Reduced Order State Space Model Research Articles

The Use of Model-Predicted Internal States from Locally Linear, Gain-Scheduled, Reduced-Order Physics-Based Models in Model-Predictive Control Strategies

The objective of the research is to develop reduced-order state-space models that can predict internal states that are not directly observable with existing sensors. In doing so, battery management systems (BMS) can make real-time control decisions based upon sensor inferences. For example, estimating local electrode Li and electrolyte Li-ion concentrations or local electrostatic potential differences between electrode and electrolyte phases offer valuable knowledge in predicting behaviors such as Li plating, dendrite growth, or electrode fracture. Such indirect predictive capabilities, for example, can enable fast-charging protocols that avoid damage mechanisms.This talk explores the mathematical feasibility of estimating local Li concentrations and electrostatic potentials using observed current-voltage sequences. The approach for determining internal battery states begins with an estimator that uses observed input-output data from a physics-based reduced-order state-space model. If the model satisfies the property of observability, a sufficiently long window of input-output data can predict unique internal-state trajectories that are compatible with the observable data. The research considers understanding the effects of sampling time and state of charge on internal-state observability.Efficient estimation algorithms must incorporate battery physics and chemistry yet be sufficiently simple as to run in real-time on a battery management system (BMS). The present research first develops a physics-based pseudo-two-dimensional (P2D) model, which describes battery thermodynamics, transport, and kinetics, results in a system of nonlinear, coupled, differential-algebraic equations that can be solved computationally, but too slow for real-time implementation. The P2D model is then reduced to a gain-scheduled, locally linear, state-space model that can be run in real-time. Both models are implemented within the MATLAB/Simulink framework that facilitates development and evaluation of control strategies.

Reduced-order state-space models for two-dimensional discrete systems via bivariate discrete orthogonal polynomials

This paper investigates the new model order reduction (MOR) methods via bivariate discrete orthogonal polynomials for two-dimensional (2-D) discrete systems. The 2-D discrete system is described by the Kurek model. First, we deduce algebraically the shift-transformation matrix of the classical discrete orthogonal polynomials of one variable. By means of the shift-transformation matrices, 2-D discrete systems are expanded in the spaces spanned by bivariate discrete orthogonal polynomials. The coefficient matrices are calculated from matrix equations. Then the reduced-order systems are produced by the orthogonal projection matrices defined by the coefficient matrices. Theoretical analysis shows that the reduced-order systems can match a certain number of coefficient vectors of the original outputs. Finally, one numerical example is simulated to demonstrate the feasibility and effectiveness of the proposed methods.

Research on the Optimal Control Strategy for the Maximum Torque per Ampere of Brushless Doubly Fed Machines

This paper presents an optimization strategy for a brushless doubly fed motor (BDFM) to achieve the maximum torque per ampere (MTPA). This method resolves the issue of high stator currents in slip frequency vector feedback linearization control (SFV-FLC) during both no-load and light-load conditions. Firstly, the paper establishes a reduced-order state-space (SS) model of the BDFM in arbitrary rotating reference coordinates. Secondly, the expression of BDFM is obtained after the control motor rotor field orientation. To ensure a minimal stator current at a specific torque, this paper constructs an auxiliary function based on Lagrange’s theorem, which forces the control motor stator current derivative to be zero, resulting in the MTPA criterion. Finally, the superiority of the MTPA optimization algorithm proposed in the paper is validated through simulation experiments.

Reduced-order state-space models of structures with imposed displacements and accelerations

Generating reduced state-space models of base-excited structures is of great help in control engineering but is not straightforward to implement in practice, and this problem is addressed in this work. Methods to impose non-homogeneous boundary displacements or accelerations are reviewed, and a novel relative acceleration method is proposed to treat the case of imposed displacements. Various construction approaches for state-space models having prescribed displacements or accelerations as input and including a static correction term are then developed. The theoretical developments are eventually illustrated with structures of increasing complexity, namely, a bar, a beam, and a multi-story building model.

Reduced-Order Identification of Aeroelastic Systems with Constrained and Imposed Poles

This work investigates the identification of reduced-order state-space models from aeroelastic simulations of the interaction of a built-in structural finite element model and a linear aerodynamic model using unsteady potential theory. The objective is to propose and compare different new frequency-based identification methods operating on frequency response associated to the inputs and outputs of interest. The first methods considered are the Loewner interpolation method and a subspace algorithm. Subsequently, the paper introduces the possibility to apply stability constraints and to impose a certain number of poles estimated beforehand by the method in order to make the identified models closer to the true aeroelastic physics. To achieve this goal, new dedicated techniques are developed and subsequently validated on data generated by random state-space models of various orders, and on aeroelastic data obtained from structural and aerodynamic aircraft models.

Frequency-Domain Lifting-Line Aerodynamic Modelling for Wing Aeroelasticity

A frequency-domain lifting-line solution algorithm for the prediction of the unsteady aerodynamics of wings is presented. The Biot–Savart law is applied to determine the normalwash generated by the wake vorticity distribution, whereas steady and unsteady airfoil theories (Glauert’s and Theodorsen’s, respectively) are used to evaluate the sectional aerodynamic loads, namely the lift and pitching moment. The wake vorticity released at the trailing edge derives from the bound circulation through the Kutta condition and is convected downstream with the velocity of the undisturbed flow. The local bound circulation is obtained by the application of the Kutta–Joukowski theorem, extended to unsteady flows. Assuming a bending and torsion wing, this paper provides the aerodynamic matrix of the transfer functions, relating the generalised aerodynamic loads to the Lagrangian coordinates of the elastic deformation. Its rational approximation yields a reduced-order state-space aerodynamic model suitable for an aeroelastic stability analysis and control purposes. The numerical investigation examines the influence of both the wake shed/trailed vorticity modelling and different approximations of the Kutta–Joukowski theorem for unsteady flows on the aerodynamic transfer functions given by the developed frequency-domain lifting-line solver. The accuracy of the solver is assessed by comparison with the predictions obtained by a three-dimensional boundary-element-method solver for potential flows. It is shown that, at least for the frequency range considered, regardless of the approximation of the Kutta–Joukowski theorem applied, the formulation based on the Theodorsen theory provides predictions that are in very good agreement with the results from the boundary element method for a slender wing. This agreement worsens as the wing aspect ratio decreases. A lower level of accuracy is obtained by the application of the sectional loads given by the Glauert theory. In this case, the predictions are more sensitive to the approximation used to express the Kutta–Joukowski theorem for unsteady flows.

A Multilayer Framework for Reliability Assessment of Shore-to-Ship Fast Charging System Design

In this article, a multilayer framework is proposed to assess the reliability of shore-to-ship charging (S2SC) systems, considering the potential for failures of the main elements, such as power converters and batteries. The proposed reliability model enables evaluating an S2SC system with redundancy, multistate system availability, and multivariable design scenarios based on a recursive algorithm. Instead of conventional Markov-chain multistate models, a reduced order state-space model is established from a set of specific subsystems with a predefined configuration complying with the design requirements. A modular approach is taken, and the probabilistic characteristics of the subsystems are established by developing a Universal Generating Function considering the available charging power and energy balance constraints. Then, the subsystems are integrated into the system configuration to assess a set of proposed application-specific system-level reliability indices. Thus, the modular approach enables expansion to the power system dimensions without extra complexity. Finally, a case study based on an operating 4-MW dc S2SC system is performed by the proposed framework, and design suggestions are given based on a Figure of Merit (FoM) defined for evaluation of reliability and energy efficiency. These design suggestions include resizing the charging system by installing onshore batteries, modularization, and introduction of redundancy in the subsystems.

Performance Comparison of Optimization Algorithm Tuned PID Controllers in Positive Output Re-Lift Luo Converter Operation for Electric Vehicle Applications

In this work, a Particle Swarm-Based optimization algorithm is proposed to optimize the PID controller used in the Positive Output Re-Lift Luo converter for the Electric Vehicle application. The novelty of the work lies in developing an improved PID controller to explore the reliability of the re-lift Luo converter. In addition, the optimization improves the effectiveness of the closed-loop system that possesses high voltage gain with esteemed power density and efficiency. With the Particle Swarm Optimization control algorithm, , , and values are tuned in such a way to obtain the appropriate steady-state condition with no peak overshoot or undershoot, much lesser settling time, no steady-state error, and improved dynamic performances. To analyze the system stability and to obtain better results, reduced-order state-space modeling of the Re-lift Luo converter is adopted, which offers high computational speed and accuracy. Initially, a PID controller is designed using the Zeigler Nicholas method, and the results are compared with the genetic algorithm and Particle Swarm Optimization algorithm. The time-domain analysis has been carried out to analyze and compare the converter’s performance. The simulation results were verified experimentally, and the results prove the robustness of the proposed converter with a PID controller optimized using the Particle Swarm Optimization algorithm.

Data-driven approximation and reduction from noisy data in matrix pencils frameworks

This work aims at tackling the problem of learning surrogate models from noisy time-domain data by means of matrix pencil-based techniques, namely the Hankel and Loewner frameworks. A data-driven approach to obtain reduced order state-space models from time-domain input-output measurements for linear time-invariant (LTI) systems is proposed. This is accomplished by combining the aforementioned model order reduction (MOR) techniques with the signal matrix model (SMM) approach. The proposed method is illustrated by a numerical example consisting of a high-order building model.

Parametric Reduced-Order Modeling of Aeroelastic Systems

In this paper, two approaches for modeling parameter-dependent unsteady aerodynamic loads for control design purposes are presented. The approaches are based on parametric Loewner frameworks, namely, transfer matrix interpolation and global basis. Combined with postprocessing techniques, these frameworks generate highly accurate, reduced-order state-space models of aerodynamic loads. The approaches are computationally efficient since they can model the entire flight envelope while requiring only a single input set of aerodynamic transfer matrix data. Additionally, constant numerical settings can be used across wide ranges of the parameter values without loss of accuracy. The proposed approaches are applied to state-space modeling of a two-dimensional aeroelastic system. The evaluated accuracy and dynamic behavior of the model show excellent agreement with the reference frequency domain solutions for both approaches.

Fast Simulation of Analog Circuit Blocks Under Nonstationary Operating Conditions

This article proposes a black-box behavioral modeling framework for analog circuit blocks (CBs) operating under small-signal conditions around nonstationary operating points. Such variations may be induced either by changes in the loading conditions or by event-driven updates of the operating point for system performance optimization, e.g., to reduce power consumption. An extension of existing data-driven parameterized reduced-order modeling techniques is proposed, which considers the time-varying bias components of the port signals as nonstationary parameters. These components are extracted at runtime by a low-pass filter and used to instantaneously update the matrices of the reduced-order state-space model realized as a SPICE netlist. Our main result is a formal proof of quadratic stability of such linear parameter varying (LPV) models, enabled by imposing a specific model structure and representing the transfer function in a basis of positive functions whose elements constitute a partition of unity. The proposed quadratic stability conditions are easily enforced through a finite set of small-size linear matrix inequalities (LMIs), used as constraints during model construction. Numerical results on various CBs, including voltage regulators, confirm that our approach not only ensures the model stability but also provides speedup in runtime up to two orders of magnitude with respect to full transistor-level circuits.

Electrochemical Model and Sigma Point Kalman Filter Based Online Oriented Battery Model

This paper presents a reduced-order electrochemical battery model designed for the online implementation of battery control systems. The model is based on porous-electrode and concentrated-solution theory frameworks and is able to predict voltage as well as the internal electrochemical variables of a battery. The reduction of the model leads to a physics-based one-dimensional discrete-time state-space reduced-order model (ROM), which is especially beneficial for online systems. Models optimized around different operational setpoints are combined to predict cell variables over a wide range of temperatures and state of charges (SOCs) using the output-blending method. A sigma-point Kalman filter is further used to manage inaccuracies generated by the reduction process and experimental-related issues such as measurement error (noise) in the current and voltage sensors. The state-estimation accuracies are measured against a full-order model (FOM) developed in COMSOL. The whole system is able to track the internal variables of the cell, as well as the cell voltage and SOC with very high accuracy, demonstrating its suitability for an online battery control system.

Reduced-order modeling of DFIG-based wind turbine connected into weak AC grid based on electromechanical time scale

Nowadays, the stability issue of the doubly fed induction generator (DFIG)-based wind turbines tied into weak AC power grid has attracted more and more attention. To analyze the stability of such cascade system, it is a common way to establish the small-signal model of the whole system. Although there have been various modeling methods, few of them can be easily utilized in practice due to the high order of the presented models. To address this issue, this paper simplifies the fast time scale subsystems and proposes a reduced-order state space model of the DFIG system based on electromechanical time scale, considering the rotor current controller (RCC), the phase-locked loop (PLL), the remained value of the grid voltage, and the impedance of the weak AC grid as well. Based on the simplified model, the stability of the wind turbine during weak AC grid voltage dips can be easily analyzed. The correctness of the integrated model and its convenience for stability analysis have been proved by eigenvalue matching method and computer simulations.

Optimal control analysis and Practical NMPC applied to refrigeration systems

This work is focused on optimal control of mechanical compression refrigeration systems. A reduced-order state-space model based on the moving boundary approach is proposed for the canonical cycle, which eases the controller design. The optimal cycle (that satisfying the cooling demand while maximizing efficiency) is defined by three variables, but only two inputs are available, therefore the controllability of the proposed model is studied. It is shown through optimization simulations how optimal cycles for a range of the cooling demand turn out not to be achieved by keeping the degree of superheating to a minimum. The Practical NMPC and a well-known feedback-plus-feedforward strategy from the literature are compared in simulation, both showing trouble in reaching the optimal cycle, which agrees with the controllability study.

Reduced-Order State Space Model for Dynamic Phasors in Active Distribution Networks

The rapid proliferation of distributed generation (DG) in active distribution networks (ADNs) increases the power system modeling complexity, which requires more efficient numerical methods to analyze dynamic interactions among DGs in ADN. This paper proposes a reduced-order dynamic phasor (DP) modeling method which is applied to large-scale ADNs with DGs for reducing the modeling scale and allowing larger time steps in ADN analyses. First, a unified state-space DP model, referred to as full-order DP model for linear circuits, is developed on the abc -axis to provide a comprehensive technique for interfacing DGs with other electric devices in large ADNs. The model order reduction (MOR) is applied to this full-order model to reduce the computation burden for large-scale ADNs. The paper develops a state-space DP model on positive and negative sequence synchronous rotating reference frames for three-phase DGs, and a stationary frame for single-phase DGs. The accuracy and the efficiency of the proposed models are verified using the modified IEEE 123-node test feeder and IEEE European low voltage test feeder with DGs.

Efficient Extraction of Electrochemical Impedance Spectra from Physical Models

Electrochemical impedance spectroscopy (EIS) is a highly valuable and widely practiced approach to assist the interpretation of electrochemical device performance. In the laboratory, the device (e.g., fuel cell or battery) is actuated with a small-amplitude harmonic signal (e.g., current) and the response (e.g., voltage or temperature) is measured. The complex impedance is represented in terms of amplitude and phase-shift differences between actuation and response as functions of actuation frequency. Most often, the measured impedance is interpreted with equivalent circuit models [1]. The present approach focuses on the use of physical models to derive and interpret the impedance spectra. Assuming that a transient physical model has been developed (e.g., protonic-ceramic fuel cell), the model could be exercised just as done in the laboratory with small-perturbation actuation [2]. This approach certainly works, but is computationally expensive, especially in the low-frequency ranges. Compared to the direct harmonic perturbation of the physical models, deriving state-space models from the physical models provides a much more efficient alternative to evaluate impedance spectra. State-space models take the form of two vector equations: dx/dt = Ax + Bu; y = Cx + Du. In these equations, x is the state vector, u is the actuation vector, and y is the vector of observables. A particular system is identified in terms of the four matrices A, B, C, and D. Once the state-space matrices are established, the complex impedance can be evaluated directly from the state-space matrices as Z(s;p) = C(sI – A)-1 B + D, where s=j, p is a vector of physical parameters, and I is the identity matrix. The complex impedance can be represented in various forms, such as in Nyquist plots. The present paper develops two alternative approaches to establish the space-state models. The first approach is based upon actuation with pseudo-random binary sequences (PRBS), seeking to identify locally linear state-space models [3-4]. The PRBS actuation is composed of small-amplitude step-change actuations of random durations, with the model-predicted responses being recorded. The relationships between the PRBS actuation and the predicted responses can be used to establish matrix parameters in a state-space model. As an alternative to PRBS identification of the state models, the physical model can be interpreted directly as a large-scale state-space model. In this case, the A, B, C, and D matrices can be evaluated by numerical differentiation of the physical model. Compared to the physical models that may have thousands of states (local composition, temperature, electrostatic potential, etc.), state-space models are typically reduced to only tens of states. In addition to evaluating the complex impedance, the reduced-order state-space models can play valuable roles in real-time model-predictive control algorithms.

Measurement-driven optimal control of utility-scale power systems: A New York State grid perspective

This paper focuses on designing and testing a supplementary controller for an ultra-large, utility-scale power system, namely the New York State (NYS) Power Grid from a completely measurement-based perspective. We present the control design using the Flexible AC Transmission System (FACTS) facility at the NYS grid. We use the utility-scale Eastern Interconnection (EI) model consisting of over 70,000 buses in the PSS/E platform for this research. The coherency structure of the NYS grid is analyzed by performing Principal Component Analysis (PCA) on frequency measurements obtained from multiple contingency simulations. Thereafter, we use the frequency measurements from PMU-enabled buses to identify a reduced-order state space model of the grid such that it matches the input-output characteristics along with identifying the inter-area modes. This model is then used to design the Linear Quadratic Gaussian (LQG) based optimal FACTS controller. Then the controller is implemented in the PSS/E model of the Eastern Interconnection (EI) as a PSS/E-FORTRAN based user defined module. The effectiveness of the control performance is shown using the non-linear simulations under different contingencies provided by the New York Independent System Operator (NYISO).

Full-scale identification of the wave forces exerted on a floating bridge using inverse methods and directional wave spectrum estimation

Abstract The dynamic behaviour of long-span bridges is governed by stochastic loads from typically ambient excitation sources. In real life, these loads cannot be measured directly at full scale. However, inverse methods can be utilised to identify these unknown forces using response measurements together with a numerical model of the relevant structure. This paper presents a case study of full-scale identification of the wave forces on the Bgsoysund bridge, a long-span pontoon bridge that has been monitored since 2013. First, a numerical model of the structure is formed, resulting in a reduced-order state-space model that takes into account the frequency-dependent hydrodynamic mass and damping from the fluid, based on fitting of rational transfer functions. Using acceleration data of the structure measured during several events of moderate and strong seas, the wave forces are identified using stochastic-deterministic methods for combined state and input estimation. In addition, a separate frequency-domain assessment of the wave forces is performed, in which the spectral density of the first-order wave forces is constructed from an estimated directional wave field model driven by wave elevation data. When compared in the frequency domain, the force estimates from the two approaches are of comparable magnitude. However, uncertainties in the assumptions and models behind the force estimates from the two approaches still play a significant role.

Full- and Reduced-Order State-Space Modeling of Wind Turbine Systems with Permanent Magnet Synchronous Generator

Full-order state-space models represent the starting point for the development of advanced control methods for wind turbine systems (WTSs). Regarding existing control-oriented WTS models, two research gaps must be noted: (i) There exists no full-order WTS model in form of one overall ordinary differential equation that considers all dynamical effects which significantly influence the electrical power output; (ii) all existing reduced-order WTS models are subject to rather arbitrary simplifications and are not validated against a full-order model. Therefore, in this paper, two full-order nonlinear state-space models (of 11th and 9th-order in the (a,b,c)- and (d,q)-reference frame, resp.) for variable-speed variable-pitch permanent magnet synchronous generator WTSs are derived. The full-order models cover all relevant dynamical effects with significant impact on the system’s power output, including the switching behavior of the power electronic devices. Based on the full-order models, by a step-by-step model reduction procedure, two reduced-order WTS models are deduced: A non-switching (averaging) 7th-order WTS model and a non-switching 3rd-order WTS model. Comparative simulation results reveal that all models capture the dominant system dynamics properly. The full-order models allow for a detailed analysis covering the high frequency oscillations in the instantaneous power output due to the switching in the power converters. The reduced-order models provide a time-averaged instantaneous power output (which still correctly reflects the energy produced by the WTS) and come with a drastically reduced complexity making those models appropriate for large-scale power grid controller design.

Investigation of a Voltage-Mode Controller for a dc-dc Multilevel Boost Converter

The multilevel dc-dc boost converter (MBC) is well known for its simple circuit structure and high voltage gain. Several controllers have been previously employed for their output voltage regulation. However, due to the non-minimum phase nature of the MBC, these controllers use both the inductor current and output voltage as feedback variables. This increases the complexity of the controller design and cost of implementation. In order to overcome these drawbacks, a voltage-mode controller for the regulation of the MBC is proposed in this brief. By using the generalized reduced-order state-space model of the converter and frequency-domain techniques, the stability of the voltage-controlled converter system is analyzed. In addition, some simulation and experimental results illustrating the ability of the proposed controller to regulate a $3{x}$ dc-dc multilevel boost converter in the presence of load and line voltage variations are provided.