Stability Enhancement Methods of Inverters Based on Lyapunov Function, Predictive Control, and Reinforcement Learning 9811971900, 9789811971907

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Xin Zhang · Jinsong He · Hao Ma · Zhixun Ma · Xiaohai Ge

Stability Enhancement Methods of Inverters Based on Lyapunov Function, Predictive Control, and Reinforcement Learning

Stability Enhancement Methods of Inverters Based on Lyapunov Function, Predictive Control, and Reinforcement Learning

Xin Zhang · Jinsong He · Hao Ma · Zhixun Ma · Xiaohai Ge

Stability Enhancement Methods of Inverters Based on Lyapunov Function, Predictive Control, and Reinforcement Learning

Xin Zhang College of Electrical Engineering Zhejiang University Hangzhou, Zhejiang, China Hao Ma College of Electrical Engineering Zhejiang University Hangzhou, Zhejiang, China Xiaohai Ge College of Electrical Engineering Zhejiang University Hangzhou, Zhejiang, China

Jinsong He Power Dispatching and Control Center of China Southern Power Grid Guangzhou, Guangdong, China School of Electrical and Electronic Engineering Nanyang Technological University Singapore, Singapore Zhixun Ma National Maglev Transportation Engineering R&D Center Tongji University Shanghai, China

ISBN 978-981-19-7190-7 ISBN 978-981-19-7191-4 (eBook) https://doi.org/10.1007/978-981-19-7191-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Power inverters become more and more important and popular in our life. For instance, in the smart grid, the power inverters transfer the solar/wind energy to the AC grid; in the more electric aircraft (MEA), the power inverters can connect the DC bus and AC bus in its hybrid AC/DC microgrid; in the shipboard power system (SPS), the power inverters can transfer the DC bus energy to the AC loads, etc. However, the power inverters still have stability problems, including the instability of the power inverter itself, as well as the instability problem due to the interaction between the inverters and the other power electronic units (i.e., weak grid, the electromagnetic interference (EMI) filter, upstream/downstream power converters). So far, most of the stability analyses and solutions are focused on small-signal stability technologies. Unfortunately, in practice, most of the real instability issues in the power inverters system are large-signal stability problem. Note that large-signal stability problems typically cause severer damage to electrical equipment compared to the small-signal counterpart. Motivated by the above discussion, this book aims to propose a family of advanced control strategies to improve the larger-signal stability of the power inverters systematically. This book first reviews the classifications of existing power inverters and their applications. Then, this book introduces a family of our proposed control strategies of power inverters for the applications of MEA, smart grid, SPS, etc. Also, the large-signal stability issues are analyzed within the scope of stand-alone inverters, grid-connected inverters, and these inverters interfaced with input EMI filters. From this book, different large-signal stability problems of the power inverters may find their corresponding control-based solutions. This book offers comprehensive self-contained knowledge on the large-signal stability study from Prof. Xin Zhang’s research group. This book introduces several modified control schemes for the power inverters to improve their large-signal stability, including the improved Lyapunov-based control method, the improved backstepping control method, the improved finite control set model predictive control method, etc. This book is suitable for all the levels of people related to power inverters. For the junior-level people who are the first time to know the power inverters, this book can help them to understand the concept, control logic, and large-signal stability v

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problem of the power inverters. For middle-level people having some background in the control and design of power inverters, this book can help them to improve their knowledge on the power inverters’ control strategy, which lets them find the tips to fix the large-signal stability problem of the power inverters from a control point of view. For the top-level people who are very familiar with power inverters, this book may offer them certain inspirations. In this case, this book can be treated as undergraduate textbooks, graduate textbooks, industrial design instruction books in the field of power electronics, power systems, control engineering, which can bring benefits to both academic and industrial sectors. Hangzhou, China

Xin Zhang

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Significance of DG in MGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Categories of MGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 AC MG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 DC MG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Hybrid MG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Cornerstone of MGs: Power Inverters . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Grid-Connected Inverters: L or LCL Filtered Inverters . . . . . 1.3.2 Standalone Inverters: LC-Filtered Inverter . . . . . . . . . . . . . . . 1.3.3 Grid-Connected and Standalone Inverters Cascaded with LC Input Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Necessity of Large-Signal Stability Analysis in Control of Inverters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Stability Problems of Inverters and the Existing Small-Signal Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 The Necessity of Large-Signal Stability Analysis . . . . . . . . . 1.4.3 Existing Large-Signal Stability Analysis of Inverters Via Lyapunov’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 The Motivation of This Book: Advanced Control Strategies for the Power Inverter to Improve Its Large-Signal Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Adaptive Backstepping Current Control of Single-Phase LCL-Grid-Connected Inverters to Improve Its Large-Signal Stability in the Presence of Parasitic Resistance Uncertainty . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Derivation of Proposed Control Scheme . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Step I: Derivation of Pseudo Reference x 2ref (t) and Adaptive Law 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.3.2 Step II: Derivation of Pseudo Reference x 3ref (t) and Adaptive Law 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Step III: Derivation of Control Law μ(t) and Adaptive Law 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 An Adaptive Dual-Loop Lyapunov-Based Control Scheme for a Single-Phase Stand-Alone Inverter to Improve Its Large-Signal Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Average Model of the Investigated System . . . . . . . . . . . . . . . 3.2.2 Load Voltage Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Current-Loop Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Model of the Load Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Proposed Adaptive Dual-Loop Lyapunov-Based Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The Proposed Lyapunov Function . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Derivation of the Adaptive Dual-Loop Control Law . . . . . . . 3.3.3 Implementation of Proposed Control Scheme . . . . . . . . . . . . 3.4 Stability Analysis and Robustness Verification . . . . . . . . . . . . . . . . . . 3.4.1 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Robustness Against Plant Parametric Variations . . . . . . . . . . 3.5 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Steady-State and Dynamic Performance Evaluation . . . . . . . 3.5.2 Overload and Recovery Scenario . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Lyapunov-Based Control of Three-Phase Stand-Alone Inverters to Improve Its Large-Signal Stability with Inherent Dual Control Loops and Load Disturbance Adaptivity . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Preliminary of the Proposed Adaptive Dual-Loop Lyapunov-Based Control: Mathematical Modelling . . . . . . . . . . . . . . 4.2.1 Average Model of the Investigated System . . . . . . . . . . . . . . . 4.2.2 Load Voltage References vodref , voqref , and Inductor Current References iLdref , iLqref . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Model of the Load Currents and Proposed Adaptive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Modified Inductor Current References iLdref , iLqref Incorporated with Adaptive Laws . . . . . . . . . . . . . . . . . . . . . . 4.3 Derivation of Proposed Adaptive Decoupled Dual-Loop Lyapunov-Based Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.3.1 Proposed Weighted All-in-One Lyapunov Function V . . . . . 4.3.2 Derivation of the Switching Functions and Adaptive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Implementation of Proposed Control Scheme and Its Resulted dq Decoupled Error Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Block Diagram of the Proposed Control Scheme . . . . . . . . . . 4.4.2 Decoupled Error Dynamics in d Frame and q Frame . . . . . . 4.4.3 Recommended Way to Set Load Voltage References . . . . . . 4.5 Stability Analysis and Controller Design Guidelines . . . . . . . . . . . . . 4.5.1 Closed-Loop System Stability Proof . . . . . . . . . . . . . . . . . . . . 4.5.2 Power Loss Analysis, Switching Frequency (f s ) Selection and Output LC Filter Design . . . . . . . . . . . . . . . . . . 4.5.3 Controller Gains Selection Via Poles Placement . . . . . . . . . . 4.6 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Performance of Proposed Approach . . . . . . . . . . . . . . . . . . . . 4.6.2 Comparisons Between the Proposed Approach and Existing Control Schemes . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 An Ellipse-Optimized Composite Backstepping Control Strategy for a Point-of-Load Inverter to Improve Its Large-Signal Stability Under Load Disturbance in the Shipboard Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Preliminary of the Ellipse-Optimized Composite Backstepping Controller: Mathematical Modelling . . . . . . . . . . . . . . 5.2.1 Dynamic Equations of the Investigated POL Inverter . . . . . . 5.2.2 Control Objectives: Load Voltage References x 1 *, x 3 * . . . . . 5.3 Recursive Derivation and Implementation of the Proposed Composite Backstepping Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Two-Step Backstepping Derivation in d Frame . . . . . . . . . . . 5.3.2 Two-Step Backstepping Derivation in q Frame . . . . . . . . . . . 5.3.3 Design of the Kalman Filter to Estimate and Feedforward the Load Currents for Load Disturbance Rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Implementation of the Proposed Composite Backstepping Controller with a Kalman Filter . . . . . . . . . . . . 5.4 Ellipse-Based Controller Gains Optimization, Feedback Gains Matrix Selection, and Robustness Analysis . . . . . . . . . . . . . . . 5.4.1 Proposed Intuitive Ellipse-Based Strategy to Optimize the Controller Parameters with Fully Consideration of ξ and ωn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.4.2 Quantitative Selection of the Feedback Gain Matrix G of the Kalman Filter Aided by Ellipse-Optimized Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Robustness Analysis of the Proposed Control Scheme Under Parametric Variations and Measurement Errors . . . . . 5.5 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Effectiveness of the Proposed Ellipse-Optimized Controller Gains Selection Strategy . . . . . . . . . . . . . . . . . . . . . 5.5.2 Robustness Tests Under Plant Parametric Variations . . . . . . . 5.5.3 Performance Evaluation Under Linear/Nonlinear Load Step, Reference Step, Overload and Recovery . . . . . . . 5.5.4 Comparisons Between Existing Lyapunov-Based Approaches and the Proposed Control Scheme . . . . . . . . . . . 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Stability Constraining Dichotomy Solution Based Model Predictive Control for the Three Phase Inverter Cascaded with Input EMI Filter in the MEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Instability Problem of the Researched AC Cascaded System in MEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Instability Problem Description . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 The Instability Reason of CPL with LC Input Filter . . . . . . . 6.3 Preliminary of the SCDS-MPC Method: Mathematical Modeling of the Researched AC Cascaded System in MEA . . . . . . . 6.3.1 Conventional Inverter Mathematical Model . . . . . . . . . . . . . . 6.3.2 Improved Mathematical Model with Consideration of the Inverter and Input EMI Filter for Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Proposed SCDS-MPC Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Conventional Model Predictive Control Scheme . . . . . . . . . . 6.4.2 Proposed Dichotomy Solution (DS) Based Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Proposed System Stability Constraining Cost Function Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Sensitivity Analysis of Model Parameters Variation . . . . . . . 6.5 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Composite-Bisection Predictive Control to Stabilize the Three Phase Inverter Cascaded with Input EMI Filter in the SPS . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Conventional FCS MPC and Problem Formulation . . . . . . . . . . . . . .

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7.4 Proposed Composite Bisection Predictive Control . . . . . . . . . . . . . . . 7.4.1 Structure of the Proposed CB-PC Scheme . . . . . . . . . . . . . . . 7.4.2 Improved Generic DC-Link Stabilization Strategy Based on Instantaneous Power Theory . . . . . . . . . . . . . . . . . . 7.4.3 Indirect Voltage Control Strategy to Achieve Better Transient Response Inspired by the Deadbeat Control . . . . . 7.4.4 Modified Bisection Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Effectiveness of the Improved Generic Stabilization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Transient Performance of the Indirect Voltage Control in Comparison with the Existing Direct Voltage Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Performance of the Proposed CB-PC Under Droop-Akin Strategy With/Without Delay Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Comparisons Between the Proposed CB-PC and Existing MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Reinforcement Learning Based Weighting Factors’ Real-Time Updating Scheme for the FCS Model Predictive Control to Improve the Large-Signal Stability of Inverters . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Weighting Factors Selection in FCS MPC Affects System Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Particular Case: WFstability Selection is a Trade-Off Between DC-Link Stabilization and Load Voltage Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Generalized Case: WFs Selection Affects System Stability (WFstability in Particular) . . . . . . . . . . . . . . . . . . . . . . . 8.3 WFs’ Real-Time Updaing Via the Reinforcement Learning-Based Approach to Improve System Stability . . . . . . . . . . 8.3.1 Structure of the Proposed Approach . . . . . . . . . . . . . . . . . . . . 8.3.2 RL Agent and Its Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 RL-Based Approach Using a DDPG Agent and Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Verification on the Particular Case: Improving Tracking Accuracy While Ensuring DC-Link Stabilization . . . . . . . . . . . . . . . . 8.4.1 Configuration of the Observation, Reward, and ANN . . . . . . 8.4.2 Parameter Settings and Training Results . . . . . . . . . . . . . . . . . 8.4.3 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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About the Authors

Xin Zhang received the Ph.D. degree in Automatic Control and Systems Engineering from the University of Sheffield, UK, in 2016 and the Ph.D. degree in Electronic and Electrical Engineering from Nanjing University of Aeronautics and Astronautics, China, in 2014. From February 2014 to December 2016, he was a Research Associate with the University of Sheffield. From January 2017 to September 2017, he was the Postdoctoral Research Fellow at the City University of Hong Kong. From September 2017 to August 2020, he was an Assistant Professor of power engineering with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. Currently, he is a Full Professor at Zhejiang University. He is generally interested in power electronics, power systems, and advanced control theory, together with their applications in various sectors. Dr. Zhang has received the Highly Prestigious Chinese National Award for Outstanding Students Aboard in 2016. He is the Associate Editor of 8 SCI journals and transactions, such as IEEE TIE/JESTPE/OJPE Access and IET Power Electronics, etc. Jinsong He received the B.Eng. degree in electrical engineering and its automation from Wuhan University, China, in 2018 and the Ph.D. degree in electrical and electronic engineering from Nanyang Technological University (NTU), Singapore, in 2022. He was an international undergraduate exchange student from 2017 to 2018 at NTU’s clean energy research lab. He is with Power Dispatching and Control Center of China Southern Power Grid, Guangzhou 510663, China, and was with the School of Electrical and Electronic Engineering, NTU, Singapore 639798. He has been an IEEE student member since 2018. He is interested in the stability analysis of power converters, microgrids, power dispatch, and control. He has published about 20 journal/conference papers in these areas. Prof. Hao Ma received the B.S., M.S. and Ph.D. degrees from Zhejiang University, Hangzhou, China, in 1991, 1994, and 1997, respectively, all in Electrical Engineering. Since 1997, he worked as a lecturer, associate professor, and professor at Zhejiang University. Currently, Prof. Ma is a professor in the College of Electrical Engineering and is served as Vice Dean of ZJU-UIUC Institute, Zhejiang University. His research xiii

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About the Authors

interests include Advanced Control in Power Electronics, Wireless Power Transfer, Fault Diagnosis of Power Electronic Circuits and Systems, and Application of Power Electronics. Prof. Ma is served as Director of Academic Committee of China Power Supply Society, Vice President and Secretary-general of Power Supply Society of Zhejiang Province, Associate Editor of Journal of Power Electronics (JPE), and Associate Editor of IEEE Journal of Emerging and Selected Topics in Power Electronics (JESTPE). Prof. Ma was served as Vice Dean of college of electrical engineering, Zhejiang University (2013–2017), AdCom member of IEEE Industrial Electronics Society (2014–2015), Technical Program Chair of IEEE PEAC 2018 and IEEE PEAC 2014, Technical Program Chair of IEEE ISIE 2012, and was served as Co-chair of Power Systems and Smart Grids, IEEE IECON 2018, Special Session Co-chair of IEEE IECON 2017, Co-chair of Power Electronics and Energy Conversion, IEEE IECON 2013, and Co-chair of Power Electronics and Renewable Energy Track, IEEE IECON 2010. Zhixun Ma received the B.S. and M.S. degrees in electrical engineering from China University of Mining and Technology, Xuzhou, China, and the Ph.D. degree in electrical engineering from Technical University of Munich, Germany, in 2006, 2009, and 2014, respectively. He is an Associate Professor at National Maglev Transportation Engineering R&D Center, Tongji University, Shanghai, China. He was also a Research Fellow at Nanyang Technological University, Singapore. He is a member of the IEEE society. His main research areas include predictive control and sensorless control of electrical drives, renewable energy systems, and FPGA-based digital control of power electronics and drive systems. Xiaohai Ge received B.S. and M.S. degrees in Electrical Engineering from Central South University, Changsha, China, in 2017 and 2020, respectively. From June 2021 to August 2022, he worked as an electrical engineer at the Hangzhou Global Scientific and Technological Innovation Center of Zhejiang University, Hangzhou, China. Currently, he is working toward the Ph.D. degree in the College of Electrical Engineering, Zhejiang University, Hangzhou, China. His main research areas include advanced control, protection, and stabilization technologies of power converters, renewable energy systems, and microgrids.

Chapter 1

Introduction

Keywords Power converters · Control schemes · Stability · Adaptive · Robustness · Microgrid operation · Efficiency · Switching loss · Loop analysis In the past few decades, the wild consumption of fossil fuels has caused severe environmental pollution, climate change, biodiversity collapse, and energy crisis. To cope with this hazard, worldwide countries/regions have announced deadlines by which they realize self-carbon–neutral, with many of them targeting the year 2050/2060 or even earlier [1]. For example, the European Union has committed to joining the global coalition for carbon neutrality by 2050. Then, Japan, the Republic of Korea, China, the United States, and more than 110 countries have participated in this global mission by the end of 2021. Moreover, governments obligated by the Paris Agreement become even more ambitious every five years and pledge the enhanced commitments, namely, Nationally Determined Contributions, showing their true momentum to achieve carbon peak and neutrality [2]. In progress with this global trend, conventional electric power systems are transforming towards the “green” and “low carbon” smart grid.

1.1 Significance of DG in MGs To enable such a transformation, the gradual replacement of fossilized fuels with renewable energy sources (RESs) is on the way. RESs generally include but are not limited to, solar photovoltaic (PV) power, onshore or offshore wind energy, fuel cells, geothermal energy, tide energy, salt gradient energy, and so forth. However, these RESs are usually located in a distributed way from a geological point of view. Fitting

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Zhang et al., Stability Enhancement Methods of Inverters Based on Lyapunov Function, Predictive Control, and Reinforcement Learning, https://doi.org/10.1007/978-981-19-7191-4_1

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1 Introduction

well with the inherent distributed feature of RESs, DG plays a critical role in achieving high penetration of RESs [3]. DG is a power-converter-based energy processing system that converts electricity from a specific voltage, current, or frequency to the other with ideally 100% efficiency and a tunable conversion ratio [4]. With further consideration of the intrinsic intermittency, low intensity, non-dispatchable characteristics of a couple of these RESs, multiple DG units are commonly integrated as a small-scale local power system, known as the MG [5]. In this way, such a comprehensive system can counteract the aforementioned case-by-case drawbacks of each DG unit and may even provide additional benefits beyond what can be achieved as individual DG units [6]. MG aims to realize a stable, reliable, and flexible utilization of RESs via seamless integration of RESs, ESs, local loads, communication devices, protection units, and the control center [7]. MG has the potential to provide a variety of ancillary services, including but not limited to peak load shaving, load shifting, and voltage support, etc. Moreover, the utilization of local RESs for local loads’ consumption also contributes to the reduction of energy losses in transmission as well as distribution process, leading to the higher efficiency of the power delivery system. When exposed to an emergency, multiple MG clusters can absorb power from their neighbors. Owning these characteristics, the evolution of MG technology is expected to achieve more economical and profitable operations with even higher reliability and flexibility compared to conventional distribution power systems [8]. In real-world applications, MG can be a university campus [9], a more electric aircraft [10], a shipboard power system [11], islanded community power system [12], etc. Recent years have witnessed a variety of R&D projects, demos of MGs all over the world. On Sep 18th, 2020, the green data center of the Zhangbei Cloud Computing Base MG, consisting of 120 MW wind power generation, 80 MW Photovoltaic power generation, and 20 MW energy storage, has been constructed by the China Energy Construction Gezhouba Power Company. The Zhangbei MG is one of the first batches of 28 new energy MG demonstration projects jointly announced by the National Development and Reform Commission and the National Energy Administration. Thanks to the MG technology, 26 venues of the Beijing 2022 Winter Olympics and Paralympics are going to be powered by 100% RESs [13]. In southeast Asia, the energy market authority of Singapore has transformed the jetty area of Pulau Ubin island into a solar-intensive MG testbench. It aims to provide more cost-effective electricity with less carbon footprint while ensuring system reliability, and scalability. Since Oct 2013, nearly 30 business participants have voluntarily joined the testbench, including restaurants, shops, telecommunications bases, government agencies, and residents [14].

1.2 Categories of MGs

3

1.2 Categories of MGs 1.2.1 AC MG Without the loss of generality, MGs can be classified into three categories from the topology (bus type) point of view: AC MG, DC MG, and hybrid MG. Most of the MGs are AC MGs because the conventional power delivery scheme is dominantly AC systems. A typical AC MG is depicted in Fig. 1.1. The AC MG generally assumes a three-phase configuration if the minimum power level reaches 100 kW. As presented in Fig. 1.1, AC MG is typically constituted by two or more parallel operated voltage source inverters (VSIs). These paralleled VSIs realize the load-power sharing and the voltage/frequency regulation of the point of common coupling (PCC) as well. For one thing, the paralleled operation provides redundancy and high reliability which is often required in some critical applications. For another, it is more economically feasible to distribute the high-power load to several VSIs and then use power devices with lower current ratings or even adapt the number of online VSIs to maximize the system efficiency. In case of an outage or low power quality of the main grid, AC MG can switch from grid-interfaced working mode to stand-alone working mode and vice versa seamlessly through on or off of the bypass switch [15]. In islanded operation mode, if the AC MG is not equipped with the firm synchronous generation, the energy storage system or solar panels together with their inverters play the role to do reactive power support and voltage support [6]. To sum up, AC MG strengthens grid resilience and helps mitigate grid disturbances, and it can act as a kind of grid resource to realize faster dynamic response and system recovery. As illustrated in Fig. 1.1, if the high bus voltage requires to step down for lowpower loads, transformers are needed [6]. Today, such transformers can be replaced by power converters, such as cycloconverters or matrix converters. However, some point-of loads, such as PCs, lighting, and telecommunication facilities, may require

AC MG PCC c

b

High-power loads

...

Lighting

PC Telecom...

...

Low-power loads

a Bypass switch

Main grid

VSIs

...

Transformer or Cycloconverters or Matrix converters

Solar panels Wind turbine

Fig. 1.1 Diagram of a typical inverter-intensive AC MG

...

Battery banks

Motor

4

1 Introduction

a DC power supply, i.e., the 48 V DC requirement in internet hosting sites approximately takes up 10% of the AC requirement. To complete such AC to DC power conversion, point-of-load (POL) rectifiers are recommended to operate in the AC MG, which are omitted in Fig. 1.1 for simplicity.

1.2.2 DC MG Nevertheless, onsite RESs generation and ESs are inherently DC. So is a variety of end-users, such as computers, building networks, telecommunication facilities, electric vehicles (EVs), and solid-state lighting, etc. Owning to these generation/consumption ends are all DC, many manufacturers, engineers, and researchers have advocated a paradigm shift from AC distribution to DC counterpart since the 1990s [16]. Figure 1.2 depicts a typical diagram of the DC MG. Dependent on application scenarios, the DC bus voltage is commonly configured as 24 V (lighting and the other low-power equipment), 48 V (telecommunication), and 380 V (high-power facilities). Their corresponding circuit breakers’ rated current drastically ranges from 15 to 2000 A [6]. DC MG realizes easier integration of RESs and ESs. Compared to AC MG, the number of back-and-forth DC/AC power conversion stages can be reduced using a DC bus, leading to less energy loss, higher efficiency, better system reliability, and less capital cost. In addition, there is no need to regulate the frequency/reactive power in DC MG, and phase unbalances and synchronization issues also get obviated. ...

... Battery Weapons (rail gun )

REs Wind turbine PV panel

AC/DC

EMI filter

DC/DC

DC/DC

AC/DC

DC/AC Fuel generator

Motor

Telecomm unication ...

DC/DC

EMI filter

DC/AC

...

Fig. 1.2 Diagram of a typical inverter-intensive DC MG [17]

DC/AC Output filter

Heating/ lighting Utilities

ESs dc bus

EMI filter

Output filter

PCs

Super C

Load

1.3 The Cornerstone of MGs: Power Inverters

5

However, the above-illustrated advantages of DC MG over AC counterpart have not taken into account the potentially higher investment in the power inverters. As depicted in Fig. 1.2, downstream load inverters having EMI filters are required to supply various AC loads in DC MG. A bidirectional power inverter is inevitable when interfacing the DC bus with AC-side utilities. Thus, it is still a debate whether DC MG is a better alternative to AC MG. For example, Schneider Electric took data centers as a case study and justified why AC power distribution still occupied a favorable position over DC counterpart for a given life-cycle cost, considering voltage levels, efficiency, cost, reliability, compatibility, and safety. Schneider Electric pointed out that the DC alternative having the best comprehensive performance is 380 V DC structure, while most the internet hosting sites end telecommunications facilities require 48 V DC. The 380 V architecture needs a new generation of information technology equipment and power facilities, while such setup is still not mainstream today [16].

1.2.3 Hybrid MG Recently, to combine the advantages of AC MG and DC MG while counteracting their limitations, the hybrid MG came into practice, as depicted in Fig. 1.3. Taking a hybrid MG at the application of more-electric aircraft (MEA), for example, it is generally fabricated by three parts. The first part is the AC sub-MG that integrates the generators, motors, and AC loads (pumps, etc.) of the MEA. The second part refers to the DC Sub-MG that interfaces the DC sources, e.g., PV panels, ESs (battery), and various DC loads, such as flight-control instruments, gyroscopes, and engine-starter generators. The last part is the bus conversion system that interlinks the DC sub-MG and AC sub-MG, which is also the feature of hybrid MG in comparison with AC MG and DC MG. Usually, an interlinking converter that allows for bidirectional power flow acts as the bus conversion system. An input filter can be cascaded ahead of the interlinking converter to deal with EMI issues.

1.3 The Cornerstone of MGs: Power Inverters According to the structure of the MG, it can be observed that power inverters function as the cornerstone of the MG. Meanwhile, power inverters are becoming more and more important and popular in our life owing to their multifunctional capability for the application of renewable energy systems. For instance, in the electrical vehicle (EV) application, the power inverters transfer the DC power to the AC power from the battery to the machine; in the PV system, the power inverters transfer the solar energy to the AC grid energy; in the wind farms, the power inverters play critical roles to integrate the kinetic energy of wind to the main grid, etc. To adapt to certain environmental conditions and cater to specific requirements, power inverters have lots

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1 Introduction

Fig. 1.3 Diagram of a typical inverter-intensive hybrid MG in the more electric aircraft [18]

of different topologies in different operation modes. Without loss of generality, these power inverters can be mainly classified into three types in terms of topologies, e.g., the grid-connected inverters, the stand-alone inverters, and their respective cascaded counterparts when cascaded with upstream power converters or LC input filters.

1.3.1 Grid-Connected Inverters: L or LCL Filtered Inverters For grid-connected applications, single-phase or three-phase power inverters coupled with L filter or LCL filter are the most popular topology, as presented in Figs. 1.4 and 1.5 respectively. For high-power applications, a simple first-order L filter leads to bulky and costly passive filters, which may slow down the dynamic response of the system if the designed converter needs to comply with the stringent grid codes specified by IEEE 519-1992 and IEC 61,000-3-12 [19]. As a more economical and volume-saving alternative, the third-order LCL filters can provide much better ripple and harmonic attenuation over the higher frequency range with even smaller passive elements. Therefore, they are more suited for high-power conversion systems. They have already been widely employed in wind farms of over hundreds of kilowatts. Nonetheless, the inclusion of LCL filters complicates the current control design compared with the L filter. The underlying reason for this is related to the theoretical fact that at certain frequencies, the resulting network may appear to have zero or very low impedance, inferring resonances, and hence closed-loop instability [20]. To address the resonance hazard of LCL-filtered grid-connected power inverters, passive damping or active damping are commonly deployed [19, 21].

1.3 The Cornerstone of MGs: Power Inverters

7

Fig. 1.4 L-filtered grid-connected inverter

Fig. 1.5 LCL-filtered grid-connected inverter

1.3.2 Standalone Inverters: LC-Filtered Inverter For DC–AC power conversion in islanded mode, single-phase or three-phase power inverters with LC output filters are the dominant topologies, as clarified in Fig. 1.6. Note that merely a single-phase LC-filtered stand-alone inverter is depicted, and the three-phase case is omitted for simplicity. The stand-alone inverter has a wide range of applications, including but not limited to DGs [22, 23] uninterruptible power supply (UPS) [24], 400-Hz ground power units for airplanes [25], electric vehicles (EVs) [26], intelligent buildings and its ventilation, etc. One significant research area of the VSI with LC output filter is the damping of LC resonance, which is very mature nowadays, including passive damping and active damping techniques [27]. Another important region is advanced control schemes of the system exposed to parametric variations, load uncertainties, and nonlinearities. In linear control design of the standalone VSI with LC filter from an impedance point of view, the key to suppressing the load disturbance lies in minimization of the inverters’ output impedance.

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1 Introduction

Fig. 1.6 LC-filtered stand-alone inverter

1.3.3 Grid-Connected and Standalone Inverters Cascaded with LC Input Filters Grid-connected inverters and standalone inverters are usually cascaded with LC input filters to deal with the electromagnetic interference (EMI) in the DC bus. However, it brings the risk of instability. As depicted in Fig. 1.2, the DC bus involves small-scale generations, energy storage, and downstream load converters. The EMI arises from the high-rate change of voltage [dv/dt] and current [di/dt] due to the fast switching of the semiconductor devices of the load converters [28]. For this reason, an electrolytic capacitor with high capacitance is commonly added ahead of a load inverter. In practice, the electrolytic capacitor can be upgraded to a dc-side LC EMI input filter using a film capacitor, so that higher power/energy density and prolonged lifespan are expected to be achieved [17]. Downstream load inverters cascaded with LC EMI filters operate as the interface between the DC bus and AC load. Nonetheless, a tightly regulated downstream load inverter is a typical type of constant power load (CPL) with a negative incremental input impedance, which tends to destabilize the system when cascaded with an LC input filter [29, 30] (Fig. 1.7). A similar cascaded structure can be found in the bus conversion system of the hybrid MGs, as presented in Fig. 1.3. The topological graph of the bus conversion

Ldc Cdc

S1

S2

S3

S4

S5

S6

r

L

Load

rdc

C n dc bus

Zout

Zin

Fig. 1.7 Stand-alone inverters cascaded with LC input filters

N

1.4 The Necessity of Large-Signal Stability Analysis in Control of Inverters

rdc

Ldc Cdc

n dc bus

Zout

S1

S2

S3

S4

S5

S6

r

9

L

Zin

Fig. 1.8 Grid-connected inverters cascaded with LC input filters

system is depicted in Fig. 1.8, which contains an EMI filter and an L-filtered gridconnected three-phase inverter. The EMI filter reduces the EMI from the inverter to the DC bus. The grid-connected inverter transfers the power from the DC bus to the AC bus if the AC microgrid needs to absorb the power from the solar panel or battery. Due to the active regulation capability (i.e., control output current) of the inverters, they can extract steady power and are thus commonly referred to as CPLs [31]. As mentioned before, the bus conversion system in the MEA also has a potential instability risk caused by the interaction between CPL and LC input filters.

1.4 The Necessity of Large-Signal Stability Analysis in Control of Inverters 1.4.1 Stability Problems of Inverters and the Existing Small-Signal Stability Analysis Currently, most of the control strategies focus on how to improve the performance of the power inverters, such as its output power quality [32], dynamic performance [33], function switching ability [34], power flow control [35], etc. Nevertheless, the power inverters still have stability problems [36, 37], including the instability problem of the power inverter itself, the instability problem between the power inverter and the other power electronic units in the system (i.e., weak grid, the EMI filter, loads, neighbours power converters). Most of the stability analyses [38] and solutions [39] are based on small-signal stability technologies. In a typical workflow of small-signal stability analysis, control of power inverters initially resorts to average switching techniques to eliminate the inherent discontinuity introduced by switching behaviours of power switches [4, 40]. The resulting averaged switching models are typically nonlinear. Then, these models are further

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1 Introduction

linearized around a specific fixed operation point using the classic small perturbation method or recent harmonic linearization method [41]. Afterward, the linearized model is investigated via the following tools, including but not limited to, bode plot, Nyquist criterion (impedance ratio, Middlebrook criterion), Roth-Hurwitz criterion, eigenvalues, condition number (sensitivity analysis) [42]. Among them, the impedance-based method has clear physical interpretations corresponding to realworld plants. Besides, the complex-torque-based method is another alternative to analyse the small-signal behaviour of a voltage source converter in a high-impedance AC grid. This method is suitable for stability analysis of the higher-order system using the state-space model. It points out that stability of the system can be ensured if the real and reactive parts of the system’s equivalent complex torques are both positive [43, 44].

1.4.2 The Necessity of Large-Signal Stability Analysis Unfortunately, the impedance-based methods are invalid if the grid cannot be modelled as a voltage source behind an inductor, and the complex-torque-based approach is inapplicable if the grid has a high penetration of power electronics [45]. Some stability criteria, e.g., the Nyquist criterion, is valid merely for a linear timeinvariant (LTI) minimum-phase system. However, the mathematical model of some power converters, e.g., boost converter, is non-minimum phase system. The inherent nonlinearity of the power inverters, such as bi-linearity, saturation, dead-zone, have been neglected in the linearized small-signal model. The control strategies using a small-signal model may confront performance degradation [46], irreversible instability due to sudden load change [47], catastrophic bifurcation due to grid voltage dip [48], and even blackout when exposed to large perturbation [49], resulting in serious damage to electrical equipment [50]. Because the elements of the smallsignal AC model are constants and functions of the large-signal DC operation point, these studies based on the small-signal model are merely valid around the vicinity of the operation point. However, various types of internal and external disturbances exist at the application of DG in MG. When exposed to large disturbance, such local stability may lose due to the sudden change of the operation point, which may even result in loss of critical load, voltage/frequency collapse. It is also hard to quantify how far away from the operation point that such linearization still has valid results [7]. As a promising alternative, these nonlinear phenomena and their resulting bifurcation/instability can be identified, explained, and even predicted via large-signal analysis. Compared to small-signal linear stability analysis, large-signal nonlinear stability analysis ensures a larger domain around the operating point within which the validity and effectiveness hold [51]. If a closed-loop system is large-signal stable, it is small-signal stable while the opposite does not definitely hold. Using large-signal stability solutions for power converters may expect greater immunity to internal and

1.4 The Necessity of Large-Signal Stability Analysis in Control of Inverters

11

external disturbances, improving the system stability, reliability with less maintenance fee as well. Therefore, investigation on the large-signal stability problems of power inverters is urgent and significant.

1.4.3 Existing Large-Signal Stability Analysis of Inverters Via Lyapunov’s Theory To study the large-signal stability of a specific system, Lyapunov’s theory is the current most popular approach [52]. Lyapunov’s theory consists of two sets: (1) the indirect method; (2) the direct method. The indirect method requires to use of the Jacobian matrix of the system and solving linear-matrix inequalities (LMIs); thus, it merely studies the local stability. In comparison, the direct method is based on the Lyapunov function, which ensures the global large-signal stability of the system. Moreover, the control design of the direct method does not need to cope with a large matrix, leading to a lower computation burden versus the direct method [53]. For this reason, Lyapunov-based analysis using the direct method is dominant for power converters application compared to the indirect method. Categories of existing methods to investigate the large-signal stability of power converters based on Lyapunov’s theory are sketched as follows. I. The direct method of Lyapunov’s theory. Note that the following methods are deemed as different approaches due to their respective ways to construct the Lyapunov function. A. (Adaptive) backstepping control [54–63]. This method presents a systematic way to build the Lyapunov function recursively. If the system has n orders, it typically formulates n Lyapunov functions via n recursive steps. When adaptive laws are incorporated to address system uncertainty, the system orders will be augmented. So is the number of required recursive Lyapunov functions. B. All-in-one Lyapunov function-based control, e.g., conventional linearquadratic Lyapunov-function-based approach, [64–67], modified dual-loop approach [68–71], and adaptive dual-loop approach [72–74]. An all-in-one Lyapunov function incorporates the weighted linear quadratic summation of tracking errors associated with capacitor voltage, inductor current, and augmented virtual state variables. C. The passivity-based control (PBC) [75], e.g., conventional PBC [76, 77], damping injection PBC [78], interconnection and damping assignment PBC [79], and adaptive PBC [80], etc. The key to this method is to transform the system mathematical model to the standard Euler–Lagrange system format. Then, a Lyapunov function can be analytically obtained using the system matrixes of the standard Euler–Lagrange system. D. …

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1 Introduction

II. The Lyapunov-based method is combined with existing control methods. A. B. C. D. E. F.

Lyapunov-based proportional-integral (PI) control [81]. Lyapunov-based proportional-resonant (PR) control [82]. Lyapunov-based H-infinity control [83]. Lyapunov-based model predictive control (MPC) [18, 84–96]. Lyapunov-based sliding-model control (SMC) [97–101]. …

Given whether the mathematical model is continuous or discrete, works on investigating the large-signal stability of power converters via Lyapunov’s direct method can be classified into two categories: I. Continuous model. A. Piecewise switching model [102, 103]. The piecewise switching model is dependent on the switching states of power converters. B. Averaging switching model [64–74]. C. Describing function model. D. Harmonic state-space model. E. … II. Discretized model [18, 52, 84–96]. The above continuous model after discretization considering the sampling interval (T s ).

1.4.4 The Motivation of This Book: Advanced Control Strategies for the Power Inverter to Improve Its Large-Signal Stability According to the above discussions, it can be summarized that the power inverters still have stability problems, including the instability problem of the power inverter itself, the instability problem between the power inverter and the other power electronic units in the system (i.e., weak grid, the EMI filter, loads, neighbours converters). Stability analyses and solutions based on small-signal stability technologies have inherent limitations in the identification, explanation of the physical origin of a variety of system bifurcation/instability phenomena, let alone provide remedy measures. This book aims to do a systematic study on the large-signal stability problem and solutions different power inverters (grid-connected inverters, stand-alone inverters, single-phase inverters, three-phase inverters, and cascaded power inverters) in practical applications. This book is intended for all researchers, engineers, and practitioners interested in power inverters. It is also suitable as auxiliary teaching material for graduate and undergraduate students in power electronics, energy, control, and related fields, as well as researchers and professional technicians in related fields. Due to the limited knowledge of the authors and the rush of time, the content of this book may still have certain errors and limitations. I sincerely seek you, experts

References

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and scholars, for your advice. We will spare no efforts to improve this book according to your feedback with the greatest sense of mission, responsibility, and gratitude.

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19. Y. Tang, P. C. Loh, P. Wang, F. H. Choo, and F. Gao, “Exploring inherent damping characteristic of LCL-filters for three-phase grid-connected voltage source inverters,” IEEE Trans. Power Electron., vol. 27, no. 3, pp. 1433–1443, 2012. 20. Y. Xiong and Y. Ye, “Physical Interpretations of Grid Voltage Full Feedforward for Grid-Tied Inverter,” IEEE Trans. Circuits Syst. II Express Briefs, vol. 66, no. 2, pp. 267–271, 2019. 21. J. Yin, S. Duan, and B. Liu, “Stability analysis of grid-connected inverter with LCL filter adopting a digital single-loop controller with inherent damping characteristic,” IEEE Trans. Ind. Informatics, vol. 9, no. 2, pp. 1104–1112, 2013. 22. T. D. Do, V. Q. Leu, Y. S. Choi, H. H. Choi, and J. W. Jung, “An adaptive voltage control strategy of three-phase inverter for stand-alone distributed generation systems,” IEEE Trans. Ind. Electron., vol. 60, no. 12, pp. 5660–5672, 2013. 23. J. He and X. Zhang, “Comparison of the back-stepping and PID control of the three-phase inverter with fully consideration of implementation cost and performance,” Chinese J. Electr. Eng., vol. 4, no. 2, pp. 82–89, 2019. 24. E. Kim, F. Mwasilu, H. H. Choi, and J. Jung, “An Observer-Based Optimal Voltage Control Scheme for Three-Phase UPS Systems,” IEEE Trans. Ind. Electron., vol. 62, no. 4, pp. 2073– 2081, 2015. 25. Z. Li, Y. Li, P. Wang, H. Zhu, C. Liu, and F. Gao, “Single-loop digital control of highpower 400-Hz ground power unit for airplanes,” IEEE Trans. Ind. Electron., vol. 57, no. 2, pp. 532–543, 2010. 26. M. I. Ghiasi, M. A. Golkar, and A. Hajizadeh, “Lyapunov Based-Distributed Fuzzy-Sliding Mode Control for Building Integrated-DC Microgrid with Plug-In Electric Vehicle,” IEEE Access, vol. 5, pp. 7746–7752, 2017. 27. Y. W. Li, “Control and resonance damping of voltage-source and current-source converters with LC filters,” IEEE Trans. Ind. Electron., vol. 56, no. 5, pp. 1511–1521, 2009. 28. A. Elrayyah, K. Mpk Namburi, Y. Sozer, and I. Husain, “An effective dithering method for electromagnetic interference (EMI) reduction in single-phase DC/AC inverters,” IEEE Trans. Power Electron., vol. 29, no. 6, pp. 2798–2806, 2014. 29. X. Wang, F. Blaabjerg, and W. Wu, “Modeling and analysis of harmonic stability in an AC power-electronics-based power system,” IEEE Trans. Power Electron., vol. 29, no. 12, pp. 6421–6432, 2014. 30. W. Du, J. Zhang, Y. Zhang, and Z. Qian, “Stability criterion for cascaded system with constant power load,” IEEE Trans. Power Electron., vol. 28, no. 4, pp. 1843–1851, 2012. 31. A. Emadi, A. Khaligh, C. H. Rivetta, and G. A. Williamson, “Constant power loads and negative impedance instability in automotive systems: definition, modeling, stability, and control of power electronic converters and motor drives,” IEEE Trans. Veh. Technol., vol. 55, no. 4, pp. 1112–1125, 2006. 32. Q.-C. Zhong and T. Hornik, Control of power inverters in renewable energy and smart grid integration, vol. 97. John Wiley & Sons, 2012. 33. E. Dos Santos and E. R. Da Silva, Advanced Power Electronics Converters: PWM Converters Processing AC Voltages. John Wiley & Sons, 2014. 34. R. Teodorescu, M. Liserre, and P. Rodriguez, Grid converters for photovoltaic and wind power systems, vol. 29. John Wiley & Sons, 2011. 35. R. D. Robinett III and D. G. Wilson, Nonlinear power flow control design: utilizing exergy, entropy, static and dynamic stability, and Lyapunov analysis. Springer Science & Business Media, 2011. 36. S. M. Sharkh, M. A. Abu-Sara, G. I. Orfanoudakis, and B. Hussain, Power electronic converters for microgrids. John Wiley & Sons, 2014. 37. H. Abu-Rub, M. Malinowski, and K. Al-Haddad, Power electronics for renewable energy systems, transportation and industrial applications. John Wiley & Sons, 2014. 38. J. Sun, “Impedance-based stability criterion for grid-connected inverters,” IEEE Trans. Power Electron., vol. 26, no. 11, pp. 3075–3078, 2011. 39. W. Cao, Y. Ma, and F. Wang, “Sequence-impedance-based harmonic stability analysis and controller parameter design of three-phase inverter-based multibus AC power systems,” IEEE Trans. Power Electron., vol. 32, no. 10, pp. 7674–7693, 2016.

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18

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Chapter 2

Adaptive Backstepping Current Control of Single-Phase LCL-Grid-Connected Inverters to Improve Its Large-Signal Stability in the Presence of Parasitic Resistance Uncertainty

Nomenclature V dc L 1 , C, L 2 r1, r2 r3 i2 (t), x 1 (t) vc (t), x 2 (t) i1 (t), x 3 (t) vpcc (t) V i (t), i=1, 2, 3. zi (t), i=1, 2, 3. r(t) x 2ref (t) x 3ref (t) μ(t) R, R+ , Z+ , C

DC-link voltage. Nominal LCL-filter-component values. The equivalent series resistance (ESR) of the filter inductors. The equivalent parallel resistance (ESR) of the filter capacitor. Grid-side inductor current (control target). Capacitor voltage. Inverter-side-inductor current. Point-of-common-coupling (PCC) voltage. Step-i Lyapunov function. Tracking error of x i (t), i=1, 2, 3. zi (t) ∈ R. Reference of x 1 (t). Pseudo tracking reference of vc (t). Pseudo tracking reference of i1 (t). Final control law (switching function). The set of reals, positive reals, positive integers, complex numbers, respectively

2.1 Introduction Grid-connected voltage source inverters are widely used in distributed generation systems. Grid-connected inverters act as the interface between the grid and a variety of renewable energy resources, such as solar energy, wind energy, energy storage systems, etc. Grid-connected inverters can be generally classified as the L-type and LCL-type categories according to the topology of the filter [1]. Compared to the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Zhang et al., Stability Enhancement Methods of Inverters Based on Lyapunov Function, Predictive Control, and Reinforcement Learning, https://doi.org/10.1007/978-981-19-7191-4_2

19

20

2 Adaptive Backstepping Current Control of Single-Phase …

L-type grid-connected inverter, the LCL-type one achieves the same harmonic attenuation capability with the much smaller size of passive components and less implementation cost as well. However, LCL-type grid-connected inverters have inherent resonance risk, which requires damping techniques, including passive damping and active damping. Passive damping techniques place additional resistors in series or parallel of LCL components. These methods are proven to be effective and robust at cost of further investment in additional hardware components. As a more economical alternative, active damping techniques realize these damping resistors virtually through adding specific state variables’ feedback. However, it complicates controller design [2]. In an attempt to realize active damping via state variables’ feedback, existing works are mature, which consist of inverter-side current feedback, grid-side current feedback, capacitor current/voltage feedback, and their combinations. Most of these works are devised based on the small-signal model of the system. For this reason, such control systems can but merely guarantee the local stability of the system around specific operation points. When exposed to external disturbance, the system may migrate to another operation point, in which case, these linear control schemes may not guarantee the stability of the system. For example, a bifurcation phenomenon emerges if the dual-loop PI-regulated L-type grid-connected inverters encounter grid voltage dip [3]. For this reason, it requires to investigate nonlinear large-signal control schemes for grid-connecter inverters application to ensure the global large-signal stability of the system. To this end, Lyapunov-function-based control schemes emerge. The conventional approach formulates a Lyapunov function (V ) as the sum of linear-quadratic tracking errors associated with the capacitor voltage and inductor current. Then, the final control law is designed to assure V˙ < 0. However, it inherently yields a single currentloop control scheme [2, 4–6]. For islanded inverter application, it leads to steadystate error and slack dynamic response [4]. For grid-connected single-phase/ threephase LCL-type grid-tied inverter application, it cannot damp the current resonance injected to the grid [2, 5]. To fix this problem, [2, 4, 7] extend the conventional singleloop control to a dual-loop counterpart via artificial importation of capacitor voltage feedback. However, this action inevitably requires afterward modification of the Lyapunov-based control law, which cannot unconditionally guarantee V˙ < 0. Largesignal stability of the system can be still guaranteed provided that the artificially imported unless the capacitor voltage feedback gain is larger than a specific lower bound. Different from the above Lyapunov-based scheme, this section explores an adaptive backstepping control scheme of a single-phase LCL-type grid-connected inverter with further consideration of parasitic resistance uncertainty. The control strategy is designed based on recursively augmented Lyapunov functions. In each step of the design process, a pseudo control reference is designed to stabilize the sub-system, and an adaptive law is selected to address the parasitic resistance uncertainty problem. This scheme rigorously guarantees the global large-signal stability of the closed-loop

2.2 Mathematical Modelling

21

system. What’s more, test results show that the proposed control scheme inherently has great background harmonics injection capability without the assistance of any auxiliary control.

2.2 Mathematical Modelling The main circuit of a single-phase LCL-type grid-connected inverter is sketched in Fig. 2.1, where V dc denotes the dc-link voltage, vab is the terminal voltage of the switching full-bridge, vg denotes the unknown grid voltage, vpcc is the measured voltage of the point-common-coupling. Besides, L 1 is the inductance of the convertersider inductor and r 1 is its equivalent series resistance (ESR). L 2 signifies the inductance of the grid-side inductor and r 2 is its ESR. C denotes the capacitance of the filter capacitor, and r 3 is its equivalent parallel resistance (r 3 ). For the convenience of representation, the state variables are denoted as: x 1 (t) = i2 (t), x 2 (t) = vC (t), x 3 (t) = i1 (t). Then, dynamic equations of the system can be expressed as x˙1 (t) =

1 1 1 x2 (t) − r2 x1 (t) − v pcc (t), L2 L2 L2

(2.1)

1 1 x2 (t) x3 (t) − x1 (t) − , C C r3

(2.2)

1 r1 Vdc μ(t) − x2 (t) − x3 (t), L1 L1 L1

(2.3)

x˙2 (t) = x˙3 (t) =

where x i (t) ∈ R. μ(t) is the control input, namely, the switching signal sent to a modulator for pulse width modulation (PWM). Control objective: design an adaptive backstepping-based current control scheme of (2.1)–(2.3) in the presence of unknown parasitic resistance so that the corresponding

Fig. 2.1 The investigated single-phase LCL-type grid-tied inverter

22

2 Adaptive Backstepping Current Control of Single-Phase …

closed-loop system is globally asymptotically stable in the sense that all the closedloop signals are bounded. Meanwhile, x 1 (t) tracks a user-defined sinusoidal signal r(t) asymptotically, limt→∞ x1 (t) − r (t) = 0. r (t) = I2 sin(ωt)

(2.4)

where I 2 ∈ R+ , ω ∈ R+ . I 2 is a user-defined constant, denoting the desired amplitude (peak-to-peak value) of grid-side-inductor current. ω is the angular frequency of the grid (ω = 100π), provided by the well-known phase-locking loop (PLL). ... Assumption 1 r(t), r˙ (t), r¨ (t), r (t), are all known, smooth, and bounded. Assumption 2 r 1 , r 2 , and r 3 are unknown positive constants, related to the environmental temperature and moisture, where r i ∈ R+ , i = 1, 2, 3 . Remark 1 Assumption 1 is needed for all classical backstepping control [8]. Assumption 1 is inherently satisfied since both I 2 and ω are known.

2.3 Derivation of Proposed Control Scheme 2.3.1 Step I: Derivation of Pseudo Reference x2ref (t) and Adaptive Law 1 From the Control objective, its corresponding tracking error is defined as z 1 (t) = x1 (t) − r (t).

(2.5)

A Lyapunov function is defined as the linear quadratic summation of (2.5), V1 (t) =

1 z 1 (t)2 . 2

(2.6)

Note that V 1 (t) and z1 (t) are denoted as V 1 and z1 for simplicity. According to (2.5), the time derivative of (2.6) can be found, V˙1 (t) = z 1 z˙ 1 = z 1 (x˙1 − r˙ ).

(2.7)

Substituting (2.1)–(2.7), it gives V˙1 (t) = z 1

[

] 1 1 1 x2 (t) − r2 x1 (t) − v pcc (t) − r˙ . L2 L2 L2

If the pseudo capacitor-voltage reference of x 2 (t) is chosen as x 2ref (t),

(2.8)

2.3 Derivation of Proposed Control Scheme

23

x2ref (t) = −k1 z 1 + r2 x1 + v pcc + L 2 r˙ ,

(2.9)

with the tracking error z2 (t), z 2 (t) = x2 (t) − x2ref (t)

(2.10)

1 k1 z1 z2 , V˙1 (t) = − z 12 + L2 L2

(2.11)

then, (2.8) is transformed to

where k 1 is a tunable constant controller gain, k 1 ∈ R+ . As in (2.9), implementation of x 2ref (t) requires the value of r 2 , while Assumption 2 tells that r2 is unknown. To solve this contraction, we now use r 2 to estimate r 2 online, where r˜2 denotes the estimation error, Δ

r˜2 = rˆ2 − r2 .

(2.12)

Δ

Meanwhile, r 2 is used to replace r 2 in (2.9), x2ref (t) = −k1 z 1 + rˆ2 x1 + v pcc + L 2 r˙ .

(2.13)

Incorporated with r˜2 , a weighted Lyapunov function is defined as 1 1 r˜22 1 , V1−1 (t) = V1 (t) + r˜22 = z 12 + 2 2 2 k1−1

(2.14)

where k 1–1 is a positive controller gain, k 1–1 ∈ R+ . From (2.8) and (2.10), the time derivative of (2.14) is given by r˜2 r˙ˆ2 V˙1−1 (t) = V˙1 (t) + k1−1 [ ] v pcc x2ref (t) + z 2 r2 r˜2 r˙ˆ2 = z1 − x1 − − r˙ + L2 L2 L2 k1−1 Substituting (2.13)–(2.15), it gives r˜2 r˙ˆ2 V˙1−1 (t) = V˙1 (t) + k1−1 ( ) −k1 z 1 + rˆ2 x1 + v pcc + z 2 v pcc r˜2 r˙ˆ2 r2 + − x1 − = z1 L2 L2 L2 k1−1 ˙ ) r˜2 rˆ2 x1 ( z1 z2 k1 rˆ2 − r2 + + = − z 12 + z 1 L2 L2 k1−1 L2

(2.15)

24

2 Adaptive Backstepping Current Control of Single-Phase …

( ) x1 k1 2 r˙ˆ2 z1 z2 = − z 1 + r˜2 z 1 + + L2 L2 k1−1 L2

(2.16)

If Adaptive law 1 is designed as (2.17) then (2.16) can be rewritten as (2.18). x1 r˙ˆ2 = −k1−1 · z 1 , L2

(2.17)

k1 z1 z2 V˙1−1 (t) = − z 12 + . L2 L2

(2.18)

2.3.2 Step II: Derivation of Pseudo Reference x3ref (t) and Adaptive Law 2 The Step-2 augmented Lyapunov function is defined as 1 V2 (t) = V1−1 (t) + z 2 (t)2 . 2

(2.19)

From (2.18), the time derivative of (2.19) is given by k1 1 z 1 z 2 + z 2 z˙ 2 . V˙2 (t) = − z 12 + L2 L2

(2.20)

Substituting (2.10)–(2.20), it gives k1 z1 z2 + z 2 [x˙2 (t) − x˙2ref (t)]. V˙2 (t) = − z 12 + L2 L2

(2.21)

According to (2.2), (2.21) can be rewritten as ) ( k1 2 z1 z2 x1 x2 x3 ˙ + z2 − x˙2ref (t) . − − V2 (t) = − z 1 + L2 L2 C C r3

(2.22)

From (2.13), (2.22) can be rewritten as (2.23), k1 z1 z2 V˙2 (t) = − z 12 + L2 L2 ⎡

⎤

) d ( x1 x2 ⎥ ⎢ x3 + z2 ⎣ − − − −k1 z 1 + rˆ2 x1 + v pcc −L 2 r¨ ⎦ C C r3 dt ' '' ' G1

(2.23)

2.3 Derivation of Proposed Control Scheme

25

If x 3 (t) tracks the pseudo inverter-side-current reference x 3ref (t), x3ref (t) = −k2 z 2 + x1 + C

dG 1 C 1 + C L 2 r¨ − z 1 + C x2 dt L2 r3

(2.24)

with the tracking error z3 (t), z 3 (t) = x3 (t) − x3ref (t),

(2.25)

1 1 1 V˙2 (t) = −k1 z 12 − k2 z 22 + z 2 z 3 . L2 C C

(2.26)

then, (2.23) is transformed to

where k 2 is a tunable constant controller gain, k 2 ∈ R+ . As in (2.24), implementation of x 3ref (t) requires the value of 1/r 3 , while Assumption 2 shows that 1/ r3 is unknown. To deal with this dilemma, we now use r 3 to estimate 1/r 3 online, where r˜3 denotes the estimation error, Δ

/ r˜3 = rˆ3 − 1 r3 .

(2.27)

Δ

Meanwhile, r 3 is used to replace 1/r 3 in (2.24), x3ref (t) = −k2 z 2 + x1 + C

dG 1 C z1 + C L 2 r¨ − + C x2 rˆ3 . dt L2

(2.28)

Incorporated with r˜3 , the other weighted augmented Lyapunov function is defined as V2−2 (t) = V2 (t) +

1 r˜32 , 2 k2−2

(2.29)

where k 2–2 is a positive controller gain, k 2–2 ∈ R+ . From (2.23) and (2.25), the time derivative of (2.29) is given by r˜3 r˙ˆ3 V˙2−2 (t) = V˙2 (t) + k2−2

[ ] z 3 + x3ref 1 k1 2 z1 z2 r˜3 r˙ˆ3 x2 dG 1 − x1 − − L 2 r¨ = − z1 + + + z2 − L2 L2 k2−2 C C r3 dt (2.30)

Substituting (2.28) and (2.13)–(2.30), it gives

26

2 Adaptive Backstepping Current Control of Single-Phase …

( ) ˙ˆ3 z k k z r 2 2 3 1 2 2 + r˜3 + z 2 x2 V˙2−2 (t) = − z 1 − z 2 + L2 C C k2−2

(2.31)

If Adaptive law 2 is designed as r˙ˆ3 = −k2−2 · z 2 x2 ,

(2.32)

k1 k2 z2 z3 . V˙2−2 (t) = − z 12 − z 22 + L2 C C

(2.33)

(2.31) is transformed to

2.3.3 Step III: Derivation of Control Law µ(t) and Adaptive Law 3 The Step-3 augmented Lyapunov function is defined as 1 V3 (t) = V2−2 (t) + z 3 (t)2 . 2

(2.34)

Using (2.33), the time derivative of (2.34) satisfies k1 k2 z2 z3 + z 3 z˙ 3 . V˙3 (t) = − z 12 − z 22 + L2 C C

(2.35)

Substituting (2.25)–(2.35), it gives k1 k2 z2 z3 + z 3 [x˙3 (t) − x˙3ref (t)] V˙3 (t) = − z 12 − z 22 + L2 C C

(2.36)

Substituting (2.3)–(2.36), it yields [ ] Vdc k2 z2 z3 x2 r1 k1 + z3 μ(t) − − x3 − x˙3ref (t) . V˙3 (t) = − z 12 − z 22 + L2 C C L1 L1 L1 (2.37) Substituting (2.28)–(2.37), it gives (2.38). k2 z2 z3 k1 V˙3 (t) = − z 12 − z 22 + L2 C C

2.3 Derivation of Proposed Control Scheme

27 ⎤

⎡

⎥ ⎢ ( ) ⎢ μVdc x2 r1 x 3 d dG 1 C z1 ...⎥ ⎥ r + z3 ⎢ − − − z + x + C + C x r ˆ −C L −k − 2 2 1 2 3 2 ⎥ ⎢ L L1 L1 dt dt L2 ⎦ ⎣ 1 '' ' ' G2

(2.38) As in (2.38), it can be transformed to (2.39) provided that μ(t) is designed as (2.40), k1 k2 k3 Vdc 2 z ≤ 0, V˙3 (t) = − z 12 − z 22 − L2 C L1 3 ] [ x2 r1 L 1 dG 2 ... z 2 + x3 + + C L2 r − , μ(t) = −k3 z 3 + Vdc Vdc Vdc dt C

(2.39) (2.40)

where k 3 is a tunable constant controller gain to provide active damping, k 3 > 0. As in (2.40), implementation of μ(t) requires knowing r 1 , which contradicts Assumption 2. To deal with this problem, we now use r 1 to estimate r 1 , where r˜1 denote its estimation error, Δ

r˜1 = rˆ1 − r1 .

(2.41)

Δ

Meanwhile, r 1 is used to replace r 1 of (2.40), ] [ x2 rˆ1 L 1 dG 2 ... z 2 r + x3 + + C L2 − . μ(t) = −k3 z 3 + Vdc Vdc Vdc dt C

(2.42)

Incorporated with r˜1 , the final all-in-one Lyapunov function V 3–3 (t) is defined as V3−3 (t) = V3 (t) +

1 r˜12 , 2 k3−3

(2.43)

where k 3–3 is a positive controller gain, k 3–3 ∈ R+ . From (2.42) and (2.38), the time derivative of (2.43) is derived as (2.44). r˜1 r˙ˆ1 V˙3−3 (t) = V˙3 (t)+ k3−3

( ) k1 2 k2 2 k3 Vdc 2 r˙ˆ1 z3 = − z1 − z2 − z + r˜1 x3 + L2 C L1 3 L1 k3−3

(2.44)

According to (2.44), if the Adaptive law 3 is designed as (2.45), (2.44) is transformed to (2.46). z3 r˙ˆ1 = −k3−3 x3 , L1

(2.45)

28

2 Adaptive Backstepping Current Control of Single-Phase …

k2 k3 Vdc 2 k1 z . V˙3−3 (t) ≤ − z 12 − z 22 − L2 C L1 3

(2.46)

2.4 Test Results According to the above illustration, the adaptive control scheme of the single-phase LCL-grid-connected inverter under parasitic resistance uncertainty is summarized as Table 2.1. Nominal system parameters are listed in Table 2.2. The design parameters are recommended to be selected as the following rule-of-thumb values: k 1 = L 2 × f sw , k 2 = C × f sw , k 3 = L 1 × f sw , k 1–1 = L 2 /f sw , k 2–2 = C/f sw , k 3–3 = L 1 /f sw . Ideal grid voltage: vg (t) = vm ·sin(π t). Nonideal grid-voltage scenario is simulated according to [9], where vg (t) is distorted by 3rd, 5th, 7th, 9th, 11th, 13th, 21st, 33rd, and 40th harmonics. The magnitudes of harmonics with respect to the grid fundamental voltage are 10%, 5%, 3%, 3%, 2%, 2%, 1%, 1%, and 0.25%, respectively, and the corresponding phases are 0°, 210°, 0°, 0°, 0°, 0°, 0°, 0°, and 0°. The following test results in this chapter are captured using the OPAL-RT 5600 and dSPACE 1202 setup. As presented in Fig. 2.2, i2 is regulated to track r in the steady-state both in ideal grid condition and non-ideal grid condition. It shows that the proposed control scheme inherently has good background harmonic rejection capability. As presented in Fig. 2.3, the proposed approach also shows good transient performance. As r steps from vm ·sin(π t) to 0.5vm ·sin(π t) and back forth, i2 tracks r instantly with a small overshoot. However, as shown in Fig. 2.4, a comparably larger overshoot of i2 can be observed under abrupt change of grid voltage (100%vm → 80%vm → 100%vm ). This phenomenon emerges for the reason that the proposed control scheme uses Table 2.1 Robust adaptive current control scheme Physical meaning

Notation

Expression

Reference of i2 (t)

r(t)

(2.4)

Pseudo reference of x 2 (t)

x 2ref (t)

(2.13)

Pseudo reference of x 3 (t)

(2.28)

Adaptive law 2

x 3ref (t) r˙2 r˙3

Adaptive law 3

r˙1

(2.45)

Design parameters

k 1 , k 2 , k 3 , k 1–1 , k 2–2 , k 3–3

R+

Final control law

μ(t)

(2.42)

Δ

Adaptive law 1

(2.17)

Δ

(2.32)

Δ

Note Active damping is tuned via k 1 , k 2 , k 3 . Dynamic performance of adaptive laws 1, 2, 3 is adjusted via k 1–1 , k 2–2 , k 3–3

2.4 Test Results

29

Table 2.2 Nominal system parameters by default Physical meaning

Notation

Value

Unit

dc-link voltage

V dc

400

Volt

The fundamental frequency of the grid

f

50

Hz

Fundamental angular frequency

ω

100π

rad/s

Switching frequency of SPWM modulator

f sw

12.5

kHz

The amplitude of i2 (t)

I2

20

Ampere

Amplitude of vg (t)

vm

230 × 1.414

Volt

Rated active power

P

5

kW

Inverter-side inductance of LCL filter

L1

1.436

mH

The capacitance of the LCL filter

C

50

μF

Grid-side inductance of LCL filter

L2

0.6867

mH

ESR of inverter-side inductor

r1

0.17

Ω

ESR of grid-side inductor

r2

0.076

Ω

EPR of the filter capacitor

r3

10

kΩ

Grid resistance

rg

0.5

Ω

Grid inductance

Lg

0.1

mH

vpcc :[200 V/div] r :[20 A/div]

i2 :[20 A/div]

[4 ms/div]

(a) vpcc :[200 V/div] r :[20 A/div]

i2 :[20 A/div]

[4 ms/div]

(b) Fig. 2.2 Steady-state control performance of the proposed control scheme under different grid conditions. a Ideal grid; b non-ideal grid

30

2 Adaptive Backstepping Current Control of Single-Phase …

50%I2

vpcc :[200 V/div]

100%I2

r :[20 A/div]

[10 ms/div]

i2 :[20 A/div]

(a) 50%I2

vpcc :[200 V/div]

100%I2

r :[20 A/div]

[10 ms/div]

i2 :[20 A/div]

(b) Fig. 2.3 Transient control performance of the proposed control scheme under reference step (I 2 → 0.5I 2 → I 2 ). a Ideal grid; b non-ideal grid

vpcc :[200 V/div]

80%vm

100%vm

r :[20 A/div]

i2 :[20 A/div]

[10 ms/div]

Fig. 2.4 Transient control performance of the proposed control scheme under grid voltage dip (100%vm → 80%vm → 100%vm )

high-order derivatives of the vpcc . When vpcc undergoes abrupt change, the derivatives of vpcc result in an infinite signal μ(t). While infinite actuation cannot be realized by a real-world controller.

2.5 Conclusion This section explores an adaptive backstepping control scheme of a single-phase LCL-type grid-connected inverter in the presence of parasitic resistance uncertainty of LCL components. The control scheme is devised using step-by-step Lyapunov

References

31

functions recursively. In each step, a pseudo control reference is derived to stabilize the system, meanwhile, an adaptive law is derived to address the parasitic resistance uncertainty problem. The proposed approach rigorously guarantees the global largesignal stability of the closed-loop system. Test results show that the proposed control scheme inherently has great background harmonics rejection capability. Moreover, the proposed approach also displays good transient performance under the scenario of reference step, however, its transient performance is not as good as before under grid voltage dip, where a comparatively larger overshoot exists. This is because this control scheme uses higher-order derivatives of vpcc . Abrupt change of vpcc leads to infinite control signal μ(t), which cannot be realized by a real-world controller. In our future work, the inherent saturation of μ(t) and external disturbance will be considered to design the adaptive backstepping control scheme.

References 1. Y. Tang, P. C. Loh, P. Wang, F. H. Choo, and F. Gao, “Exploring inherent damping characteristic of LCL-filters for three-phase grid-connected voltage source inverters,” IEEE Trans. Power Electron., vol. 27, no. 3, pp. 1433–1443, 2012. 2. I. Sefa, S. Ozdemir, H. Komurcugil, and N. Altin, “An Enhanced Lyapunov-Function Based Control Scheme for Three-Phase Grid-Tied VSI with LCL Filter,” IEEE Trans. Sustain. Energy, vol. 3029, no. 2, pp. 504–513, 2018. 3. M. Huang, Y. Peng, C. K. Tse, Y. Liu, J. Sun, and X. Zha, “Bifurcation and Large-Signal Stability Analysis of Three-Phase Voltage Source Converter under Grid Voltage Dips,” IEEE Trans. Power Electron., vol. 32, no. 11, pp. 8868–8879, 2017. 4. H. Komurcugil, N. Altin, S. Ozdemir, and I. Sefa, “An Extended Lyapunov-function-based Control Strategy for Single-phase UPS Inverters,” IEEE Trans. Power Electron., vol. 30, no. 7, pp. 3976–3983, 2015. 5. I. Sefa, S. Ozdemir, H. Komurcugil, and N. Altin, “Comparative study on Lyapunov-functionbased control schemes for single-phase grid-connected voltage-source inverter with LCL filter,” IET Renew. Power Gener., vol. 11, no. 11, pp. 1473–1482, 2017. 6. H. Komurcugil, S. Member, N. Altin, and S. Ozdemir, “Lyapunov-Function and ProportionalResonant-Based Control Strategy for Single-Phase Grid-Connected VSI With LCL Filter,” IEEE Trans. Ind. Electron., vol. 63, no. 5, pp. 2838–2849, 2016. 7. S. Bayhan, S. S. Seyedalipour, H. Komurcugil, and H. Abu-Rub, “Lyapunov Energy Function Based Control Method for Three-Phase UPS Inverters With Output Voltage Feedback Loops,” IEEE Access, vol. 7, pp. 113699–113711, 2019. 8. J. Zhou and C. Wen, Adaptive backstepping control of uncertain systems: Nonsmooth nonlinearities, interactions or time-variations. Springer, 2008. 9. D. Yang, X. Ruan, and H. Wu, “Impedance shaping of the grid-connected inverter with LCL filter to improve its adaptability to the weak grid condition,” IEEE Trans. Power Electron., vol. 29, no. 11, pp. 5795–5805, 2014.

Chapter 3

An Adaptive Dual-Loop Lyapunov-Based Control Scheme for a Single-Phase Stand-Alone Inverter to Improve Its Large-Signal Stability

Nomenclature E fs ω L C vref P iLref iL io e1 e2 ε˜ Z V μ

Dc-link voltage Switching frequency Fundamental angular frequency Nominal filter inductance Nominal filter capacitance Load voltage reference Output active power Inductor current reference Inductor current Load current Load voltage tracking error Inductor current tracking error Virtual load estimating error Load impedance (complex) Lyapunov function Switching function

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Zhang et al., Stability Enhancement Methods of Inverters Based on Lyapunov Function, Predictive Control, and Reinforcement Learning, https://doi.org/10.1007/978-981-19-7191-4_3

33

34

3 An Adaptive Dual-Loop Lyapunov-Based Control Scheme …

3.1 Introduction Single-phase inverter with LC output filter has been widely utilized in uninterruptible power supply (UPS) systems. In order to ensure pure sinusoidal output voltage with great power quality and fast transient response under load disturbance, numerous linear and nonlinear control schemes are intensively studied. Cascaded PI/PR control are commonly used linear control approaches. According to the internal model principle, PI control in the stationary reference frame follows the sinusoidal reference accompanied by steady-state errors. While PR control could track the sinusoidal command accurately. Deadbeat control ensures fast dynamic response while it is inherently sensitive to parametric variations [1]. Sliding-mode control is robust against parametric variations but suffers from the chattering phenomenon [2]. Repetitive control is expert at tracking/rejecting periodic reference/disturbance but it has comparatively large inter-sample ripples [3]. Model predictive control achieves great transient response at the expense of a relatively large computation burden and varying switching frequency [4]. In [5], a hybrid dualloop control is proposed, embedded with PI control, repetitive control, and feedforward control. This approach requires complicated individual design stages. In [6], H-infinity control is investigated, which could theoretically ensure control robustness under specific premises. The above control strategies have case-by-case strengths and limitations, but one limitation they have in common is that, i.e., global large-signal stability of the system cannot be rigorously guaranteed. In fact, it is a concern for the system when exposed to large perturbations away from the operating point. To this end, Lyapunov-based control schemes emerge. The conventional approach formulates a Lyapunov function (V ) as the sum of linear-quadratic tracking errors associated with the capacitor voltage and inductor current. Then, the final control law is selected to assure the negative definiteness of dV /dt. However, it inherently yields out a single current-loop control scheme [7–10]. For islanded inverter applications, it leads to steady-state error and slack dynamic response [9]. To fix this problem, [7, 9, 11] extend the conventional single-loop control to a dual-loop counterpart via artificial importation of capacitor voltage feedback. However, this action inevitably requires afterward modification of the Lyapunov-based control law, which cannot rigorously guarantee the negative definiteness of dV /dt unconditionally. This strategy also needs one load current sensor for practical implementation. As an economical alternative, the load current sensor is replaced by observers [4, 12]. While observer and controller design indeed require two separate design stages. Observer optimization complicates the controller design, which correlates with Riccati equation solving, bandwidth allocation based on Kalman filter theory. Therefore, current research on Lyapunov-based control schemes for the system is still on the way. Aiming to address the above obstacles, an adaptive dual-loop Lyapunov-based control strategy is proposed in this chapter. The proposed approach inherently has dual control loops with no need to import the voltage control loop artificially. Thus, the global large-signal stability of the system can be rigorously

3.2 Mathematical Modelling S1

E

35 S3

iL

µ

Z

L

µE S2

io

C

S4

vo

PWM

Fig. 3.1 Investigated single-phase UPS inverter

guaranteed. Meanwhile, load disturbance can be suppressed adaptively without any additional load-current sensors or observers. The test results show that the proposed control scheme brings better state-steady and dynamic performance to the inverter with good robustness against the variations of the parameters.

3.2 Mathematical Modelling 3.2.1 Average Model of the Investigated System According to Kirchhoff’s voltage law and Kirchhoff’s current law (KCL and KVL), the average model of the system, as shown in Fig. 3.1, can be represented as, L di L /dt = μ · E − vo

(3.1)

C dvo /dt = i L − i o

(3.2)

where system parameters are denoted as: filter inductance L, filter capacitance C, dc-link voltage E, switching function μ, inductor current iL , load voltage vo , and load current io .

3.2.2 Load Voltage Reference Investigated system is bound to reach the equilibrium point in steady-state, where vo will track vref , given by vr e f = vm sin(ω t) Here, vm denotes the load voltage reference amplitude.

(3.3)

36

3 An Adaptive Dual-Loop Lyapunov-Based Control Scheme …

3.2.3 Current-Loop Reference Substituting vref into (3.2), expression of the current-loop reference iref can be found, ir e f = ωCvm cos(ωt) + i o

(3.4)

3.2.4 Model of the Load Current As shown in (3.4), load current io is indispensable to configure iref . Accurate io can be expressed as i o = vo /Z → vr e f /Z

(3.5)

where Z is the load impedance. Since Z is an unknown complex, susceptible to uncertainty and nonlinearity, using (3.5) to calculate io is impossible in practice. Load current in this chapter is estimated online via following adaptive law, iˆo = vr e f εˆ

(3.6)

where εˆ is defined to replace the unknown 1/Z for load disturbance suppression, whose detailed derivation will be given in the next chapter.

3.3 Proposed Adaptive Dual-Loop Lyapunov-Based Control Scheme 3.3.1 The Proposed Lyapunov Function An all-in-one modified Lyapunov function is proposed to derive out the final control law μ and expression of εˆ , given by V =

) 1 1( 2 Le1 + Ce22 + ε˜ 2 /γ 2 2

(3.7)

where e1 is the current-loop tracking error and e2 is the voltage-loop tracking error. e1 and e2 are defined in (3.8) and (3.9) respectively. In addition, ε˜ is defined as the virtual tracking error resulting from load disturbance, which is defined in (3.10). γ is an artificially imported controller parameter (0 < γ < 1).

3.3 Proposed Adaptive Dual-Loop Lyapunov-Based Control Scheme

37

e1 = i L − ir e f

(3.8)

e2 = vo − vr e f

(3.9)

/ ε˜ = εˆ − 1 Z

(3.10)

3.3.2 Derivation of the Adaptive Dual-Loop Control Law If (3.7) reached the equilibrium, (3.10), (3.8), (3.9) would have converged to zero, whose behaviour around the equilibrium point (e1 , e2 , ε˜ = 0) is investigated via Lyapunov’s direct method. It points out that the equilibrium point is globally asymptotically stable unless V satisfies the following three premises: • I: V ≥ 0, where V = 0 if and only if e1 , e2 , ε˜ = 0; • II: V → ∞ if any of e1 , e2 , ε˜ → ∞; • III: dV /dt < 0 for all points except the equilibrium point. Premises I, II can be inherently fulfilled according to the positive definite expression of (3.7). To exam whether Premise III holds or not, expression dV/ dt requires to be derived out, dV /dt = Le1 e˙1 + Ce2 e˙2 + ε˜ ε˙˜ /γ

(3.11)

To represent (3.11), derivatives of (3.8)–(3.10) should be figured out. According to (3.1), (3.2), (3.5), it is not far to seek out that [ ] e˙1 = μ · E − (vr e f + e2 ) /L − i˙r e f

(3.12)

) ] [ ( e˙2 = ir e f + e1 − vr e f + e2 /Z /C − ωvm cos(ωt)

(3.13)

ε˙˜ = d˜ε /dt = ε˙ˆ = dˆε/dt

(3.14)

As shown in (3.12), expression of diref /dt is further required. To this end, substituting (3.6) into (3.4). Then, the modified iref incorporated with εˆ will be represented as, ir e f = ωCvm cos(ωt) + vr e f εˆ

(3.15)

38

3 An Adaptive Dual-Loop Lyapunov-Based Control Scheme …

Taking first-order time derivative of (3.15), it gives ) ( dir e f /dt = i˙r e f = vr e f ε˙ˆ − ω2 C + ωvm cos(ωt)ˆε

(3.16)

Subsequently, substituting (3.12)–(3.16) into (3.11), it yields out ( ) dV /dt = −σ e12 − 1/Z · e22 + vr e f e2 + ε˙ˆ /γ ε˜ ' '' ' ' '' ' [

α

(

β

)

] + μE − 1 + ω LC − L ε˙ˆ vr e f − ωLvm cos(ωt)ˆε e1 +σ e12 ' '' ' 2

(3.17)

χ

which demonstrates that dV /dt = α + β + χ, where α, β, χ are expressed as α = −σ e12 − 1/Z · e22 < 0

(3.18)

( ) β = vr e f e2 + ε˙ˆ /γ ε˜

(3.19)

[ ( ) ] χ = μE − 1 + ω2 LC − L ε˙ˆ vr e f − ωLvm cos(ωt)ˆε + σ e1 e1

(3.20)

As shown in (3.17), –σ e1 2 is contained in α, whose counterpart + σ e1 2 is embedded in χ, so that dV /dt remain unchanged. This action leads to the importation of –σ e1 in later (3.23). σ is another controller parameter (σ > 0). Since α < 0 has been inherently guaranteed, premise III can be satisfied if β = 0 and χ = 0, which leads to dV /dt = α < 0. By setting β = 0, it gives ε˙ˆ = −γ vr e f e2

(3.21)

Thus, the formula of the adaptive term εˆ can be finally expressed as ∫ εˆ =

ε˙ˆ dt =

∫ −γ vr e f e2 dt

(3.22)

∫ The above integrator ( ) will not stop working until the equilibrium point has been reached, where e2 → 0. According to (3.20), χ = 0 can be guaranteed if the final control law is selected as ) ] [( (3.23) μ = 1 − ω2 LC + L ε˙ˆ vr e f + ωLvm cos(ωt)ˆε − σ e1 /E

3.4 Stability Analysis and Robustness Verification

39

Fig. 3.2 Proposed adaptive dual-loop Lyapunov-based control scheme

3.3.3 Implementation of Proposed Control Scheme According to (3.21)–(3.23), the control block of the proposed control strategy is sketched in Fig. 3.2. As seen, the proposed control scheme inherently has dual control loops (e1 , e2 are contained). In addition, none of the additional load-current sensors or observers is required in Fig. 3.2. Note that the proposed adaptive Lyapunov-based control is specially devised for single-phase UPS applications. When extended for grid-connected application and multiple inverters’ parallel operation, separate derivation for the topology of a specific system is required, where a seamless transition technique is commonly used [13].

3.4 Stability Analysis and Robustness Verification 3.4.1 Stability Analysis Substituting (3.21)–(3.23), into (3.12)–(3.14), the error dynamics of the regulated closed-loop system would be expressed as x˙ = A(t)x where x = [e1 , e2 , ε˜ ]T , and

(3.24)

40

3 An Adaptive Dual-Loop Lyapunov-Based Control Scheme …

⎛

⎞ −σ/L −1/L 0 A(t) = ⎝ 1/C −1/ZC vm sin(ωt)/C ⎠ 0 0 −γ vm sin(ωt)

(3.25)

It demonstrates that A(t) = A(t + T ) where T = 0.02 s. Thus, (3.24) is a linear time periodic (LTP) system [14]. Taking equivalent transformation of A(t), expressed as ⎛ −1 ⎞ ⎞⎛ 0 0 L −σ −1 0 A(t) = ⎝ 0 C −1 0 ⎠ ⎝ 1 (3.26) vm sin(ωt) ⎠ −Z −1 0 0 0 γ 0 −vm sin(ωt) '' ' ' M

Taking a linear equivalent transformation of x gives y = PT x

(3.27)

where PT is a non-singular diagonal matrix, given by ⎛

⎞ 0 L 1/2 0 PT = ⎝ 0 C 1/2 0 ⎠ 0 0 γ −1/2

(3.28)

Then, (3.24) will be transformed to y˙ = PT · A(t) · PT−1 y

(3.29)

Since matrix PT is non-singular, the stability of the system (3.29) is equivalent to that of (3.24). Substituting (3.26) into (3.29), it gives y˙ = PT−1 · M · PT−1 y

(3.30)

A Lyapunov function V 2 is formulated to prove the large-signal stability of the system, given by V2 (y) = y H y

(3.31)

where yH is the conjugate transpose matrix of y. Derivative of (3.31) is expressed as dV2 /dt = y H y˙ + y˙ H y Substituting (3.30) into (3.32), it yields out ( ) ( )H dV2 /dt = y H PT−1 · M · PT−1 y + PT−1 · M · PT−1 y y

(3.32)

3.4 Stability Analysis and Robustness Verification

41

[ )H )H ] ( ( y = y H PT−1 · M · PT−1 + PT−1 · M H · PT−1

(3.33)

According to (3.28), (PT −1 )H = PT −1 , thus (3.33) can be written as ( ) dV2 /dt = y H PT−1 M + M H PT−1 y

(3.34)

Substituting M, PT into (3.34), M + M H cancel out all the off-diagonal entries of M and M H , which gives ⎛

⎞ 0 0 −2σ L −1 [ ] dV2 /dt = y H ⎝ 0 − 1/Z + (1/Z ) H C −1 0 ⎠ y 0 0 0

(3.35)

Note that 1/Z is the generalized load admittance, denoted as 1/Z = G + B j

(3.36)

where G > 0, and B is arbitrary. Substituting (3.36) into (3.35), it yields out that ⎛

⎞ 0 0 −2σ L −1 dV2 /dt = y H ⎝ 0 −2GC −1 0 ⎠ y < 0 0 0 0

(3.37)

It shows that the filter will not affect the global large-signal stability using the proposed control method provided that σ > 0 and G > 0. In fact, this condition can be easily satisfied, and σ, L, C, Z could vary at a rather large range without deteriorating the system stability. Owning to γ , vm are located at the off-diagonal entries of A(t), they mainly affect the system damping. While L, C, Z, σ are located at the diagonal entries of (3.35), they will affect the dynamic response of the system. Larger σ will result in a higher convergence speed of the error dynamics, leading to better dynamic response. Smaller γ will provide more damping for the system. For system parameters specified in Table 3.1, σ (20–2000) and γ (0.01–0.05) are their recommended selection range in this example case. Table 3.1 Nominal system parameters E

ω

fs

L

C

P

vm

350 V

100π

10 kHz

1 mH

10 μF

5 kW

220 × 1.414 V

42

3 An Adaptive Dual-Loop Lyapunov-Based Control Scheme …

3.4.2 Robustness Against Plant Parametric Variations To evaluate the robustness of the proposed approach, simulations are conducted under the reference step and parametric variations where σ and γ are set as 200 and 0.05 respectively. As shown in Fig. 3.3, the stability of the system does not get deteriorated even under such a large scale of parametric variations. Estimated load currents in Fig. 3.3a–d are just slightly different, and the load voltage always remains pure sinusoidal.

Fig. 3.3 Robustness against reference steps and plant parametric variations. a −50%L, −50%C; b + 50%L, −50%C; c + 50%L, + 50%C; d −50%L, + 50%C

3.5 Test Results

43

4.7 mH

200 μF R load

R Linear resistive load

Nonlinear rectifier load

Fig. 3.4 Linear and nonlinear load used to do experiments

3.5 Test Results A single-phase UPS inverter with an LC output filter is fabricated to do tests. Plant parameters are listed as Table 3.1.

3.5.1 Steady-State and Dynamic Performance Evaluation Experiments are conducted to test the performance of the proposed approach when supplying the linear resistive load and nonlinear rectifier load, as shown in Fig. 3.4. The test results of the single-phase UPS inverter have been shown in Fig. 3.5. In Fig. 3.5a, linear resistive load steps from 500 W to 5 kW. In Fig. 3.5b, load voltage reference firstly steps from 220 V RMS to 110 V RMS with 10 Ω linear resistive load. Then, it is restored to 220 V RMS again. In Fig. 3.5c, nonlinear rectifier load steps from 200 W to 2 kW. As seen, the proposed control scheme regulates the load voltage with great steady-state and dynamic response under the scenario of, linear load step, reference step, and nonlinear rectifier load step.

3.5.2 Overload and Recovery Scenario As shown in Fig. 3.6, the single-phase UPS inverter initially operates with a 100%P linear resistive load. Then, 200%P overload happens. After two fundamental periods, the system goes into the recovery process (200%P → 100%P). The regulated load voltage almost remains sinusoidal even under the scenario of overload and recovery, which shows that the proposed adaptive control has great robustness against load disturbance.

44

3 An Adaptive Dual-Loop Lyapunov-Based Control Scheme …

Fig. 3.5 Experimental waveforms with the proposed control method. a Resistive load step; b voltage reference step with resistive load; c nonlinear rectifier load step

3.6 Conclusion In this chapter, an adaptive Lyapunov-based control scheme is proposed for a single-phase UPS inverter, which not only has inherent dual control loops to ensure better steady-state and dynamic performance but also can rigorously guarantee the global large-signal stability. The detailed control law derivation is presented. The corresponding stability analysis and robustness verification are also discussed in this chapter. Furthermore, with this control scheme, the load disturbance can be suppressed adaptively without any additional load-current sensors or observers. The

References

45

Fig. 3.6 Experimental waveforms with the proposed control method under overload and recovery process

experiment results eventually verify the effectiveness of the proposed control scheme. In our future work, the proposed control scheme is expected to be extended for grid-connected applications and multiple inverters’ parallel operation.

References 1. M. Pichan, H. Rastegar, and M. Monfared, “Deadbeat Control of the Stand-Alone Four-Leg Inverter Considering the Effect of the Neutral Line Inductor,” IEEE Trans. Ind. Electron., vol. 64, no. 4, pp. 2592–2601, 2017. 2. A. Marcos-Pastor, E. Vidal-Idiarte, A. Cid-Pastor, and L. Martinez-Salamero, “Interleaved Digital Power Factor Correction Based on the Sliding-Mode Approach,” IEEE Trans. Power Electron., vol. 31, no. 6, pp. 4641–4653, 2016. 3. G. Escobar, A. A. Valdez, J. Leyva-Ramos, and P. Mattavelli, “Repetitive-based controller for a UPS inverter to compensate unbalance and harmonic distortion,” IEEE Trans. Ind. Electron., vol. 54, no. 1, pp. 504–510, 2007. 4. P. Cortés, G. Ortiz, J. I. Yuz, J. Rodríguez, S. Vazquez, and L. G. Franquelo, “Model predictive control of an inverter with output LC filter for UPS applications,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1875–1883, 2009. 5. Y. G. Gao, F. Y. Jiang, J. C. Song, L. J. Zheng, F. Y. Tian, and P. L. Geng, “A novel dual closed-loop control scheme based on repetitive control for grid-connected inverters with an LCL filter,” ISA Trans., vol. 74, pp. 194–208, 2018. 6. M. A. Chowdhury, “Dual-loop H ∞ controller design for a grid-connected single-phase photovoltaic system,” Sol. Energy, vol. 139, pp. 640–649, 2016. 7. I. Sefa, S. Ozdemir, H. Komurcugil, and N. Altin, “An Enhanced Lyapunov-Function Based Control Scheme for Three-Phase Grid-Tied VSI with LCL Filter,” IEEE Trans. Sustain. Energy, vol. 3029, no. 2, pp. 504–513, 2018. 8. I. Sefa, S. Ozdemir, H. Komurcugil, and N. Altin, “Comparative study on Lyapunov-functionbased control schemes for single-phase grid-connected voltage-source inverter with LCL filter,” IET Renew. Power Gener., vol. 11, no. 11, pp. 1473–1482, 2017. 9. H. Komurcugil, N. Altin, S. Ozdemir, and I. Sefa, “An Extended Lyapunov-function-based Control Strategy for Single-phase UPS Inverters,” IEEE Trans. Power Electron., vol. 30, no. 7, pp. 3976–3983, 2015. 10. H. Komurcugil, S. Member, N. Altin, and S. Ozdemir, “Lyapunov-Function and ProportionalResonant-Based Control Strategy for Single-Phase Grid-Connected VSI With LCL Filter,” IEEE Trans. Ind. Electron., vol. 63, no. 5, pp. 2838–2849, 2016.

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11. S. Bayhan, S. S. Seyedalipour, H. Komurcugil, and H. Abu-Rub, “Lyapunov Energy Function Based Control Method for Three-Phase UPS Inverters With Output Voltage Feedback Loops,” IEEE Access, vol. 7, pp. 113699–113711, 2019. 12. E. Kim, F. Mwasilu, H. H. Choi, and J. Jung, “An Observer-Based Optimal Voltage Control Scheme for Three-Phase UPS Systems,” IEEE Trans. Ind. Electron., vol. 62, no. 4, pp. 2073– 2081, 2015. 13. D. S. Ochs, B. Mirafzal, and P. Sotoodeh, “A method of seamless transitions between grid-tied and stand-alone modes of operation for utility-interactive three-phase inverters,” IEEE Trans. Ind. Appl., vol. 50, no. 3, pp. 1934–1941, 2014. 14. “Beijing 2022 Olympics to be powered by 100% renewable energy.” [Online]. Available: https://www.powerengineeringint.com/renewables/beijing-2022-olympics-to-be-powered-by100-renewable-energy/. [Accessed: 15-Nov-2021].

Chapter 4

Lyapunov-Based Control of Three-Phase Stand-Alone Inverters to Improve Its Large-Signal Stability with Inherent Dual Control Loops and Load Disturbance Adaptivity

Nomenclature E L, C r voabc iLabc ioabc µabc vodref , voqref vod , voq iLdref , iLqref iLd , iLq iod , ioq i od i oq 1/Z μd , μq e1 , e3 e2 , e4 εd εq ∼ ∼ εd εq k, ρ, γ T −1 Δ

Δ

Δ

Δ

Dc-link voltage Nominal filter inductance, capacitance The series resistance of the filter inductor Load-voltage vector: [voa vob voc ]T Inductor-current vector: [iLa iL b iLc ]T Load-current vector: [ioa iob ioc ]T Switching-function vector: [μa μb μc ]T Load voltage references in d, q frames Load voltages in d, q frames Inductor current references in d, q frames Inductor currents in d, q frames Load currents in d, q frames Estimated Load currents in d, q frames Generalized load admittance 1/Z = G + Bj Derived switching functions in d, q frames Load voltage tracking errors in d, q frames Inductor current tracking errors in d, q frames Derived adaptive laws in d, q frames Virtual estimating errors in d, q frames Controller gains (k, ρ, γ > 0) The inverse matrix of T (non-singular matrix)

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Zhang et al., Stability Enhancement Methods of Inverters Based on Lyapunov Function, Predictive Control, and Reinforcement Learning, https://doi.org/10.1007/978-981-19-7191-4_4

47

48

4 Lyapunov-Based Control of Three-Phase Stand-Alone Inverters …

4.1 Introduction Voltage source inverter (VSI) with LC output filter has been widely implemented for stand-alone power conversion applications where the utility grid is not available. This off-grid system draws dc power from energy storage devices charged by PV array or the other renewable resources and converts it to pure sinusoidal ac output, which suits for distributed generation (DG) [1], uninterruptible power supply (UPS) system [2], 400-Hz ground power units for airplanes [3], etc. One significant research area of the stand-alone inverter system is the damping of LC resonance [4], which is mature nowadays. Another intensively studied region lies in the load-voltageoriented advanced control strategies under the distortions of parametric mismatch, load uncertainties, and nonlinearities. Conventional linear control has been widely utilized for stand-alone inverter regulation, such as PI/PR control [5]. They are typically arranged as a multi-loop cascaded control structure [6]. Their disturbance rejecting ability can be further enhanced when combined with full-feedforward control. Feedforward control has been applied to single-phase/three-phase LCL-type inverters for grid voltage distortions compensation [7]. They are also devised for LC-type stand-alone inverters application for load disturbance rejection [8]. However, the performance of the full-feedforward control may fade out during load change since the required infinite feedforward signals are only possible in mathematics but not in practice [7]. Besides, linear control does not keep the inherent discrete nonlinear traits of a power inverter system, which means that it fails to thoroughly manipulate the system’s full potential. To this end, nonlinear control strategies emerge. Conventional deadbeat control has a good dynamic response at the expense of sensitivity to parametric mismatch. Recently, the robustness of deadbeat control gets greatly enhanced in [9]. Model predictive control achieves great dynamic performance at the expense of a comparatively high computation burden [10, 11]. Terminal slidingmode control contributes to limiting the sliding surface in finite time and realizing better dynamic response when combined with Lyapunov’s direct method [12]. In [13], repetitive control devised in three d-q references is proposed to address periodic load disturbance and phase unbalance. H-infinity control can be applied to a multi-variable cross-coupling system, however, it only achieves the optimal under specific prescribed cost function [14]. The aforementioned linear/nonlinear control schemes have case-by-case strengths and defects. But one limitation they have in common is that they cannot rigorously guarantee the global large-signal stability of the closed-loop system, which is of great significance when exposed to large perturbations away from the operation point. To bridge this gap, Lyapunov-based control schemes appear. The conventional Lyapunov-based approach formulates a Lyapunov function (V ) as the sum of linearquadratic tracking errors associated with the capacitor voltage and inductor current. Then, the duty cycle is derived out to assure V˙ < 0. This approach has been applied to a wide range of power electronics systems, including but not limited to, dc-dc converters [15, 16], rectifiers [17, 18], micro-grids [19], stand-alone and grid-tied

4.1 Introduction

49

inverters [20, 21]. Recently, Lyapunov-based control has been extended for torque control of a quad-rotorcraft to avoid singularity during attitude tracking [22], load position control of a crane to damp out the pendulum motions of the load [23], and so forth. However, conventional Lyapunov-based control inherently yields a single currentloop control scheme. For grid-connected single-phase/three-phase LCL-type gridtied inverter application, it cannot damp the current resonance injected to the grid [24, 25]. For an LC-type stand-alone inverter application, it leads to sluggish transient response and steady-state errors. To fix these problems, [20] extends the conventional single-loop control to a dual-loop counterpart via afterward importation of state variables’ feedback. However, it inevitably modifies the Lyapunov-based control law, which may sacrifice the negative definiteness of V˙ . While large-signal stability of the system can be still guaranteed provided that the artificially imported positive controller gain is larger than a specific lower bound. Besides, existed Lyapunov-based control for single-phase stand-alone inverter needs one load-current sensor to achieve load disturbance rejection [20]. When extended for a three-phase application, the current-sensor cost penalty would get tripled. As an economical alternative, the real-world sensors are replaced by observers [16, 26], including but not limited to a third-order observer in abc frame [27], and fourth-order observer in d-q frame [28]. The omission of sensors improves system reliability. However, observer and controller design are indeed two different separate processes. Additional observer design complicates the control scheme design, which is involved with Riccati equation solving, bandwidth allocation based on Kalman filter theory [29]. In summary, conventional Lyapunov-based control for stand-alone inverters is inherently a single current-loop control scheme having steady-state errors. Modified dual-loop control achieves less steady-state error and better dynamic performance via artificial importation of the capacitor voltage feedback. The large-signal stability of the modified approach can be assured on the condition that the imported controller gain is large enough. Load-current sensors or observers are indispensable in these approaches. As a better alternative, the proposed dual-loop Lyapunov-based control scheme can rigorously guarantee the global large-signal stability of the system with lower implementation cost and minimized steady-state errors, whose contributions can be mainly summarized as follows: (1) The proposed approach inherently has dual control loops without any afterward modification. Stability analysis proves that the global large-signal stability of the system can be rigorously guaranteed unconditionally both for linear load and nonlinear load without any premises. (2) The load disturbance is compensated adaptively, saving three load-current sensors or observers. (3) The proposed approach inherently has integrators, which can theoretically achieve zero steady-state-error regulation according to the internal model principle.

50

4 Lyapunov-Based Control of Three-Phase Stand-Alone Inverters …

4.2 Preliminary of the Proposed Adaptive Dual-Loop Lyapunov-Based Control: Mathematical Modelling 4.2.1 Average Model of the Investigated System The topology of the investigated three-phase stand-alone inverter is sketched in Fig. 4.1, where L, C denote the nominal filter inductance and capacitance. r is the equivalent series resistance of the filter inductor. E denotes the dc-link voltage. According to KCL and KVL, dynamic equations of the stand-alone inverter in abc frame are given by Ldi Labc /dt + r i Labc = μabc · 0.5E − voabc

(4.1)

Cdvoabc /dt = i Labc − ioabc

(4.2)

where iLabc , voabc , ioabc , µabc denote the vectors of the filter inductor currents [iLa iLb iLc ]T , filter capacitor voltages [voa vob voc ]T , load currents [ioa iob ioc ]T and switching functions [μa μb μc ]T . Since it is more accurate to regulate dc elements in dq frame compared with their sinusoidal counterparts in abc frame. With abc-dq coordinate transformation, (4.1) and (4.2) can be transformed to

S1

Ldi Ld /dt + ri Ld = μd · 0.5E − vod + ωLi Lq

(4.3)

Ldi Lq /dt + ri Lq = μq · 0.5E − voq − ωLi Ld

(4.4)

Cdvod /dt = i Ld − i od + ωCvoq

(4.5)

Cdvoq /dt = i Lq − i oq − ωCvod

(4.6)

S2

S3

r

E S4

S5

S6

L iLa iLb iLc

voa

ioa vob

iob voc ioc

Nonlinear...

C dc bus

PWM

a

b

c

N

Fig. 4.1 Investigated three-phase stand-alone inverter with LC filter

Linear load

4.2 Preliminary of the Proposed Adaptive Dual-Loop Lyapunov-Based …

51

where ω is equal to 100 π. In the d-q frame, μd , μq denote the switching functions, vod , voq signify the load voltages, iLd , iLq represent the inductor currents, and iod , ioq denote the load currents. μd , μq are derived out to ensure V˙ < 0.

4.2.2 Load Voltage References vodref , voqref , and Inductor Current References iLdref , iLqref To realize voltage-oriented control objective, load voltages vod , voq are enforced to track their dc references vodref , voqref , expressed as ⎧

vod → vodr e f voq → voqr e f

(4.7)

where vodref , voqref are desired quiescent dc constants. Substituting(4.7)–(4.5) and (4.6), it yields out the expression of inductor current tracking references iLdref , iLqref , given by ⎧

i Ld → i Ldr e f = i od − ωCvoq i Lq → i Lqr e f = i oq + ωCvod

(4.8)

where −ωCvoq , + ωCvod are kept in iLdref and iLqref on the objective of dq decoupling.

4.2.3 Model of the Load Currents and Proposed Adaptive Laws According to (4.8), iod , ioq are indispensable to generate iLdref , iLqref to achieve load disturbance rejection. Instead of sampling load currents directly via current sensors of observers, iod , ioq are estimated online using the Lyapunov-based method. To this end, they are preliminarily modeled as ⎧

i od =vodr e f /Z i oq =voqr e f /Z

(4.9)

where Z denotes the unknown load impedance, which is regarded as the external disturbance, susceptible to load distortion due to load uncertainty or nonlinearity. To address the load disturbance, two adaptive terms εˆ d , εˆ q are defined in d frame and q frame respectively to replace the unknown 1/Z, transforming (4.9) to ⎧

iˆod = vodr e f · εˆ d iˆoq = voqr e f · εˆ q

(4.10)

52

4 Lyapunov-Based Control of Three-Phase Stand-Alone Inverters …

Detailed derivation and analytic expression of the adaptive laws εˆ d , εˆ q will be given in the following chapter.

4.2.4 Modified Inductor Current References iLdref , iLqref Incorporated with Adaptive Laws After incorporation of the adaptive terms εˆ d , εˆ q described by (4.10), inductor current reference in (4.8) will be transformed to ⎧

i Ld → i Ldr e f = iˆod − ωCvoq = vodr e f εˆ d − ωCvoq i Lq → i Lqr e f = iˆoq + ωCvod = voqr e f εˆ q + ωCvod

(4.11)

4.3 Derivation of Proposed Adaptive Decoupled Dual-Loop Lyapunov-Based Control Scheme 4.3.1 Proposed Weighted All-in-One Lyapunov Function V An adaptive weighted Lyapunov function (V ) is proposed in this chapter, which is the key to deriving the adaptive laws (ˆεd , εˆ q ) and switching functions (μd , μq ) in d-q frame. Defined V not only contains current-loop errors (e1 , e3 ) and voltage-loop errors (e2 , e4 ) but also contain virtual tracking errors (˜εd , ε˜ q ), V =

) 1 ( ) 1 ( ) 1 ( 2 L e1 + e32 + Cρ e22 + e42 + γ ε˜ d2 + ε˜ q2 2 2 2

(4.12)

where ρ, γ are two weighting factors, providing to degrees of freedom to adjust the controller’s performance (ρ > 0, γ > 0). Here, e1 , e3 denote inductor current-loop tracking errors, and e2 , e4 denote load voltage-loop tracking errors, defined as ⎧

⎧

e1 = i Ld − i Ldr e f e3 = i Lq − i Lqr e f

e2 = vod − vodr e f e4 = voq − voqr e f

(4.13)

ε˜ d , ε˜ q represent virtual load estimating errors, defined as ⎧

Δ

~ εd = ε d − 1/Z ~ εq = εq − 1/Z Δ

(4.14)

4.3 Derivation of Proposed Adaptive Decoupled Dual-Loop …

53

If ε˜ d , ε˜ q are regulated to approach zero, load disturbance will have been compensated.

4.3.2 Derivation of the Switching Functions and Adaptive Laws Equations (4.13)–(4.14) would have been regulated to be zero if (4.12) reaches the equilibrium point, expressed as [e1 , e2 , ε˜ d , e3 , e4 , ε˜ q ]T = [0, 0, 0, 0, 0, 0]T . According to the direct method of Lyapunov’s theory, investigated equilibrium point is globally asymptotically stable unless V satisfies the following three premises: (1) V ≥ 0, and V = 0 if and only if e1 , e2 , e3 , e4 , ε˜ d , ε˜ q = 0; (2) V → ∞ if any of e1 , e2 , e3 , e4 , ε˜ d , ε˜ q → ∞; (3) dV/ dt < 0 for all points except the equilibrium point. Since V in (4.12) is inherently positive definite, premise (1), (2) are fulfilled automatically. Premise III is the key to designing the control law (μd , μq ) and adaptive laws (ˆεd , εˆ q ). According to (4.12), analytic expression of V˙ turns out to be V˙ = Le1 e˙1 +Cρe2 e˙2 + γ ε˜ d ε˙˜ d + Le3 e˙3 + Cρe4 e˙4 + γ ε˜ q ε˙˜ q

(4.15)

Taking first-order derivatives of e1 , e3 defined in (4.11). Then, substituting (4.3), (4.4), (4.13) to their derivatives, it gives ⎧

( ) e˙1 = (0.5μd E − vod + ωLi Lq − ri Ld )/L − vodr e f ε˙˜ d e˙3 = 0.5μq E − voq − ωLi Ld − ri Lq /L − voqr e f ε˙˜ q

(4.16)

Here, load voltages (vod , voq ) can be regarded as dc constants when calculating de1 /dt, de3 /dt since current-loop dynamics are much faster than those of the voltageloop counterparts. Similarly, according to (4.5), (4.6) and (4.13), voltage-loop error dynamics can be found, ⎧

) ] [ ( e˙2 = [i Ld − (e2 + vodr e f )/Z + ωCvoq ]/C e˙4 = i Lq − e4 + voqr e f /Z − ωCvod /C

(4.17)

Taking first-order derivatives of (4.14), it yields out the dynamics associated with load estimating errors, given by ⎧

ε˙˜ d = d˜εd /dt = dˆεd /dt ε˙˜ q = d˜εq /dt = dˆεq /dt

(4.18)

Substituting derived error dynamics (4.16)–(4.18) to (4.15), it yields out the expression of V˙ , given by

54

4 Lyapunov-Based Control of Three-Phase Stand-Alone Inverters …

( ) V˙ = e1 μd E/2 − vod + ωLi Lq − ri Ld − Lvodr e f ε˙˜ d [ ) ] ( + ρe2 i Ld − e2 + vodr e f /Z + ωCu oq + γ ε˜ d ε˙˜ d ( ) +e3 μq E/2 − voq − ωLi Ld − ri Lq − Lvoqr e f ε˙˜ q [ ) ] ( + ρe4 i Lq − e4 + voqr e f /Z − ωCvod + γ ε˜ q ε˙˜ q

(4.19)

Expanding all the state variables in (4.19) to be the summation of the references plus their respective tracking errors using (4.13), it yields out that ( ) V˙ = e1 μd E/2 − vodr e f − e2 + ωLi Lq − ri Ld − Lvodr e f ε˙˜ d [ ) ] ( + ρe2 i Ldr e f + e1 − e2 + vodr e f /Z + ωCvoq + γ ε˜ d ε˙˜ d ( ) +e3 μq E/2 − voqr e f − e4 − ωLi Ld − ri Lq − Lvoqr e f ε˙˜ q [ ) ] ( + ρe4 i Lqr e f + e3 − e4 + voqr e f /Z − ωCu od + γ ε˜ q ε˙˜ q

(4.20)

Substituting iLdref , iLqref as shown in (4.11)–(4.20) and arrange, it contributes to import the adaptive terms εˆ d , εˆ q into V˙ , which gives ( ) V˙ = e1 μd E/2 − vodr e f − e2 + ωLi Lq − ri Ld − Lvodr e f ε˙˜ d [ ) ] ( + ρe2 vodr e f εˆ d + e1 − e2 + vodr e f /Z + γ ε˜ d ε˙˜ d ( ) +e3 μq E/2 − voqr e f − e4 − ωLi Ld − ri Lq − Lvoqr e f ε˙˜ q [ ) ] ( + ρe4 voqr e f εˆ q + e3 − e4 + voqr e f /Z + γ ε˜ q ε˙˜ q

(4.21)

By substituting (4.14)–(4.21), expression of the adaptive terms εˆ d , εˆ q are further expanded out, transforming (4.21) to ( ) V˙ = e1 μd E/2 − vodr e f − e2 + ωLi Lq − ri Ld − Lvodr e f ε˙˜ d ] [ + ρe2 vodr e f ε˜ d + e1 − e2 /Z + γ ε˜ d ε˙˜ d ( ) +e3 μq E/2 − voqr e f − e4 − ωLi Ld − ri Lq − Lvoqr e f ε˙˜ q [ ] + ρe4 voqr e f ε˜ q + e3 − e4 /Z + γ ε˜ q ε˙˜ q

(4.22)

After collecting and merging the similar items in (4.22), the negative definite part, −ρ(e2 2 + e4 2 )/Z, and the other sign-uncertain part can be separated out, given by ( ) ( ) ) ( V˙ = −ρ e22 + e42 /Z +˜εd ρe2 vodr e f + γ ε˙ˆ d + ε˜ q ρe4 voqr e f + γ ε˙ˆ q ' ' '' 0, and B is an arbitrary real number. Thus, 1/Z + (1/Z)H = 2G > 0, transforming (35) to ⎛

⎞ −2k/L 0 0 dVd /dt = y H ⎝ 0 −2G/C 0 ⎠ y < 0 0 0 0

(4.45)

which shows L, C will not affect the system stability if k > 0 and G > 0, which can be easily satisfied. And k, L, C, Z could vary at a large range without deteriorating the system stability.

4.5.2 Power Loss Analysis, Switching Frequency (f s ) Selection and Output LC Filter Design In the investigated system, the conduction loss is affected by the load current and duty cycle. The forward cut-off loss is affected by the inherent characteristic of the semiconductors. The switching loss of the IGBT and diode is determined by their inherent property, environmental temperature, switching frequency, and dclink voltage. The driver loss contains the IGBT gate charging loss and integrated circuit loss. The proposed method calculates the switching functions (μa , μb , μc ) and compares them with the triangular carrier wave to generate the control signals (s1 ~ s6 ) for six power switches. Higher f s will yield out lower THD at the expense of more power loss and lower efficiency. However, f s is determined by the frequency of the triangular carrier wave, rather than the switching functions. Therefore, the proposed control method itself does not affect the power loss or efficiency. The proposed approach realizes load voltage control with low THD due to its inherent load disturbance adaptivity. The well-known triple harmonic injection technique can be further used for THD suppression and dc voltage utilization rate improvement. Since it modifies the switching functions sent to the PWM modulator, conduction loss and system efficiency will be affected. The selection of f s is demand-dependent, which relates to EMI standard, acoustic concern, THD of load voltage, etc. Higher f s generally requires more advanced semiconductor devices/digital processors, better thermal management, while it leads to smaller and lighter LC components requirements. Inspired by [20, 24], f s is selected as 10 kHz. It strikes a trade-off between √ lower THD and higher efficiency. √ The cut-off frequency f 0 = 1/(2π LC) and quality factor Q = Z C/L) are used for the LC filter design [30]. −40 dB attenuation can be realized at f s if f 0 is selected as 0.1 f s . Q is set as 0.707 to achieve an optimal system damping with a 10 ⎧ kW load. √ Desired LC parameters are calculated out to be: L = 2Z√/2π f 0 ≈ 0.9 × 10−3 H C = 1/2 2Z π f ≈ 28.2 × 10−6 F

4.5 Stability Analysis and Controller Design Guidelines

61

Table 4.1 Nominal system parameters E

fs

r

L

C

vm

400 V

10 kHz

0.096 Ω

1 mH

30 μF

115 V

√ 2

ω 100 π

LC components having the nearest value to calculated parameters are used for prototyping, as seen in Table 4.1. Note that the proposed approach is a general approach that is feasible for plants with different LC parameters.

4.5.3 Controller Gains Selection Via Poles Placement The characteristic equation of the system (4.32) is given by det(s I − A) = as 3 + bs 2 + cs + d = 0

(4.46)

where ‘I’ denotes an identity matrix, and ‘s’ denotes eigenvalue. The coefficients of (4.46) is given by a = 1, b =

k ZC + L ρ Z Lvm2 + 2Zργ + 2kγ ρkvm2 , c= , d= . ZC L 2γ L ZC 2γ LC

Through proper controller gains selection, (4.46) can be designed to have the following equivalent form: ) ( det(s I − A) = (s + mξ ωn ) s 2 + 2ξ ωn s + ωn2 = 0

(4.47)

which has one real pole (s1 ) and two conjugated poles (s2 , 3 ), s1 = −mξ ωn ,

√ s2,3 = −ξ ωn ± jωn 1 − ζ 2 ,

(4.48)

where ξ and ωn denote the damping ratio and natural frequency of the conjugated poles, and m is a positive constant. Equation (4.46) can be simplified to an equivalent second-order system if s2 , 3 are allocated to be the dominant poles. To this end, the absolute value of the real part of the constant m in s1 is designed to be 5 times as large as that of s2 ,3 [31]. Namely, √ system damping 2/2. (4.48) is set as 5. ξ is fixed to the optimal √ k, ρ, γ satisfying m = 5 and ξ = 2/2 can be solved out by forcing the coefficients of (4.46) to follow (4.47), which gives √ k = 7 2ωn L/2 − L/(ZC)

(4.49)

( √ ) / ρ = 6LCωn2 − 5ωn3 ZC 2 L/ 7ZCωn − 2 − k Z

(4.50)

62

4 Lyapunov-Based Control of Three-Phase Stand-Alone Inverters …

[ ( )] ) √ ( γ = ρ 7/ 10ωn2 C − 2/ 10ωn3 ZC 2 vm2

(4.51)

According to the well-known Nyquist-Shannon sampling theorem, ωn /2π should be strictly less than f s /2. For practical application, ωn /2π is generally set as 1/6 of f s f or a switched power supply [6], which yields out that ωn = 2π f s /6

(4.52)

Equation (4.52) also shows that the proposed control method can flexibly operate with different f s . The controller gains are selected under the harshest condition by setting Z as + ∞ so that enough system damping can be ensured regardless of load variation. Substituting parameters in Table 4.1 to (4.52), (4.49)–(4.51) consecutively, the controller gains are calculated out to be: k = 51.8, ρ = 17.4, γ = 97.8.

4.6 Test Results A 2 kW three-phase stand-alone inverter is used to do tests. Used system parameters are specified in Table 4.1. Specifications of the linear load and nonlinear load are shown in Fig. 4.3. Equation (4.30) is used to set references by default.

4.6.1 Performance of Proposed Approach Simulation results in Table 4.2 show that the THD of load voltage for phase B gradually reduces from 0.46 to 0.36% as the linear load power level increases from 0 to 16 kW. Table 4.3 demonstrates that THD of load voltage tends to have a proportional relationship to the nonlinear load power levels. As seen in Fig. 4.4, linear load firstly steps from no load to 100% P (2 kW) and then restores to no-load after 40 ms, where the regulated load voltage is almost immune to such a scale of load distortion, which verifies the great load disturbance adaptivity of proposed approach.

Ld

Ld = 4.7 mH

R

Linear Resistive Load

Cd

Cd = 200 F

Rload

Nonlinear Rectifier Load

Fig. 4.3 Linear load and nonlinear load used to do case studies

4.6 Test Results

63

Table 4.2 THD % of the load voltage for each phase when supplying linear resistive load THD % for each phase R(Ω)

Power (W)

A

B

C

2.5

16,000

0.37

0.36

0.37

5

8000

0.41

0.42

0.41

10

4000

0.44

0.45

0.45

20

2000

0.44

0.45

0.44

40

1000

0.47

0.46

0.47

80

500

0.47

0.47

0.46

160

250

0.5

0.5

0.47

320

125

0.49

0.51

0.49

∞

0

0.46

0.46

0.47

Table 4.3 THD % of the load voltage for each phase when supplying nonlinear rectifier load THD % for each phase Rload (Ω)

A

B

15

3.54

3.55

C 3.53

30

2.11

2.13

2.13

60

1.52

1.52

1.5

120

1.26

1.26

1.24

240

0.86

0.89

0.84

480

0.63

0.63

0.62

960

0.53

0.56

0.55

∞

0.44

0.45

0.45

Fig. 4.4 Performance of the proposed approach under the linear load step

64

4 Lyapunov-Based Control of Three-Phase Stand-Alone Inverters …

Fig. 4.5 Performance of the proposed control method under nonlinear rectifier load step

In Fig. 4.5, the nonlinear load step is considered. As Rload steps from 60 Ω to noload, THD of load voltage reduces from approximately 1.5–0.45%, where the load voltage always remains sinusoidal both in steady states and transient states, which shows that the proposed control approach also has good load disturbance adaptivity when supplying nonlinear rectifier load. The proposed control scheme is robust against source variation, as shown in Fig. 4.6a, b, dc-link voltage is stepped from 400 to 300 V, and then restored to 400 V after 40 ms. In this case, the system stability and power quality are almost immune to such a scale of source variation both for nonlinear rectifier load and no-load. To test the robustness of the proposed approach against LC variation, 10 ~ 1000% L and 10 ~ 1000% C scale of plant parametric mismatch have been considered by changing the filter inductance/capacitance of the main circuit in Simulink (Simscape) environment. As seen in Fig. 4.7a, when supplying the linear load, an increase of L or C all contribute to suppressing THD. In the harshest condition, 40% C and 40% L yield out about 4.5% THD, and100% C and 10% L produce about 4% THD. As seen in Fig. 4.7b, when supplying nonlinear rectifier load, an increase of L or C will do help to suppress THD until it reaches around 1.5%. Under the harsher scenario, 40% C and 40% L yield out less than 6% THD, and 100% C together with 10% L produces less than 4.5% THD. Under the condition of 100% parametric matching, THD of the load voltage reaches less than 0.5% when supplying a 2 kW linear load, while this value is around 1.5% when supplying nonlinear rectifier load. As shown in Fig. 4.8, voqref is set as 0 V, while the d-frame reference vodref initially steps from vm to 0 V and then restores to vm after two fundamental periods when supplying 2 kW linear resistive load. Every time vodref step happens, vod tracks vodref instantly, while voq always remains 0 V with negligible distortion. It demonstrates that the proposed control has realized d-q decoupled control of the load voltages. As shown in Fig. 4.9, investigated three-phase stand-alone inverter initially operates with a 100%P linear resistive load. Then, 200%P overload happens. After two fundamental periods, the system goes into the recovery process, expressed as 200%P → 100%P. As seen, the regulated load voltages remain sinusoidal with negligible disturbance even under the process of overload and recovery.

4.6 Test Results

65

Fig. 4.6 Performance of the proposed control scheme under the dc-link voltage step. a Nonlinear rectifier load (Rload = 60 Ω); b no load

4.6.2 Comparisons Between the Proposed Approach and Existing Control Schemes As seen in Fig. 4.2, the implementation of the proposed approach does not require any load-current sensors or observers, which are indispensable in the other control scheme. The THD of load voltage using the proposed approach is 0.46%, which is lower than that in the other control schemes. Both PI control and the proposed control have the internal model of the dc references: integrators. The steady-state error of the proposed approach and PI control reaches 0.017% and 0.123% respectively, much smaller than that in the other approaches. However, PI control merely ensures local stability at a specific operating point. The conventional Lyapunov-based approach can rigorously guarantee the global large-signal stability of the closed-loop system. However, its steady-state error reaches 17.036% due to the missing of voltage feedback loop. The modified dual-loop Lyapunov-based approach reduces the steady-state error to 1.046% via additional importation of capacitor voltage feedback. Large-signal stability of the system can be ensured on the premise that the imported voltage feedback gain is larger than a specific lower bound. As a better alternative, the

66

4 Lyapunov-Based Control of Three-Phase Stand-Alone Inverters … 60%L

100%L

THD (%)

THD(%) Linear Load

40%L

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 40%C

60%C

150%L 4 3.5 3 2.5 2 1.5 1 0.5 0

200%L

300%L

500%L

1000%L 100%C

100%C 150%C 200%C 300%C 500%C 1000%C

(a) 40%L

60%L

100%L

150%L

THD (%)

THD(%) Nonlinear Load

6 5 4 3

200%L

300%L

500%L

4 3.5 3 2.5 2 1.5 1 0.5 0

1000%L 100%C

2 1 0 40%C

60%C

100%C 150%C 200%C 300%C 500%C

1000%C

(b) Fig. 4.7 Total harmonic distortion (THD %) of the load voltage when supplying linear and nonlinear load under plant parametric mismatch. a Linear load (R = 20 Ω); b nonlinear rectifier load (Rload = 60 Ω)

Fig. 4.8 Performance of the proposed control method under d frame load voltage reference step

4.7 Conclusion

67

Fig. 4.9 Performance of the proposed control method under overload and recovery scenario

proposed approach inherently has dual control loops, which can rigorously guarantee the global large-signal stability of the system without any premises. Meanwhile, it displays better dynamic performance compared to the conventional approach and PI control. MPC approach achieves great dynamic performance at the expense of heavy computation burden and a varying f s . The computation cost of the proposed approach is lower than MPC, while higher than PI control. The proposed approach is generally implemented on a floating-point processor. Table 4.4 gives a summary of the comparisons between the existing control schemes and the proposed control approach of this chapter for the three-phase standalone inverters. It can be seen that, the proposed control approach of this chapter has obvious advantages on large-signal stability, load-current sensors, switching frequency, steady-state error, dynamic performance and THD performance.

4.7 Conclusion This chapter proposes a decoupled Lyapunov-based control strategy for a three-phase stand-alone inverter in the d-q frame. The proposed control scheme inherently has dual control loops without any artificial configuration. It can rigorously guarantee the global large-signal stability of the system both for linear load and nonlinear load. The load disturbance is suppressed adaptively, saving three load-current sensors or observers compared to existing Lyapunov-based control methods. The adaptive laws are embedded with integrators, which can theoretically track dc references with zero steady-state errors according to the internal model principle. In experiments, he steady-state error of the proposed approach is 0.017% ≈ 0. The proposed control method can operate with different f s flexibly. Then, 3 controller gains are tuned via explicit formulas quantitatively. The proposed control method itself does not affect the power loss or efficiency. It realizes low THD of the load voltage due to its inherent

68

4 Lyapunov-Based Control of Three-Phase Stand-Alone Inverters …

Table 4.4 Comparisons between existed control schemes and proposed approach for three-phase stand-alone inverters References

[8]

[27]

[3, 28, 32, 33]

This chapter

Control schemes

Feedforward PI control

MPC approach

Conventional approach

Feedback loops

Single loop

None

Single loop

Design guideline

Impedance shaping and bode plot

Model-based prediction and cost function

Large-signal stability

No (local stability)

Not reported

Yes

Yes (conditional)

Yes

Computation burden

Low

Very High

Medium

Medium

Medium

Load-current sensors

3

0 (require observers)

3

3

0

Switching frequency f s

Fixed

Varying

Fixed

Fixed

Fixed

Fund. 50 Hz (Volt)

162.4

156.7

134.9

160.9

162.6

Steady-state error (%)

0.123

3.629

17.036

1.046

0.017

Dynamic response

Fair

Best

Fair

Better

Better

THD% of load voltage

0.48

2.31

2.86

0.49

0.46

3rd harmonic %

0.0229

0.1403

0.6070

0.0228

0.0223

5th harmonic %

0.0166

0.2474

0.6549

0.0179

0.0061

7th harmonic %

0.0493

0.1272

0.4154

0.0599

0.0434

Modified approach

Proposed approach

Modified dual Inherent loop dual loop ˙ Lyapunov function (V ), V < 0, and closed-loop pole placement

load disturbance adaptivity. As the linear load power level increases from 0 to 16 kW, THD of load voltage gradually reduces from 0.46 to 0.36%. When supplying the nonlinear load, as Rload steps from 60 Ω to no-load, THD of load voltage reduces from 1.5 to 0.45%. The proposed approach also has displayed good robustness, where 10–1000% scale of LC variation, 300–400 V scale of source variation has been considered. How to extend the proposed approach to different types of inverters e.g., Z-source inverters, has been incorporated into our future work.

References

69

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Chapter 5

An Ellipse-Optimized Composite Backstepping Control Strategy for a Point-of-Load Inverter to Improve Its Large-Signal Stability Under Load Disturbance in the Shipboard Power System

Nomenclature E ω = 2πf L, C vod , voq (x 1 , x 3 ) iLd , iLq (x 2 , x 4 ) iod , ioq x1 * , x3 * vm x2 * , x4 * z1 , z3 z2 , z4 V 1, V 2 V 3, V 4 V˙ (dV /dt) μd , μq k1, k2, k3, k4 k1*, k2* λ1 , λ2 ωn ξ S MNOP A, B, C G V o, V c

Dc-link voltage Fundamental angular frequency (100 π) Nominal filter inductance, capacitance Load voltages in d, q frames Inductor currents in d, q frames Load currents in d, q frames Load-voltage references in d, q frames Desired load-voltage amplitude Inductor-current references in d, q frames Load voltage tracking errors in d, q frames Current tracking errors in d, q frames Defined Lyapunov functions in d frame Defined Lyapunov functions in q frame First-order time derivative of V Switching functions in d, q frames Controller gains (> 0) Optimized controller gains Solutions of the characteristic equation The equivalent natural resonance frequency Equivalent damping ratio The area of the rectangle ⛛MNOP . System matrixes of the state-space model Feedback gain matrix of the Kalman filter Lyapunov functions for robustness analysis

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Zhang et al., Stability Enhancement Methods of Inverters Based on Lyapunov Function, Predictive Control, and Reinforcement Learning, https://doi.org/10.1007/978-981-19-7191-4_5

71

72

5 An Ellipse-Optimized Composite Backstepping Control Strategy …

5.1 Introduction Electrification of the shipboard power system (SPS) has started ever-faster change both for surface and submarine applications with different uses and tonnages recently [1]. The hybrid MG technology serves as the most promising technique to realize such up-gradation with improved system efficiency, reliability, and resilience [2, 3]. As shown in Fig. 5.1, the hybrid MG in the SPS is an islanded mobile entity that integrates DG, energy storage systems (ESs), and consumptive ends. In the hybrid MG, the bus conversion stage interlinks the ac MG subsystem and the dc MG subsystem together [4]. Such a dual-bus comprehensive system achieves redundancy design and combines the strength of ac MG and dc MG as well [5]. In the hybrid MG of the SPS, sensitive ac loads exist, such as telecommunication facilities, computers, which cannot operate normally unless load voltage with high power quality is supplied. For this reason, they can hardly be driven by the ac bus directly due to the voltage sag/unbalance, harmonics, and electromagnetic interference (EMI), that promote the birth of the point-of-load inverter (POL) system [6].

Fuel generator

Wind turbine

...

AC DC

DC AC Solar panels

Motor On/off

DC

AC

DC

DC Bus conversion

...

AC

DC

AC bus Point-of-load inverter

...

Utilities

DC

DC

DC PC

ESs Battery banks Bidirectional

AC

Sensitive load

...

DGs Rail gun, radar, laser...

Telecom...

DC DC DC bus

Fig. 5.1 Investigated hybrid MG in the shipboard power systems (SPS)

Super C

...

5.1 Introduction S1

E

73 S2

S3

Cdc S4

S5

S6

L iLa iLb iLc

voa

ioa vob

iob voc ioc

Nonlinear...

C dc bus

PWM

μ a μ b μc

N

Linear load

Fig. 5.2 Investigated POL inverter system in the hybrid MG of the SPS

As depicted in Fig. 5.2, the POL inverter is susceptible to load disturbance due to load uncertainty/nonlinearity. Meanwhile, it operates in complex and harsh environmental conditions. The varying temperatures, moistures, and air salinity inevitably cause plant parametric variations. To realize load voltage-oriented control objective under load disturbance and plant parametric uncertainty, advanced control schemes should be devised with full consideration of system stability, reliability, power density, and the implementation cost [7]. Conventional linear control methods, including proportional integral derivative (PID) control and proportional resonant (PR) control, have been widely used both in academia and industry in the past decades [8, 9]. They have valid mature tools for controller gains selection, including but not limited to the Nyquist criterion, gain margin, and phase margin based on bode plot [10]. However, linear approaches are based on the small-signal model of the system in the vicinity of specific operation points [11]. Thus, only the local stability of the closed-loop system can be guaranteed around the operation point [12]. Once the POL inverter system is exposed to large perturbations away from the operation point, system stability may deteriorate. Furthermore, linear approaches merely expect linear behavior of the closed-loop system that does not thoroughly leverage the system’s potential, leading to limited dynamic response [13]. To achieve a better dynamic response, nonlinear control methods emerge. In [14], conventional integer PID control is elevated to fractional-order PID control with greatly enhanced tracking accuracy and dynamic response. However, tuning of the controller gains is the current obstacle due to the mutual coupling effect. Model predictive control (MPC) is known for its great dynamic performance at the cost of a comparatively high computation burden [15, 16]. Apart from this, MPC lacks a commonly acknowledged effective method to tune the weighting factor so far. Deadbeat control is devised in a discrete-time domain, which could theoretically ensure zero steady-state error regulation with minimum rise time by assigning its poles at the origin of the z plane. However, it is comparatively sensitive to plant parameter uncertainty with an inherent two-step digital delay [17]. Sliding-mode control is robust

74

5 An Ellipse-Optimized Composite Backstepping Control Strategy …

against plant parametric variations but suffers from the chattering phenomenon [18]. Repetitive control is an expert at tracking/rejecting periodic reference/disturbance, but it is accompanied by relatively long internal time delay [19]. The above nonlinear approaches are devised from different perspectives with caseby-case advantages and defects. While they have two limitations in common. For one thing, they fail to rigorously guarantee the large-signal stability of the system. For another, it lacks effective valid guidelines for controller parameter selection, not to mention their optimization. To ensure the large-signal stability of the system, Lyapunov-based control emerges, which is designed based on an all-in-one Lyapunov function (V ) according to the direct method of Lyapunov’s theory. In conventional Lyapunov-based control, the pseudo control variables’ reference and switching function are selected to ensure dV /dt < 0 [20]. However, the derived final control law merely has a single current control loop [21, 22]. When extended for the POL inverter regulation, the load voltage is accompanied by steady-state error and sluggish dynamic response. To address this problem, [23] artificially imports the filter capacitor voltage feedback so that a dualloop control scheme can be realized. However, this action fails to rigorously guarantee the negative definiteness of dV /dt, which means the large-signal stability cannot be always ensured. It also requires load-current sensors for practical implementation. Recently, adaptive control is reported in [24]. It inherently has dual control loops to achieve better steady-state and transient performance, which could also rigorously guarantee the large-signal stability unconditionally. Meanwhile, it obviates the requirement of load-current sensors. However, the design of V in the above Lyapunov-based control has no explicit rules. Comparatively, backstepping control provides a systematic way to construct the Lyapunov function, which contributes to deriving the pseudo control variables’ reference and final control law within finite recursive steps [25]. Besides, the controller gains selection guideline in [1–3, 5–11, 13–24, 26–84] are just qualitatively illustrated without quantitative formulas. Currently, methods to optimize the controller gains in nonlinear control are still on the way. The trial-and-error method is straightforward but onerous and time-consuming [85]. After linearization around the equilibrium point, conventional linear tools can be implemented, such as eigenvalue analysis, root-locus technique, and bode plot [84]. Artificial intelligence (AI) aide method is a promising technique, involved with advanced algorithms and complicated online or off-line training processes [86], such as particle swarm optimization (PSO) algorithm, multivariable multi-objective genetic algorithm (MMGA) [14], artificial neural network (ANN) [87], etc. However, the above approaches do not provide explicit formulas to select the controller gains. When extended for systems with different parameters, the whole tuning process requires to be repeated. Motivated by the limitations of the above methods, this chapter proposes an ellipse-optimized composite backstepping control scheme to regulate the POL inverter in the SPS, whose main contributions can be summarized as three points:

5.2 Preliminary of the Ellipse-Optimized Composite Backstepping …

75

(1) The composite backstepping controller inherently has decoupled dual control loops, that could rigorously guarantee the large-signal stability of the system. (2) To compensate for the load disturbance, a Kalman filter is designed to estimate and feedforward the load currents to the backstepping controller in the replacement of three current sensors, resulting in minimized implementation cost and enhanced system reliability. (3) An intuitive ellipse-based strategy with explicit formulas is proposed to optimize the controller gains, which not only theoretically achieves the optimal system damping and maximized dynamic response, but also helps select the feedback gain matrix of the Kalman filter quantitatively. Robustness analysis of the proposed control scheme has been incorporated both theoretically and experimentally.

5.2 Preliminary of the Ellipse-Optimized Composite Backstepping Controller: Mathematical Modelling 5.2.1 Dynamic Equations of the Investigated POL Inverter As shown in Fig. 5.2, according to KCL and KVL, the dynamic equation of the three-phase balanced system in vector-based form can be expressed as ⎧

Labc L didt = 21 μabc E − voabc dvoabc C dt = i Labc − ioabc

(5.1)

where iLabc , voabc , ioabc , μabc denote the vectors of the filter inductor currents [iLa iLb iLc ]T , filter capacitor voltages [voa vob voc ]T , load currents [ioa iob ioc ]T and switching functions [μa μb μc ]T , composed of balanced three-phase individual quantities. In comparison with the three-phase ac quantifies in abc frame, compatible state variables in the synchronously rotating reference frame (d-q frame) are dc quantities, which is much easier to be regulated. With the help of abc-dq coordinate transformation, (5.1) is transformed to d-q-frame counterpart, expressed as ⎧ ⎪ L di Ld ⎪ ⎪ ⎨ didtLq L dt dvod ⎪ C ⎪ dt ⎪ ⎩ C dvoq dt

= 21 μd E − vod + ωLi Lq = 21 μq E − voq − ωLi Ld = i Ld − i od + ωCvoq = i Lq − i oq − ωCvod

where ω = 2πf = 100 π denotes the fundamental angular frequency.

(5.2)

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5 An Ellipse-Optimized Composite Backstepping Control Strategy …

For the convenience of representation and derivation, the following equivalent substitution is performed, given by x1 = vod x2 = i Ld x3 = voq x4 = i Lq

(5.3)

C x˙1 = x2 − i od + ωC x3

(5.4)

L x˙2 = 0.5μd E − x1 + ωL x4

(5.5)

C x˙3 = x4 − i oq − ωC x1

(5.6)

L x˙4 = 0.5μq E − x3 − ωL x2

(5.7)

which transforms (5.2) into

5.2.2 Control Objectives: Load Voltage References x1 *, x3 * The control objective is to regulate the three-phase load voltage to be purely sinusoidal. To fulfill this objective, x 1 , x 3 are forced to track their dc references (x 1 * , x 3 * ) respectively, ⎧

x1 → x1∗ = vm x3 → x3∗ = 0

(5.8)

5.3 Recursive Derivation and Implementation of the Proposed Composite Backstepping Controller 5.3.1 Two-Step Backstepping Derivation in d Frame 5.3.1.1

Step One: Derivation of Inductor Current Reference x2 *

Initially, load voltage tracking error in d frame is defined as z 1 = x1∗ − x1

(5.9)

5.3 Recursive Derivation and Implementation of the Proposed Composite …

77

The first Lyapunov function associated with the defined load voltage tracking error in the d frame is defined as 1 2 z 2 1

(5.10)

V˙1 = z 1 z˙ 1

(5.11)

V1 = Differentiating (5.10), it yields out that

According to (5.4) and (5.9), the detailed expression of (5.11) can be derived out, expressed as ) ( ( ∗ ( x2 − i od + ωC x3 ∗ ˙ V1 = z 1 x˙1 − x˙1 = z 1 x˙1 − C

(5.12)

The pseudo-current-control-loop reference x 2 * is selected to assure V˙1 ≤ 0, given by x2∗ = i od − ωC x3 + k1 C z 1 + C x˙1∗

(5.13)

where k 1 is an imported positive constant, serving as the first controller parameter (k 1 > 0). If x 2 is regulated to follow x 2 * , (5.12) will be transformed to V˙1 = −k1 z 12 ≤ 0

(5.14)

which is always negative definite.

5.3.1.2

Step Two: Derivation of the d-Frame Switching Function

The pseudo inductor current tracking error is defined as z 2 = x2∗ − x2

(5.15)

The second Lyapunov function is defined as the summation of V 1 and linearquadratic pseudo-current-loop tracking error z2 , expressed as V2 =

1 2 1 2 z + z 2 1 2 2

(5.16)

The time derivative of V 2 is given by V˙2 = z 1 z˙ 1 + z 2 z˙ 2

(5.17)

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5 An Ellipse-Optimized Composite Backstepping Control Strategy …

According to (5.12), (5.15), detailed expression of (5.17) can be found, given by [ V˙2 =

z 1 x˙1∗

] ( ( ( x2∗ − z 2 − i od + ωC x3 − + z 2 x˙2∗ − x˙2 C (

(5.18)

Substituting (5.13) and (5.5)–(5.18), it gives ) ( 0.5μd E − x1 + ωL x4 z1 z2 ∗ 2 ˙ V2 = −k1 z 1 + + z 2 x˙2 − C L

(5.19)

The d-frame switching function μd is selected to guarantee V˙2 ≤ 0, expressed as ( ) 2 L ∗ μd = x1 − ωL x4 + L x˙2 + z 1 + k2 z 2 E C

(5.20)

where k 2 is another imported positive constant, and as the second controller parameter (k 2 > 0). (5.20) transforms (5.19) into the negative definite format, given by 2

z V˙2 = −k1 z 12 − k2 2 ≤ 0 L

(5.21)

5.3.2 Two-Step Backstepping Derivation in q Frame 5.3.2.1

Step One: Derivation of Inductor Current Reference x4 *

Similarly, load voltage tracking error in q frame is defined as z 3 = x3∗ − x3

(5.22)

Then, the third Lyapunov function associated with the load voltage tracking error in q frame is defined as 1 2 z 2 3

(5.23)

V˙3 = z 3 z˙ 3

(5.24)

V3 = Differentiating (5.23), it yields out that

According to (5.6) and (5.22), the detailed expression of (5.24) can be found, expressed as

5.3 Recursive Derivation and Implementation of the Proposed Composite …

) ( ( ( x4 − i oq − ωC x1 V˙3 = z 3 x˙3∗ − x˙3 = z 3 x˙3∗ − C

79

(5.25)

Then, the pseudo internal current-loop reference x 3 * is designed to ensure V˙3 ≤ 0, given by x4∗ = i oq + ωC x1 + C x˙3∗ + k3 C z 3

(5.26)

where k 3 is an imported positive constant, serving as the third controller parameter (k 3 > 0). If x 4 is regulated to follow x 4 * , (5.25) will be transformed to V˙3 = −k3 z 32 ≤ 0

5.3.2.2

(5.27)

Step Two: Derivation of the q-Frame Switching Function

Internal inductor-current-loop tracking error is defined as z 4 = x4∗ − x4

(5.28)

The fourth Lyapunov function is defined as the summation of V 3 , and linearquadratic current-control-loop tracking error, V4 =

1 2 1 2 z + z 2 3 2 4

(5.29)

The first-order time derivative of (5.29) turns out to be V˙4 = z 3 z˙ 3 + z 4 z˙ 4

(5.30)

According to (5.25), (5.28), V˙4 can be rewritten as [ V˙4 =

z 3 x˙3∗

] ( ( ( x4∗ − z 4 − i oq − ωC x1 − + z 4 x˙4∗ − x˙4 C (

(5.31)

Substituting (5.26) and (5.7)–(5.31), it gives ) ( 0.5μq E − x3 − ωL x2 z3 z4 ∗ 2 ˙ V4 = −k3 z 3 + + z 4 x˙4 − C L

(5.32)

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5 An Ellipse-Optimized Composite Backstepping Control Strategy …

There exists such a q-frame switching function (μq ) that could unconditionally guarantee V˙4 ≤ 0, given by ( ) 2 L ∗ x3 + ωL x2 + L x˙4 + z 3 + k4 z 4 μq = E C

(5.33)

where k 4 is another imported positive constant, serving as the fourth controller parameter (k 4 > 0). Equation (5.33) transforms (5.32) into the negative definite format, expressed as k4 V˙4 = −k3 z 32 − z 42 ≤ 0 L

(5.34)

5.3.3 Design of the Kalman Filter to Estimate and Feedforward the Load Currents for Load Disturbance Rejection Equations (5.13) and (5.26) demonstrate that iod , ioq are indispensable to generate the pseudo references x 1 * , x 3 * on the objective of load disturbance suppression. To this end, three load-current sensors and additional abc-dq coordinate transformation can be added to generate iod , ioq . However, it increases the implementation cost and sacrifices the system’s reliability as well. As a better alternative, a Kalman filter can be designed to estimate iod , ioq online [81]. Together with the backstepping control law, it constitutes the composite nonlinear controller. It is known that the variation of load current can be neglected within one sampling interval [88], expressed as di od /dt = di oq /dt = 0

(5.35)

Combining (5.35) with (5.4) and (5.6), the augmented state-space model of the system can be formulated as ⎧

x˙ = Ax + Bγ y = Cn x

where ⎛

0 ⎜ −ω A=⎜ ⎝ 0 0

ω 0 0 0

⎞ ⎛ ⎞ ⎛1 ⎞ − C1 0 0 vod C ⎟ ⎜ ⎜ 1⎟ 0 − C1 ⎟ ⎟, B = ⎜ 0 C ⎟, x = ⎜ voq ⎟ ⎠ ⎝ ⎠ ⎝ i od ⎠ 0 0 0 0 i oq 0 0 0 0

(5.36)

5.3 Recursive Derivation and Implementation of the Proposed Composite …

81

POL inverter regulated by backstepping controller

⎛ iLd ⎞ ⎜ ⎟ ⎝ iLq ⎠

y y − yˆ +

G

γ

+ dxˆ dt + +

B

∫ A

−

xˆ

C

⎛ vod ⎞ ⎜ ⎟ ⎝ voq ⎠

yˆ ˆ

ˆ

i ,i ⎛ 0 0 1 0 ⎞ od oq ⎜ ⎟ ⎝0 0 0 1 ⎠

Fig. 5.3 Block diagram of the Kalman filter

) Cn =

( ) ( ) ( 1000 i Ld vod ,γ = ,y = 0100 i Lq voq

The investigated system has been proved to be observable since its observability matrix has full rank, given by rank (Cn Cn A Cn A2 Cn A3 )T = 4

(5.37)

Thus, a Kalman filter can be configured to realize real-time estimation of iod , ioq , as shown in Fig. 5.3. Its compatible analytic formula is given by ⎧

xˆ˙ = A xˆ + Bγ − G(y − yˆ ) yˆ = Cn xˆ

(5.38)

Δ

where x denotes the estimated state variables (x). G is the feedback gain matrix, whose selection is the key for Kalman filter design. Note that the use of observers (e.g., Luenberger observers) is typically restricted to the deterministic case. The Kalman filter is preferable for the stochastic case. The key difference between Luenberger observers and Kalman filter lies in how to update the G. Selection of G in the Kalman filter has taken account of estimation error variance and Gaussian noises, while the Luenberger observers may not. The investigated POL inverter operates on the SPS that is susceptible to stochastic parametric variation/measuring errors, making it suitable for the application of the Kalman filter. To some extent, the Kalman filter could be understood as an advanced Luenberger observes with better performance. Perhaps it can explain why some literature calls such state estimation technique an observer, while the others call it a Kalman filter. These sorts of literature aim to realize the same state estimation objective while having different relative emphases. How to distinguish the difference between the above two technologies goes beyond the focus of this book.

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5 An Ellipse-Optimized Composite Backstepping Control Strategy …

5.3.4 Implementation of the Proposed Composite Backstepping Controller with a Kalman Filter According to (5.13), (5.26), (5.20), (5.33), a block diagram of the POL inverter in the SPS regulated by the proposed composite backstepping controller is depicted in Fig. 5.4. Inductor currents are regarded as the pseudo control variables, and x 2 *, x 4 * denote their tracking references. The designed controller has both voltage controlloop errors (z1 , z3 ), and pseudo-current-loop errors (z2 , z4 ), which means that it inherently has dual control loops. k 1 , k 3 are voltage-loop controller gains, and k 2 , k 4 are current-loop controller gains. μd , μq will be compared with the triangular carrier signals in the SPWM modulator, where ‘on’ and ‘off’ signals (s1 ~ s6 ) are generated for the six switches. The inductor currents (iLa , iLb , iLc ) and load voltages (voa , vob , voc ) are measured by sensors. The Kalman filter estimates and feedforward iod , ioq to generate x 2 *, x 4 *, leading to the omission of load-current sensors compared to [23].

Fig. 5.4 Block diagram of the POL inverter in the SPS regulated by the proposed composite backstepping controller

5.4 Ellipse-Based Controller Gains Optimization, Feedback Gains Matrix …

83

5.4 Ellipse-Based Controller Gains Optimization, Feedback Gains Matrix Selection, and Robustness Analysis 5.4.1 Proposed Intuitive Ellipse-Based Strategy to Optimize the Controller Parameters with Fully Consideration of ξ and ωn To tune the controller gains (k 1 , k 2 , k 3 , k 4 ) quantitatively, dynamics associated with the closed-loop errors (z1 , z2 , z3 , z4 ) should be derived out at first. To this end, (5.15) and (5.28) are substituted to (5.4) and (5.6) respectively, which gives ⎧

C x˙1 = x2∗ − z 2 − i od + ωC x3 C x˙3 = x4∗ − z 4 − i oq − ωC x1

(5.39)

The internal current-loop reference (5.13), (5.26) are substituted into (5.39), which yields out that ⎧

z˙ 1 = −k1 z 1 + z 2 /C z˙ 3 = −k3 z 3 + z 4 /C

(5.40)

Similarly, substituting the switching functions (5.20), (5.33)–(5.5), (5.7) respectively, it produces that ⎧

z˙ 2 = −z 1 /C − k2 z 2 /L z˙ 4 = −z 3 /C − k4 z 4 /L

(5.41)

Equations (5.40) and (5.41) demonstrate the error dynamics in d frame (z1 , z2 ) and q frame (z3 , z4 ) are inherently decoupled using the proposed approach. d-frame error dynamics will have the symmetric dynamic property to the q-frame counterparts provided that k 1 = k 3 > 0 and k 2 = k 4 > 0. Under this premise, only k 1 and k 2 are required to be tuned, and (5.40), (5.41) can be rewritten as z˙ d = T z d , z˙ q = T z q

(5.42)

where zd = (z1 , z2 )T , zq = (z3 , z4 )T , and ) T =

−k1 1/C −1/C −k2 /L

( k1 > 0, k2 > 0

(5.43)

The characteristic polynomial of matrix T is given by ( ) k1 k2 1 k2 λ+ + 2 det(λI − T ) = λ2 + k1 + L L C

(5.44)

84

5 An Ellipse-Optimized Composite Backstepping Control Strategy …

Since the closed-loop error dynamics have been reduced to a typical second-order system, its compatible damping ratio (ξ ) and natural resonant frequency (ωn ) could serve as the physical indicators to do the controller gains’ optimization. It is known that ξ, ωn directly manifests the steady-state and dynamic characteristics of the closed-loop system. Larger ωn means the faster dynamic response of the system, while larger ξ implies more damping can be provided for the system. To be more specific, larger ωn signifies that the state variables will be regulated to track their references with less convergence time. Larger ξ leads to less total harmonic distortion in steady-state and less overshoot during transients. Initially, (5.44) is written as the characteristic-polynomial format of a typical second-order system, given by det(λI − T ) = λ2 + 2ξ ωn λ + ωn2 det(λI -T ) = 0 has two solutions, denoted as λ1 ,2 : √ λ1,2 = −ξ ωn ± ωn ξ 2 − 1

(5.45)

(5.46)

where ξ, ωn are derived out to be C Lk1 + Ck2 , ξ = √ 2 k1 k2 LC 2 + L 2

/ ωn =

1 k1 k2 + 2 L C

(5.47)

Figure 5.5a, b depict the scenario where ξ, ωn vary as the increase of k 1 , k 2 based on (5.47). It shows that ωn monotonically increases as k 1 or k 2 increases. However, ξ has a comparatively non-monotonic relationship to k 1 or k 2. Although k 1 , k 2 can be roughly selected using Fig. 5.5, it is hard to take ξ, ωn into account simultaneously, no less to say the parameter optimization As a better alternative, this chapter proposes an intuitive ellipse-based optimization strategy for k 1 and k 2 from a geometrical point of view, which could ensure optimal system damping and maximized dynamic response. To strike a trade-off between better dynamic response and overshoot during transients, ξ is fixed to be the optimal damping ratio [20, 83]: √ 2 C Lk1 + Ck2 ≈ 0.707 = ξ = √ 2 2 2 2 k1 k2 LC + L

(5.48)

After mathematical transformation (5.48) can be rewritten as k12 k22 (√ )2 + (√ )2 = 1 (k1 > 0, k2 > 0) 2/C 2/C

(5.49)

5.4 Ellipse-Based Controller Gains Optimization, Feedback Gains Matrix …

85

10×104 9×104 8×104 7×104 6×104 5×104 4×104

k1

k2

(a)

1.5

1 0.707 0.5

k2

k1

(b) Fig. 5.5 Variation of the physical indicators as controller parameters increase (k 1 > 0, k 2 > 0). a ωn as k 1 and k 2 increase; b ξ as k 1 and k 2 increase

Note that L is no less than 1 H (1000 mH) in a real-world physical system. Therefore, (5.50) always holds, and (5.49) turns out to be a standard equation of an ellipse in the x–y Cartesian coordinate system where the two focuses of the ellipse are located on the x-axis, depicted as Fig. 5.6.

86

5 An Ellipse-Optimized Composite Backstepping Control Strategy …

Fig. 5.6 An intuitive interpretation of the controller gains’ optimization process from a geometric point of view

√ √ 2/C > 2L/C > 0

(5.50)

As seen in Fig. 5.6, the controller parameters’ combination M(k 1 , k 2 ) that guarantees the optimal damping ratio ξ = 0.707 are located at the first quadrant part of the ellipse (marked in red) since k 1 > 0 and k 2 > 0. Without loss of generality, a mathematically equivalent parametric equation of (5.49) can be expressed as √

2 cos θ, k2 = k1 = C

√ [ π] 2L sin θ, where θ ∈ 0, C 2

(5.51)

According to (5.47), ωn has a proportional relationship to the area of the rectangle, denoted as S MNOP = k 1 k 2 . Using (5.51), S MNOP can be rewritten as √ √ L 2 2L cos θ · sin θ = 2 sin 2θ S M N O P = k1 k2 = C C C / 1 SM N O P 1√ + 2 = sin 2θ + 1 ωn = L C C

(5.52)

(5.53)

Equation (5.53) shows that the maximum of S MNOP and ωn can be achieved if and only if θ = π/4, which gives max S M N O P

| | L L = 2 sin 2θ || = 2 C C θ= π 4

⇒

√ 2 max ωn = C

(5.54)

5.4 Ellipse-Based Controller Gains Optimization, Feedback Gains Matrix …

87

Under this premise, the optimal controller parameters (k 1 * , k 2 * ) can be determined accordingly, given by ⎧ | √ | ⎪ ⎨ k1∗ = 2 cos θ/C | π = 1/C θ|= 4 √ | ⎪ ⎩ k2∗ = 2L sin θ/C | π = L/C

(5.55)

θ= 4

5.4.2 Quantitative Selection of the Feedback Gain Matrix G of the Kalman Filter Aided by Ellipse-Optimized Strategy A well-acknowledged classical approach selects the observer feedback gain matrix G via the following formula, G = −PCn−1 R −1

(5.56)

where matrix P can be obtained by solving the Riccati equation, given by P A T +A P−PC T R −1 C P + Q = 0

(5.57)

Here, Q and R are two design matrixes. Q is positive definite and R is semi-positive definite, whose selection depends on trial and errors [88]. By (5.46), ξ ωn measures the distance between λ1 ,2 and the imaginary axis. Using the proposed ellipse-optimized strategy, ξ ωn has been maximized to (Ψ C ), given by √ / ψC = 0.707 × max ωn = 0.707 2 C

(5.58)

Closed-loop Poles of the Kalman filter can be calculated out by solving det (sIA-GC n ) = 0. The dominant poles of the Kalman filter should have such real parts (Ψ O ) that are (1 ~ 2) times of Ψ C , which provides a quantitative guideline for Q, R selection. In Fig. 5.7, the tracking performance of the Kalman filter is considered under x 1 * step when supplying a linear resistive load. It demonstrates that the selection of Ψ O is a trade-off between convergence speed and disturbance susceptibility. When Ψ O = 0.1Ψ C or 0.2Ψ C , it takes a long time to converge. If Ψ O = 2Ψ C , the estimated iod tracks the iod measured by current sensors instantly, while it is undesirably disturbed by circuit noise.

88

5 An Ellipse-Optimized Composite Backstepping Control Strategy …

vm

sensor

iod :[2A/div]

x1*: 0

2 ψC 1 ψC 0.2 ψC 0.1 ψC Time:[0.4 ms/div] Fig. 5.7 The tracking performance of the Kalman filter as ψ O varies

5.4.3 Robustness Analysis of the Proposed Control Scheme Under Parametric Variations and Measurement Errors Matrix A, B in (5.38) are dependent on the system parameters that are susceptible to measurement errors and plant parametric variation for a real-world system. To take account of this effect (5.38) is rewritten as ⎧

xˆ˙ = A xˆ + Bγ − G(y − yˆ ) + ┌ xˆ yˆ = Cn xˆ

(5.59)

where Γ is an unknown 4-by-4 matrix, modelling the lumped uncertainty due to parametric variation and measurement errors. According to (5.36) and (5.59), the error dynamics of the Kalman filter considering external disturbance is given by z˙ = ( A+┌ + GCn )z

(5.60)

Δ

where z denotes the state estimation error: z = x - x. Inspired by Kim et al. [88], a Lyapunov function V o is formulated to investigate the stability of the Kalman filter, given by Vo = z T P −1 z

(5.61)

whose first-order time derivative is derived as V˙o = 2z T P −1 z˙

(5.62)

5.4 Ellipse-Based Controller Gains Optimization, Feedback Gains Matrix …

89

Substituting (5.60) and (5.56)–(5.62), it gives ( ( V˙o (z) = 2z T P −1 A + ┌ − PCn−1 R −1 Cn z ( ( = 2z T P −1 A P + ┌ P − PCn−1 R −1 Cn P P −1 z ( ( = z T P −1 A P + P A T + ┌ P + P┌ T − 2PCn−1 R −1 Cn P P −1 z

(5.63)

According to (5.57), (5.63) can be rewritten as ( ( V˙o = z T P −1 −Q + ┌ P + P┌ T − PCn−1 R −1 Cn P P −1 z

(5.64)

Negative definiteness of (5.64) can be rigorously guaranteed one the premise that: ┌ P + P┌ T < Q + PCn−1 R −1 Cn P

(5.65)

Therefore, for any scale of parametric mismatch and measurement errors satisfying (5.65), V˙o < 0 can be still ensured, and z will exponentially convergence to zero. Another Lyapunov function V c is formulated to prove the stability of the closedloop system (5.42), given by Vc = z dH z d + z qH z q

(5.66)

where (•)H denotes the conjugate transpose matrix of (•). The time derivative of (3.31) is given by V˙c = z˙ dH z d + z dH z˙ d + z˙ qH z q + z qH z˙ q

(5.67)

Substituting (5.42) to (3.32), it gives (H ( V˙c = (T z d ) H z d + z dH T z d + T z q z q + z qH T z q ( ( ( ( = z dH T H + T z d + z qH T H + T z q

(5.68)

where T H + T cancels out all their off-diagonal entries. According to (5.43), (5.68) can be rewritten as ) ) ( ( 0 0 H −2k1 H −2k1 ˙ zd + zq zq < 0 Vc = z d 0 −2k2 /L 0 −2k2 /L

(5.69)

which demonstrates that L, C does not affect the system stability if k 1 , k 2 > 0. Meanwhile, L, C, k 1 , k 2 can vary at a large range without deteriorating the stability of the closed-loop system.

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5 An Ellipse-Optimized Composite Backstepping Control Strategy …

5.5 Test Results A 1 kW-rated POL inverter is used to do tests. System parameters are listed in Table 5.1. According to (5.55), the optimal controller gains are calculated out to be: k 1 * = k 3 * = 3.33 × 104 , k 2 * = k 4 * = 33.33. Ψ O is selected as around 1Ψ C , where G is selected as G = 104 × (6.664, -0.0314, -3.333, 0; 0.0314, 6.666, 0, -3.3326)T .

5.5.1 Effectiveness of the Proposed Ellipse-Optimized Controller Gains Selection Strategy In Fig. 5.8a, k1 and k2 are both selected smaller than the optimal controller gains. According to Fig. 5.5b, inadequate controller gains will lead to poor system damping, complying well with the experimental results shown in Fig. 5.8a. In Fig. 5.8b, k1 increase to 10k 1 * , and k 2 remain unchanged compared to the counterparts in Fig. 5.8a. Figure 5.5b predicts that an increase of k 1 would lead to an increase of damping ratio, which also fulfills the experimental result shown in Fig. 5.8b. The designed optimal controller gains can theoretically fix the damping ratio to 0.707. Under this premise, the natural frequency has been maximized to achieve the best dynamic performance. Figure 5.8c presents the experimental results with the optimal controller gains. Compared with Fig. 5.8a, c has effectively suppressed the oscillation and overshoot during transients. In comparison with Fig. 5.8b, c takes less transient time to track the reference step (x 1 *: 0 → vm ).

5.5.2 Robustness Tests Under Plant Parametric Variations To test the robustness of the proposed control scheme against parametric variations and reference step, ± 50% L and ± 50% C scale of filter parametric variation have been considered, where x 1 * steps from 0 to vm and x 3 * is fixed to 0. As seen in Fig. 5.9, the variation of LC parameters mainly affects the THD of the load voltage, while the stability of the closed-loop system does not get deteriorated. A larger L or C contributes to suppressing the THD of the load voltage. Every time x 1 * steps from 0 to vm , the load voltage (x 1 ) soon tracks x 1 * with negligible rise time and overshoot. To summarize, Fig. 5.9 demonstrates that the proposed control scheme can at least withstand ± 50% L, C scale of plant parametric variation. Table 5.1 Nominal system parameters E

ω

fs

L

C

vm

C dc

Z

350 V

100π

10 kHz

1 mH

30 μF

115 V

470 μF

Ω

5.5 Test Results

91

C1,2,3 voabc :[50V/div]

C4

vod :[50V/div]

C5

voq :[50V/div]

[2 ms/div]

(a) C1,2,3

voabc :[50V/div]

C4

vod :[50V/div]

C5

voq :[50V/div]

[2 ms/div]

(b) C1,2,3

voabc :[50V/div]

C4

vod :[50V/div]

C5

voq :[50V/div]

[2 ms/div]

(c) Fig. 5.8 Control performance of the proposed approach under reference step (x 1 *: 0 → vm ) with different set of controller gains. a k 1 = 0.1k 1 * , k 2 = 0.1k 2 * ; b k 1 = 0.1k 1 * , k 2 = 10k 2 * ; c k 1 = k1*, k2 = k2*

5.5.3 Performance Evaluation Under Linear/Nonlinear Load Step, Reference Step, Overload and Recovery Four case studies have been conducted to evaluate the performance of the proposed control scheme. As seen in Figure 5.10a–c, the POL inverter is under linear load step, nonlinear load step from 100 W to 1 kW, and overload condition respectively,

92

5 An Ellipse-Optimized Composite Backstepping Control Strategy …

v oabc :[50V/ div]

v oabc :[50V/ div] C1,2,3

C1,2,3 THD: 0.88 %

THD: 1.70 % C4 v od :[50V/ div] C5 v oq :[50V/ div]

[2 ms/div]

C4 v od :[50V/ div] C5 v oq :[50V/ div]

(a)

[2 ms/div]

(b)

v oabc :[50V/ div]

v oabc :[50V/ div]

C1,2,3

C1,2,3 THD: 0.55 %

C4 v od :[50V/ div] C5 v oq :[50V/ div]

[2 ms/div] (c)

THD: 0.78 % C4 v od :[50V/ div] C5 v oq :[50V/ div]

[2 ms/div]

(d)

Fig. 5.9 Experimental results of the robustness tests. a −50%L, −50%C; b + 50%L, −50%C; c + 50%L, + 50%C; d −50%L, + 50%C

while the three-phase load voltage always remains pure sinusoidal. It shows that the proposed controller is capable to suppress such a scale of load disturbance due to load uncertainty and nonlinearity. Figure 5.10d demonstrates the great tracking performance of the controller under x 1 * steps up/down. It always shows that d-q decoupled control has been realized.

5.5.4 Comparisons Between Existing Lyapunov-Based Approaches and the Proposed Control Scheme Existing Lyapunov-based approaches and the proposed control scheme are compared in Table 5.2 and Fig. 5.11. Komurcugil et al. [23] is devised based on the conventional Lyapunov function (V ), where V is formulated as the linear quadratic tracking errors associated with the inductor current and capacitor voltage. Komurcugil et al. [23] is an extended dual-loop control scheme where the capacitor voltage feedback is artificially imported. Global large-signal stability of the system can be ensured on the premise that the imported voltage feedback gain is larger than a specific lower bound. The load-current sensor is indispensable in [23]. Both [24] and the proposed approach are inherently dual-loop control schemes, that can rigorously guarantee the global large-signal stability of the system. The key for [24] design is based on an adaptive weighted Lyapunov function, whose design requires genuine expertise. In comparison, the proposed approach selects

5.5 Test Results

93

Fig. 5.10 Performance evaluation of the proposed control scheme. a linear load step; b nonlinear load step; c overload and recovery; d reference step

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5 An Ellipse-Optimized Composite Backstepping Control Strategy …

Table 5.2 Comparisons between existed Lyapunov-based control schemes and the proposed approach References

[23]

[24]

This chapter

States’ feedbacks

Extended dual loops

Inherent dual loops

Inherent dual loops

Lyapunov function (V)

Conventional

Adaptive weighted

Recursive

Large-signal stability

Ensured (conditional)

Ensured

Ensured

Load-current sensors

Required

0

0

Controller tuning

Root locus

Jacobian matrix A(t)

Explicit formulas

Controller gains

K i = 2 × 103 K v = 4.9 × 103

σ = 100 γ = 0.1

k 1 * = 3.33 × 104 , k 2 * = 33.33

THD %

1.16

0.97

0.79

3rd harmonic %

0.07

0.10

0.03

5th harmonic %

0.07

0.09

0.11

7th harmonic %

0.05

0.14

0.03

Robustness against LC mismatch

Not reported

At least ± 50%

At least ± 50%

the Lyapunov function step by step in a recursive manner using the backstepping approach, which is systematic. With an adaptive law, [24] has saved load-current sensors. However, as seen in Fig. 5.11, the load voltage undergoes a certain amount of voltage sag during the load step. The proposed approach obviates the load-current sensor with the help of the Kalman filter. One common obstacle of [23, 24] is that the controller gains selection is indeed a trial-and-error process. Although the root-locus technique in [23], the eigenvalues of the Jacobian matrix in [24] may provide certain help for controller gains selection, it lacks explicit formulas to tune the controller gains. In comparison, the optimal controller gains in this chapter are quantitatively calculated out, which is straightforward and time-saving for practical implementation. THD of the load voltage using the proposed approach is around 0.79%, which is lower than 1.16% using [23] and 0.97% using [24].

5.6 Conclusion The proposed composite backstepping controller inherently has dual control loops to achieve both great steady-state and dynamic performance. Meanwhile, it can rigorously guarantee the large-signal stability of the POL inverter with great robustness against at least a ± 50% scale of plant parametric variations. A Kalman filter is further designed to estimate the load currents and feedforward them to the backstepping controller, leading to the omission of load-current sensors compared to [23]. The proposed strategy realizes d-q decoupled control, resulting in a dimension-reduced second-order system. Komurcugil et al. [23] and He et al. [24] do not present any

5.6 Conclusion

95

Fig. 5.11 Performance evaluation under linear load step. a [23]; b [24]; c the proposed approach

explicit formulas for controller gains selection, where trial-and-error methods are inevitable. In this chapter, the optimal controller gains are quantitatively calculated out via explicit formulas to achieve the optimal system damping and maximized dynamic response, which can be intuitively interpreted via an ellipse-based strategy from a geometrical point of view. Rigorous stability proof and robustness analysis of the system is also provided. THD % of the load voltage using the proposed control scheme is around 0.79%, which is lower than that using [23, 24].

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5 An Ellipse-Optimized Composite Backstepping Control Strategy …

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76. R. Gavagsaz-Ghoachani, M. Phattanasak, J. P. Martin, B. Nahid-Mobarakeh, S. Pierfederici, and P. Riedinger, “Observer and Lyapunov-Based Control for Switching Power Converters with LC Input Filter,” IEEE Trans. Power Electron., vol. 34, no. 7, pp. 7053–7066, 2019. 77. N. Kashyap, M. Narimani, and J. Bauman, “Modified Lyapunov-Based Model Predictive Direct Power Control of an AC-DC Converter with Power Ripple Reduction,” IEEE Int. Symp. Ind. Electron., vol. 2019-June, pp. 926–931, 2019. 78. H. Kömürcügil and O. Kükrer, “A novel current-control method for three-phase PWM AC/DC voltage-source converters,” IEEE Trans. Ind. Electron., vol. 46, no. 3, pp. 544–553, 1999. 79. R. Dasgupta, S. Basu Roy, and S. Bhasin, “Lyapunov-based hierarchical control design of a quad-rotorcraft with singularity avoidance,” 2019 18th Eur. Control Conf. ECC 2019, no. 1, pp. 2707–2712, 2019. 80. C. O. Tysse and O. Egeland, “Crane load position control using lyapunov-based pendulum damping and nonlinear MPC position control,” 2019 18th Eur. Control Conf. ECC 2019, pp. 1628–1635, 2019. 81. D. Simon, Optimal State Estimation-Kalman, H-infinity, and Nonlinear Approaches. Cleveland State University: John Wiley & Sons, 2006. 82. Texas Instruments, “LC Filter Design Application Report,” Texas Instruments Inc., vol. SLAA701A, no. October, pp. 8–10, 2016. 83. W. S. Levine, The Control Handbook (three volume set). CRC Press, 2018. 84. D. Karimipour and F. R. Salmasi, “Stability Analysis of AC Microgrids With Constant Power Loads Based on Popov’s Absolute Stability Criterion,” IEEE Trans. Circuits Syst. II Express Briefs, vol. 62, no. 7, pp. 696–700, 2015. 85. Q. Xu, C. Zhang, C. Wen, and P. Wang, “A Novel Composite Nonlinear Controller for Stabilization of Constant Power Load in DC Microgrid,” IEEE Trans. Smart Grid, vol. 10, no. 1, pp. 752–761, 2019. 86. B. Zhao, X. Zhang, and J. Huang, “AI algorithm-based two-stage optimal design methodology of high-efficiency CLLC resonant converters for the hybrid AC-DC microgrid applications,” IEEE Trans. Ind. Electron., vol. 66, no. 12, pp. 9756–9767, 2019. 87. T. Dragiˇcevi´c and M. Novak, “Weighting Factor Design in Model Predictive Control of Power Electronic Converters: An Artificial Neural Network Approach,” IEEE Trans. Ind. Electron., vol. 66, no. 11, pp. 8870–8880, 2019. 88. E. Kim, F. Mwasilu, H. H. Choi, and J. Jung, “An Observer-Based Optimal Voltage Control Scheme for Three-Phase UPS Systems,” IEEE Trans. Ind. Electron., vol. 62, no. 4, pp. 2073– 2081, 2015.

Chapter 6

Stability Constraining Dichotomy Solution Based Model Predictive Control for the Three Phase Inverter Cascaded with Input EMI Filter in the MEA

6.1 Introduction More electric aircraft (MEA) are becoming more popular as operations costs and gas emissions are being reduced as well as providing comfortable and safe services [1–3]. MEA presents many challenges for the design of the onboard electrical power system due to a substantially increased power demand [3], as well as the impact on the distribution system. There are many possible distribution system structures, including AC, DC, and hybrid AC/DC microgrids [3]. Among them, hybrid AC/DC microgrid architecture has attracted a lot of research attention since it has the advantages of both AC and DC microgrids, such as facilitating the connection of renewable AC and DC sources and loads [4, 5]. Figure 1.3 presents a hybrid schematic diagram of the AC/DC MG-based MEA power system [6, 7]. This system is composed of three parts: (a) the DC microgrid which connects the DC sources (i.e., solar panel and battery) and DC loads (i.e., instruments, gyroscopes.); (b) the AC microgrid which connects the generators, motors, and AC loads of the MEA; (c) the bus conversion system which connects the DC bus and AC bus. The purpose of this chapter is to analyse the bus conversion system’s stability. As shown in Fig. 1.3, the bus conversion system is a typical AC cascaded system that contains Electro-Magnetic Interference (EMI) filters and threephase inverters. The EMI filter is used to reduce the electromagnetic interference from the inverter to the DC bus. When the AC microgrid needs to absorb the power from the solar panel or battery, the inverter transfers the power from the DC bus to the AC bus. Having active regulation capabilities (i.e., control output current), inverters can extract steady power and are thus commonly referred to as constant power loads (CPLs) [8]. Interestingly, CPLs have a negative incremental input impedance. As a result, they tend to destabilize the system to which they are connected [9, 10]. Thus, the CPL also poses a risk of instability for the bus conversion system in the MEA. The above stability problem has been known for a long time and is often solved by the Nyquist criterion [11, 12], as well as various stabilization methods. While passive © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Zhang et al., Stability Enhancement Methods of Inverters Based on Lyapunov Function, Predictive Control, and Reinforcement Learning, https://doi.org/10.1007/978-981-19-7191-4_6

101

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6 Stability Constraining Dichotomy Solution Based Model Predictive …

stabilization strategies such as supplementing of necessary capacitors and/or resistors [13] or design of the LC filters [14] have the advantage of simplicity and effectiveness, they are costly and could result in extra power loss. Using active damping methods, the system can be stabilized through modification control loops of the CPL to eliminate [15] or partially eliminate [16] its negative resistor characteristics. In comparison with passive stabilization methods, active stabilization methods are more flexible and can reduce additional power losses, making them increasingly popular. Nevertheless, the majority of active stabilization methods use Nyquist criteria and small-signal models. In spite of their ability to ensure small signal stability around the steady-state, they may become unsuitable for larger signals [17]. In this chapter, a large signal stability constraining dichotomy solution (SCDS) based model predictive control (MPC) has been proposed for the MEA power system to take the large signal stability issues. The SCDS-MPC uses a model-based stabilization method that will stabilize the system in a large signal sense in accordance with Lyapunov criteria [18–20]. A backstepping method is used in our work in order to derive the Lyapunov function. It offers the following benefits: Firstly, the proposed SCDS-MPC method specifies clearly the large-signal constraining cost function for the MPC. Unlike traditional MPC methods, the proposed SCDS-MPC scheme can provide analytical guidance as to how to add stability to MPC cost functions rather than relying on the experience-based weight method [21]; Secondly, The proposed SCDS-MPC method is an improvement over the existing FCS-MPC method, which keeps many of the traditional FCS-MPC advantages, while introducing several new ones [22–26]. For instance, constant switching frequency and better steady-state/dynamic current control performance can be realized for the inverter. As well as that, the ‘dichotomy solution’ concept was originally proposed in the previous publication of the authors as a way to improve the FCS-MPC ability in motor control [26]. Our previous work, however, cannot be used to directly control the inverter in the bus conversion system at MEA, nor can it be used to stabilize the AC cascaded system at MEA. Therefore, although this chapter shares the dichotomy solution concept with our previous work, in reality they are two completely different works. Currently, a number of MPC methods that consider stability constraints have already been implemented in single power converter applications, such as motor drives. In [27, 28], the finite control set prediction and the Lyapunov theory are combined to minimize the torque ripple. In order to calculate the desired duty cycle for each voltage vector, the dominant part of the cost function is used as a Lyapunov function. In light of this, the contribution of this work is to minimize torque ripple using duty cycle calculation based on Lyapunov theory. It can be used to implement a predictive control system with a fixed switching frequency. In [29], a uniform global stability approach for convex control set (CCS) and finite control set (FCS) MPC is proposed. The control Lyapunov function (CLF) constraint is used to obtain stability. It has been shown that this method has robust stability properties and that the resulting system should be asymptotically stable. In [30], It is proposed to develop

6.2 Instability Problem of the Researched AC Cascaded System in MEA

103

a Lyapunov function-based model predictive direct current control (MP-DCC) algorithm for PMSG, which consists of two modes, namely tracking mode and minimization of switching losses mode. The algorithm considers eight voltage vectors for each sampling interval and provides simulation results. In [31], FCS-MPC for power converters is studied from the viewpoint of practical stability, i.e., that is, studying the convergence of state variables to a bounded invariant set. The focus of the work is on the development of cost functions based on the Lyapunov stability concept to ensure sufficient conditions for local stability. A variable switching frequency is used in the cost function, and eight voltage vectors are taken into consideration. Nevertheless, these existing works combine MPC with the Lyapunov function only for a single power converter. As mentioned earlier, a cascaded AC-AC system consisting of an inverter and an EMI input filter may suffer from instabilities in the MEA. Unfortunately, there have been few studies that combine MPC with Lyapunov function to improve cascaded system performance. Therefore, attention should be paid to this aspect of the problem. Therefore, for this work, we are incorporating the Lyapunov stability criteria into the cost function for SCDS-MPC to ensure the stability of the AC cascaded system. Moreover, a dichotomy solution algorithm includes a complete consideration of every voltage vector in the voltage vector plain, as well as the option of selecting and calculating various voltage vectors. Consequently, the proposed algorithm has a fast constraint, with a constant switching frequency. The contribution of this work can be concluded as: (a) Stability constraining dichotomy solution-based model predictive control (SCDS-MPC) is proposed to control the AC cascaded system including EMI input filter and inverter in MEA application. In addition to maintaining good dynamic and static characteristics, it is worth pointing out that the proposed control algorithm guarantees full stability of the AC cascaded system. Traditional MPC only considers the stability of the inverter itself, without considering the stability of the entire system; (b) The proposed stability constraint cost function is constructed using Lyapunov stability criteria; (c) the entire control method SCDS-MPC is implemented on an FPGA controller.

6.2 Instability Problem of the Researched AC Cascaded System in MEA 6.2.1 Instability Problem Description In a conventional three-phase inverter system, the input typically cascades a large DC bus capacitor to smooth DC bus ripples and to meet EMI requirements. The capacitance of this DC bus capacitor is relatively high since it is normally electrolytic in nature. There is, however, a problem with electrolytic capacitors, since they require a lot of space, and their lifetime usually limits the reliability of an inverter system. With the aim of improving power density and extending the lifetime of the

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6 Stability Constraining Dichotomy Solution Based Model Predictive …

Fig. 6.1 A constant power load (CPL) case study: Three-phase inverter fed from a DC link and supplying an AC grid

inverter system, an LC EMI filter, consisting of an inductor and a small capacitance film capacitor, will replace the large capacitance electrolytic DC bus capacitor, as shown in Fig. 6.1. As a result, this structure can contribute to increased reliability and a longer lifespan of the three-phase inverter. However, this approach also causes instability problems for the three-phase inverter system. To demonstrate the above-mentioned phenomena, A three-phase inverter with L filter and different DC bus capacitance controlled by the conventional proportionalintegral (PI) controller was simulated with MATLAB/Simulink. The parameters of the inverter system are presented in Table 6.1. In Fig. 6.2a, a single three-phase inverter only with a battery is simulated, which shows this inverter is stable individually. In Fig. 6.2b, a three-phase inverter with a large capacitance electrolytic DC bus capacitor, whose value is 1200μF, was simulated. As we can see, both the DC bus voltage and the three-phase output current are stable. In Fig. 6.2c, a three-phase inverter with an LC EMI filter was simulated as well. It can be seen that large oscillation happens in both the DC bus voltage and three-phase inverter output current. Therefore, the system is unstable in this scenario. Table 6.1 Parameters of the grid-connected Inverter Parameter

Value

Rated power PN

1.5 kW

DC bus voltage Udc

300 V

AC bus voltage

115 V (RMS)

Filter inductance L

1 mH

Grid fundamental frequency

50 Hz

DC filter inductance L f

10 mH

DC filter capacitance C f

100 μF

Line resistance R

0.01 Ω

6.2 Instability Problem of the Researched AC Cascaded System in MEA

105

Fig. 6.2 DC bus voltage and three-phase output current performance with different DC bus capacitance: a a single three-phase inverter; b conventional large capacitance electrolytic capacitor with stable performance; c film capacitor with small capacitance for unstable performance

6.2.2 The Instability Reason of CPL with LC Input Filter As is widely known, the closed-loop controlled inverter can be regarded as a constant power load (CPL). When cascaded with the input LC filter, the CPL may experience instability. In this section, this instability reason will be carefully analyzed. It is known that the CPL input resistance looks negative, so the CPL with its LC input filter can be presented in Fig. 6.3. Here, L f and C f represent the inductor and capacitor

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6 Stability Constraining Dichotomy Solution Based Model Predictive …

Fig. 6.3 The CPL with its LC input filter

of the LC input filter, respectively, and the transfer function of LC input filter can be described using the well-known second-order function,

T (s) =

1 s2 L

fCf

L

f +1 + s Rin,inc

(6.1)

The following format can be obtained by introducing a resonant frequency ω0 and damping factor ζ : T (s) =

1 s2 ω02

+ 2ζ ωs0 + 1

(6.2)

where 1 LfCf

(6.3)

L f ω0 2Rin,inc

(6.4)

ω0 = √ ζ =

As a result, if the input is excited by a unit-step function, we can obtain the following Laplace equation: Y (s) =

1 s2 ω02

1

+ 2ζ ωs0 + 1 s

(6.5)

As a means of obtaining the response in the time domain, the inverse Laplace transform (6.5) is used. The following relationship can then be obtained: e−ζ ω0 t sin(ωd t + θ ) f or ζ < 1 Y (t) = 1 − √ 1 − ζ2

(6.6)

where √ ωd = ω0 1 − ζ 2

(6.7)

6.3 Preliminary of the SCDS-MPC Method: Mathematical Modeling …

θ = cos−1 (ζ )

107

(6.8)

It is known that the poles affect (6.2), denoted ρ1 and ρ2 , represent the denominator roots for which s 2 + 2ζ ω0 s + ω02 = 0. Depending on the ζ value, these poles affect the stability of the system described by (6.5): • ζ < 0: In this case, poles affecting (6.5) feature a positive real portion. Whatever the excitation level, the transient response diverges. • ζ = 0: This particular case implies two pure imaginary poles ρ1,2 = ± j ω0 , making the system output permanently oscillating (no decay). • ζ > 0: The two poles now have a real portion (ohmic losses), and the system exhibits different responses depending on whether ζ > 1 (overdamping), ζ = 1 (critical damping), or 0 < ζ < 1 for which we obtain a decaying oscillating response. According to (6.4), if Rin,inc < 0, then ζ < 0, and therefore, in this case, the whole system will become unstable, and it will diverge at any level of excitation, which is the reason why the LC input filter of the CPL will become unstable.

6.3 Preliminary of the SCDS-MPC Method: Mathematical Modeling of the Researched AC Cascaded System in MEA 6.3.1 Conventional Inverter Mathematical Model Figure 6.1 depicts the topology of a three-phase inverter. The DC grid and AC grid are both bidirectionally connected to the inverter. According to Fig. 6.1, an AC microgrid (ea , eb , ec ) is connected to the inverter’s AC side through filter inductances L and resistances R to reduce switching harmonics. Consequently, the current dynamic equations can be summarized as ⎛ dia ⎞ ⎝

dt di b dt di c dt

⎞ ⎛ ⎛ ⎞⎛ ⎞ 100 u a N − ea − u n N ia R 1 ⎠ = − ⎝ 0 1 0 ⎠⎝ i b ⎠ + ⎝ u bN − eb − u n N ⎠ L L ic u cN − ec − u n N 001

(6.9)

where i a , i b , i c are the output currents of the inverter, u a N , u bN , u cN are the output voltages of the inverter, u n N is the voltage between the neutral point of the grid voltage and the negative point of dc-link, ea , eb , ec are the three-phase grid voltages, respectively. Suppose that the three-phase voltage is balanced, that is ea +eb +ec = 0, the current dynamics can be described in the stationary αβ frame by the vector equation

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6 Stability Constraining Dichotomy Solution Based Model Predictive …

L

dis = us − Ris − es dt

(6.10)

where i s is the current vector, us is the voltage generated by the inverter, es is the AC grid line voltage. The current vector is related to the phase currents by the equation is =

) 2( i a + ai b + a2 i c 3

(6.11)

where a = e j (2π/3) . The voltage vector is defined in a similar way us =

) 2( u a N + au bN + a2 u cN 3

(6.12)

The voltage us is determined by the switching state of the inverter and the dc-link voltage, and can be also described by the equation us = SUdc

(6.13)

where Udc is the dc-link voltage and S is the switching state vector of the inverter defined as S=

) 2( Sa + aSb + a2 Sc 3

(6.14)

where Sa , Sb , Sc are the switching states of each inverter leg, as shown in Fig. 6.1, and take the value of 0 if Sx is OFF, or 1 if Sx is ON (x = a, b, c).

6.3.2 Improved Mathematical Model with Consideration of the Inverter and Input EMI Filter for Stability Analysis After elaborating the mathematical relationship in the AC grid filter side, the DC side dynamics equations were also examined since we need to build the mathematical model of the entire system in order to use the control strategy and stability analysis. According to Fig. 6.1, the filter inductance L f in series with the resistance Rdc is connected between the DC voltage source vsdc and the DC filter capacitance Cdc which are composed of the input filter. The following is a description of the modelling of DC bus input filter dynamics in this work: Cf

dvdc = i dc − i pol dt

(6.15)

6.3 Preliminary of the SCDS-MPC Method: Mathematical Modeling …

109

As shown in Fig. 6.1, the equation, i dc , that flows through the DC-link inductance L dc , is considered an external disturbance. It is only true when the DC-link inductance is large enough and the continuous nature of the DC source voltage and the DC-link voltage so that the current between each sampling interval can be assumed constant. i pol is the total current fed into the inverter and can be described as the following equation: i pol = Sa i a + Sb i b + Sc i c

(6.16)

According to previous stationary αβ frame equations, the Eq. (6.16) can be expressed as a vector equation. i pol

[ ] ] −1 i α = Sa Sb Sc T iβ [

(6.17)

The following discrete-time equation can be used to approximate the Euler’s formula derivation of the predicted current is (k + 1) = (1 −

Ts RTs )is (k) + [us (k) − es (k)] L L

(6.18)

which is obtained from the discretization (6.10) with a sampling time of Ts . In this discretization, the derivative is approximated as the difference over a single sampling interval is (k + 1) − is (k) dis ≈ dt Ts

(6.19)

Using the following discretization equation, the DC-link capacitance dynamics are approximated to estimate voltage variation in each sampling interval for charging and discharging currents of the capacitance: vdc (k + 1) = vdc (k) +

i pol,i + i pol, f 1 )Ts (i dc − Cdc 2

(6.20)

where i pol,i and i pol, f are the initial and final currents flowing into the inverter system during the following time interval, respectively. As we can see from the equation, both currents depend on the future applied voltage vector. Therefore, the calculated average current can be used to predict the energy input on the dc side with greater accuracy. Consequently, a more precise DC voltage can be obtained. The calculation will grow exponentially if the prediction horizon is two or more steps. However, a one-step prediction usually meets the majority of the requirements of most cases. Therefore, one-step prediction is used in this work.

110

6 Stability Constraining Dichotomy Solution Based Model Predictive …

6.4 The Proposed SCDS-MPC Method 6.4.1 Conventional Model Predictive Control Scheme The conventional finite-state model predictive control (FCS-MPC) scheme is shown in the block diagram in Fig. 6.4. The whole control strategy works in the stationary αβ frame. So the measured grid side three phase voltage and current are firstly transformed to the stationary αβ frame using Clarke transformation. The block “Predictive Model” includes the discrete-time model of (6.18) and (6.20). The value of the current in the next sampling interval i s (k + 1) for each of the different eight voltage vectors is calculated using this block. Therefore, eight current values are predicted one sampling step ahead. Then, on the block “Minimization of cost function”, a cost function G 1 , defined in (6.21), calculates the absolute error between the reference and predicted currents in the next sampling interval. | | | | G 1 = |i α∗ − i α (k + 1)| + |i β∗ − i β (k + 1)|

Fig. 6.4 Conventional FCS-MPC block diagram of a three-phase grid-connected inverter

(6.21)

6.4 The Proposed SCDS-MPC Method

111

A voltage vector with the lowest current error is selected, and a switching state is applied to the inverter accordingly. The FCS-MPC for the current control strategy presents a variable switching frequency that depends on the sampling frequency and the operating conditions. The performance of this control strategy mainly depends on the sampling time. In this control strategy, the absolute current error is reduced with an increase in sampling frequency. However, an increase in switching frequency increases the absolute current error. In industrial applications, the switching frequency is limited by the power semiconductors. Therefore, the sampling frequency in the controller is usually selected based on the maximum switching frequency. As a result, the basic predictive current control tends not to perform as well as some conventional fixed switching frequency control schemes on the performance, especially in the control of grid-connected inverters. Furthermore, a mathematical approach is not available for ensuring stability for FCS-MPCs.

6.4.2 Proposed Dichotomy Solution (DS) Based Model Predictive Control The control scheme of the proposed stability constraining dichotomy solution-based model predictive control (DS-MPC) for grid-connected inverters is presented in Fig. 6.5. Firstly, Clarke transformation is used to transform the three-phase grid voltages and currents to Cartesian coordinates, and then model predictive equations are used to predict the currents. Then, using the dichotomy cost function solution algorithm, the optimal voltage vector is selected and sent to the PWM modulator after comparison with the reference current. Finally, the inverter receives the switching state and regulates the currents accordingly. In the case of current control, the basic cost function can be defined the same Eq. (6.13). The original idea of the cost function optimization algorithm is a dichotomy. As shown in Fig. 6.6, the voltage vector search area is the circular plane with a radius of Um –the maximum available voltage. In each switching interval, with Um /2 amplitude and π/4 phase different with each other, eight voltage vectors are selected as the vector candidates at the initialization. According to the cost function, one of the eight vectors is selected as the optimal vector in the first step of the iteration. Next, using the optimal vector selected in the last step, 18 new vectors are selected, whose amplitude and phase angle are added or subtracted by half of the optimal vector from the previous step. Hence, the search area decreases to half of what was searched in the last step. After 14 steps of iteration, the optimal vector is selected as the reference vector for the PWM modulator.

112

6 Stability Constraining Dichotomy Solution Based Model Predictive …

Fig. 6.5 Block diagram of the proposed SCDS-MPC for the grid-connected inverter

6.4.3 Proposed System Stability Constraining Cost Function Definition The flow chart that follows explains how the proposed stability constraining cost function operates. In this work, the cost function is generally split into two parts. The first part is stability constrain. All the generated voltage vectors are firstly judged by the stability constrain. Only when the voltage vector satisfies this criterion, it could be considered further. In the second part, the fundamental cost function is used, which is mainly used for current control. It includes both current protection and current control cost function. The flow chart of the proposed control algorithm is presented in Fig. 6.7a, and the detailed voltage vector selection process is shown in Fig. 6.7b, correspondingly. We can see the voltage vectors can be chosen in the whole voltage vector plain according to the algorithm. When stability constraining G3 (red rectangle part in Fig. 6.7a) is disabled, G3 will be bypassed in the algorithm. The system may be unstable as shown in Fig. 6.15a. While G3 is enabled, the voltage vectors are firstly evaluated by the stability constrain. If the voltage vector could not satisfy the stability criteria, it will be not selected, and the other voltage vectors will be evaluated by G3 further. Only the one which satisfies the stability criteria can be considered for control of the system. Therefore, the system can be controlled and stable when G3 is enabled as presented in Fig. 6.16.

6.4 The Proposed SCDS-MPC Method

113

Fig. 6.6 Dichotomy solution algorithm of the cost function

Cost function I: Stability Constrain Part To facilitate the stability derivation, the analysis will be conducted in the rotating d-q frame. Therefore, combining (6.10) and (6.15), we can achieve ⎧ ⎨ x˙1 = a0 x2 /x1 + λ x˙ = a1 x2 + ωx3 + u 1 ⎩ 2 x˙3 = a1 x3 − ωx2 + u 2

(6.22)

114

6 Stability Constraining Dichotomy Solution Based Model Predictive …

Fig. 6.7 Stability constraining cost function solution flow chart: a Tope level flow chart; b voltage vector vi selection process flow chart

6.4 The Proposed SCDS-MPC Method

115

where x 1 = vdc , x 2 = id , x 3 = iq , λ = idc /Cdc , a0 = − ed /Cdc , a1 = − R /L, u1 = (ud − ed )/L, u2 = (uq − eq )/L, ω = 2πf , and f is the fundamental frequency of the AC bus. Step 1. Supposing the desired DC capacitor voltage to be x 1ref , the first tracking error is thus defined as z 1 = x1ref − x1

(6.23)

Our task in this step is to achieve the tracking task x 1 → x 1ref , i.e., z1 → 0, by considering the Lyapunov function V1 = 0.5z 12

(6.24)

a0 x 2 − λ) V˙1 = z 1 z˙ 1 = z 1 (x˙1ref − x1

(6.25)

Then, the derivative of V 1 is

Therefore, a virtual control x 2ref can be defined as x2ref =

x1 (x˙1ref − λ + k1 z 1 ) a0

(6.26)

where k 1 > 0 is a design parameter. Based on (6.26), if x 2 = x 2ref holds, we have V˙1 = −k1 z 12 ≤ 0

(6.27)

Step 2. By defining ⎧

z 2 = x2ref − x2 z 3 = x3ref − x3

(6.28)

z1 → 0 can be guaranteed by z2 → 0. To ensure system stability, the Lyapunov function V 2 is chosen as V2 = 0.5z 12 + 0.5z 22 + 0.5z 32

(6.29)

According to (6.22), (6.26) and (6.28), the resulting derivative of V2 is V˙2 = −k1 z 12 + a1 (z 22 + z 32 ) + γ

(6.30)

k1 z 2 x 1 a0 a0 z 1 z 2 − z 2 u 1 − z 3 u 2 − a1 z 2 x2ref +ωz 3 x2ref − (λ + x2 ) x1 a0 x1

(6.31)

where γ =

116

6 Stability Constraining Dichotomy Solution Based Model Predictive …

which is achieved by utilizing x 3ref = 0 to ensure the unity power factor and x˙2ref =

∂ x2ref k1 x 1 a0 z˙ 1 = (x˙1ref − λ − x2 ) ∂z 1 a0 x1

(6.32)

Since a1 < 0, V˙2 ≤ 0 can be achieved by γ ≤0

(6.33)

Therefore, (6.33) can be regarded as the stability constrain of the MPC approach. Cost function II: Fundamental Cost Function Part In this work, the control system is responsible for keeping the current following the reference current. In addition, it must have an acceptable steady-state distortion and dynamic response. The cost function G 1 , defined in (6.34), calculates the absolute error between the reference and predicted currents during the next sampling interval in order to meet this control quality. | | | | G 1 = |i α∗ − i α (k + 1)| + |i β∗ − i β (k + 1)|

(6.34)

Additionally, the control system should be able to prevent overcurrent. This can be achieved by defining a cost function of the following type: ⎧ G2 =

0 i s < Is. max ∞ i s ≥ Is. max

(6.35)

where Is.max is the current protection value. As mentioned above, keeping the system stable is the prerequisite to achieving the control task. The stability constraint has been derived mathematically for the above section. So, we can define a stability cost function element easily in the following way: ⎧ G3 =

0 γ ≤0 ∞γ >0

(6.36)

According to the earlier analysis, the ultimate cost function for realizing the control task and guaranteeing system stability could be summarized as follows: G = G1 + G2 + G3

(6.37)

The voltage vector which minimizes the cost function G will be selected as the optimal voltage vector. Then, it will be modulated with the PWM technique and applied to the inverter.

6.4 The Proposed SCDS-MPC Method

117

Fig. 6.8 Three-dimensional curve V2 vs. z2 and z3

The system stability will be analyzed in this sub-section. By storing the value of z1 , z2, and z3 , we can plot the system Lyapunov function V 2 shown in Fig. 6.8, where z1 is not presented as according to (6.27) and (6.28), the stability of z1 can be guaranteed if z2 → 0. It can be observed for the final state, z2 → 0, z3 → 0, and V 2 → 0. Regardless of the initial state, the final state is the same. This indicates the DC voltage is stable at the desired level, and the AC current is stable at the desired level. As a result, the proposed approach effectively guarantees system stability.

6.4.4 Sensitivity Analysis of Model Parameters Variation It is necessary to investigate parameter variation sensitivities for the proposed SCDSMPC, as it is based on an inverter mathematical model for all prediction and cost function solutions in this work. Simulation of the effect of parameter variation is done by Matlab/Simulink to avoid unnecessary interference and clarify the effect of parameter variation. In the simulation, the PWM switching frequency is set to 20 kHz, and the frequency of the power grid is set to a typical 400 Hz. The parameters of the inverter system can be seen in Table 6.2. In the simulation, the reference three-phase current amplitude is 4A before 0.005 s, then it steps to 8 A. the DC-link voltage is 300 V. Figure 6.9a shows the current and voltage performance without parameter variation. With 5 times grid side resistance variation, the control performance is

118

6 Stability Constraining Dichotomy Solution Based Model Predictive …

presented in Fig. 6.9b. As we can see, despite the resistance being five times higher, the control performance is not affected much. It is estimated that the resistance of the proposed SCDS-MPC is much more robust to grid side resistance variations in the real system because the resistance could only increase by a maximum of two times due to temperature rise. The grid side inductance variation test results are presented in Fig. 6.10. Figure 6.10a, b are the control performance with grid side inductance variations of 0.5 and 2 times, respectively. From the results, we can see the current ripples appear in Fig. 6.10b. There is a good chance that the actual inductance variation could be between 0.5 and 1.5 times. However, even though the inductance variation will degrade the control performance, the system could still operate stably. Figures 6.11 and 6.12 present control results with DC-link inductance and capacitance variation, respectively. From all the results, we get to the conclusion that, despite parameter variations having some relative impact on DC-link voltage performance, the proposed SCDS-MPC is resilient to DC-link inductance and capacitance fluctuation. Table 6.2 FPGA resources utilization of proposed SCDS-MPC implementation Resource

Amount used/total available

Logic elements

14,367/114,480 (13%)

Combinational functions

13,257/114,480 (12%)

Dedicated logic registers

7,028/114,480 (6%)

Pins

61/529 (12%)

Memory bits

241,934/3,981,312 (6%)

Embedded Multiplier 9-bit elements

114/532 (21%)

PLLs

1/4 (25%)

Fig. 6.9 Grid side resistance variation test (simulation). a no parameter variation; b Rˆ = 5R

6.4 The Proposed SCDS-MPC Method

Fig. 6.10 Grid side inductance variation test (simulation). a Lˆ = 0.5L; ; b Lˆ = 2L

Fig. 6.11 DC-link inductance variation test (simulation). a Lˆ f = 0.5L f ; ; b Lˆ f = 1.5L f

Fig. 6.12 DC-link capacitance variation test (simulation). a Cˆ f = 0.5C f ; b Cˆ f = 1.5C f

119

120

6 Stability Constraining Dichotomy Solution Based Model Predictive …

6.5 Test Results In this part, an actual AC cascaded system, whose architecture and characteristics have already been provided in Fig. 6.1 and Table 6.2, is used in this section to experimentally verify the proposed SCDS-MPC. The experimental testbench is shown in Fig. 6.13. From the testbench, we can see it is a hardware system with an FPGA-based controller. The FPGA-based controller is composed of the Altera DE2-115 development board and A/D converter and PWM interface board. Without clear emphasis, this testbench is modified in part to verify the performance of control schemes in the other chapters. The FPGA controller can be readily replaced by commercial digital signal processors (DSP), dSPACE, and other processors. The FPGA-based controller implements the entire control algorithm. The pipeline-parallel architecture is used to implement the control algorithm. With the help of the CORDIC (Coordinate Rotation Digital Computer) algorithm, complex trigonometric and quadratic functions are calculated. The clock frequency for this SCDS-MPC implementation design is 80 MHz, and each voltage vector assessment calls for a 40-step pipeline. Thus, this procedure takes 0.5S. In this study, the iteration step is set to 14, and each iteration evaluates 18 voltage vectors. As a result, 10.15 μS is required for this proposed SCDS-MPC in this design totally. The resource utilization of FPGA EP4CE115F29C7 is described in the following Table 6.2. First, the behavior of the conventional FCS-MPC and the proposed SCDS-MPC for the inverter was evaluated and compared by a 1.5 kW grid-connected threephase inverter. The maximum switching frequency is limited to 20 kHz by the FCS-sampling MPC’s frequency setting of 40 kHz. In industrial applications, this switching frequency is often the highest switching frequency for IGBTs. The PWM switching frequency for SCDS-MPC is set at 10 kHz. An active current step test is used to assess both the steady-state and dynamic performance of both control techniques.

Fig. 6.13 Experimental testbench

6.5 Test Results

121

Fig. 6.14 Experimental waveforms of the three-phase inverter in the MEA with the traditional FCS-MPC and the proposed SCDS-MPC methods: a With the traditional FCS-MPC method; b with the proposed SCDS-MPC method

As shown in Fig. 6.14a, b, in the beginning, the active and reactive current values are set to 4 A and 0 A, respectively. The reactive current remains zero while the active current is increased by one step to 6 A. The experimental findings show that both control systems function well under dynamic conditions, with the SCDSMPC method having a smaller current ripple than the more conventional FCS-MPC method. In addition, Fig. 6.14 presents that the current (total harmonics distortion) THD of traditional FCS-MPC and the proposed SCDS-MPC is 4.71% and 2.88%, respectively. The THD of SCDS-MPC at this method is significantly lower than the conventional one. In addition, the harmonics of the contemporary SCDS-MPC are regularly distributed around the switching frequency, whereas the harmonics of the traditional FCS-MPC are dispersed throughout the entire switching frequency range. Therefore, the filter design of the SCDS-MPC is easier than the typical FCS-MPC to remove the harmonic from the current distribution controlled by the latter. On the researched AC cascaded system, the proposed SCDS-MPC method’s stabilizing function was then proven to be effective. In Fig. 6.15, the derived stability constraints (i.e., G3 at (37)) are removed from the cost function of the SCDS-MPC method. According to Fig. 6.15a, this AC cascaded system is unstable without the recommended stability requirements because the

122

6 Stability Constraining Dichotomy Solution Based Model Predictive …

Fig. 6.15 Instability problem of the AC cascaded system when removing the stability constraining element G3 from the cost function of SCDS-MPC: a Instability experimental waveforms; b the zoomed instability experimental waveforms of (a)

DC bus voltage is clearly oscillating. If the oscillation waveforms are zoomed at Fig. 6.15b, it can be seen that this oscillation frequency is around 150 Hz which is very close to the resonant frequency of the L f and C f of the input EMI filter. According to this phenomenon, the AC cascaded system indeed has instability problems due to interaction between the EMI filter and the three-phase inverter [32, 33]. In Fig. 6.16, the stability constraining element G3 is enabled in the unstable AC cascaded system in Fig. 6.15. It can be seen that the unstable AC cascaded system stabilizes once the SCDS-MPC approach activates the stability restricting factor G3. As a result, the SCDS-MPC method’s derived stability constraint element G3 can successfully address the instability problems with the AC cascaded system. According to Figs. 6.14 , 6.15 and ~ 6.16, the effectiveness of the proposed SCDSMPC method has been verified.

6.6 Conclusion This chapter proposes an SCDS-MPC method to solve the stability problem of the AC cascaded system in the MEA. According to the Lyapunov criteria, this SCDS-MPC

References

123

Fig. 6.16 Experimental waveforms of the AC cascaded system when enabling the stability constraining element G3 in the cost function of SCDS-MPC. a The whole process of the system is from unstable to stable; b the transient process of (a)

derives the large-signal stability constrain clearly and transfers it to the analytical results, as well as adding it to the cost function of the SCDS-MPC method. Given the clear stability constraints of the proposed SCDS-MPC, this proposed method does not need to entangle how to select the weight of the stability in the cost function compared to the existing MPC method. It is therefore one contribution of SCDS-MPC to tell users how to make their MPC stable in a logical and mathematical manner, as opposed to the traditional ‘experience’ method. Additionally, SCDS-MPC enjoys the same advantages as FCS-MPC, including a constant switching frequency and better state/dynamic performance. Finally, a real prototype is built at the lab to verify the effectiveness of the proposed SCDS-MPC method.

References 1. P. Wheeler and S. Bozhko, “The more electric aircraft: Technology and challenges.,” IEEE Electrif. Mag., vol. 2, no. 4, pp. 6–12, 2014. 2. X. Roboam, B. Sareni, and A. De Andrade, “More electricity in the air: Toward optimized electrical networks embedded in more-electrical aircraft,” IEEE Ind. Electron. Mag., vol. 6, no. 4, pp. 6–17, 2012.

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3. B. Sarlioglu and C. T. Morris, “More electric aircraft: Review, challenges, and opportunities for commercial transport aircraft,” IEEE Trans. Transp. Electrif., vol. 1, no. 1, pp. 54–64, 2015. 4. Z. Xu, D. Zhang, F. Wang, and D. Boroyevich, “A unified control for the combined permanent magnet generator and active rectifier system,” IEEE Trans. Power Electron., vol. 29, no. 10, pp. 5644–5656, 2014. 5. H. Tian, Y. W. Li, and P. Wang, “Hybrid AC/DC system harmonics control through grid interfacing converters with low switching frequency,” IEEE Trans. Ind. Electron., vol. 65, no. 3, pp. 2256–2267, 2017. 6. M. Tariq, A. I. Maswood, C. J. Gajanayake, and A. K. Gupta, “Aircraft batteries: current trend towards more electric aircraft,” IET Electr. Syst. Transp., vol. 7, no. 2, pp. 93–103, 2017. 7. T. Wu, S. V Bozhko, and G. M. Asher, “High speed modeling approach of aircraft electrical power systems under both normal and abnormal scenarios,” in 2010 IEEE International Symposium on Industrial Electronics, 2010, pp. 870–877. 8. A. Emadi, A. Khaligh, C. H. Rivetta, and G. A. Williamson, “Constant power loads and negative impedance instability in automotive systems: definition, modeling, stability, and control of power electronic converters and motor drives,” IEEE Trans. Veh. Technol., vol. 55, no. 4, pp. 1112–1125, 2006. 9. X. Wang, F. Blaabjerg, and W. Wu, “Modeling and analysis of harmonic stability in an AC power-electronics-based power system,” IEEE Trans. Power Electron., vol. 29, no. 12, pp. 6421–6432, 2014. 10. W. Du, J. Zhang, Y. Zhang, and Z. Qian, “Stability criterion for cascaded system with constant power load,” IEEE Trans. Power Electron., vol. 28, no. 4, pp. 1843–1851, 2012. 11. R. D. Middlebrook, “Input filter considerations in design and application of switching regulators,” IAS’76, 1976. 12. X. Zhang, X. Ruan, and C. K. Tse, “Impedance-based local stability criterion for DC distributed power systems,” IEEE Trans. Circuits Syst. I Regul. Pap., vol. 62, no. 3, pp. 916–925, 2015. 13. A. Kwasinski and C. N. Onwuchekwa, “Dynamic behavior and stabilization of DC microgrids with instantaneous constant-power loads,” IEEE Trans. power Electron., vol. 26, no. 3, pp. 822– 834, 2010. 14. M. Cespedes, L. Xing, and J. Sun, “Constant-power load system stabilization by passive damping,” IEEE Trans. Power Electron., vol. 26, no. 7, pp. 1832–1836, 2011. 15. A. M. Rahimi and A. Emadi, “Active damping in DC/DC power electronic converters: A novel method to overcome the problems of constant power loads,” IEEE Trans. Ind. Electron., vol. 56, no. 5, pp. 1428–1439, 2009. 16. X. Zhang, Q. C. Zhong, V. Kadirkamanathan, J. He, and J. Huang, “Source-side Series-virtualimpedance Control to Improve the Cascaded System Stability and the Dynamic Performance of Its Source Converter,” IEEE Trans. Power Electron., vol. 34, no. 6, pp. 5854–5866, 2019. 17. Q. Xu, C. Zhang, C. Wen, and P. Wang, “A Novel Composite Nonlinear Controller for Stabilization of Constant Power Load in DC Microgrid,” IEEE Trans. Smart Grid, vol. 10, no. 1, pp. 752–761, 2019. 18. J. Yao, Z. Jiao, and D. Ma, “Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping,” IEEE Trans. Ind. Electron., vol. 61, no. 11, pp. 6285–6293, 2014. 19. S. Oucheriah and L. Guo, “PWM-based adaptive sliding-mode control for boost DC–DC converters,” IEEE Trans. Ind. Electron., vol. 60, no. 8, pp. 3291–3294, 2012. 20. Y. Sun, M. Su, X. Li, H. Wang, and W. Gui, “Indirect Four-leg Matrix Converter Based on Robust Adaptive Back-stepping Control,” IEEE Trans. Ind. Electron., vol. 58, no. 9, pp. 4288–4298, 2011. 21. T. Dragiˇcevi´c, “Dynamic Stabilization of DC Microgrids with Predictive Control of Point-ofLoad Converters,” IEEE Trans. Power Electron., vol. 33, no. 12, pp. 10872–10884, 2018. 22. P. Acuna, R. P. Aguilera, A. M. Y. M. Ghias, M. Rivera, C. R. Baier, and V. G. Agelidis, “Cascade-free model predictive control for single-phase grid-connected power converters,” IEEE Trans. Ind. Electron., vol. 64, no. 1, pp. 285–294, 2016.

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Chapter 7

Composite-Bisection Predictive Control to Stabilize the Three Phase Inverter Cascaded with Input EMI Filter in the SPS

7.1 Introduction Nowadays, the electrification of shipboard power systems (SPS) is proceeding at an ever-faster pace [1]. In order to meet future energy requirements, it allows the integration of a variety of renewable energy sources (REs) [2]. In the SPS, all conventional hydraulic, pneumatic, and mechanical subsystems have been replaced with power converter-intensive counterparts in order to increase system efficiency and reliability [3]. An SPS will operate as a mobile island-based dc microgrid. This will eliminate the need for frequency regulation, reactive power regulation, phase unbalance, and synchronization required for AC microgrids [4]. Meanwhile, dc MG has fewer power conversion stages [5]. As illustrated in Fig. 7.1, the dc MG based SPS consists of three parts: (1) There are small-scale generators, energy storage systems, and downstream load converters involved in the DC bus. The electromagnetic interference (EMI) arises from the high-rate change of voltage [dv/dt] and current [di/dt] due to the fast switching of the semiconductor devices of the load converters [6]. (2) Load distortions are caused by a variety of loads, including heating, lighting, motors, and PCs. (3) A cascade of downstream load inverters operating with LC EMI filters acts as the interface between dc bus and ac load. To cope with the EMI at dc bus, an electrolytic capacitor with high capacitance (C ele ) and equivalent series resistance (ESR) is commonly added ahead of load inverters. The capacitors act as low-impedance paths for high-frequency signal switching currents. As a result, it reduces voltage ripples of the DC-link voltage and stores energy [7]. To regulate such systems, numerous approaches are investigated: PI/PR control [8], backstepping control [2, 9], sliding-mode control [10], deadbeat control [11], MPC [12], Lyapunov-based control [13], repetitive control [14], event-triggered control [15], adaptive control [16], etc. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Zhang et al., Stability Enhancement Methods of Inverters Based on Lyapunov Function, Predictive Control, and Reinforcement Learning, https://doi.org/10.1007/978-981-19-7191-4_7

127

128

7 Composite-Bisection Predictive Control to Stabilize the Three Phase …

...

Weapons ... Battery (rail gun...)

REs Wind turbine PV panel

AC/DC

EMI filter

DC/DC

DC/DC

AC/DC

DC/DC

EMI filter

DC/AC

Super C

ESs dc bus

EMI filter

DC/AC

DC/AC ...

Fuel generator Motor

Telecomm unication ...

Output filter

Output filter Heating/ lighting

PCs Utilities

Load Investigated system

Fig. 7.1 Investigated cascaded system in a shipboard power system (SPS)

In practice, the dc-side LC EMI input filter, shown in Fig. 7.2, can provide higher power/energy density and a longer lifespan, if the electrolytic capacitor is replaced by a film capacitor. It reduces the electromagnetic interference from the inverter to the dc bus as a result of interconnecting the dc bus and the dc-link voltage of the inverter through the LC EMI input filter [17]. As a result, the above control schemes are applied to downstream inverters cascaded with LC input filters, resulting in oscillations in the voltages at the load and dc-link in this scenario. Tightly regulated inverters behave like a typical type of CPL and exhibit negative incremental resistance, which is well known [18]. Using the Middlebrook criterion, the CPL problem arises as a result of an impedance intersection between the source-converter output impedance (Z out ) and load-converter input impedance (Z in ). The LC input filter is generally regarded as a type of source converter. Several linear techniques exist to stabilize cascaded systems, such as passive damping or active damping in which Z out and Zin are reshaped to avoid impedance section problems [19]. Passive damping suppresses the resonant peak of Z out with additional RC/RL/C dampers [20]. They cost more in terms of additional hardware and power loss but are sturdy and effective. In order to completely or partially eliminate the negative incremental impedance feature within a certain frequency range, active damping shapes impedance using control loops [19]. However, it changes the load converter’s internal structure, which might be bad for performance.

7.1 Introduction

129

via E

Cdc

vdc S4

S2

vib S5

dc bus

Zout

vic S6

r

L i voa ioa La v iLb iob ob voc ioc iLc ESR

EMI filter

n

S3

Upgrades Zin

Cele

Load

S1

rdc Ldc idc i pol

C N

Fig. 7.2 The downstream load inverters cascaded with LC EMI input filters

Several improvements to FCS MPC are discussed in [21]. This scheme optimizes load-side performance while stabilizing dc-link oscillation using penalties. As such, it is especially suitable for FCS MPC without PWM, in which on/off states of switches and switching vectors have an inherent mapping relationship. It should be noted that modulated MPC is lacking such a mapping relationship, which may lead to the stabilizing method not performing to expectations. Moreover, modulated MPC has a varying switching frequency (f s ). Zheng et al. [22] realizes a fixed f s via continuous control set (CCS) MPC with PWM. Nonetheless, the cost function in [22–25] are all load-voltage-oriented that does not have an additional item to improve stability, which cannot address the CPL problem of the cascaded system. Ma et al. [26] combines the modulated FCS MPC with a novel dichotomy algorithm to achieve better steady-state performance at the cost of more computing resources. On the other hand, Ma et al. [17] proposes stability-constraining FCS MPC to stabilize the dc-link oscillation. The constraining condition (γ ≤ 0) is derived to ensure the convergence of a Lyapunov function (V) [9]. It is a model-based approach, though, and it is currently only applicable to grid-tied inverters of the L type. The case-by-case derivation is unavoidable when applied to many topologies, although the design of V is very specialized. More than one γ exists to ensure the convergence of V. It is hard to say which one is better. Also, the parameter tunning guideline in γ is not given. Additionally, each sample interval of the dichotomy algorithm used in [17] results in the calculation of 252 vectors (T s ). Its larger application is likewise hampered by this computationally expensive trait. We propose the CB-PC scheme to stabilize and indirectly regulate the examined cascaded system in the SPS as a result of the foregoing discussions on MPC techniques. This chapter’s primary contributions can be summed up as follows: (1) System-level Stability with Wider Applicable Scenarios—The suggested CB-PC not only guarantees the inverter’s stability but also stabilizes the entire cascaded system. It is suggested to stabilize dc-link oscillation using a better general technique that is applicable to all filter types. The instantaneous power theory, which is applicable to FCS MPC with or without PWM, is also a foundation for this technique.

130

7 Composite-Bisection Predictive Control to Stabilize the Three Phase …

(2) Improved Transient Response—By keeping track of the inductor-current vector iLref (k + 1), which is obtained offline by deadbeat control, the load voltage management is indirectly achieved. The transient time required to stabilize the dc-link voltage is shorter than with direct voltage management. (3) Modified Bisection Algorithm with Tunable Computing Burden—When compared to the dichotomy algorithm, the improved bisection technique reduces computation overhead by around 66.7% per iteration. The THD of the load voltage can be reduced by altering the iteration number (n), which permits flexible use of computational resources.

7.2 Mathematical Modeling As shown in Fig. 7.2, according to KCL/KVL, the ac-side dynamics of the investigated system can be modeled as ⎛ di La ⎞ ⎝

dt di Lb dt di Lc dt

⎞ ⎞ ⎛ ⎞⎛ 0 via − voa i La 1 0 ⎠⎝ i Lb ⎠ + ⎝ vib − vob ⎠ L i Lc vic − voc 1 ⎞ ⎛ i La − i oa 1 ⎠ = ⎝ i Lb − i ob ⎠ C i Lc − i oc

⎛ 10 r ⎠ = − ⎝0 1 L 00 ⎛ dvoa ⎞ dt

⎝ dvob dt dvoc dt

(7.1)

(7.2)

The gating signals of the three-phase bridges are modeled as: ⎧ Sa =

1 s1 on & s4 off 0 s1 off & s4 on

⎧ Sb =

1 s2 on & s5 off 0 s2 off & s5 on

⎧ Sc =

1 s3 on & s6 off 0 s3 off & s6 on

The switching states are represented in vector format, S = Sα + j · Sβ =

) 2( Sa + aSb + a2 Sc 3

(7.3)

√ where a = e j (2π/3) = −1/2 + j 3/2. Therefore, the three-phase bridges’ terminal switching vector can be expressed as follows: vi = vi α + j · viβ = S · E

(7.4)

Thus, we can determine the filter current vector, load voltage vector, and load current vector, given by: i L = i Lα + j · i Lβ =

) 2( i La + ai Lb + a2 i Lc 3

7.3 Conventional FCS MPC and Problem Formulation

131

) 2( voa + avob + a2 voc 3 ) 2( = i oa + ai ob + a2 i oc 3

vo = voα + j · voβ = io = i oα + j · i oβ

(7.5)

Using (7.5), dynamic Eqs. (7.1) and (7.2) can be represented as r iL + L C

di L = vi − vo dt

(7.6)

dvo = i L − io dt

(7.7)

As a result of considering the LC EMI filter further, the dc-link dynamic equation of the investigated system can be modeled as follows: Cdc

dvdc = i dc − i pol dt

(7.8)

7.3 Conventional FCS MPC and Problem Formulation As shown in Fig. 7.3, the three-phase bridge inherently has eight possible switching states (S a , S b , S c ), producing eight terminal switching vectors, in which v0 (000) and v7 (111) are equivalent to each other. Conventional FCS MPC is based on the enumeration of v0 –v7 , which directly manipulates the switches s1 –s6 without a PWM modulator. Firstly, (7.6)–(7.7) are discretized using a zero-order holder. Then, v0 –v7 are enumerated to predict vo (k + 2), the specific vi that minimizes (7.9) will be executed [23]. | | g = 0.5|vor e f (k+2) − vo (k+2)|

(7.9)

where the well-known two-step prediction method has been implemented for control delay compensation [27]. Fig. 7.3 Enumerated eight switching states of three-phase bridges

v3(010) v4(011) v0(000)

v5(001)

β

v2(110)

v7(111) α v1(100) v6(101)

132

7 Composite-Bisection Predictive Control to Stabilize the Three Phase …

However, conventional FCS MPC only considers the stability of the downstream inverter itself, not the entire cascade. As shown in Fig. 7.4, in a cascaded system, conventional FCS MPC results in severe dc-link oscillations, which get worse as the load resistance increases. A high dc-link capacitor current results from the dclink voltage oscillation. The capacitor’s lifetime is shortened as a result of internal self-heating brought on by the dc-link current, which also raises the capacitor’s ESR and speeds up electrolyte evaporation [28, 29]. The root cause of this oscillation problem has been analyzed in [17]. The LC input filter can be sized to get rid of the oscillations at the cost of a bulky capacitor [30]. Table 7.1 records the nominal system parameters, where vm denotes the desired load voltage amplitude. According to [17, 21], L dc is selected larger than its real-world industrial value to instigate severer dc-link oscillation, making it convenient for observation via an oscilloscope. Additionally, traditional FCS MPC results in variable f s . Simulink 2020a simulation results show that as the sampling interval (T s ) is reduced from 100 to 10 μs, the average fs steadily increases from 3.14 to 24.26 kHz.

300

[20ms/div]

100

voabc (V)

THD:0.47%

500

vdc(V)

dc-link electrolytic Cele

0

[20ms/div]

-200 -200

Cascaded system

300

[20ms/div]

100

THD:0.43%

200

THD:7.63%

voabc (V)

vdc(V)

500

0

[20ms/div]

-200 -200

1kW 2kW

10 0

[20ms/div]

ioabc (A)

ioabc (A)

1kW 2kW

-10

THD:14.01%

200

10 0

[20ms/div]

-10

(a)

(b)

Fig. 7.4 Conventional FCS MPC regulated downstream load inverter (T s = 20 μs). a With a dc-link electrolytic capacitor; b cascaded with an LC EMI filter

Table 7.1 Nominal system parameters E

L

C

r

C ele

300 V

6.3 mH

50 μF

0.096 Ω

2000 μF

vm

C dc

L dc

r dc

f

115 × 1.414 V

100 μF

10 mH

0.12 Ω

50 Hz

7.4 Proposed Composite Bisection Predictive Control

133

7.4 Proposed Composite Bisection Predictive Control 7.4.1 Structure of the Proposed CB-PC Scheme The proposed CB-PC solution tries to combine the benefits of deadbeat control with FCS MPC. To achieve multi-control goals, an all-in-one cost function (G) imitates the FCS MPC. Firstly, based on instantaneous power theory, an improved method for stabilizing DC-link oscillation is proposed, which is valid for FCS MPC with or without PWM. Then, load voltage control is realized indirectly by tracking a deadbeat-like vector, leading to faster dc-link voltage stabilization compared to the direct voltage control using (7.9). The proposed modified bisection algorithm uses a tunable iteration number to optimize computation resources, resulting in lower load voltage THD. In order to achieve multiple control objectives, G is formulated as the sum of three sub-cost functions: G = G1 + G2 + G3.

(7.10)

Sub-cost function G1 penalizes future dc-link voltage variations in order to suppress dc-link voltage oscillations, G 1 = ζ [vdc (k + 1) − E]2 ,

(7.11)

where ζ is a weighting factor (ζ > 0), and the subscript k + 1 denotes the (k + 1)th discrete timestep. G1 serves as the active damper to suppress the dc-link voltage oscillation. G2 is defined to realize three-phase load voltage control, which is realized indirectly by enforcing iL (k + 1) to track the inductor-current vector iLref (k + 1), given by | | G 2 = 0.5|i Lr e f (k + 1) − i L (k + 1)|.

(7.12)

The dynamic vector iLref (k + 1) is shown in (7.25), which is derived offline inspired by deadbeat control in [11, 31]. G3 imports the over-current protection ability, expressed as ⎧ G3 =

|i L (k + 1)| < i L max 0 , 6 10 (≈ +∞) |i L (k + 1)| ≥ i L max

(7.13)

where iLmax is the maximum allowed inrush inductor current threshold, which is determined considering the characteristics of feeding loads and semiconductors switching devices. The inductor-current vector iL (k + 1) is the transportation hub of the proposed control scheme, which is predicted by discretizing (7.6), yielding that

134

7 Composite-Bisection Predictive Control to Stabilize the Three Phase …

Fig. 7.5 Proposed droop-akin strategy to adjust the weighing factor (ζ )

(c0, ess0)

c

c = - k ess +co (c*, ess*) (c1, ess1)

(0, 0)

i L (k + 1) =

Ts [vi (k) − vo (k) − r i L (k)] + i L (k). L

ess (%) (7.14)

By designing G properly, harmonic elimination, switching loss minimization, and common-mode voltage minimization can also be achieved [32], which is mature but beyond the focus of this chapter. Thus, it is omitted here. It is a trade-off between load voltage tracking and dc-link voltage stabilization that determines the weighting factor ζ in G1 . To quantify this effect, the steady-state error of load voltage (ess ) and dc-link voltage (edc ) are expressed in percentage with respect to their references (vm , E) respectively, given by | | ess = 100|vor e f − vo |/vm , edc = 100|vdc − E|/E .

(7.15)

According to (7.11) and (7.15), larger ζ tends to yield smaller edc at the expense of increasing ess . To strike a trade-off between them, a droop-akin method is proposed to update the ζ during operation. As illustrated in Fig. 7.5, if the controlled system undergoes a sudden increase in load power level, the operation point may move from (ζ 0 , ess 0 ) to (ζ 1 , ess 1 ), where ess 1 > ess 0 and ζ 1 < ζ 0 . With a smaller ζ, ess reduces gradually. Finally, the system will reach another operation point (ζ * , ess * ). ζ in G1 is set as 0.5 initially, which is equal to the weighting factor in G2 , so that dc-link voltage stabilization and load voltage tracking control objectives are deemed as equally important.

7.4.2 Improved Generic DC-Link Stabilization Strategy Based on Instantaneous Power Theory To predict vdc (k + 1), (7.8) is discretized as (7.16) using forward Euler approximation, given by vdc (k+1) =

] Ts [ i dc − i pol + vdc (k), Cdc

where idc and vdc (k) are measured by sensors.

(7.16)

7.4 Proposed Composite Bisection Predictive Control

135

In [21], ipol is calculated via the following formula, i pol = Sa i La + Sb i Lb + Sc i Lc ,

(7.17)

which demonstrates that ipol is essential for interfacing the dc-side dynamic of the LC EMI filter with the ac-side subsystem. The working mechanism of the stabilization method can be intuitively interpreted as: vdc (k + 1) is penalized in G → ipol dynamic is considered → S a –S c selection gets optimized. Nevertheless, (7.17) is particularly devised for FCS MPC without PWM since each vi and S a –S c has a pre-known mapping relationship, as illustrated in Table 7.2 of [21]. However, implementation (7.17) has difficulty for MPC having a PWM stage. For example, in this chapter, vi is generated by the modified bisection algorithm, while S a –S c is generated by PWM. Such a mapping relationship between S a –S c and vi does not exist. Table 7.2 Comparisons between existing MPC and the proposed approach for downstream load inverters having LC output filters MPC schemes

[23]

[22]

[21]

[25]

This chapter

Category of the predictive Conventional CCS MPC control OSV MPC

Improved FCS (OSS) Bisection + FCS MPC deadbeat + (OSV) OSV MPC MPC

Application LC-filtered Stable scenarios downstream inverters

Stable

Stable

Stable

Oscillation

Oscillation

Stable

Oscillation Stable

Applicable scope of the stabilization method

–

–

Without PWM

–

With/without PWM

Load voltage control

Direct

Direct

Direct

Direct

Indirect

Need offline solving of the vector reference

No

Yes

No

Yes

Yes

Transient time for dc-link stabilization

–

–

≈ 35 ms

–

≈ 5 ms

Switching frequency f s

Varying

Fixed

Varying

Fixed

Fixed

THD of load voltage when T s : 100 μs (simulation results)

> 10%

> 10%

2.45%

> 10%

0.68%

The number of calculated vi per T s

8

1

8

–

6(n + 1)

Computation burden

Medium

Low

Medium

> Medium Tunable

Integration of system constraints

Flexible

Online post Flexible correction

If cascaded with LC input filters

Flexible

Stable

Flexible

136

7 Composite-Bisection Predictive Control to Stabilize the Three Phase …

To address this issue and achieve dc-link voltage stabilization, this chapter bridges the gap between ipol and vi using instantaneous power theory from a power balance point of view. Since vi can be expressed as vi = viα + jviβ , the active power (P) and reactive power (Q) injected from the ac-side subsystem can be approximated as ⎧

] [ P = 1.5 [viα · i Lα (k + 1) + viβ · i Lβ (k + 1) ] , Q = 1.5 vi α · i Lβ (k + 1) − viβ · i Lα (k + 1)

(7.18)

where the switching vector vi and its resulting inductor-current vector iL (k + 1) is involved. Here, iL (k + 1) is predicted via (7.14). The amplitude-invariant Clarke coordinate transformation (T 3/2 ) is utilized to convert three-phase state variables in abc frame to the stationary αβ reference frame. In this case, the coefficient in (7.18) is set as 1.5. If T 3/2 is a power-invariant counterpart, the coefficient should be set as 1. According to the power balance, dc-side power is equal to ac-side power. As a result of the system’s topology, the reactive power Q has a negligible amplitude compared to its active counterpart P. ipol can be approximated as i pol ≈

√

P 2 + Q 2 /vdc (k).

(7.19)

The working mechanism of the improved stabilization method can be intuitively interpreted as: vdc (k + 1) is penalized in G → ipol dynamic is considered → vi selection gets optimized. The improved generic stabilization method using (7.11), (7.16), (7.18), (7.19) is a general method to suppress dc-link voltage oscillation, which is applicable for FCS MPC with or without modulators.

7.4.3 Indirect Voltage Control Strategy to Achieve Better Transient Response Inspired by the Deadbeat Control The first step in predicting load voltage is to discretize (7.7) using explicit Euler approximation, resulting in vo (k + 1) =

Ts [i L (k) − io (k)] + vo (k). C

(7.20)

In the next instant (k + 1), the predicted load voltage vo (k + 1) is expected to track its reference voltage voref (k + 1), where the tracking error is expected to be zero. e(k + 1) = vo (k + 1) − vor e f (k + 1) = 0.

(7.21)

7.4 Proposed Composite Bisection Predictive Control

137

According to (7.20), (7.21) always holds if the inductor-current vector iLref (k) in charge at the next instant is selected as i Lr e f (k) =

] C[ vor e f (k + 1) − vo (k) + io (k), Ts

(7.22)

Using second-order Lagrange extrapolation, voref (k + 1) is predicted as follows: vor e f (k + 1) = 3vor e f (k) − 3vor e f (k − 1) + vor e f (k − 2).

(7.23)

Nevertheless, (7.22) does not yet consider delay compensation, which will adversely affect control performance. For digital delay compensation, two-step prediction is performed by shifting (7.22) one step forward, resulting in i Lr e f (k + 1) =

] C[ vor e f (k + 2) − vo (k + 1) + io (k + 1). Ts

(7.24)

Substituting (7.20)–(7.24), the deadbeat-like dynamic vector iLref (k + 1) considering delay compensation becomes i Lr e f (k + 1) =

[ ] C Ts vor e f (k + 2) − [i L (k) − io (k)] + vo (k) + io (k + 1), Ts C (7.25)

where voref (k + 2) is predicted using Lagrange extrapolation expressed in (7.26), and io (k + 1) can be approximated to be io (k) within one sampling interval. As seen in (7.25), iLref (k + 1) incorporates the future voltage reference voref (k + 2). Because of this inherent deadbeat-like property, faster voltage stabilization can be achieved in dc-links than in direct voltage control (7.9). vor e f (k + 2) = 9vor e f (k) − 11vor e f (k − 1) + 3vor e f (k − 2).

(7.26)

Using higher-order Lagrange extrapolation, voref (k + 2) in (7.26) can be predicted more accurately if a digital processor has sufficient storage capacity.

7.4.4 Modified Bisection Algorithm To improve steady-state control performance, we present a modified bisection algorithm with tunable iteration numbers. The candidate set of switching vectors is extended from v1–7 of the hexagon to the circular plane. As illustrated in Fig. 7.6, the initial switching vector V 1 ∠θ 1 remains the same when each iteration is launched. Switching vectors have maximum amplitudes determined by the radius of the plane (V m ), where V m = 2E/3. In the proposed CB-MPC, the optimal switching vector

138

7 Composite-Bisection Predictive Control to Stabilize the Three Phase …

Fig. 7.6 Explanation of the modified bisection algorithm to search the optimal switching vector V opt ∠θ

(V opt ∠θ ) per T s is determined by n + 1 times of iterations. Figure 7.7 illustrates the flowchart of the proposed CB-MPC. In each iteration, dθ phase difference and dV amplitude difference would be the step size to filter and select the optimal switching vector V opt ∠θ. In each iteration, dθ will be divided into two parts equally, resulting in three switching vectors that determine the optimal angle. As dV is bisected continuously, 3 switching vectors are generated to determine the optimal amplitude. After completing each iteration, both phase and amplitude step sizes (dθ, dV ) respectively decrease to their halves, expressed as dθ = 0.5dθ, dV = 0.5dV. Initially, dθ = 2π/3, dV = V m /2 in the first iteration. At the nth-time iteration, dθ = 2π/(3 × 2n−1 ) and dV = V m /(2n ). After n + 1 times of iteration, the specific switching vector that minimizes G is regarded as V opt ∠θ, which is sent to a PWM modulator to attain a fixed f s . It is important to note that n is relaxed to a tunable positive integer on purpose, which allows processors to exploit some of their computing power. The resolutions of V opt and θ are deemed as dθ and dV at the (n + 1)th iteration, which are functions of n,

7.4 Proposed Composite Bisection Predictive Control

139

Fig. 7.7 Pseudocode of the proposed CB-PC to stabilize and indirectly regulate a downstream load inverter cascaded with an LC input filter in the SPS

7 Composite-Bisection Predictive Control to Stabilize the Three Phase …

3.5 %

54

2.5 %

38

1.5 %

22

0.5 %

1

3 4 5 6 7 The iteration number (n)

2

8

6

Number of vi

Votlage THD

140

Fig. 7.8 Relationship between load-voltage THD, the number of evaluated switching vectors per T s as the increase of n

⎧

dθ = 2π/(3 × 2n ) . dV = Vm /(2n+1 )

(7.27)

Meanwhile, the number of calculated switching vectors (N m ) is also a function of n, Nm = 6(n + 1).

(7.28)

(7.27) and (7.28) demonstrate that larger n achieves higher resolutions of V opt and θ at the expense of increased computation cost. Simulation tests in Fig. 7.8 show that the THD of load voltage decreases as the increase of n. Since n + 1 times of iterations require to be completed within 1/f s , the selection of n relates to f s , computation resources, and user needs. In this chapter, f s of the SPWM modulator is 10 kHz. If n is set as 5, load-voltage THD is low enough that corresponds to standard IEC 62,040-3. In addition, experiments using commercial FPGA processors confirm that such computation requirements can be met easily. The implementation schematic of the proposed CB-PC is presented in Fig. 7.9. V opt ∠θ is normalized with respect to half of the dc-link voltage (E/2) before PWM. Inductor currents, load voltages, and load currents are sampled and converted to Cartesian coordinates via amplitude-invariant Clarke transformations. Three loadcurrent sensors can be replaced by observers to reduce implementation costs [23].

7.5 Test Results 7.5.1 Effectiveness of the Improved Generic Stabilization Method To test the downstream load inverter, a downscale prototype cascaded with an LC input filter is used. Nominal system parameters are referred to in Table 7.1. Due to the limited number of channels of the oscilloscope, voa or ioa is shown as a representative of three-phase elements. By default, the digital delay has been compensated without clear emphasis.

7.5 Test Results

141

Fig. 7.9 Implementation of the proposed CB-PC scheme on a downstream load inverter cascaded with LC input filters in the SPS

As presented in Fig. 7.10a, G1 is initially disabled by setting ζ to be 0, where severe dc-link voltage oscillation and distorted three-phase load voltage can be observed. Then, G1 is enabled by setting ζ to be 0.5, which effectively suppresses the dc-link oscillation within 4 ms. While this is occurring, the three-phase load voltage is being regulated to be sinusoidal. As shown in Fig. 7.10b, followed by the disabling of G1 , there is a spontaneous oscillation in the dc-link voltage.

7.5.2 Transient Performance of the Indirect Voltage Control in Comparison with the Existing Direct Voltage Control To justify the superiority of the proposed indirect voltage control method over the direct counterpart, (7.9) and (7.12) are implemented to control the cascaded system respectively when supplying 2 kW resistive load (T s : 100 μs). In Fig. 7.11a, it takes 40 ms to stabilize the dc-link oscillation via direct voltage control. In comparison, as shown in Fig. 7.11b, it is reduced to 10 ms after using the proposed indirect approach. A comparison of the proposed indirect voltage control with direct voltage control shows that it has achieved faster stabilization of dc-links.

142

7 Composite-Bisection Predictive Control to Stabilize the Three Phase …

Fig. 7.10 The cascaded system under the proposed approach with a 2 kW resistive load where T s = 100 μs and ζ = 0.5. a G1 off → on; b G1 on → off

7.5.3 Performance of the Proposed CB-PC Under Droop-Akin Strategy With/Without Delay Compensation According to the simulation results presented in Fig. 7.12, when connected to a 2 kW load, as ζ increases from 0.025 to 1, edc decreases from 0.27 to 0.185%, while the ess grows from 1.1 to 18%. The slope (−k) of the linear droop curve (ζ =−kess + ζ o ) in Fig. 7.5 is set as the negative reciprocal of the slope (17.455) in Fig. 7.12a. And ζ o is a constant ranging from 0 to 1. As seen in Fig. 7.10a, when ζ is initially set as 0.5, the resulted ess % is 11.5% and edc % is equal to 0.22% when supplying 2 kW load. In comparison, as in Fig. 7.13b, after using the proposed droop-akin method to update ζ, the operation point (ζ, ess %, edc %) reaches (0.07, 2.2%, 0.26%) in steady-state. The analytic mathematical expression of the used droop curve is ζ =−ess /17.455 + 0.2. A saturation block is added for ζ generation whose [lower limit, upper limit] is set as [0.05, 0.5]. Compared to setting ζ as 0.5, the droop-akin strategy has effectively reduced ess % from 11.5 to 2.2% at the expense of slightly enlarged edc % and increased transient time in terms of dc-link stabilization. Comparing Fig. 7.13a and b demonstrates that faster dc-link stabilization and less load-voltage distortions during transients get achieved after control delay compensation.

7.5 Test Results

143

Fig. 7.11 Transient performance comparisons when T s = 100 μs, ζ = 0.15. a Direct voltage control using (7.9); b proposed indirect voltage control using (7.12)

7.5.4 Comparisons Between the Proposed CB-PC and Existing MPC Table 7.2 records comparisons among the proposed CB-PC and existing MPC schemes for three-phase inverters. Ma et al. [17] proposes a backstepping-based stability-constraining FCS MPC for an L-type grid-connected bus conversion system using a dichotomy algorithm. The initial phase searching step of the existing dichotomy algorithm is π/2, where 18 switching vectors are computed per iteration. In comparison, the initial phase searching step gets extended to 2π/3 in the presented modified bisection algorithm. By evaluating only six switching vectors, approximately 66.7% of the computation burden has been saved per iteration as compared to the existing algorithm. Besides, the number of iterations in [17] is fixed to 14. By relaxing this parameter to a tunable positive integer, surplus computing resources can be more effectively utilized to enhance steady-state performance. Optimal switching vector (Optimal switching vector (OSV)) and optimal switching sequence (OSS) MPC are two subsets of FCS MPC. It is worth noting that a fixed f s is merely an additional advantage of the proposed CB-PC. The OSS MPC also has a fixed f s [25]. However, it typically leads to a higher computation burden compared to the utilized OSV MPC [33]. Comparatively, the proposed CB-PC can flexibly adjust its computation burden depending on computing resources.

7 Composite-Bisection Predictive Control to Stabilize the Three Phase …

ess (%)

144 18 16 14 12 10 8 6 4 2 0

200W

0.025 0.05 0.1

0.2

800W

1600W 2000W ess = 17.455 + 1.7489

0.3 0.4 0.5 0.6 0.7 Weighting Factor

0.8

0.9

1

0.9

1

(a) 200 W

800 W

1600 W

2000 W

0.25

edc (%)

0.2 0.15 0.1 0.05 0 0.025 0.05 0.1

0.2 0.3 0.4 0.5 0.6 0.7 Weighting Factor

0.8

(b) Fig. 7.12 Adjustment of the weighing factor (ζ ) in G3 where T s = 100 μs: a trade-off between load voltage tracking and dc-link voltage stabilization. a ess (%) as the increase of ζ; b edc (%) as the increase of ζ

An optimal switching vector is solved in [22] via offline analytic derivation. It is inherently a continuous control set (CCS) MPC approach, which enjoys a much lower computation burden compared to FCS MPC. It also achieves a fixed f s via a PWM modulator. System limitations may be flexibly included in the FCS MPC by simply adding them to the cost function, whereas the CCS MPC has to develop an “online post-correction method.” Besides, [22] is particularly designed for singleobjective voltage control of LC-type downstream load inverters, which may not be appropriate for the stabilization of the investigated cascaded system. The conventional FCS MPC is improved in [21] with artificial importation of dc-link voltage stabilization ability. Improved FCS MPC has variable f s . In order to compare the performance of the proposed CB-PC and the improved FCS MPC, the same cascaded system is controlled with the same ζ of 0.5. As presented in

7.5 Test Results

145

Fig. 7.13 Regulated cascaded system operating with the droop-like strategy, T s : 100 μs. a Without delay compensation (7.22); b with delay compensation (7.25)

Fig. 7.14a and b, the THD of improved FCS MPC is around 2.57%. The proposed CB-PC reduces this to approximately 0.94%, which indicates that it can achieve better steady-state performance under the same ζ. There has been evidence that the improved FCS MPC is robust to the large scale of parametric variation in [21]. Using Simulink 2020a (Simscape), the following simulation results demonstrate the proposed approach has great robustness, which is comparable to improved FCS MPC. As seen in Fig. 7.15a–c, two approaches are deployed to control the same cascaded system with 2 kW load (ζ: 0.5), where 25%– 1000%C dc , 20%–1000%C and 20%–1000%L scale of plant parametric variation have been considered. The scenario of L dc variation is not shown since the value of L dc is not utilized in the proposed CB-PC. T s of the proposed approach are fixed to 100 μs. As shown, with the same T s , THD using improved FCS MPC is higher than that of the proposed approach no matter whether the parametric mismatch is considered or not. If T s of improved FCS MPC are reduced to 50 μs, in most cases, its THD is still higher than that of the proposed method under the same scale of parametric mismatch. As presented in Fig. 7.16a, as C dc varies, the THD of the proposed approach fluctuates from 0.68 to 1.52%, where the THD of improved FCS MPC with T s :

146

7 Composite-Bisection Predictive Control to Stabilize the Three Phase …

Fig. 7.14 Steady-state control performance comparisons supplying a 2 kW resistive load (ζ = 0.5). a Improved FCS MPC; b the proposed approach

20 μs changes from 0.27 to 0.55%. In Fig. 7.15b, from 20%C to 1000%C, THD of the proposed approach monotonically decreases from 3.17 to 0.37% while THD of improved FCS MPC with T s being 20 μs fluctuates between 0.27 and 3.07%. During 20%L–1000%L, THD of the proposed approach varies from 0.68 to 6.79%, where THD of improved FCS MPC with T s being 20 μs ranges from 0.27 to 7.8%. For a fair comparison, T s of improved FCS MPC are reduced to 20 μs to do experiments, where its average f s is approximately 11 kHz, while T s of the proposed approach remain 100 μs. As seen in Fig. 7.16a and b, according to the proposed approach, the power quality of the load voltage using improved FCS MPC is comparable, which means that improved FCS MPC achieves similar steady-state performance at the expense of 4 times higher sampling frequency compared to the proposed CB-PC. The proposed approach employs indirect load voltage control, while the improved FCS MPC implements direct voltage control. As shown in Fig. 7.16a, it takes improved FCS MPC approximately 35 ms to stabilize the dc-link voltage oscillation resulting from the load resistance step, while such transient time is 5 ms using the proposed approach. Meanwhile, the load voltage in Fig. 7.16b displays fewer distortions in comparison with Fig. 7.16a during the transient process.

7.5 Test Results

147

Fig. 7.15 Simulation results of proposed approach and improved FCS MPC under parametric variations. a 25%–1000%C dc ; b 20%–1000%C; c 20%–1000%L

148

7 Composite-Bisection Predictive Control to Stabilize the Three Phase …

Fig. 7.16 Transient response of the cascaded system under load resistance step (ζ = 0.4). a Improved FCS MPC T s : 20 μs; b the proposed approach, T s : 100 μs

7.6 Conclusion In this chapter, a CB-PC scheme is proposed for the control of load voltage and the stabilization of dc-link voltage of downstream load inverters cascaded with LC input filters. To stabilize the dc-link oscillation, an improved generic method based on instantaneous power theory has been devised that is applicable to FCS MPC with or without modulators. To reduce the transient time of dc-link stabilization, the load voltage control is realized indirectly by tracking an inductor-current vector iLref (k + 1), which is derived out offline via deadbeat control considering delay compensation. Compared to the direct voltage control, such transient time gets reduced from 40 to 10 ms after using the indirect voltage control. A droop-akin strategy updates ζ online, which strikes a trade-off between the above two control objectives. A modified bisection algorithm with a tunable iteration number (n) is presented to improve steady-state performance. It enables flexible exploitation of the surplus computing resources to achieve THD suppression. The proposed algorithm reduces computation costs by approximately 66.7% over the existing dichotomy algorithm. It also exhibits good robustness to parametric variations.

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19. X. Zhang, Q. C. Zhong, V. Kadirkamanathan, J. He, and J. Huang, “Source-side Series-virtualimpedance Control to Improve the Cascaded System Stability and the Dynamic Performance of Its Source Converter,” IEEE Trans. Power Electron., vol. 34, no. 6, pp. 5854–5866, 2019. 20. M. Cespedes, L. Xing, and J. Sun, “Constant-power load system stabilization by passive damping,” IEEE Trans. Power Electron., vol. 26, no. 7, pp. 1832–1836, 2011. 21. T. Dragiˇcevi´c, “Dynamic Stabilization of DC Microgrids with Predictive Control of Point-ofLoad Converters,” IEEE Trans. Power Electron., vol. 33, no. 12, pp. 10872–10884, 2018. 22. C. Zheng, T. Dragicevic, B. Majmunovic, and F. Blaabjerg, “Constrained Modulated-Model Predictive Control of an LC-Filtered Voltage Source Converter,” IEEE Trans. Power Electron., vol. 8993, no. c, pp. 1–1, 2019. 23. P. Cortés, G. Ortiz, J. I. Yuz, J. Rodríguez, S. Vazquez, and L. G. Franquelo, “Model predictive control of an inverter with output LC filter for UPS applications,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1875–1883, 2009. 24. T. Dragicevic, “Model Predictive Control of Power Converters for Robust and Fast Operation of AC Microgrids,” IEEE Trans. Power Electron., vol. 33, no. 7, pp. 6304–6317, 2018. 25. C. Zheng, T. Dragicevic, Z. Zhang, J. Rodriguez, and F. Blaabjerg, “Model Predictive Control of LC-Filtered Voltage Source Inverters With Optimal Switching Sequence,” IEEE Trans. Power Electron., vol. 36, no. 3, pp. 3422–3436, 2020. 26. Z. Ma, S. Saeidi, and R. Kennel, “FPGA implementation of model predictive control with constant switching frequency for PMSM drives,” IEEE Trans. Ind. Informatics, vol. 10, no. 4, pp. 2055–2063, 2014. 27. P. Cortes, J. Rodriguez, C. Silva, and A. Flores, “Delay compensation in model predictive current control of a three-phase inverter,” IEEE Trans. Ind. Electron., vol. 59, no. 2, pp. 1323– 1325, 2012. 28. H. Wang, P. Davari, H. Wang, D. Kumar, F. Zare, and F. Blaabjerg, “Lifetime Estimation of DC-Link Capacitors in Adjustable Speed Drives under Grid Voltage Unbalances,” IEEE Trans. Power Electron., vol. 34, no. 5, pp. 4064–4078, 2019. 29. H. Wang et al., “Lifetime Prediction of DC-Link Capacitors in Multiple Drives System Based on Simplified Analytical Modeling,” IEEE Trans. Power Electron., vol. 36, no. 1, pp. 844–860, 2021. 30. A. S. and J. V. K. S. Rajashekara, V. Rajagopalan, “DC Link Filter Design Considerations in Three-Phase Voltage Source Inverter-Fed Induction Motor Drive System,” IEEE Trans. Ind. Appl., no. 4, pp. 673–680, 1987. 31. P. Akter, S. Mekhilef, S. Member, N. Mei, L. Tan, and H. Akagi, “Modified Model Predictive Control of a Bidirectional AC – DC Converter Based on Lyapunov Function for Energy Storage Systems,” IEEE Trans. Ind. Electron., vol. 63, no. 2, pp. 704–715, 2016. 32. M. Novak, U. M. Nyman, T. Dragicevic, and F. Blaabjerg, “Analytical design and performance validation of finite set MPC regulated power converters,” IEEE Trans. Ind. Electron., vol. 66, no. 3, pp. 2004–2014, 2019. 33. S. Vazquez, J. Rodriguez, M. Rivera, L. G. Franquelo, and M. Norambuena, “Model Predictive Control for Power Converters and Drives: Advances and Trends,” IEEE Trans. Ind. Electron., vol. 64, no. 2, pp. 935–947, 2017.

Chapter 8

Reinforcement Learning Based Weighting Factors’ Real-Time Updating Scheme for the FCS Model Predictive Control to Improve the Large-Signal Stability of Inverters

8.1 Introduction The previous chapter demonstrates that the finite control set (FCS) model predictive control (MPC) of downstream load inverters cascaded with LC input filters realizes load voltage regulation and dc-link stabilization simultaneously. However, it imports weighting factors (WFs) to balance the relative importance of regulation and stabilization. Selection of the WFs is directly linked to regulation accuracy and dc-link voltage stabilization, while it lacks an analytic or numeric approach to set the WFs, let alone real-time updating of them. To bridge this gap, this chapter presents a generalized approach based on reinforcement learning (RL), which also has the potential to improve the system stability and strike a better trade-off between multiple control objectives. In the past decades, since the development of digital processors, FCS MPC has been widely applied to power converter applications. With an all-in-one cost function, it integrates multiple control objectives and intuitively incorporates system constraints [1]. In the cost function (G), it is the WFs that balance the relative priority of multiple control objectives. However, there is no commonly acknowledged analytic or numerical method in academia or industry for determining weighting factors [2, 3]. Besides, a similar parameter tuning problem also exists in the other nonlinear control schemes of power converters, such as the Lyapunov-based control [4–6], adaptive control [7], backstepping control [8, 9], repetitive control [10], and so forth.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Zhang et al., Stability Enhancement Methods of Inverters Based on Lyapunov Function, Predictive Control, and Reinforcement Learning, https://doi.org/10.1007/978-981-19-7191-4_8

151

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8 Reinforcement Learning Based Weighting Factors’ Real-Time …

It requires numerous simulations and a lot of time to select the WFs using the existing method of trial-and-error. Cortés et al. [11] summarizes different tuning steps for different types of cost functions and WFs based on empirical experience to reduce tuning effort. As an automated alternative for WFs’ selection, In order to automate WFs’ selection [12] devises a genetic algorithm-based heuristic multiobjective optimization method. Each time the design objective changes, it requires a new set of iterative simulations. Davari et al. [13] proposes an analytical WFs optimization strategy for induction motor applications. During each control interval, the specific WF that zeros the derivative of torque ripple is calculated. It should be noted, however, that this method is not a general one, and when extended to systems with different topologies, it requires different modeling and derivations. Devised from another perspective [14] obviates WFs selection requirements using a fuzzy multi-criteria decision-making strategy. However, it introduces membership functions with priority coefficients that are difficult to select. This is a state-of-the-art approach for generating WFs for two-level three-phase uninterruptible power supplies (UPS) and three-level neutral point clamped (NPC) converter based on an offline artificial neural network (ANN) [15, 16]. Nevertheless, this approach is inherently offline, so training data must be prepared beforehand. Furthermore, the training process does not take into account changes in load power levels or plant parametric mismatches. It is theoretically impossible for this approach to adjust the WFs online once these environmental conditions vary dramatically. In order to make sure that the WFs are selected optimally, another offline training process should be conducted with new system parameters. These parameters are hard to predict in practice. What’s more, WFs selection also has a coupling effect with system stability, while existing works to improve the system stability are mainly by adding stability-relating constraints or stability-oriented sub-objective function (Gstability ) [17, 18]. It is still open to improving the system stability by updating the WFs. Inspired by the above discussion, this chapter proposes a general method based on RL. By using the proposed approach, FCS MPC-regulated power converters can behave like intelligent agents with memory to learn from their own experiences, where the optimal WFs are selected automatically. If the load changes, parameters mismatch due to aging issues, or the environment fluctuates, the online RL process can be reactivated at any time. Using FCS MPC regulated stand-alone inverters cascaded with LC input filters, this idea is verified. Using FCS MPC regulated stand-alone inverters cascaded with LC input filters, this idea is verified. The self-taught policy learns a trade-off between dc-link stabilization and load voltage tracking via artificial neural networks (ANNs) and the deep deterministic policy gradient (DDPG) algorithm with full consideration of user-defined criteria. According to simulation results, the proposed RL approach effectively suppresses the load-voltage tracking error while maintaining DC-link voltage stabilization.

8.2 Weighting Factors Selection in FCS MPC Affects System Stability

153

8.2 Weighting Factors Selection in FCS MPC Affects System Stability 8.2.1 Particular Case: WFstability Selection is a Trade-Off Between DC-Link Stabilization and Load Voltage Tracking In this chapter, the FCS MPC of a stand-alone inverter having an LC input filter in [19] acts as the particular case for easy comprehension. As presented in Fig. 8.1, voa , vob , voc denote the three-phase load voltages. ioa , iob , ioc are three-phase load currents. iLa , iLb , iLc are three-phase inductor currents. According to [17–19] FCS MPC of the investigated cascaded system is inherently a dual-objective control scheme, requiring simultaneous dc-link voltage stabilization and load voltage tracking. In order to implement three-phase load voltage tracking, a cost function (G1 ) is typically formulated as follows: | | G 1 = |vor e f − vo (k+1)|,

(8.1)

where voref is the load-voltage reference in the αβ stationary reference frame: voref = voαref + jvoβ ref . The dc-link voltage oscillation is suppressed by penalizing the dc-link voltage variation at the next instant in Gstability , G stability = |vdc (k + 1) − E|.

(8.2)

rdc Ldc idc ipol s1

Environment

s2

s3 ioa iLa voa iLb vob iob iLc voc ioc

via E

n dc bus

vdc

Cdc s4

vib s5

vic s6

r

L C

s1 ~ s6 WFs RL

FCS MPC Controller

Refences

N

Sensors

Fig. 8.1 Particular case: schematic of the FCS MPC regulated three-phase stand-alone inverter cascaded with an LC input filter

154

8 Reinforcement Learning Based Weighting Factors’ Real-Time …

The all-in-one cost function (G) is formulated to integrate the above dual control objectives balanced by the weighting factor (WF stability ) of Gstability , G = G 1 + W Fstability · G stability .

(8.3)

where WF is an artificially imported weighting factor to balance the relative importance of (7.9) and (8.2). As for control input, a finite control set of switching vectors is enumerated to calculate (8.3) in each sampling interval (T s ). The specific vector that minimizes (8.3) will be executed. For practical application, the well-known two-step prediction method is used for delay compensation [20]. As illustrated in [18], the selection of WFstability is a trade-off between loadvoltage tracking and DC-link voltage stabilization, which lacks a universal approach for tuning.

8.2.2 Generalized Case: WFs Selection Affects System Stability (WFstability in Particular) The above-illustrated coupling effect between WFstability selection and system stability does not merely exist in the particular case. This section analyzes the generalized case. It can be found that WFs selection affects system stability in multi-objective FCS MPC of power converters, especially WFstability selection. As presented in (8.4), the all-in-one cost function (G) in multi-objective FCS MPC can be generalized to the weighted summation of the sub-objective functions, G = G 1 + W F1 · G 2 + W F2 · G 3 + · · · + W Fn−1 · G n + W Fstability · G stability , (8.4) where each sub-objective function represents one control objective or system constraint (G1 ~ Gn ), including but not limited to switching loss minimization, harmonic spectrum shaping, over-current protection and system stability (Gstability ), etc. WFn−1 denotes the (n−1)th weighting factor for the nth sub-cost function (Gn ). Therefore, WFs play a critical role to balance the relative importance between these sub-control objectives. In this case, without loss of generality, approaches to improve the stability of FCS MPC-regulated power converters via cost function design can be classified into two categories: (1) By adding stability-relating constraints or stability-oriented sub-objective function Gstability . (2) By updating the WFs, especially WFstability . In this book, works reported in Chaps. 6 and 7 belong to category (1). To be more specific, (6.36) is the corresponding Gstability in Chap. 6, and (7.11) act as the WFstability ·Gstability in Chap. 7.

8.3 WFs’ Real-Time Updaing Via the Reinforcement Learning-Based …

155

So far, most of works to improve system stability belong to category (1). While category (2) still lacks a generalized approach that update of WFs during real-time operation. The next chapter aims to bridge this gap via reinforcement learning-based approach.

8.3 WFs’ Real-Time Updaing Via the Reinforcement Learning-Based Approach to Improve System Stability 8.3.1 Structure of the Proposed Approach Deep RL-based methods are generally applicable to updating WFs online, regardless of converter topology. The proposed deep RL-based approach is a general method to update the WFs online, independent of the topology of the power converters [21]. As shown in Fig. 8.2, the FCS MPC controller and its regulated power converters as a whole are deemed as the environment. (1) The Observation may include but is not limited to: • the measured signals from the environment, such as voltages, currents, specific constants, temperatures, power ratings, and tracking references. • as well as their integration or differentiation, the value of the above signals after addition, subtraction, multiplication, and division. (2) The scalar Reward signals guide the learning process of RL. The reward signals are artificially defined by users, which measures how well the updated WFs satisfy the control performance. Namely, the Reward acts as a performance indicator of executing a specific WF. Generally, positive sub-rewards are added to the Reward to encourage a certain WF, while negative sub-rewards discourage it.

WFn-1, WFstability)

diag(WF1, WF2 To improve system stabiltiy?

WFs

Update RL Policy Algorithm Agent

References FCS MPC Controller

Power Converters Control input

Incorporate userdefined stabilityrelating indicators to the Reward.

Reward

Observation Environment

Fig. 8.2 The structure of the proposed RL-based approach

Sensors

156

8 Reinforcement Learning Based Weighting Factors’ Real-Time …

(3) The policy maps from received observations to WFs to maximize the reward, where the environment changes due to operation point change, reference step, load resistance step, and plant parametric mismatch can be fully considered. (4) The RL algorithm updates the policy according to the Observation, Reward, as well as WFs. As a response to the WFs, the environment receives observations from the agent. Meanwhile, the WFs are rewarded according to how well they contribute to satisfying desirable performance goals. To improve system stability, we only need to incorporate user-defined stability-relating indicators to the Reward .

8.3.2 RL Agent and Its Selection As presented in Fig. 8.2, the RL agent consists of two components: a policy and a learning algorithm [22]. In this chapter, (.) agent denotes an agent where the (.) RL algorithm is implemented. (1) The policy is a tunable function approximator that maps observations to WFs, such as decision trees, ANNs, nearest neighbors, linear basis functions, lookup tables, and Fourier/wavelet bases. (2) The RL algorithm continuously updates the policy based on WFs, observations, and rewards. It aims to find an optimal policy that maximizes the cumulative reward. For an RL agent to train its policy, it requires *1 function approximators. Depending on the learning algorithm, the function approximators can be categorized into two types: actor and critic. As explained in Fig. 8.3, an actor generates the WFs that maximize the expected long-term future reward as a function of the given observation. Based on the observations and WFs, critics approximate the future rewards that will be expected in the long run [22]. Recently, some commercial software has provided a variety of built-in RL agents, e.g., Reinforcement Learning Toolbox of MATLAB. An appropriate RL agent should be selected before training. In this chapter, as presented in Fig. 8.2, the observation taken from the environment are generated by real-world sensors, such as voltage/current sensors, and thermal sensors. The FCS MPC WFs are Fig. 8.3 Two types of function approximators in RL agent: actor and critic

Critic

Expected long-term future reward

Actor

WFs

Observation

Function approximators in agent

8.3 WFs’ Real-Time Updaing Via the Reinforcement Learning-Based …

157

tunable constants, which indicates that both the observation and action spaces have continuous working spaces. In an environment with a continuous action space as well as a continuous observation space, the deep deterministic policy gradient (DDPG) agent is the simplest compatible agent. DDPG agents compute an optimal policy that maximizes long-term rewards by employing actor-critic RLs [23]. If the DDPG cannot satisfy desirable requirements, more complicated RL agents can be used. In general, the section of RL agents starts from the simplest one: DDPG, followed by TD3, PPO, SAC, until desirable performance is achieved. Among these agents, SAC is considered an advanced version of DDPG, which provides a stochastic policy, and PPO generally provides the most stable updates at the expense of longer training time [22].

8.3.3 RL-Based Approach Using a DDPG Agent and Artificial Neural Networks In this chapter, as shown in Fig. 8.4, a DDPG agent is used for implementation since both WFs and Observation are all continuous. Meanwhile, a deep artificial neural network (ANN) is selected as the policy since it can approximate any given linear or nonlinear input/output data relationship with arbitrary precision [24]. For a specific ANN of the Actor, the number of neurons in the input layer and output layer is selected according to the following guideline: the number of neurons in the input layer is equal to the number of observations, while the number of neurons in the output layer is equal to the number of WFs. As for the hidden layers and their Expected long-term future reward

Critic Input l.

...

Agent

WF1

...

WF2

Output l.

... WFs

Observation

WFn

1

WFstability Parameter Update

Hidden l.

WFs

DDPG learning algorithm Reward Environment

Fig. 8.4 Proposed RL-based general approach to update WFs using the DDPG algorithm

158

8 Reinforcement Learning Based Weighting Factors’ Real-Time …

neurons, it still lacks a commonly acknowledged way to select their optimal number. In general, an ANN that has a greater number of hidden layers or neurons in each layer is more capable of mimicking. They are increased until the ANN reaches desirable performance, where a certain trial-and-error process is required.

8.4 Verification on the Particular Case: Improving Tracking Accuracy While Ensuring DC-Link Stabilization 8.4.1 Configuration of the Observation, Reward, and ANN As in (8.1), voαref and voβ ref are generated via dq0-αβ coordinate transformation, (

voαr e f voβr e f

⎛ ⎞ ) [ ] ref cos(ωt) sin(ωt) 0 ⎝ = 0 ⎠. − sin(ωt) cos(ωt) 0 0

(8.5)

where ref is a user-defined positive constant, which is equal to the desired amplitude of the load voltage. ω is equal to 100π, and t is the clock of the controller. To illustrate the above dual control objectives quantitatively, load-voltage tracking error (eac ) and dc-link voltage regulating error (edc ) are defined as follows: eac = 100 · (vod − r e f )/r e f,

(8.6)

edc = 100 · (vdc − E)/E.

(8.7)

where vod is the d-frame load voltage in a synchronous rotating frame (d-q frame). Three-phase load-voltage vector [voa , vob , voc ]T is transformed to their dc counterparts [vod , voq ]T using the well-known abc-dq coordinate transformation (T abc-dq ), [

vod voq

]T

[ ]T =Tabc−dq voa vob voc ,

where Tabc−dq =

) ( ) ] ( [ 2 cos(ωt) cos ωt − 2π cos (ωt + 2π 3 ) 3 ) . ( − sin ωt + 2π 3 − sin(ωt) − sin ωt − 2π 3 3

(8.8)

8.4 Verification on the Particular Case: Improving Tracking Accuracy While …

159

As presented in Fig. 8.5, the Observation vector taken from the Environment consists of five elements. The thd is one of the elements that measure the total harmonics distortion (THD) of voa . thd is added to consider power quality while updating WFstability . eac , edc and their integral counterparts are incorporated to capture the characteristics of load-voltage tracking accuracy and dc-link stabilization in response to the WFstability . The adjustable reference (ref ) is relevant to the operation point, thus, it may be a helpful observation. The structure of the ANNs in Actor and Critic is shown in Fig. 8.6.

voa

THD

100

thd

ref thd

eac

edc

KTs

Observation

z–1 KTs

z–1 Discrete integrator Fig. 8.5 Observation

Actor Fig. 8.6 Structure of the ANNs in actor and critic

Critic

160

8 Reinforcement Learning Based Weighting Factors’ Real-Time …

Rewar d = 10 × (|thd| < 2) − 1 × (|thd| ≥ 2) + 10 × (|eac | < 5) − 1 × (|eac | ≥ 2) + 10 × (|edc | < 0.35) − 1 × (|edc | ≥ 0.35) − 100{(|thd| ≥ 4)||(|eac | ≥ 5)||(|edc | ≥ 0.35)}

(8.9)

As illustrated before, the Reward contains user-defined criteria to quantify how well the DDPG agent achieves the tasks. Meanwhile, eac and edc also functions a stability-relating indicators. A surge or severe fluctuation of eac and edc indicates system instability, and smaller eac and edc indicates better system stability. Therefore, to improve the system stability, eac and edc are expected to be penalized in the Reward. To be more specific, in this chapter, as present in (8.9), (|thd| < 2), (|eac | < 5), (|edc | < 0.35) are encouraged, while their opposite is discouraged. A severe penalty (−100) is given to the Reward if (|thd| ≥ 4) or (|eac | ≥ 6) or (|edc | ≥ 0.4). In practical application, these constants (such as the piece-wise thresholds of these stability-relating indicators) can be set according to particular IEEE standards or user-defined requirements.

8.4.2 Parameter Settings and Training Results The nominal system parameters are shown in Table 8.1. Specifications of the DDPG agent are presented in Table 8.2. The training process of the DDPG agent is presented in Fig. 8.7. The training settings sketched in Fig. 8.7 show that the Episode Reward tends to perturb significantly during the training process. To better present the changing tendency of the Episode Reward, the moving average of Episode Reward with the 10-window length is presented, which is denoted as Average Reward. According to Fig. 8.7, the DDPG agent does not converge after more than 250 episodes. As can also be observed, the DDPG agent at the 12th episode is sometimes the only one with an Episode Reward greater than zero. Thus, the DDPG agent in the 12th episode is the specific agent that maximizes the Reward among these 250 episodes. Thus, the DDPG agent in the 12th episode is implemented to update the WF in the simulation experiment.

8.4.3 Test Results As presented in Fig. 8.8a, if the WF is set as a constant, eac % is around 8%, and edc % is around 0%, which means that the dc-link oscillation is effectively suppressed accompanied by 8% of load-voltage tracking error. In comparison, as shown in Fig. 8.8b, if the WF is updated via the RL-based approach, the WF is automatically adapted,

8.4 Verification on the Particular Case: Improving Tracking Accuracy While …

161

Table 8.1 Nominal system parameters Physical meaning

Notation

Value

Unit

dc-link voltage

E

350

Volt

Fundamental frequency

f

50

Hz

Reference of the load voltage

|voref |

1.414 × 115

Volt

Rated active power

P

2.5

kW

Sampling interval

Ts

100

μs

Inductance of LC output filter

L

6.3

mH

ESR of the output-filter inductor

r

0.096

Ω

The capacitance of the LC output filter

C

50

μF

Inductance of LC input filter

L dc

10

mH

ESR of the input-filter inductor

r dc

0.12

Ω

Load resistance

R

65

Ω

Table 8.2 Specifications of the DDPG agent using reinforcement learning toolbox Physical meaning

Value

Sample time

20 μs

Simulation length

0.08 s

Max steps per episode

4000

Target smooth factor

1 × 10–3

Experience buffer length

1 × 106

Discount factor

0.99

Mini batch size

256

Reset experience buffer before training

False

Save experience buffer with agent

True

AgentOptions.NoiseOptions.Variance

1 × 10–1

AgentOptions.NoiseOptions.VarianceDecayRate

1 × 10–6

Hidden layer size

20

Learning rate of critic

1 × 10–3

Gradient threshold of critic

1

Learning rate of actor

1 × 10–4

Gradient threshold of actor

1

attenuating eac % from 8 to 1% without sacrificing the dc-link voltage stabilization. It is conceded that, when Time is around 0.25, the WF suddenly drops from 0.9 to 0, which regulates eac % from 8% to −5.2% where the dc-link voltage undergoes certain disturbance. This phenomenon is common in RL because RL requires to accumulate its experience from the ‘bad scenario’ to avoid it in the future. This undesirable

162

8 Reinforcement Learning Based Weighting Factors’ Real-Time …

10 5

Episode Reward

0

Episode Reward

Average Reward

This agent is used to do simuation

-0.5 -1

Episode Q0

-1.5 -2 -2.5 -3 -3.5 -4 -4.5

0 12

50

100 150 Episode Number

200

250

Fig. 8.7 The training process of the DDPG agent

edc

edc

eac

eac

WFstability

WFstability

transient oscillation is expected to be avoided with the proper design of Reward and Observation.

(a)

(b)

Fig. 8.8 Simulation results of the FCS MPC regulated three-phase stand-alone inverter cascaded with an LC input filter. a The WF is set 1; b the WF is updated via the trained DDPG agent at the 12th episode

References

163

8.5 Conclusion This chapter reports that it is a general phenomenon WFs selection affects system stability in FCS MPC of power converters, especially WFstability . Research on realtime updating WFs while considering the improvement of system stability is still open. As a promising candidate to bridge this gap, this chapter presents an RL-based approach to updating the WFs. The WFs’ selection is self-taught online. The trained policy is deployed in real-time to update the WFs. The scalar Reward signals guide the learning process of RL. Reward quantifies how well the updated WFs satisfy the desired control performance. System stability improvement is realized by incorporating user-defined stability-relating indicators into the Reward. The proposed approach has been verified on FCS MPC of stand-alone inverters cascaded with LC input filters. Compared to the conventional way that sets WFstability as a constant, the proposed approach updates WFstability in real-time operation. Meanwhile, it contributes to reducing eac % from 8 to 1% while ensuring DC-link voltage stabilization. Our future work will involve implementing the RL-based method on a real-world prototype to update WFs. Systematic ways to design the Reward, Observation, and ANN structure will be explored.

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