Region Of Attraction Estimation Research Articles

Estimating the region of attraction of wind integrated power systems based on improved expanding interior algorithm

AbstractThis paper proposes an improved expanding interior algorithm (EIA) to estimate the region of attraction (ROA) of power systems with wind power generation based on sum of squares (SOS) programming. An ordinary differential equation (ODE) model is derived for the doubly‐fed induction generator‐based wind turbine (DFIGWT), which is named as an enhanced synchronous‐generator‐mimicking (ESGM) model. The ESGM model bridges the gap between the requirement of an ODE model in ROA estimation and the conventional differential‐algebraic equation (DAE) model of the DFIGWT system. The ESGM model is able to accurately reflect the low frequency dynamics of the DFIGWT. Moreover, an improved EIA is designed to estimate the ROA based on SOS programming, which has higher efficiency than the existing ROA estimation algorithms based on SOS programming. It is able to adaptively search for the Lyapunov function and obtain an optimal estimation of the ROA in an iterative process. The accuracy and efficiency of this algorithm are verified in three test systems composed of DFIGWTs and synchronous generators (SGs). The morphological changes in the ROA of the test systems caused by the penetration of DFIGWT are examined.

LyZNet with Control: Physics-Informed Neural Network Control of Nonlinear Systems with Formal Guarantees

Optimal control for high-dimensional nonlinear systems remains a fundamental challenge. One bottleneck is that classical approaches for solving the Hamilton-Jacobi-Bellman (HJB) equation suffer from the curse of dimensionality. Recently, physics-informed neural networks have demonstrated potential in overcoming the curse of dimensionality in solving certain classes of PDEs, including special cases of HJB equations. However, one perceived limitation of neural networks is their lack of formal guarantees in the solutions they provide. To address this issue, we have built LyZNet, a Python tool that combines physics-informed learning with formal verification. The previous version of the tool demonstrated the capability for stability analysis and region of attraction estimates. In this paper, we present the tool for solving optimal control problems. We expand the functionalities of the tool to support the formulation and solving of optimal control problems for control-affine systems via physics-informed neural network policy iteration (PINN-PI). We outline the methodology that enables the learning and verification of PINN for optimal stabilization tasks. We demonstrate with a classical control example that the learned optimal controller indeed has significantly improved performance and verifiable regions of attraction.

Estimation and Expansion of Vehicle Stability Region With Sums of Squares Programming

The vehicle stability region can be used to evaluate vehicle stability performance and lay the foundation for vehicle safety improvement. It can be obtained via the region of attraction (RoA) based on sums of squares (SOS) programming. However, it is usually conservative, and the computational burden is large. This article focuses on the above issues and proposes a well-developed method for RoA estimation, with application to the estimation and expansion of the vehicle stability region. For RoA estimation, an improved SOS program is formulated, and its optimization objective combined with the customized algorithm significantly reduces conservatism. In addition, the feasibility prejudgment strategy and the dynamic search range are proposed in the algorithm. As a result, the computational burden is greatly reduced, which also indirectly helps reduce conservatism. Then, a more accurate vehicle stability region is obtained with less time consumption, and a map-based controller is proposed to improve vehicle stability. Better stability performance and larger stability regions are guaranteed under different scenarios due to the adaptive adjustment of controller gains. Finally, simulations and hardware-in-the-loop experiments with an embedded vehicle controller are performed to verify the effectiveness of our method.

Effects of PLL Frequency Limiters on Synchronization Stability of Grid Connected VSC and Strategy to Realize Global Stability

This paper investigates the synchronization stability of voltage source converter (VSC) connected to infinite bus system with frequency limiter of phase-locked loop (PLL). First, three different types of PLL frequency limiters are modelled based on piecewise-smooth dynamical system (PSDS). Then the effects of different frequency limiters and frequency limit values on the region of attraction (ROA) of the system are studied and the properties of the PSDSs are analyzed. It is found that PLL with anti-windup limiter with back calculation can achieve global stability with appropriate limit value, and the principle of setting limit value based on a specific ROA estimation is proposed. Based on ROA estimation with Lyapunov function, a method of setting frequency limit value which guarantees global stability is proposed. The validity of the proposed strategy is verified through MATLAB/Simulink simulation.

A Lyapunov approach based on gravitational search algorithm for transient stability assessment of AC/DC systems with wind power

AbstractWith large‐scale wind farms connected to AC/DC network, the transient stability assessment (TSA) of the power system becomes more and more difficult. Among them, estimating the region of attraction (ROA) of the equilibrium point is a traditional but still challenging problem. Based on the Lyapunov stability theory, this paper proposes a new approach to obtain the enlarged estimation of the ROA. The optimization and updating strategy of shape function in the sum of squares (SOS) optimization problem are studied to reduce the conservatism of estimation result. In the proposed method, the gravitational search algorithm (GSA) is employed to optimize the coefficients of initial shape function to improve estimation performance. Based on the time‐domain simulation (TDS) of expected faults, the fitness value for shape function optimization is calculated using the values of state variables at the fault clearing time and the system stability information. Furthermore, the optimal Lyapunov function is computed by introducing the update condition of shape function and adjusting the iteration strategy of the ROA estimation algorithm. Finally, the proposed method is applied to a two‐machine‐infinite‐bus system and a more complex nine‐bus AC/DC system with wind power. And the effectiveness of the proposed method is verified by comparing with the existing ROA estimation methods.

Combining Trajectory Data With Analytical Lyapunov Functions for Improved Region of Attraction Estimation

The increasing uptake of inverter based resources (IBRs) has resulted in many new challenges for power system operators around the world. The high level of complexity of IBR generators makes accurate classical model-based stability analysis a difficult task. This letter proposes a novel methodology for solving the problem of estimating the Region of Attraction (ROA) of a nonlinear system by combining classical model based methods with modern data driven methods. Our method yields certifiable inner approximations of the ROA, typical to that of model based methods, but also harnesses trajectory data to yield an improved accurate ROA estimation. The method is carried out by using analytical Lyapunov functions, such as energy functions, in combination with data that is used to fit a converse Lyapunov function. Our methodology is independent of the function fitting method used. In this letter, for implementation purposes, we use Bernstein polynomials to function fit. Several numerical examples of ROA estimation are provided, including the Single Machine Infinite Bus (SMIB) system, a three machine system and the Van-der-Pol system.

Kernel Machine to Estimate a Lyapunov Function and Region of Attraction (ROA) for Nonlinear Systems

The Lyapunov theory establishes the fundamental details to determine the stability and the Lyapunov function associated with an equilibrium point of a nonlinear system. Sum-of-square (SOS) optimization is a well-known technique to estimate the region of attraction (ROA) from the Lyapunov function. But if some of the system’s parameters and the controller are unknown, the Lyapunov function becomes unknown, and the ROA estimation becomes quite impossible. In this article, a methodology is proposed to use Machine Learning Kernel Method to parameterize Lyapunov function candidates, and a two steps optimization approach is adopted to determine the Lyapunov function and the system’s parameter. The objective function of the optimization is set so that (i) the Lyapunov function satisfies all the fundamental properties of the Lyapunov theory and (ii) it learns the unknown parameters of the system. The proposed methodology also finds a larger ROA than the existing methods. We provide numerical experiments on how the optimization technique works and how it can obtain high-quality solutions for challenging control problems.

Estimating the Region of Attraction on Controlled Complex Networks With Time-Varying Delay

The importance of estimating the region of attraction (ROA) on complex networks has drawn attention very recently. However, challenging problems arise when applying the existing theory to networks with delay. In this article, we estimate the ROA on controlled complex networks with time-varying delay. A delay-independent ROA estimation method is derived at first. This method can deal with general delay and is convenient to use due to its independence of the delay. Accordingly, it may cause conservativeness. To further reduce the conservativeness, delay-dependent ROA estimation is established for networks with small delay by developing a new technique of dealing with the delay. Numerical simulations are provided to verify the theoretical results.

Estimation of non-symmetric and unbounded region of attraction using shifted shape function and R-composition

Sum-of-squares programming is widely used for region of attraction (ROA) estimations of asymptotically stable equilibrium points of nonlinear polynomial systems. However, existing methods yield conservative results, especially for non-symmetric and unbounded regions. In this study, a cost-effective approach for ROA estimation is proposed based on the Lyapunov theory and shape functions. In contrast to existing methods, the proposed method iteratively places the center of a shifted shape function (SSF) close to the boundary of the acquired invariant subset. The set of obtained SSFs yields robust ROA subsets, and R-composition is employed to express these independent sets as a single but richer-shaped level set. Several benchmark examples show that the proposed method significantly improves ROA estimations, especially for non-symmetric or unbounded ROA without a significant computational burden.

Region of Attraction Estimation for DC Microgrids With Constant Power Loads Using Potential Theory

The stability issues of DC power grids are attracting researchers' attention, especially with the increasing adoption of power electronic devices and nonlinear loads. Large-signal stability analysis is required to detect and avoid large disturbance and destabilization, which can cause detrimental effects on DC power grids. However, the issue is still unsolved due to the complicated dynamics of large-scale power grids. This paper develops a novel method for estimation of the region of attraction (ROA) with less conservativeness using the Brayton-Moser mixed potential theory. This reliable and robust ROA estimation method provides useful insights into the stable operation of DC power grids. Moreover, this paper reveals the weak correlation between the state variables <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> of branch currents and system stability. It makes it possible to reduce computational cost and lessen the curse of dimensionality by separating these state variables. The case study shows that the proposed approach can obtain a much less conservative ROA compared to traditional methods such as Lyapunov's method.

A General Framework to Increase Safety of Learning Algorithms for Dynamical Systems Based on Region of Attraction Estimation

Although the state-of-the-art learning approaches exhibit impressive results for dynamical systems, only a few applications on real physical systems have been presented. One major impediment is that the intermediate policy during the training procedure may result in behaviors that are not only harmful to the system itself but also to the environment. In essence, imposing safety guarantees for learning algorithms is vital for autonomous systems acting in the real world. In this article, we propose a computationally effective and general safe learning framework, specifically for complex dynamical systems. With a proper definition of the safe region, a supervisory control strategy, which switches the actions applied on the system between the learning-based controller and a predefined corrective controller, is given. A simplified system facilitates the estimation of the safe region for the high-dimensional dynamical system. During the learning phase, the belief of the safe region is updated with the actual execution results of the corrective controller, which in turn enables the learning-based controller to have more freedom in choosing its actions. Two examples are given to demonstrate the performance of the proposed framework, one simple inverted pendulum to illustrate the online adaptation method, and one quadcopter control task to show the overall performance.

Synchronizing Stability Analysis and Region of Attraction Estimation of Grid-Feeding VSCs Using Sum-of-Squares Programming

Grid-synchronising Stability (GSS) is an emerging issue that exists in grid-feeding voltage-source converters (VSCs). Its occurrence is primarily related to the nonlinear dynamics of a type of vastly applied synchronization units - Phase-locked Loops (PLL). Dynamic characterisation and modelling for the GSS analysis can be achieved by using a simplified system model, which is a second-order and autonomous nonlinear equation; however, with the presence of an indefinite damping term. As revealed and demonstrated in this work, this indefinite damping effect can result in an inaccurate Region-of-Attraction (ROA) estimation of the traditional Equal Area Principle (EAP)-based method. To overcome this issue and achieve a valid ROA estimation, this paper adopts the sum-of-squares (SOS) programming technique, which is a numeric optimisation method with SOS relaxations. The development and implementation of the SOS program for ROA estimation are presented. Numerical case studies and time-domain verifications demonstrate that this method is valid for GSS analysis, and an almost precise estimation is achieved in the first quadrant. This evidence makes the SOS method a promising tool for GSS analysis because the GSS problem is most concerned with the stability within the first swing cycle, i.e., in the first quadrant.

A new control approach for automated drifting in consideration of the driving characteristics of an expert human driver

This paper presents a new control approach for automated drifting in consideration of how expert drivers conduct drifting. The expert drivers stabilize their vehicles at high sideslip angle using not prior knowledge of tire dynamics and equilibrium but current vehicle states. By mimicking how these expert drivers control their vehicles at high sideslip angle, the proposed controller is able to control the vehicle at high sideslip angle using only current vehicle states, i.e without prior knowledge of tire and drift equilibria. Specifically, the proposed controller can perform steady state drifting without the knowledge of the equilibrium of the vehicle system. The proposed controller consists of three consecutive parts. First, the supervisor determines the desired yaw rate and rear longitudinal slip ratio. Second, the upper-level controller calculates the desired front lateral force and rear longitudinal force to track the desired motions. Third, the lower-level controller converts the upper-level controller’s desired forces into steering wheel angle and gas pedal input, which are actual control inputs of vehicles. The proposed algorithm has been investigated via vehicle tests. A rear wheel driven mid-sized vehicle with limited slip differential and internal combustion engine was utilized for testbed. The test results indicate that the proposed algorithm can successfully conduct automated drift maneuvers. Furthermore, the stability of the closed-loop drift control system has been proved by using Lyapunov stability analysis and estimating Region of Attraction (RoA). The vehicle test results are superimposed on the estimate of RoA, and it has been shown that vehicle states within the estimate of RoA converge well to the desired motions. In addition, the comparison of the proposed algorithm with the previously developed drifting control algorithm was conducted. In contrast to the drifting control algorithm that was based on drift equilibrium, the proposed control algorithm is capable of conducting drift maneuvers on both high friction roads and low friction roads without prior knowledge of the tire model and road friction coefficient.

Region of attraction analysis for nonlinear vehicle lateral dynamics using sum-of-squares programming

ABSTRACTIn this study we will estimate the region of attraction (RoA) of the lateral dynamics of a nonlinear single-track vehicle model. The tyre forces are approximated using rational functions that are shown to capture the nonlinearities of tyre curves significantly better than polynomial functions. An existing sum-of-squares (SOS) programming algorithm for estimating regions of attraction is extended to accommodate the use of rational vector fields. This algorithm is then used to find an estimate of the RoA of the vehicle lateral dynamics. The influence of vehicle parameters and driving conditions on the stability region are studied. It is shown that SOS programming techniques can be used to approximate the stability region without resorting to numerical integration. The RoA estimate from the SOS algorithm is compared to the existing results in the literature. The proposed method is shown to obtain significantly better RoA estimates.

Region of attraction estimation using invariant sets and rational Lyapunov functions

This work addresses the problem of estimating the region of attraction (RA) of equilibrium points of nonlinear dynamical systems. The estimates we provide are given by positively invariant sets which are not necessarily defined by level sets of a Lyapunov function. Moreover, we present conditions for the existence of Lyapunov functions linked to the positively invariant set formulation we propose. Connections to fundamental results on estimates of the RA are presented and support the search of Lyapunov functions of a rational nature. We then restrict our attention to systems governed by polynomial vector fields and provide an algorithm that is guaranteed to enlarge the estimate of the RA at each iteration.

Stability of Nonlinear Systems with Unknown Time-varying Feedback Delay

This paper considers the problem of stabilizing a class of nonlinear systems with unknown bounded delayed feedback wherein the time-varying delay is 1) piecewise constant 2) continuous with a bounded rate. We also consider application of these results to the stabilization of rigid-body attitude dynamics. In the first case, the time-delay in feedback is modeled specifically as a switch among an arbitrarily large set of unknown constant values with a known strict upper bound. The feedback is a linear function of the delayed states. In the case of linear systems with switched delay feedback, a new sufficiency condition for average dwell time result is presented using a complete type Lyapunov-Krasovskii (L-K) functional approach. Further, the corresponding switched system with nonlinear perturbations is proven to be exponentially stable inside a well characterized region of attraction for an appropriately chosen average dwell time. In the second case, the concept of the complete type L-K functional is extended to a class of nonlinear time-delay systems with unknown time-varying time-delay. This extension ensures stability robustness to time-delay in the control design for all values of time-delay less than the known upper bound. Model-transformation is used in order to partition the nonlinear system into a nominal linear part that is exponentially stable with a bounded perturbation. We obtain sufficient conditions which ensure exponential stability inside a region of attraction estimate. A constructive method to evaluate the sufficient conditions is presented together with comparison with the corresponding constant and piecewise constant delay. Numerical simulations are performed to illustrate the theoretical results of this paper.

On the stability of receding horizon control with a general terminal cost

We study the stability and region of attraction properties of a family of receding horizon schemes for nonlinear systems. Using Dini's theorem on the uniform convergence of functions, we show that there is always a finite horizon for which the corresponding receding horizon scheme is stabilizing without the use of a terminal cost or terminal constraints. After showing that optimal infinite horizon trajectories possess a uniform convergence property, we show that exponential stability may also be obtained with a sufficient horizon when an upper bound on the infinite horizon cost is used as terminal cost. Combining these important cases together with a sandwiching argument, we are able to conclude that exponential stability is obtained for input-constrained receding horizon schemes with a general nonnegative terminal cost for sufficiently long horizons. Region of attraction estimates are also included in each of the results.