different settings are discussed. First, we consider optimal strategy of the ensemble aggregator balancing between minimization of the cost of operations and minimization of the ensemble welfare penalty, where the latter is represented as a KL-divergence between actual and normal probability distributions of the ensemble. Then, second, we shift to the demand response setting modeling the aggregator's task to minimize the welfare penalty under the condition that the aggregated consumption matches the targeted time-varying consumption requested by the system operator. We discuss a modification of both settings aimed at encouraging or constraining the transitions between different states. The dynamic programming feature of the resulting modified MDPs is always preserved; however, `linear solvability' is lost fully or partially, depending on the type of modification. We also conducted some (limited in scope) numerical experimentation using the formulations of the first setting. We conclude by discussing future generalizations and applications.

Konstantin Turitsyn, Scott Backhaus, Maxim Ananyev et al.

We introduce and analyze Markov Decision Process (MDP) machines to model individual devices which are expected to participate in future demand-response markets on distribution grids. We differentiate devices into the following four types: (a) optional loads that can be shed, e.g. light dimming; (b) deferrable loads that can be delayed, e.g. dishwashers; (c) controllable loads with inertia, e.g. thermostatically-controlled loads, whose task is to maintain an auxiliary characteristic (temperature) within pre-defined margins; and (d) storage devices that can alternate between charging and generating. Our analysis of the devices seeks to find their optimal price-taking control strategy under a given stochastic model of the distribution market.

Deepjyoti Deka, Scott Backhaus, Michael Chertkov

Distribution grids refer to the part of the power grid that delivers electricity from substations to the loads. Structurally a distribution grid is operated in one of several radial/tree-like topologies that are derived from an original loopy grid graph by opening switches on some lines. Due to limited presence of real-time switch monitoring devices, the operating structure needs to be estimated indirectly. This paper presents a new learning algorithm that uses only nodal voltage measurements to determine the operational radial structure. The algorithm is based on the key result stating that the correct operating structure is the optimal solution of the minimum-weight spanning tree problem over the original loopy graph where weights on all permissible edges/lines (open or closed) is the variance of nodal voltage difference at the edge ends. Compared to existing work, this spanning tree based approach has significantly lower complexity as it does not require information on line parameters. Further, a modified learning algorithm is developed for cases when the input voltage measurements are limited to only a subset of the total grid nodes. Performance of the algorithms (with and without missing data) is demonstrated by experiments on test cases.

René Pfitzner, Konstantin Turitsyn, Michael Chertkov

We study the evolution of fast blackout cascades in the model of the Polish (transmission) power grid (2700 nodes and 3504 transmission lines). The cascade is initiated by a sufficiently severe initial contingency tripping. It propagates via sequential trippings of many more overheated lines, islanding loads and generators and eventually arriving at a fixed point with the surviving part of the system being power-flow-balanced and the rest of the system being outaged. Utilizing an improved form of the quasi-static model for cascade propagation introduced in our earlier study (Statistical Classification of Cascading Failures in Power Grids, IEEE PES GM 2011), we analyze how the severity of the cascade depends on the order of tripping overheated lines. Our main observation is that the order of tripping has a tremendous effect on the size of the resulting outage. Finding the "best" tripping, defined as causing the least damage, constitutes a difficult dynamical optimization problem, whose solution is most likely computationally infeasible. Instead, here we study performance of a number of natural heuristics, resolving the next switching decision based on the current state of the grid. Overall, we conclude that controlled intentional tripping is advantageous in the situation of a fast developing extreme emergency, as it provides significant mitigation of the resulting damage.

Ali Hassan, Yury Dvorkin, Deepjyoti Deka et al.

Distribution systems are undergoing a dramatic transition from a passive circuit that routinely disseminates electric power among downstream nodes to the system with distributed energy resources. The distributed energy resources come in a variety of technologies and typically include photovoltaic (PV) arrays, thermostatically controlled loads, energy storage units. Often these resources are interfaced with the system via inverters that can adjust active and reactive power injections, thus supporting the operational performance of the system. This paper designs a control policy for such inverters using the local power flow measurements. The control actuates active and reactive power injections of the inverter-based distributed energy resources. This strategy is then incorporated into a chance-constrained, decentralized optimal power flow formulation to maintain voltage levels and power flows within their limits and to mitigate the volatility of (PV) resources.

Michael Chertkov, Vladimir Chernyak

Thermostatically Controlled Loads (TCL), e.g. air-conditioners and heaters, are by far the most wide-spread consumers of electricity. Normally the devices are calibrated to provide the so-called bang-bang control of temperature -- changing from on to off, and vice versa, depending on temperature. Aggregation of a large group of similar devices into a statistical ensemble is considered, where the devices operate following the same dynamics subject to stochastic perturbations and randomized, Poisson on/off switching policy. We analyze, using theoretical and computational tools of statistical physics, how the ensemble relaxes to a stationary distribution and establish relation between the relaxation and statistics of the probability flux, associated with devices' cycling in the mixed (discrete, switch on/off, and continuous, temperature) phase space. This allowed us to derive and analyze spectrum of the non-equilibrium (detailed balance broken) statistical system and uncover how switching policy affects oscillatory trend and speed of the relaxation. Relaxation of the ensemble is of a practical interest because it describes how the ensemble recovers from significant perturbations, e.g. forceful temporary switching off aimed at utilizing flexibility of the ensemble in providing "demand response" services relieving consumption temporarily to balance larger power grid. We discuss how the statistical analysis can guide further development of the emerging demand response technology.

Line Roald, Sidhant Misra, Michael Chertkov et al.

Over the past years, the share of electricity production from wind power plants has increased to significant levels in several power systems across Europe and the United States. In order to cope with the fluctuating and partially unpredictable nature of renewable energy sources, transmission system operators (TSOs) have responded by increasing their reserve capacity requirements and by requiring wind power plants to be capable of providing reserves or following active power set-point signals. This paper addresses the issue of efficiently incorporating these new types of wind power control in the day-ahead operational planning. We review the technical requirements the wind power plants must fulfill, and propose a mathematical framework for modeling wind power control. The framework is based on an optimal power flow formulation with weighted chance constraints, which accounts for the uncertainty of wind power forecasts and allows us to limit the risk of constraint violations. In a case study based on the IEEE 118 bus system, we use the developed method to assess the effectiveness of different types of wind power control in terms of operational cost, system security and wind power curtailment.

Michael Chertkov, Deepjyoti Deka, Yury Dvorkin

Flexible loads, e.g. thermostatically controlled loads (TCLs), are technically feasible to participate in demand response (DR) programs. On the other hand, there is a number of challenges that need to be resolved before it can be implemented in practice en masse. First, individual TCLs must be aggregated and operated in sync to scale DR benefits. Second, the uncertainty of TCLs needs to be accounted for. Third, exercising the flexibility of TCLs needs to be coordinated with distribution system operations to avoid unnecessary power losses and compliance with power flow and voltage limits. This paper addresses these challenges. We propose a network-constrained, open-loop, stochastic optimal control formulation. The first part of this formulation represents ensembles of collocated TCLs modelled by an aggregated Markov Process (MP), where each MP state is associated with a given power consumption or production level. The second part extends MPs to a multi-period distribution power flow optimization. In this optimization, the control of TCL ensembles is regulated by transition probability matrices and physically enabled by local active and reactive power controls at TCL locations. The optimization is solved with a Spatio-Temporal Dual Decomposition (ST-D2) algorithm. The performance of the proposed formulation and algorithm is demonstrated on the IEEE 33-bus distribution model.

Vladimir Frolov, Priyanko Guha Thakurta, Scott Backhaus et al.

Decentralized electricity markets and more integration of renewables demand expansion of the existing transmission infrastructure to accommodate inflected variabilities in power flows. However, such expansion is severely limited in many countries because of political and environmental issues. Furthermore, high renewables integration requires additional reactive power support, which forces the transmission system operators to utilize the existing grid creatively, e.g., take advantage of new technologies, such as flexible alternating current transmission system (FACTS) devices. We formulate, analyze and solve the challenging investment planning problem of installation in an existing large-scale transmission grid multiple FACTS devices of two types (series capacitors and static VAR compensators.) We account for details of AC character of the power flows, probabilistic modeling of multiple-load scenarios, FACTS devices flexibility in terms of their adjustments within the capacity constraints, and long term practical tradeoffs between capital vs operational expenditures (CAPEX vs OPEX). It is demonstrated that proper installation of the devices allows to do both - extend or improve feasibility domain for the system and also decrease long term power generation cost (make cheaper generation available). Nonlinear, nonconvex, and multiple-scenario-aware optimization is resolved through an efficient heuristic algorithm consisting of a sequence of quadratic programmings solved by CPLEX combined with exact AC PF resolution for each scenario for maintaining feasible operational states during iterations. Efficiency and scalability of the approach is illustrated on the IEEE 30-bus model and the 2736-bus Polish model from Matpower.

Deepjyoti Deka, Michael Chertkov, Scott Backhaus

Optimal operation of distribution grid resources relies on accurate estimation of its state and topology. Practical estimation of such quantities is complicated by the limited presence of real-time meters. This paper discusses a theoretical framework to jointly estimate the operational topology and statistics of injections in radial distribution grids under limited availability of nodal voltage measurements. In particular we show that our proposed algorithms are able to provably learn the exact grid topology and injection statistics at all unobserved nodes as long as they are not adjacent. The algorithm design is based on novel ordered trends in voltage magnitude fluctuations at node groups, that are independently of interest for radial physical flow networks. The complexity of the designed algorithms is theoretically analyzed and their performance validated using both linearized and non-linear AC power flow samples in test distribution grids.

Robert Ferrando, Laurent Pagnier, Robert Mieth et al.

This paper addresses the challenge of efficiently solving the optimal power flow problem in real-time electricity markets. The proposed solution, named Physics-Informed Market-Aware Active Set learning OPF (PIMA-AS-OPF), leverages physical constraints and market properties to ensure physical and economic feasibility of market-clearing outcomes. Specifically, PIMA-AS-OPF employs the active set learning technique and expands its capabilities to account for curtailment in load or renewable power generation, which is a common challenge in real-world power systems. The core of PIMA-AS-OPF is a fully-connected neural network that takes the net load and the system topology as input. The outputs of this neural network include active constraints such as saturated generators and transmission lines, as well as non-zero load shedding and wind curtailments. These outputs allow for reducing the original market-clearing optimization to a system of linear equations, which can be solved efficiently and yield both the dispatch decisions and the locational marginal prices (LMPs). The dispatch decisions and LMPs are then tested for their feasibility with respect to the requirements for efficient market-clearing results. The accuracy and scalability of the proposed method is tested on a realistic 1814-bus NYISO system with current and future renewable energy penetration levels.

David Métivier, Michael Chertkov

We pose an engineering challenge of controlling an Ensemble of Energy Devices via coordinated, implementation-light and randomized on/off switching as a problem in Non-Equilibrium Statistical Mechanics. We show that Mean Field Control} with nonlinear feedback on the cumulative consumption, assumed available to the aggregator via direct physical measurements of the energy flow, allows the ensemble to recover from its use in the Demand Response regime, i.e. transition to a statistical steady state, significantly faster than in the case of the fixed feedback. Moreover when the nonlinearity is sufficiently strong, one observes the phenomenon of "super-relaxation" -- where the total instantaneous energy consumption of the ensemble transitions to the steady state much faster than the underlying probability distribution of the devices over their state space, while also leaving almost no devices outside of the comfort zone.

Saurav Talukdar, Deepjyoti Deka, Michael Chertkov et al.

In this article, we present a method to reconstruct the topology of a partially observed radial network of linear dynamical systems with bi-directional interactions. Our approach exploits the structure of the inverse power spectral density matrix and recovers edges involving nodes up to four hops away in the underlying topology. We then present an algorithm with provable guarantees, which eliminates the spurious links obtained and also identifies the location of the unobserved nodes in the inferred topology. The algorithm recovers the exact topology of the network by using only time-series of the states at the observed nodes. The effectiveness of the method developed is demonstrated by applying it on a typical distribution system of the electric grid.

Krishnamurthy Dvijotham, Michael Chertkov, Scott Backhaus

Deviations of grid frequency from the nominal frequency are an indicator of the global imbalance between genera- tion and load. Two types of control, a distributed propor- tional control and a centralized integral control, are cur- rently used to keep frequency deviations small. Although generation-load imbalance can be very localized, both controls primarily rely on frequency deviation as their in- put. The time scales of control require the outputs of the centralized integral control to be communicated to distant generators every few seconds. We reconsider this con- trol/communication architecture and suggest a hybrid ap- proach that utilizes parameterized feedback policies that can be implemented in a fully distributed manner because the inputs to these policies are local observables at each generator. Using an ensemble of forecasts of load and time-intermittent generation representative of possible fu- ture scenarios, we perform a centralized off-line stochas- tic optimization to select the generator-specific feedback parameters. These parameters need only be communi- cated to generators once per control period (60 minutes in our simulations). We show that inclusion of local power flows as feedback inputs is crucial and reduces frequency deviations by a factor of ten. We demonstrate our con- trol on a detailed transmission model of the Bonneville Power Administration (BPA). Our findings suggest that a smart automatic and distributed control, relying on ad- vanced off-line and system-wide computations commu- nicated to controlled generators infrequently, may be a viable control and communication architecture solution. This architecture is suitable for a future situation when generation-load imbalances are expected to grow because of increased penetration of time-intermittent generation.

Roman Pop, Ali Hassan, Kenneth Bruninx et al.

This paper describes a step-by-step procedure that converts a physical model of a building into a Markov Process that characterizes energy consumption of this and other similar buildings. Relative to existing thermo-physics-based building models, the proposed procedure reduces model complexity and depends on fewer parameters, while also maintaining accuracy and feasibility sufficient for system-level analyses. Furthermore, the proposed Markov Process approach makes it possible to leverage real-time data streams available from intelligent data acquisition systems, which are readily available in smart buildings, and merge it with physics-based and statistical models. Construction of the Markov Process naturally leads to a Markov Decision Process formulation, which describes optimal probabilistic control of a collection of similar buildings. The approach is illustrated using validated building data from Belgium.

Michael Chertkov, Sidhant Misra, Marc Vuffray et al.

In this manuscript we review new ideas and first results on application of the Graphical Models approach, originated from Statistical Physics, Information Theory, Computer Science and Machine Learning, to optimization problems of network flow type with additional constraints related to the physics of the flow. We illustrate the general concepts on a number of enabling examples from power system and natural gas transmission (continental scale) and distribution (district scale) systems.

Scott Backhaus, Russell Bent, Daniel Bienstock et al.

Direct methods can provide rapid screening of the dynamical security of large numbers fault and contingency scenarios by avoiding extensive time simulation. We introduce a computationally-efficient direct method based on optimization that leverages efficient cutting plane techniques. The method considers both unstable equilibrium points and the effects of additional relay tripping on dynamical security\cite{01SH}. Similar to other direct methods, our approach yields conservative results for dynamical security, however, the optimization formulation potentially lends itself to the inclusion of additional constraints to reduce this conservatism.

Laurent Pagnier, Robert Ferrando, Yury Dvorkin et al.

This paper seeks to design a machine learning twin of the optimal power flow (OPF) optimization, which is used in market-clearing procedures by wholesale electricity markets. The motivation for the proposed approach stems from the need to obtain the digital twin, which is much faster than the original, while also being sufficiently accurate and producing consistent generation dispatches and locational marginal prices (LMPs), which are primal and dual solutions of the OPF optimization, respectively. Availability of market-clearing tools based on this approach will enable computationally tractable evaluation of multiple dispatch scenarios under a given unit commitment. Rather than direct solution of OPF, the Karush-Kuhn-Tucker (KKT) conditions for the OPF problem in question may be written, and in parallel the LMPs of generators and loads may be expressed in terms of the OPF Lagrangian multipliers. Also, taking advantage of the practical fact that many of the Lagrangian multipliers associated with lines will be zero (thermal limits are not binding), we build and train an ML scheme which maps flexible resources (loads and renewables) to the binding lines, and supplement it with an efficient power-grid aware linear map to optimal dispatch and LMPs. The scheme is validated and illustrated on IEEE models. We also report a trade of analysis between quality of the reconstruction and number of samples needed to train the model.

Changhong Zhao, Michael Chertkov, Scott Backhaus

We consider joint control of a switchable capacitor and a D-STATCOM for voltage regulation in a distribution circuit with intermittent load. The control problem is formulated as a two-timescale optimal power flow problem with chance constraints, which minimizes power loss while limiting the probability of voltage violations due to fast changes in load. The control problem forms the basis of an optimization problem which determines the sizes of the control devices by minimizing sum of the expected power loss cost and the capital cost. We develop computationally efficient heuristics to solve the optimal sizing problem and implement real-time control. Numerical experiments on a circuit with high-performance computing (HPC) load show that the proposed sizing and control schemes significantly improve the reliability of voltage regulation on the expense of only a moderate increase in cost.

Hamidreza Behjoo, Michael Chertkov

Computing the partition function, $Z$, of an Ising model over a graph of $N$ \enquote{spins} is most likely exponential in $N$. Efficient variational methods, such as Belief Propagation (BP) and Tree Re-Weighted (TRW) algorithms, compute $Z$ approximately by minimizing the respective (BP- or TRW-) free energy. We generalize the variational scheme by building a $λ$-fractional interpolation, $Z^{(λ)}$, where $λ=0$ and $λ=1$ correspond to TRW- and BP-approximations, respectively. This fractional scheme -- coined Fractional Belief Propagation (FBP) -- guarantees that in the attractive (ferromagnetic) case $Z^{(TRW)} \geq Z^{(λ)} \geq Z^{(BP)}$, and there exists a unique (\enquote{exact}) $λ_*$ such that $Z=Z^{(λ_*)}$. Generalizing the re-parametrization approach of \citep{wainwright_tree-based_2002} and the loop series approach of \citep{chertkov_loop_2006}, we show how to express $Z$ as a product, $\forall λ:\ Z=Z^{(λ)}{\tilde Z}^{(λ)}$, where the multiplicative correction, ${\tilde Z}^{(λ)}$, is an expectation over a node-independent probability distribution built from node-wise fractional marginals. Our theoretical analysis is complemented by extensive experiments with models from Ising ensembles over planar and random graphs of medium and large sizes. Our empirical study yields a number of interesting observations, such as the ability to estimate ${\tilde Z}^{(λ)}$ with $O(N^{2::4})$ fractional samples and suppression of variation in $λ_*$ estimates with an increase in $N$ for instances from a particular random Ising ensemble, where $[2::4]$ indicates a range from $2$ to $4$. We also discuss the applicability of this approach to the problem of image de-noising.

Hamidreza Behjoo, Michael Chertkov

In this manuscript, we present a novel approach for sampling from a continuous multivariate probability distribution, which may either be explicitly known (up to a normalization factor) or represented via empirical samples. Our method constructs a time-dependent bridge from a delta function centered at the origin of the state space at $t=0$, optimally transforming it into the target distribution at $t=1$. We formulate this as a Stochastic Optimal Control problem of the Path Integral Control type, with a cost function comprising (in its basic form) a quadratic control term, a quadratic state term, and a terminal constraint. This framework, which we refer to as Harmonic Path Integral Diffusion (H-PID), leverages an analytical solution through a mapping to an auxiliary quantum harmonic oscillator in imaginary time. The H-PID framework results in a set of efficient sampling algorithms, without the incorporation of Neural Networks. The algorithms are validated on two standard use cases: a mixture of Gaussians over a grid and images from CIFAR-10. The transparency of the method allows us to analyze the algorithms in detail, particularly revealing that the current weighted state is an order parameter for the dynamic phase transition, signaling earlier, at $t<1$, that the sample generation process is almost complete. We contrast these algorithms with other sampling methods, particularly simulated annealing and path integral sampling, highlighting their advantages in terms of analytical control, accuracy, and computational efficiency on benchmark problems. Additionally, we extend the methodology to more general cases where the underlying stochastic differential equation includes an external deterministic, possibly non-conservative force, and where the cost function incorporates a gauge potential term.

Nikolay Stulov, Dejan J Sobajic, Yury Maximov et al.

In this work we investigate approaches to reconstruct generator models from measurements available at the generator terminal bus using machine learning (ML) techniques. The goal is to develop an emulator which is trained online and is capable of fast predictive computations. The training is illustrated on synthetic data generated based on available open-source dynamical generator model. Two ML techniques were developed and tested: (a) standard vector auto-regressive (VAR) model; and (b) novel customized long short-term memory (LSTM) deep learning model. Trade-offs in reconstruction ability between computationally light but linear AR model and powerful but computationally demanding LSTM model are established and analyzed.

Hamidreza Behjoo, Michael Chertkov

We investigate diffusion models generating synthetic samples from the probability distribution represented by the Ground Truth (GT) samples. We focus on how GT sample information is encoded in the Score Function (SF), computed (not simulated) from the Wiener-Ito (WI) linear forward process in the artifical time $t\in [0\to \infty]$, and then used as a nonlinear drift in the simulated WI reverse process with $t\in [\infty\to 0]$. We propose U-Turn diffusion, an augmentation of a pre-trained diffusion model, which shortens the forward and reverse processes to $t\in [0\to T_u]$ and $t\in [T_u\to 0]$. The U-Turn reverse process is initialized at $T_u$ with a sample from the probability distribution of the forward process (initialized at $t=0$ with a GT sample) ensuring a detailed balance relation between the shorten forward and reverse processes. Our experiments on the class-conditioned SF of the ImageNet dataset and the multi-class, single SF of the CIFAR-10 dataset reveal a critical Memorization Time $ T_m $, beyond which generated samples diverge from the GT sample used to initialize the U-Turn scheme, and a Speciation Time $ T_s $, where for $ T_u > T_s > T_m $, samples begin representing different classes. We further examine the role of SF non-linearity through a Gaussian Test, comparing empirical and Gaussian-approximated U-Turn auto-correlation functions, and showing that the SF becomes effectively affine for $ t > T_s $, and approximately affine for $t\in [T_m,T_s]$.

Michael Chertkov

Most modern bridge-diffusion methods achieve finite-time transport by specifying an interpolation, Schrödinger-bridge, or stochastic-control objective and then learning the associated score or drift field with a neural network. In contrast, we identify a restricted but sufficiently broad and analytically solvable class in which the score, intermediate marginals, and protocol gradients are available in closed form without inner stochastic simulation loops and without neural networks in the optimization loop. We recast the classical linear--quadratic--Gaussian (LQG) stochastic-control structure as a transport problem of the Path Integral Diffusion (PID) type. In classical LQG control, linear dynamics, Gaussian noise, and quadratic costs lead to Riccati equations and closed-form optimal feedback. In LQ-GM-PID, we retain the linear--quadratic stochastic-control backbone, but replace terminal state regulation by a prescribed terminal probability density and allow both the initial and terminal laws to be Gaussian Mixtures (GM). Moreover, LQ-GM-PID turns bridge diffusion from a tool for terminal target matching alone into a tool for path shaping. We demonstrate this on a 2D corridor task, a 2D multi-entrance transport task, and a high-dimensional scaling study with $d=32$ and $M=16$ Gaussian-mixture terminal modes, all with sub-50\,ms analytic precompute on a laptop. We position LQ-GM-PID as an analytically solvable reference model for the state-of-the-art neural bridge-diffusion and generative-transport methods: a controlled setting in which neural approximations, score estimates, path-shaping objectives, and protocol-learning procedures can be tested against exact quantities.

Michael Chertkov

The paper reflects on the future role of AI in scientific research, with a special focus on turbulence studies, and examines the evolution of AI, particularly through Diffusion Models rooted in non-equilibrium statistical mechanics. It underscores the significant impact of AI on advancing reduced, Lagrangian models of turbulence through innovative use of deep neural networks. Additionally, the paper reviews various other AI applications in turbulence research and outlines potential challenges and opportunities in the concurrent advancement of AI and statistical hydrodynamics. This discussion sets the stage for a future where AI and turbulence research are intricately intertwined, leading to more profound insights and advancements in both fields.

Michael Chertkov, Sungsoo Ahn, Hamidreza Behjoo

In this manuscript, we introduce a novel Decision Flow (DF) framework for sampling decisions from a target distribution while incorporating additional guidance from a prior sampler. DF can be viewed as an AI-driven algorithmic reincarnation of the Markov Decision Process (MDP) approach in stochastic optimal control. It extends the continuous-space, continuous-time Path Integral Diffusion sampling technique of [Behjoo, Chertkov 2025] to discrete time and space, while also generalizing the Generative Flow Network (GFN) framework of [Bengio, et al 2021]. In its most basic form an explicit formulation that does not require Neural Networks (NNs), DF leverages the linear solvability of the underlying MDP [Todorov, 2007] to adjust the transition probabilities of the prior sampler. The resulting Markov process is expressed as a convolution of the reverse-time Green's function of the prior sampling with the target distribution. We illustrate the DF framework through an example of sampling from the Ising model -- compare DF to Metropolis-Hastings to quantify its efficiency, discuss potential NN-based extensions, and outline how DF can enhance guided sampling across various applications.

Hamidreza Behjoo, Michael Chertkov

In this study, we introduce a novel method for generating new synthetic samples that are independent and identically distributed (i.i.d.) from high-dimensional real-valued probability distributions, as defined implicitly by a set of Ground Truth (GT) samples. Central to our method is the integration of space-time mixing strategies that extend across temporal and spatial dimensions. Our methodology is underpinned by three interrelated stochastic processes designed to enable optimal transport from an easily tractable initial probability distribution to the target distribution represented by the GT samples: (a) linear processes incorporating space-time mixing that yield Gaussian conditional probability densities, (b) their diffusion bridge analogs that are conditioned to the initial and final state vectors, and (c) nonlinear stochastic processes refined through score-matching techniques. The crux of our training regime involves fine-tuning the nonlinear model, and potentially the linear models -- to align closely with the GT data. We validate the efficacy of our space-time diffusion approach with numerical experiments, laying the groundwork for more extensive future theory and experiments to fully authenticate the method, particularly providing a more efficient (possibly simulation-free) inference.

Michael Chertkov

An agent that operates sequentially must incorporate new experience without forgetting old experience, under a fixed memory budget. We propose a framework in which memory is not a parameter vector but a stochastic process: a Bridge Diffusion on a replay interval $[0,1]$, whose terminal marginal encodes the present and whose intermediate marginals encode the past. New experience is incorporated via a three-step \emph{Compress--Add--Smooth} (CAS) recursion. We test the framework on the class of models with marginal probability densities modeled via Gaussian mixtures of fixed number of components~$K$ in $d$ dimensions; temporal complexity is controlled by a fixed number~$L$ of piecewise-linear protocol segments whose nodes store Gaussian-mixture states. The entire recursion costs $O(LKd^2)$ flops per day -- no backpropagation, no stored data, no neural networks -- making it viable for controller-light hardware. Forgetting in this framework arises not from parameter interference but from lossy temporal compression: the re-approximation of a finer protocol by a coarser one under a fixed segment budget. We find that the retention half-life scales linearly as $a_{1/2}\approx c\,L$ with a constant $c>1$ that depends on the dynamics but not on the mixture complexity~$K$, the dimension~$d$, or the geometry of the target family. The constant~$c$ admits an information-theoretic interpretation analogous to the Shannon channel capacity. The stochastic process underlying the bridge provides temporally coherent ``movie'' replay -- compressed narratives of the agent's history, demonstrated visually on an MNIST latent-space illustration. The framework provides a fully analytical ``Ising model'' of continual learning in which the mechanism, rate, and form of forgetting can be studied with mathematical precision.

Michael Chertkov

Independent sample generation is the prevailing paradigm in modern diffusion-based generative models of AI. We ask a different question: can samples \emph{coordinate} through shared population statistics to transport probability mass more efficiently? We introduce Mean-Field Path-Integral Diffusion (MF-PID), a framework in which samples are promoted to interacting agents whose drift depends self-consistently on the evolving population density. The coupling converts distribution matching into a McKean--Vlasov extension of the stochastic optimal transport problem, unifying generative modeling and multi-agent control under the same Hamilton--Jacobi--Bellman/Kolmogorov--Fokker--Planck duality. We identify two analytically tractable regimes: a Linear--Quadratic--Gaussian (LQG) benchmark in which the infinite-dimensional mean-field system reduces to a finite set of Riccati and linear ODEs, and a Gaussian-mixture regime governed by a piecewise-constant protocol that preserves closed-form solvability. For a quadratic interaction potential with schedule $β_t$ and zero base drift we prove that the self-consistent MF guidance is the \emph{exact} linear interpolant between initial and target global means -- a result that holds for arbitrary initial and target densities and any $β_t$. Applied to demand-response control of energy systems, where agents aggregated into an ensemble are energy consumers (e.g.\ thermal zones within a building), MF-PID achieves 19--24\% reductions in cumulative control energy over independent-agent baselines while matching the prescribed terminal distribution exactly, and reveals how coordination redistributes actuation effort across heterogeneous sub-populations.

Michael Chertkov

We introduce Guided Harmonic Path-Integral Diffusion (GH-PID), a linearly-solvable framework for guided Stochastic Optimal Transport (SOT) with a hard terminal distribution and soft, application-driven path costs. A low-dimensional guidance protocol shapes the trajectory ensemble while preserving analytic structure: the forward and backward Kolmogorov equations remain linear, the optimal score admits an explicit Green-function ratio, and Gaussian-Mixture Model (GMM) terminal laws yield closed-form expressions. This enables stable sampling and differentiable protocol learning under exact terminal matching. We develop guidance-centric diagnostics -- path cost, centerline adherence, variance flow, and drift effort -- that make GH-PID an interpretable variational ansatz for empirical SOT. Three navigation scenarios illustrated in 2D: (i) Case A: hand-crafted protocols revealing how geometry and stiffness shape lag, curvature effects, and mode evolution; (ii) Case B: single-task protocol learning, where a PWC centerline is optimized to minimize integrated cost; (iii) Case C: multi-expert fusion, in which a commander reconciles competing expert/teacher trajectories and terminal beliefs through an exact product-of-experts law and learns a consensus protocol. Across all settings, GH-PID generates geometry-aware, trust-aware trajectories that satisfy the prescribed terminal distribution while systematically reducing integrated cost.

Michael Chertkov, Hamidreza Behjoo

Diffusion-based samplers -- Score Based Diffusions, Bridge Diffusions and Path Integral Diffusions -- match a target at terminal time, but the real leverage comes from choosing the schedule that governs the intermediate-time dynamics. We develop a path-wise schedule -- selection gramework for Harmonic PID with a time-varying stiffness, exploiting Piece-Wise-Constant(PWC) parametrizations and a simple hierarchical refinement. We introduce schedule-sensitive Quality-of-Sampling (QoS) diagnostics. Assuming a Gaussian-Mixture (GM) target, we retain closed-form Green functions' ration and numerically stable, Neural-Network free oracles for predicted-state maps and score. Experiments in 2D show that QoS driven PWC schedules consistently improve early-exit fidelity, tail accuracy, conditioning of the dynamics, and speciation (label-selection) timing at fixed integration budgets.

Christopher Koh, Laurent Pagnier, Michael Chertkov

Turbulent diffusion causes particles placed in proximity to separate. We investigate the required swimming efforts to maintain an active particle close to its passively advected counterpart. We explore optimally balancing these efforts by developing a novel physics-informed reinforcement learning strategy and comparing it with prescribed control and physics-agnostic reinforcement learning strategies. Our scheme, coined the actor-physicist, is an adaptation of the actor-critic algorithm in which the neural network parameterized critic is replaced with an analytically derived physical heuristic function, the physicist. We validate the proposed physics-informed reinforcement learning approach through extensive numerical experiments in both synthetic BK and more realistic Arnold-Beltrami-Childress flow environments, demonstrating its superiority in controlling particle dynamics when compared to standard reinforcement learning methods.

Laurent Pagnier, Michael Chertkov, Julian Fritzsch et al.

This manuscript reports the first step towards building a robust and efficient model reduction methodology to capture transient dynamics in a transmission level electric power system. Such dynamics is normally modeled on seconds-to-tens-of-seconds time scales by the so-called swing equations, which are ordinary differential equations defined on a spatially discrete model of the power grid. Following Seymlyen (1974) and Thorpe, Seyler, and Phadke (1999), we suggest to map the swing equations onto a linear, inhomogeneous Partial Differential Equation (PDE) of parabolic type in two space and one time dimensions with time-independent coefficients and properly defined boundary conditions. We illustrate our method on the synchronous transmission grid of continental Europe. We show that, when properly coarse-grained, i.e., with the PDE coefficients and source terms extracted from a spatial convolution procedure of the respective discrete coefficients in the swing equations, the resulting PDE reproduces faithfully and efficiently the original swing dynamics. We finally discuss future extensions of this work, where the presented PDE-based modeling will initialize a physics-informed machine learning approach for real-time modeling, $n-1$ feasibility assessment and transient stability analysis of power systems.

Michael Woodward, Yifeng Tian, Criston Hyett et al.

Building efficient, accurate and generalizable reduced order models of developed turbulence remains a major challenge. This manuscript approaches this problem by developing a hierarchy of parameterized reduced Lagrangian models for turbulent flows, and investigates the effects of enforcing physical structure through Smoothed Particle Hydrodynamics (SPH) versus relying on neural networks (NN)s as universal function approximators. Starting from Neural Network (NN) parameterizations of a Lagrangian acceleration operator, this hierarchy of models gradually incorporates a weakly compressible and parameterized SPH framework, which enforces physical symmetries, such as Galilean, rotational and translational invariances. Within this hierarchy, two new parameterized smoothing kernels are developed in order to increase the flexibility of the learn-able SPH simulators. For each model we experiment with different loss functions which are minimized using gradient based optimization, where efficient computations of gradients are obtained by using Automatic Differentiation (AD) and Sensitivity Analysis (SA). Each model within the hierarchy is trained on two data sets associated with weekly compressible Homogeneous Isotropic Turbulence (HIT): (1) a validation set using weakly compressible SPH; and (2) a high fidelity set from Direct Numerical Simulations (DNS). Numerical evidence shows that encoding more SPH structure improves generalizability to different turbulent Mach numbers and time shifts, and that including the novel parameterized smoothing kernels improves the accuracy of SPH at the resolved scales.

Andrei Afonin, Michael Chertkov

We consider a power transmission system monitored with Phasor Measurement Units (PMUs) placed at significant, but not all, nodes of the system. Assuming that a sufficient number of distinct single-line faults, specifically pre-fault state and (not cleared) post-fault state, are recorded by the PMUs and are available for training, we, first, design a comprehensive sequence of Neural Networks (NNs) locating the faulty line. Performance of different NNs in the sequence, including Linear Regression, Feed-Forward NN, AlexNet, Graphical Convolutional NN, Neural Linear ODE and Neural Graph-based ODE, ordered according to the type and amount of the power flow physics involved, are compared for different levels of observability. Second, we build a sequence of advanced Power-System-Dynamics-Informed and Neural-ODE based Machine Learning schemes trained, given pre-fault state, to predict the post-fault state and also, in parallel, to estimate system parameters. Finally, third, and continuing to work with the first (fault localization) setting we design a (NN-based) algorithm which discovers optimal PMU placement.

Laurent Pagnier, Michael Chertkov

Modern state and parameter estimations in power systems consist of two stages: the outer problem of minimizing the mismatch between network observation and prediction over the network parameters, and the inner problem of predicting the system state for given values of the parameters. The standard solution of the combined problem is iterative: (a) set the parameters, e.g. to priors on the power line characteristics, (b) map input observation to prediction of the output, (c) compute the mismatch between predicted and observed output, (d) make a gradient descent step in the space of parameters to minimize the mismatch, and loop back to (a). We show how modern Machine Learning (ML), and specifically training guided by automatic differentiation, allows to resolve the iterative loop more efficiently. Moreover, we extend the scheme to the case of incomplete observations, where Phasor Measurement Units (reporting real and reactive powers, voltage and phase) are available only at the generators (PV buses), while loads (PQ buses) report (via SCADA controls) only active and reactive powers. Considering it from the implementation perspective, our methodology of resolving the parameter and state estimation problem can be viewed as embedding of the Power Flow (PF) solver into the training loop of the Machine Learning framework (PyTorch, in this study). We argue that this embedding can help to resolve high-level optimization problems in power system operations and planning.

Francesco Concetti, Michael Chertkov

We propose a new iterative optimization method for the {\bf Data-Fitting} (DF) problem in Machine Learning, e.g. Neural Network (NN) training. The approach relies on {\bf Graphical Model} (GM) representation of the DF problem, where variables are fitting parameters and factors are associated with the Input-Output (IO) data. The GM results in the {\bf Belief Propagation} Equations considered in the {\bf Large Deviation Limit} corresponding to the practically important case when the number of the IO samples is much larger than the number of the fitting parameters. We suggest the {\bf Message Passage Descent} algorithm which relies on the piece-wise-polynomial representation of the model DF function. In contrast with the popular gradient descent and related algorithms our MPD algorithm rely on analytic (not automatic) differentiation, while also (and most importantly) it descents through the rugged DF landscape by \emph{making non local updates of the parameters} at each iteration. The non-locality guarantees that the MPD is not trapped in the local-minima, therefore resulting in better performance than locally-updated algorithms of the gradient-descent type. We illustrate superior performance of the algorithm on a Feed-Forward NN with a single hidden layer and a piece-wise-linear activation function.

Laurent Pagnier, Michael Chertkov

Parameter Estimation (PE) and State Estimation (SE) are the most wide-spread tasks in the system engineering. They need to be done automatically, fast and frequently, as measurements arrive. Deep Learning (DL) holds the promise of tackling the challenge, however in so far, as PE and SE in power systems is concerned, (a) DL did not win trust of the system operators because of the lack of the physics of electricity based, interpretations and (b) DL remained illusive in the operational regimes were data is scarce. To address this, we present a hybrid scheme which embeds physics modeling of power systems into Graphical Neural Networks (GNN), therefore empowering system operators with a reliable and explainable real-time predictions which can then be used to control the critical infrastructure. To enable progress towards trustworthy DL for PE and SE, we build a physics-informed method, named Power-GNN, which reconstructs physical, thus interpretable, parameters within Effective Power Flow (EPF) models, such as admittances of effective power lines, and NN parameters, representing implicitly unobserved elements of the system. In our experiments, we test the Power-GNN on different realistic power networks, including these with thousands of loads and hundreds of generators. We show that the Power-GNN outperforms vanilla NN scheme unaware of the EPF physics.

Nikolay Stulov, Michael Chertkov

In the past decades, great progress has been made in the field of optical and particle-based measurement techniques for experimental analysis of fluid flows. Particle Image Velocimetry (PIV) technique is widely used to identify flow parameters from time-consecutive snapshots of particles injected into the fluid. The computation is performed as post-processing of the experimental data via proximity measure between particles in frames of reference. However, the post-processing step becomes problematic as the motility and density of the particles increases, since the data emerges in extreme rates and volumes. Moreover, existing algorithms for PIV either provide sparse estimations of the flow or require large computational time frame preventing from on-line use. The goal of this manuscript is therefore to develop an accurate on-line algorithm for estimation of the fine-grained velocity field from PIV data. As the data constitutes a pair of images, we employ computer vision methods to solve the problem. In this work, we introduce a convolutional neural network adapted to the problem, namely Volumetric Correspondence Network (VCN) which was recently proposed for the end-to-end optical flow estimation in computer vision. The network is thoroughly trained and tested on a dataset containing both synthetic and real flow data. Experimental results are analyzed and compared to that of conventional methods as well as other recently introduced methods based on neural networks. Our analysis indicates that the proposed approach provides improved efficiency also keeping accuracy on par with other state-of-the-art methods in the field. We also verify through a-posteriori tests that our newly constructed VCN schemes are reproducing well physically relevant statistics of velocity and velocity gradients.