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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Lyapunov stability of smart inverters using linearized distflow approximation

Fast-acting smart inverters that utilize preset operating conditions to determine real and reactive power injection/consumption can create voltage instabilities (over-voltage, voltage oscillations and more) in an electrical distribution network if set-points are not properly configured. In this work, linear distribution power flow equations and droop-based Volt–Var and Volt–Watt control curves are used to analytically derive a stability criterion using Lyapunov analysis that includes the network operating condition. The methodology is generally applicable for control curves that can be represented as Lipschitz functions. The derived Lipschitz constants account for smart inverter hardware limitations for reactive power generation. A local policy is derived from the stability criterion that allows inverters to adapt their control curves by monitoring only local voltage, thus avoiding centralized control or information sharing with other inverters. The criterion is independent of the internal time-delays of smart inverters. Simulation results for inverters with and without the proposed stabilization technique demonstrate how smart inverters can mitigate voltage oscillations locally and mitigate real and reactive power flow disturbances at the substation under multiple scenarios. The study concludes with illustrations of how the control policy can dampen oscillations caused by solar intermittency and cyberattacks.

42 ENGINEERING↗
DOE OSTI2021

A Model-Reference Adaptive Direct-Power Control Scheme for Grid-Interactive Inverters

This paper develops an adaptive direct power control scheme for grid-interactive inverters while the control parameters are adaptively adjusted as the grid and filter parameters vary under different conditions. To demonstrate the efficacy and robustness of the adaptive operation, the performance of the adaptive controller is compared with a typical direct power controller by estimating the filter and grid impedances using a model reference parameter estimation technique with small signal injection. The simulation results verify that the developed methodology can track the reference power setpoints under output circuit parameter uncertainties while the typical design leads to instability. To improve the performance of the adaptive technique and eliminate the need for filter parameter estimation, a Lyapunov stability criterion-based model-reference adaptive direct power controller is designed to track the filter impedance. The developed method was tested using MATLAB/Simulink software to demonstrate the effectiveness of the proposed method. The results verify that adaptively tuning the control parameter can still maintain stable operation if the control gains are poorly selected while applying the classical method leads to abnormal operation or instability.

Gursoy, Mehmetcan↗
DOE OSTI2020

Multiresolution GPC-Structured Control of a Single-Loop Cold-Flow Chemical Looping Testbed

Chemical looping is a near-zero emission process for generating power from coal. It is based on a multi-phase gas-solid flow and has extremely challenging nonlinear, multi-scale dynamics with jumps, producing large dynamic model uncertainty, which renders traditional robust control techniques, such as linear parameter varying H ∞ design, largely inapplicable. This process complexity is addressed in the present work through the temporal and the spatiotemporal multiresolution modeling along with the corresponding model-based control laws. Namely, the nonlinear autoregressive with exogenous input model structure, nonlinear in the wavelet basis, but linear in parameters, is used to identify the dominant temporal chemical looping process dynamics. The control inputs and the wavelet model parameters are calculated by optimizing a quadratic cost function using a gradient descent method. The respective identification and tracking error convergence of the proposed self-tuning identification and control schemes, the latter using the unconstrained generalized predictive control structure, is separately ascertained through the Lyapunov stability theorem. The rate constraint on the control signal in the temporal control law is then imposed and the control topology is augmented by an additional control loop with self-tuning deadbeat controller which uses the spatiotemporal wavelet riser dynamics representation. The novelty of this work is three-fold: (1) developing the self-tuning controller design methodology that consists in embedding the real-time tunable temporal highly nonlinear, but linearly parametrizable, multiresolution system representations into the classical rate-constrained generalized predictive quadratic optimal control structure, (2) augmenting the temporal multiresolution loop by a more complex spatiotemporal multiresolution self-tuning deadbeat control loop, and (3) demonstrating the effectiveness of the proposed methodology in producing fast recursive real-time algorithms for controlling highly uncertain nonlinear multiscale processes. The latter is shown through the data from the implemented temporal and augmented spatiotemporal solutions of a difficult chemical looping cold flow tracking control problem.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

A Neural Lyapunov Approach to Transient Stability Assessment in Interconnected Microgrids

We propose a neural Lyapunov approach to assessing transient stability in power electronic-interfaced microgrid interconnections. The problem of transient stability assessment is cast as one of learning a neural network-structured Lyapunov function in the state space. Based on the function learned, a security region is estimated for monitoring the security of interconnected microgrids in real-time operation. The efficacy of the approach is tested and validated in a grid-connected microgrid and a three-microgrid interconnection. A comparison study suggests that the proposed method can achieve a less conservative characterization of the security region, as compared with a conventional approach.

Huang, Tong↗
DOE OSTI2023

Large-Signal Stability of Phase-Balanced Equilibria in Single-Phase Grid-Forming Inverter Systems

This article explores the setup where large numbers of single-phase grid-forming inverters with droop control across distribution networks self-organize into a stable and balanced system with 120° phase offsets across aggregates in the absence of balanced three-phase generating resources or external communication. A suite of circuit- and system-theoretic notions are leveraged to derive a dynamical model for phase-angle differences across aggregates of inverters connected in the three phases. Focusing on this model, large-signal stability is established and the region of attraction of the phase-balanced equilibria is determined with the aid of a Lyapunov function. Experimental validation for a bench-top prototype network is included to support the analytical developments. Altogether, the effort supports the vision of facilitating balanced operation of distribution networks with grid-forming inverters during service disruptions at the bulk transmission network.

Droop control↗
DOE OSTI2021

Model Reference Adaptive Control (MRAC) for Additive Manufacturing (AM)

Model Reference Adaptive Control (MRAC) is based on the fundamental concept that the process under investigation is to be controlled to follow or “track” a reference system (model) characterized by a state/input/output model employing an adaptive optimization algorithm to adjust the controller parameters in real-time. The generic structure of the MRAC is shown in Fig. 1 consisting of the following primary components: Reference model, Process (system) model, controller and the adaption algorithm. The basic structure of the controller is specified by a linear construct with the corresponding real-time adaption algorithms given by a gradient-type (so-called MIT rule) or based on stability theory (Lyapunov, hyperstability). This approach to adaptive control is termed “direct”, since the controller (parameters) are adjusted based on the component models/algorithm in contrast to the “indirect” approach that adjusts the process model parameters applying real-time system identification techniques.

42 ENGINEERING↗
DOE OSTI2021

Neural Lyapunov Control for Power System Transient Stability: A Deep Learning-Based Approach

We report that power system control and transient stability analysis play essential roles in secure system operation. Control of power systems typically involves highly nonlinear and complex dynamics. Most of the existing works address such problems with additional assumptions in system dynamics, leading to a requirement for a complete and general solution. This paper, therefore, proposes a novel control framework for various power system control and stability problems leveraging a learning-based approach. The proposed framework includes a two-module structure that iteratively and jointly learns the candidate Lyapunov function and control law via deep neural networks in a learning module. Meanwhile, it guides the learning procedure towards valid results satisfying Lyapunov conditions in a falsification module. The introduced termination criteria ensure provable system stability. This control framework is verified through several studies handling different types of power system control problems. The results show that the proposed framework is generalizable and can simplify the control design for complex power systems with the stability guarantee and enlarged region of attraction.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2021

Blockchain-Enabled Cyber-Secure Microgrid Control Using Consensus Algorithm

This paper proposes a distributed control method based on a consensus algorithm for distributed energy resources (DERs) using blockchain as a secure communication medium for cyber resilience. Each DER communicates with a local blockchain server that is maintained by the distributed ledger technology to allow for securely sharing local measurements among neighbouring assets to achieve the global control objectives, i.e., voltage and frequency regulation as well as accurate power sharing among the DERs, including collective grid-forming capability. To prove that the distributed control can retain system stability under a blockchain-induced variable communication delay, Lyapunov function-based stability analysis is carried out. This paper demonstrates the concept on an 11-bus test case developed in MATLAB Simulink, which has been modified from the IEEE 9-bus test case, to study dynamic operations of the five inverter-based DERs working with a blockchain-induced variable delay. The results validate the superior performance of the proposed control method, compared to heavily compromised operations of the other test cases without the security measure, affected by heavy communication delays and communication interruptions.

blockchains↗
DOE OSTI2023

Out-of-time-order correlators and Lyapunov exponents in sparse SYK

We use a combination of analytical and numerical methods to study out-of-time order correlators (OTOCs) in the sparse Sachdev-Ye-Kitaev (SYK) model. We find that at a given order of N, the standard result for the q-local, all-to-all SYK, obtained through the sum over ladder diagrams, is corrected by a series in the sparsity parameter, k. We present an algorithm to sum the diagrams at any given order of 1/(kq) n . We also study OTOCs numerically as a function of the sparsity parameter and determine the Lyapunov exponent. We find that numerical stability when extracting the Lyapunov exponent requires averaging over a massive number of realizations. This trade-off between the efficiency of the sparse model and consistent behavior at finite N becomes more significant for larger values of N.

2D gravity↗
DOE OSTI2020

Data-driven modeling and control of dynamical systems using Koopman and Perron-Frobenius operators

This dissertation studies the data-driven modeling and control problem of nonlinear systems by exploiting the linear operator theoretic framework involving Koopman and Perro-Frobenius operator. A systematic linear-operator based controller design procedure has been established, which can be used to solve a variety of nonlinear control problems, including feedback stabilization using control Lyapunov functions, optimal quadratic regulation using Koopman eigenfunctions and convex optimization formulation of optimal control problem using P-F and Koopman operator approximation. As the core of data-driven modeling, we first propose a new algorithm for the finite-dimensional approximation of the linear transfer Koopman and Perron-Frobenius operator from time-series data. We argue that the existing approach for the finite-dimensional approximation of these transfer operators such as Dynamic Mode Decomposition (DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two important properties of these operators, namely positivity and Markov property. The algorithm we propose preserves these two properties. We call the proposed algorithm as naturally structured DMD (NSDMD) since it retains the inherent properties of these operators. Naturally structured DMD algorithm leads to a better approximation of the steady-state dynamics of the system regarding computing Koopman and Perron- Frobenius operator eigenfunctions and eigenvalues. However, preserving positivity property is critical for capturing the real transient dynamics of the system. This positivity property of the transfer operators and it's finite-dimensional approximation play an important role for controller and estimator design of nonlinear systems. To solve the feedback stabilization problem for nonlinear control systems, we tried to take advantage of the Koopman operator framework. The Koopman operator approach provides a linear representation for a nonlinear dynamical system and a bilinear representation for a nonlinear control system. The problem of feedback stabilization of a nonlinear control system is then transformed to the stabilization of a bilinear control system. We propose a control Lyapunov function (CLF)-based approach for the design of stabilizing feedback controllers for the bilinear system. The search for finding a CLF for the bilinear control system is formulated as a convex optimization problem. This leads to a schematic procedure for designing CLF-based stabilizing feedback controllers for the bilinear system and hence the original nonlinear system. Another advantage of the proposed controller design approach outlined in this dissertation is that it does not require explicit knowledge of system dynamics. In particular, the bilinear representation of a nonlinear control system in the Koopman eigenfunction space can be obtained from time-series data. Next, we study the optimal quadratic regulation problem for nonlinear systems. The linear operator theoretic framework involving the Koopman operator is used to lift the dynamics of nonlinear control system to an infinite-dimensional bilinear system. The optimal quadratic regulation problem for nonlinear system is formulated in terms of the finite-dimensional approximation of the bilinear system. A convex optimization-based approach is proposed for solving the quadratic regulator problem for bilinear system. We applied a variety of examples and compared the simulation results between our framework and conventional LQR control using linearized model. For more general optimal control problems, we provide a density-function based convex formulation for the optimal control problem of the nonlinear system. The convex formulation relies on the duality result in the stability theory of a dynamical system involving density function and Perron-Frobenius operator. The optimal control problem is formulated as an infinite-dimensional convex optimization program. The finite-dimensional approximation of the optimization problem relies on the recent advances made in the data-driven computation of the Koopman operator, which is dual to the Perron-Frobenius operator. Simulation results are presented to demonstrate the application of the developed framework.

Huang, Bowen↗
DOE OSTI2021

Blockchain-Enabled Cyber-Secure Microgrid Control Using Consensus Algorithm: Preprint

This paper proposes a distributed control method based on a consensus algorithm for distributed energy resources (DERs) using blockchain as a secure communication medium for cyber resilience. Each DER communicates with a local blockchain server that is maintained by the distributed ledger technology to allow for securely sharing local measurements among neighboring assets to achieve the global control objectives, i.e., voltage and frequency regulation as well as accurate power sharing among the DERs, including collective grid-forming capability. To prove that the distributed control can retain system stability under a blockchain-induced variable communication delay, Lyapunov function-based stability analysis is carried out. This paper demonstrates the concept on an 11-bus test case developed in MATLAB Simulink, which has been modified from the IEEE 9-bus test case, to study dynamic operations of the five inverter-based DERs working with a blockchain-induced variable delay. The results validate the superior performance of the proposed control method, compared to heavily compromised operations of the other test cases without the security measure, affected by heavy communication delays and communication interruptions.

blockchain↗
DOE OSTI2025

Secure State Estimation with Asynchronous Measurements for Coordinated Cyber Attack Detection in Active Distribution Systems

Coordinated cyber attacks tamper with measurement data to disrupt the situational awareness of active distribution systems. Various sensors report measurements asynchronously at different rates, which introduces challenges during state estimation. In addition, this forces cyber intruders to exert greater effort to compromise multiple communication channels and launch coordinated attacks. Therefore, multi-channel and asynchronous measurements could be harnessed to develop more secure cyber defense strategies. In this paper, a prediction-correction-based multi-rate observer is designed to exploit the value of asynchronous measurements for the detection of coordinated false data injection (FDI) attacks. First, a time-function-dependent prediction-correction strategy is proposed to adjust the sampling interval for each sensor’s measurement. Then, an observer is designed based on the trade-off between estimation error and the optimal period of the most recent sampling instant, with the convergence of estimation error with the maximum permitted sampling interval. Moreover, the conditions for exponential stability are developed using the Lyapunov–Krasovskii functional technique. Next, a coordinated FDI attack detection strategy is developed based on the dual nonlinear minimization problem. The proposed attack detection and secure state estimation strategies are tested on the IEEE 13-node system. Simulation results show that these schemes are effective in enhancing attack detection based on asynchronous measurements or compromised data.

asynchronous measurements↗
DOE OSTI2021

A Convex Data-Driven Approach for Nonlinear Control Synthesis

We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Multistage economic MPC for systems with a cyclic steady state: A gas network case study

Multistage model predictive control (MPC) provides a robust control strategy for dynamic systems with uncertainties and a setpoint tracking objective. Moreover, extending MPC to minimize an economic cost instead of tracking a pre-calculated optimal setpoint improves controller performance. This paper presents a novel multistage economic nonlinear model predictive control (E-NMPC) framework for dynamic systems operating under uncertainty, with specific application to natural gas transmission networks. A key innovation lies in the integration of cyclic steady-state (CSS) constraints within the multistage MPC formulation, enabling the controller to manage periodic operating conditions commonly observed in energy systems. A Lyapunov-based descent condition is enforced to ensure robust stability of the controller. The multistage economic MPC framework is validated on two gas pipeline case studies, where it successfully minimizes net energy consumption, respects operational constraints under uncertain demand profiles, and guides the network to optimal cyclic operation. The Lyapunov function remains bounded in both case studies, validating the robust stability of multistage E-NMPC.

03 NATURAL GAS↗
DOE OSTI2022

A Lyapunov-Based Generalized Dc-Side Controller Design for PV-Connected Systems: Preprint

The objective of this paper is to realize a universal dc-side controller for photovoltaic (PV) systems where the control is agnostic to the downstream converter configuration. To achieve this, the downstream power converter and its controls are manipulated into an effective power control loop that is then cast into a generalized multi-loop design framework. On the dc side, a nonlinear small-signal model of the PV input is realized exclusively in terms of PV datasheet parameters (i.e., open-circuit voltage, short-circuit current, and maximum power point). Finally, a linear controller is used to modulate the dc-side PV system with the generic downstream power controller. A Lyapunov candidate is proposed to analyze the stability of the interconnected system and provide a streamlined approach for the controller design. The proposed design is validated on a 1-kVA experimental setup that interfaces a PV module to the grid.

dc-dc control↗
DOE OSTI2025

Network analysis of memristive device circuits: dynamics, stability and correlations

Abstract Networks with memristive devices are a potential basis for the next generation of computing devices. They are also an important model system for basic science, from modeling nanoscale conductivity to providing insight into the information-processing of neurons. The resistance in a memristive device depends on the history of the applied bias and thus displays a type of memory. The interplay of this memory with the dynamic properties of the network can give rise to new behavior, offering many fascinating theoretical challenges. But methods to analyze general memristive circuits are not well described in the literature. In this paper we develop a general circuit analysis for networks that combine memristive devices alongside resistors, capacitors and inductors and under various types of control. We derive equations of motion for the memory parameters of these circuits and describe the conditions for which a network should display properties characteristic of a resonator system. For the case of a purely memresistive network, we derive Lyapunov functions, which can be used to study the stability of the network dynamics. Surprisingly, analysis of the Lyapunov functions show that these circuits do not always have a stable equilibrium in the case of nonlinear resistance and window functions. The Lyapunov function allows us to study circuit invariances, wherein different circuits give rise to similar equations of motion, which manifest through a gauge freedom and node permutations. Finally, we identify the relation between the graph Laplacian and the operators governing the dynamics of memristor networks operators, and we use these tools to study the correlations between distant memristive devices through the effective resistance.

97 MATHEMATICS AND COMPUTING↗