Modal Series Method Research Articles

Effect of Modal Interaction on Transient Behavior in Double-Input SEPIC DC–DC Converters

In this paper, we exploit an idea of nonlinear modal series method to investigate the effects of modal interaction in the one-cycle controlled (OCC) double-input SEPIC DC–DC converter. Based on the proposed nonlinear averaged model, the analytical approximate solutions are obtained to characterize the dynamic response characteristic of the transient behaviors in the double-input SEPIC converters. The fundamental modal analysis is utilized to identify the dominant oscillation modes and discover the relationship between parameters, fundamental modes and state variables. Furthermore, the second-order interaction indices are proposed to uncover the underlying mechanism of nonlinear interaction behaviors. In particular, the correlation between parameters and modal interaction are derived to optimize the transient process of the double-input SEPIC converters. Finally, numerical simulations are performed to verify the theoretical analysis.

Transfer Function with Nonlinear Characteristics Definition Based on Multidimensional Laplace Transform and its Application to Forced Response Power Systems

In this research, the concept of nonlinear transfer function with nonlinear characteristics is introduced through the multidimensional Laplace transform and modal series (MS) method. The method of modal series is applied to the power systems dynamics analysis in order to consider nonlinear oscillations and modal interactions, which contribute to the response of the system's dynamic following disturbances. The method of MS allows the inclusion of input excitation functions obtained as Laplace domain kernels superposed to obtain a transfer function. Applying the Volterra series expansion through kernels decomposition, a transfer function with nonlinear characteristics is obtained which incorporates some of the main modal characteristics of the nonlinear system. Following the same schematic procedure, it is possible to determine second and higher order transfer functions. Once the transfer functions both linear and with nonlinear characteristics are determined, a time domain and frequency response analyses can be performed. The methodology is exemplified by denoting the numerical and analytical properties with the application to a synchronous machine-infinite busbar test power system and to a three synchronous machines–nine buses test power system. Bode and Nyquist analysis are utilized to demonstrate the transfer functions accuracy and frequency response.

Optimal control of four-scroll chaotic system using modal series method

Optimal control of four-scroll chaotic system using modal series method

A new switched off‐line NMPC approach for nonlinear systems with a switching performance index using an extended modal series method

SummaryThis paper presents a new switched nonlinear model predictive control (NMPC) approach for continuous‐time affine‐input nonlinear systems with a number of different cost functions where switching occurs between them in order to improve the performance. In this approach, the NMPC‐related nonlinear two‐point boundary value problem derived from Pontryagin's maximum principle is solved by the extended modal series method. The resulting suboptimal control law as to each of the cost functions is feasible and has an explicit form. In order to guarantee closed‐loop stability, certain assumptions are considered in the NMPC literature and in the switched systems literature, such as finding an invariant terminal region and a feasible solution for the NMPC and considering a certain average dwell time for switching signals. In this paper, we consider switching among different cost functions using the average dwell‐time approach. Since, in our proposed method, the NMPC problem solution obtained by the extended modal series method is feasible and since the invariance condition for the terminal region is satisfied, the common assumptions for the stability of the switched NMPC can be established. Furthermore, we show that this method guarantees the stability of the entire closed‐loop system in the presence of unknown persistent disturbances. The applicability and effectiveness of the proposed approach are illustrated by two numerical examples.

Optimal control of nonlinear dynamical systems based on a new parallel eigenvalue decomposition approach

SummaryThis manuscript aims to investigate a new approach based on the modal series method and eigenvalue decomposition technique to solve a class of nonlinear optimal control problems. For this purpose, a sequence of decoupled linear two‐point boundary value problems is solved iteratively instead of solving the coupled nonlinear two‐point boundary value problem derived from the maximum principle. The convergence analysis of the suggested technique is also investigated. In addition, the problem that needs to be solved at each iteration is composed of lower‐order decoupled subproblems; hence, they can be solved in parallel. Thus, the new scheme has a parallel computing property improving its computational effectiveness. Numerical simulations and comparative results show that the proposed approach is efficient and provides satisfactory results.

A Closed Normal Form Solution Under Near-Resonant Modal Interaction in Power Systems

Normal form (NF) is an effective tool to quantitatively analyze nonlinear modal interaction, which is believed to contribute to the complex nonlinear dynamics in power systems. However, in conventional NF analysis, the solution under resonance condition cannot be expressed in a closed form. Hence the NF analysis of power system higher order modal interaction was limited to nonresonant cases, or directly neglecting the resonant terms. In this paper, the NF solutions under modal resonance are derived, by means of the polynomial vector space decomposition. The obtained solution is in a simple closed form, including both the nonresonant part and the resonant part. Numerical simulations are performed to verify the effectiveness of the proposed approach. The results show that, under modal resonant conditions, neglecting the resonant terms may cause significant errors to the obtained numerical solutions, whereas including modal resonant terms in NF may increase the accuracy of the analysis. Finally, the approach proposed in this paper is compared with the modal series method, which, unlike NF method, is not limited by resonant condition. The results show that the proposed method in this paper has better performance in both accuracy and computing time.

Nonlinear Modal Analysis of Transient Behavior in Cascade DC–DC Boost Converters

This paper deals with the transient characteristics of the cascade DC–DC Boost converter under a large disturbance by using nonlinear modal series method. Based on the power balance principle, a nonlinear averaged model is derived to describe the nonlinear dynamics of the cascade converter. And then, the modal series method is described in detail and the second order approximate solutions are derived by this method. Furthermore, the simulation results and the theoretical analysis demonstrate that there are plenty of modal interactions, particularly the stronger second order modal interactions, which can affect the transient behavior significantly in the disturbed transient process. The modified approximate solutions considering the dominant nonlinear interaction modes of some state variables are subsequently obtained. In addition, by selecting appropriate system parameters, we can improve the transient behavior of the system under disturbance so that the amplitude and duration of the oscillation can be effectively reduced to satisfy the requirements of the system tolerance during the transient process. Moreover, the dominant oscillation modes of each state variable are also studied, which will help us improve the understanding of the salient transient behaviors in DC–DC cascade converters with disturbance. Finally, PSpice circuit experiments are performed to verify the above theoretical and numerical results.

Non‐linear modal analysis of transient interaction behaviours in SEPIC DC–DC converters

This study investigates the transient interaction behaviours in SEPIC DC-DC converters by using non-linear modal series method. Based on the derived non-linear model, the analytical approximate solutions are obtained to characterise the transient behaviours of the converters. Then, fundamental modal analysis is performed to identify the correlation among system parameters, fundamental modes and state variables so that system key parameters are subsequently acquired. In addition, the second-order interaction indices are evaluated in order that the underlying mechanism of non-linear interaction behaviours is uncovered. Finally, theoretical analysis is verified by circuit experiments. These results are beneficial to the improvement of the transient behaviours and the system reliability.

An efficient parallel processing optimal control scheme for a class of nonlinear composite systems

This article presents an efficient parallel processing approach for solving the optimal control problem of nonlinear composite systems. In this approach, the original high-order coupled nonlinear two-point boundary value problem (TPBVP) derived from the Pontryagin's maximum principle is first transformed into a sequence of lower-order decoupled linear time-invariant TPBVPs. Then, an optimal control law which consists of both feedback and forward terms is achieved by using the modal series method for the derived sequence. The feedback term specified by local states of each subsystem is determined by solving a matrix Riccati differential equation. The forward term for each subsystem derived from its local information is an infinite sum of adjoint vectors. The convergence analysis and parallel processing capability of the proposed approach are also provided. To achieve an accurate feedforward-feedback suboptimal control, we apply a fast iterative algorithm with low computational effort. Finally, some comparative results are included to illustrate the effectiveness of the proposed approach.

Systematic higher order nonlinear analysis of power systems operating under perturbation conditions

In this paper, the modal series method is revisited and an extension on the method to include higher order terms is proposed. This proposal is based on the introduction of the multidimensional Laplace transform and association of variables theorems to deduce the analytical closed-form solution when it is applied to the analysis of a nonlinear power system model. The method is systematic and can incorporate higher order terms to the modal analysis to determine nonlinear modal interaction. When the power system is operating under stressed conditions, such as an increase in load demand, it results very important to consider the oscillations due to its nonlinear nature. Thus, the method is carefully exemplified with the application to the synchronous machine-infinite busbar power system operating under stress conditions. The oscillations produced during changes in its operation are analyzed as well as the nonlinear interaction through nonlinear indices and nonlinear participation factors. The time domain responses are compared between linear approximation, modal series, normal forms method and the direct numerical full solution of the nonlinear power system model.

Second-order oscillation mode study of hydropower system based on linear elastic model and modal series method

Summary Low-frequency oscillation occurs easily in a hydropower system with weak connected grid. Some oscillation behaviors cannot be explained by linear analysis method. According to unexplained oscillation frequency of hydropower stations, second-order oscillation mode of complex hydropower stations with weak grid is studied based on linear elastic model and modal series analysis method in this paper. Firstly, detailed hydropower station models with hydro-mechanical-electrical coupling are introduced. Especially, a new linear elastic model is used to model hydraulic conduit system. Secondly, hydro-mechanical-electrical coupling characteristics of hydropower stations are studied based on the whole model. Finally, the second-order oscillation mode and associated index of hydropower stations are calculated and studied by modal series method. Through numerical simulation of a single unit finite bus system, some useful results are achieved. Copyright © 2016 John Wiley & Sons, Ltd.

Closed‐form analytical characterisation of non‐linear oscillations in power systems incorporating unified power flow controller

The unified power flow controller (UPFC) is a component that belongs to the flexible AC transmission systems devices family which has been deeply studied under different conditions of operation within the power network. One of its main applications is on damping oscillations in power systems. In this research, the UPFC non-linear interaction with the power system is studied through the use of modal series method, which is a kind of analytical tool that has the ability to characterise the non-linear effects because of small perturbations. The analysis is based on non-linear contribution of UPFC variables when controller parameters are varied; time-domain analysis and non-linear interaction indices are calculated in order to quantify the non-linear effect of UPFC connected to a synchronous machine-infinite busbar test system.

Non‐linear analysis of DFIG based wind farm in stressed power systems

This study deals with non-linear interaction problems of DFIG (doubly fed induction generator) based wind farm in a stressed power system. The study is performed on an IEEE modified test system using the non-linear analysis based on the modal series (MS) method. As the installed capacity of renewable energy, especially the wind-energy increases, the wind farm connection to the grid should be studied from different viewpoints. The dynamic performance of this composite power system is an important issue and should be investigated carefully especially in stressed conditions and different operating points of system controllers. The non-linear interactions will complicate the dynamic behaviour of power systems. If such interaction exists between wind farm and other system controllers and is quantitatively significant, the stability of wind plant is jeopardised. To verify the analytical results of MS method, the necessary comparisons have been made with the results of the non-linear simulation and approximate linear methods.

An off-line NMPC strategy for continuous-time nonlinear systems using an extended modal series method

This paper presents a new off-line nonlinear model predictive control (NMPC) approach for continuous-time affine-input nonlinear systems. In this approach, the NMPC-related nonlinear two-point boundary value problem derived from the Pontryagin’s maximum principle is solved by the extended modal series method. The resulting suboptimal control law explicitly depends on the initial conditions and is updated by replacing the initial conditions with the new state measurements in future sampling instants. Therefore, there is no need to repeat the recursive online optimization process in each sampling instant. Since the applicability of NMPC is generally restricted by computational burden of the online optimization, we propose an NMPC scheme, which not only reduces the online computational burden significantly, but also can be applied to fast dynamic systems with short prediction horizons. An efficient algorithm is presented which approximates the order of the modal series such that feasibility of the optimization problem is guaranteed. Closed-loop stability of the proposed NMPC approach is shown using the off-line terminal region calculations suggested in quasi-infinite horizon NMPC scheme. The applicability and effectiveness of the proposed approach are illustrated by two numerical examples.

Transient analysis of low-voltage ride-through in three-phase grid-connected converter with LCL filter using the nonlinear modal series method

From a theoretical perspective, this paper develops a systematic approach to analyze the transient characteristics of the low-voltage ride-through (LVRT) in three-phase grid-connected converter with LCL filter by using the nonlinear modal series method. Firstly, the mathematical model and the corresponding second order modal series model of the system are described. And then, the simulation results and the theoretical analysis manifest that there are abundant modal interactions, particularly the stronger second order modal interaction, in the LVRT transient process. The revised expressions considering the nonlinear terms for the state variables are subsequently obtained. By selecting the appropriate parameters to weaken second order modal interaction, we can effectively reduce the amplitude and the duration of the oscillation to satisfy the requirements of the system tolerance during LVRT process. Additionally, the dominant oscillation modes of each state variable are also studied. Finally, the second order quasi-resonance boundaries are defined to facilitate the selection of practical parameters for maintaining normal transient behavior.

ANALYTIC-APPROXIMATE SOLUTION FOR A CLASS OF NONLINEAR OPTIMAL CONTROL PROBLEMS BY HOMOTOPY ANALYSIS METHOD

In this paper, we apply the homotopy analysis method (HAM) for nonlinear quadratic optimal control problems (OCP's). This method is employed to solve extreme conditions obtained from the Pontryagin's maximum principle (PMP). Applying the HAM, we in essence transfer a nonlinear two-point boundary value problem (TPBVP), into an infinite number of linear sub-problems, and then use the sum of the solutions of its first several sub-problems to approximate the exact solution. Note that such a kind of transformation does not need the existence of any small or large parameters in governing equation and initial/boundary conditions. The comparison of the HAM results with the Measure theory method, Modal series and collocation method solutions are made. Some illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method.

Nonlinear modal analysis of interaction between torsional modes and SVC controllers

This paper deals with the use of Modal Series (MS) method to study the nonlinear interaction between the torsional modes of a turbine-generator set and Static VAr Compensator (SVC) in the stressed power systems. The observations resulting from nonlinear simulations made incentive for suggesting the need for nonlinear analysis of torsional interaction phenomenon in stressed conditions. The obtained results from linear modal analysis were not consistent with those of nonlinear simulations and could not quantify the problem. While the results of nonlinear analysis MS method reveal that, the nonlinear interactions exist between shaft sections and SVC controllers. In the stressed conditions, these nonlinear interactions become more pronounced and it may be a warning for severe damage of shaft segments. By using the indices correspond to the MS method, the nonlinear interactions of torsional modes are evaluated for different conditions. To demonstrate the physical effects of nonlinear interaction of torsional modes with SVC controllers, the torsional torques imposed on shaft sections are provided using the nonlinear simulations. Finally, based on MS method a supplementary controller is designed for SVC to remedy the problem. The proposed approach is implemented on the IEEE 4-machine 11-bus test system.

An approximate-analytical solution for the Hamilton–Jacobi–Bellman equation via homotopy perturbation method

In this paper, we give an analytical-approximate solution for the Hamilton–Jacobi–Bellman (HJB) equation arising in optimal control problems using He’s polynomials based on homotopy perturbation method (HPM). Applying the HPM with He’s polynomials, solution procedure becomes easier, simpler and more straightforward. The comparison of the HPM results with the exact solution, Modal series and measure theoretical method are made. Some illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method.

Solving infinite horizon nonlinear optimal control problems using an extended modal series method

This paper presents a new approach for solving a class of infinite horizon nonlinear optimal control problems (OCPs). In this approach, a nonlinear two-point boundary value problem (TPBVP), derived from Pontryagin’s maximum principle, is transformed into a sequence of linear time-invariant TPBVPs. Solving the latter problems in a recursive manner provides the optimal control law and the optimal trajectory in the form of uniformly convergent series. Hence, to obtain the optimal solution, only the techniques for solving linear ordinary differential equations are employed. An efficient algorithm is also presented, which has low computational complexity and a fast convergence rate. Just a few iterations are required to find an accurate enough suboptimal trajectory-control pair for the nonlinear OCP. The results not only demonstrate the efficiency, simplicity, and high accuracy of the suggested approach, but also indicate its effectiveness in practical use.

Study of fault location effect on the inter-area oscillations in stressed power systems using modal series method

Heavily loaded, stressed power systems exhibit complex dynamic behavior such as auto and hetero parametric resonances and inter-area oscillations when subjected to a disturbance. These behaviors cannot be analyzed by conventional linearization-based methods. Nonlinear tools, such as normal form and modal series methods can be utilized for studying and understanding these complex behaviors. In this paper, using modal series technique, the effect of fault location and duration on the dynamic performance of stressed power systems is investigated. Three indices are defined and used to explain the effect of different fault scenarios on the transient behaviors of the system. Also, a qualitative analysis of plant mode and inter-area mode type behaviors are presented and typical differences are highlighted. The numerical simulations on the IEEE 50-Generator test system demonstrate validity of the study carried out by modal series analysis.