根據國際IEEE標準，併網型逆變器在傳送電力時須連接LCL濾波器以維持電力品質。然而，正是低通濾波器降低了併網逆變器的功率密度，使得摩爾定律不適用於功率轉換裝置。工業界已對於併網型逆變器的功率密度有了緩慢且穩定地改善，它們藉由採用高階濾波器(市電連接LCL)以及使用有著軟飽和特性的鐵芯材料(例如: 粉末金屬)。 LCL濾波器具有比較好的開關漣波衰減，可以降低成本、體積和功率損耗，粉末金屬芯具有較低的磁芯損耗和更高的飽和磁通密度。然而， LCL濾波器中的諧振問題以及粉末金屬芯的非線性電流相關電感特性，增加了對併網逆變器控制的整體複雜性。
本論文將探討具有寬電感變化的LCL濾波器之併網型逆變器設計、控制及穩定度分析三個方面。本論文之主要貢獻概述如下：
• 提出了一種基於諧振頻率的LCL濾波器設計方法，該方法考慮了市電電壓諧波和控制穩定邊界，以確保在寬範圍的感性線路阻抗和濾波電感變化下的穩定度。
• 提出一種基於可變電感估測模型的補償型直接數位控制方法，使系統對電網阻抗變化具有強健性，改善大信號瞬態響應。
• 提出了一種常規穩定度分析的多維參數方法，來分析寬範圍的感性電網阻抗和濾波電感變化下的穩定度。
本論文探討了基於諧振頻率的LCL濾波器設計方法，因為它是控制穩定度和注入電網電流的重要因素。諧振頻率在運行期間受內部和外部因素影響的事實使其更加關鍵。因此，在所提出的設計方法中，首先通過考慮電網電壓諧波和控制穩定度邊界來識別諧振頻帶。 接著考慮由感性阻抗所引起的諧振頻率變化，以及電感鐵芯的磁導率，來定義對LCL參數的限制。最後有系統地從限制中導出LCL參數。
由於直接數位控制方法簡單易行，本論文做了深入的研究，不需涉及調節控制增益的複雜程序，並且將非線性電感值變化直接包含在控制法則中。將用於L和LC濾波器的現有直接數位控制方法，擴展並且應用於逆變器具有LCL濾波器，同時考慮逆變器側和電網側電感的寬廣變化。此控制方法使用非滯回電感估計模型來估計每個開關週期中的濾波器電感值。此外，本論文提出使用變結構電感估計模型，以處理大信號瞬變，其中實際濾波電感和估計濾波電感之間的明顯不匹配是不可避免的。本論文還介紹消除濾波電容效應的工作週期補償方法，透過採用主動阻尼控制以改善動態性能，並使系統對電網阻抗變化具有強健性。
通常基於濾波器電感標稱值的穩定度研究不能預測整個電感變化範圍內的不穩定度。 因此，本論文探討了傳統穩定度分析方法的參數變化，該方法具有由實際和估測電感變化的參數空間。 透過將電網電感阻抗作為參數空間中的附加維度，還使用基於阻抗的穩定度標準來研究電網阻抗對穩定度的影響。 瞭解到由電感變化和電網阻抗所引起穩定裕度的模式，並且辨別出參數空間的頂點或沿著邊緣的穩定裕度，足以預測整體穩定度。
本論文所提出的設計、控制和穩定度分析方法，已經由MATLAB Simulink的模擬和5 kW單相併網LCL逆變器的實驗結果得到驗證。另外，也驗證所提出設計方法比起傳統設計方法更有效。所有提出的控制補償，也經由具電網阻抗和電壓諧波考量的5 kw系統之實驗結果得到驗證。由分析結果與實驗結果的吻合度，證實參數方法在濾波電感變化下對傳統穩定度分析方法的適用性。

Interfacing low pass filters is essential for grid-connected inverters to deliver power with quality in accordance to the regulatory standards. However, it is the low-pass filters that lower the power density of grid-connected inverters making Moore’s law inapplicable to power conversion devices. The industry has been making slow and steady progress in improving the power density of grid-connected inverters by adopting higher-order filters such as LCL at grid interface and by using core materials having soft saturation characteristics such as powder metals. LCL filters have better switching-ripple attenuation at reduced cost, volume and power loss, and powder metal cores have lower core losses and higher saturation flux density. However, the inherent issue of resonance in LCL filters along with non-linear current-dependent inductance characteristics of powder metal cores, adds a whole level of complexity to the control of grid-connected inverters.
In this dissertation, the three aspects of design, control and stability analysis of grid-connected inverters with LCL filters having wide inductance variation will be explored. The major contributions are summarized as follows:
A resonant frequency based LCL filter design method is proposed that takes grid-voltage harmonics and control stability boundary into account to ensure stability under wide range of inductive line impedance and filter inductance variation.
A variable inductance estimation model based compensated direct digital control method is proposed to make the system robust to grid impedance variation and improve large signal transient response.
A multidimensional parametric approach to conventional stability analysis is proposed to analyze the stability under a wide range of inductive line impedance and filter inductance variation.
The resonant frequency based LCL filter design method is explored in this dissertation because it is an important factor for control stability and grid-injected current quality. The fact that resonant frequency is affected by both internal and external factors during operation, makes it even more critical. Hence, in the proposed design method, a resonant frequency band is first identified by considering grid-voltage harmonics and control stability boundary. The variations in resonant frequency due to inductive line impedance as well as magnetic permeability of inductor core are then taken into account to define the constraints on LCL parameters. The LCL parameters are then systematically derived from the identified constraints.
Direct digital control method is explored in this dissertation because of its simplicity. There is usually no complex procedure involved for tuning the control gains and the non-linear inductance variations are directly included in the control law. The existing direct digital control method for L and LC filters has been extended for application to LCL filters with due consideration to wide variation in both inverter-side and grid-side inductances. The control method typically relies on an anhysteretic inductance estimation model for estimating the filter inductance in each switching cycle. The variable structure inductance estimation model is proposed in this dissertation to deal with large-signal transients where a significant mismatch between the actual and estimated filter inductances is inevitable. The method of duty ratio compensation to cancel the effect of filter capacitor is also introduced in this dissertation. This basically improves its dynamic performance by acting like an active damper in the control and makes the system robust to grid impedance variation.
Investigation of stability which is conventionally based on nominal values of filter inductors cannot predict instabilities over the entire range of inductance variation. Hence, the parametric approach to conventional stability analysis methods with a parameter space defined by variation in actual and estimated inductances is explored in this dissertation. The effect of line impedance on stability is also investigated with impedance-based stability criterion by considering line inductance as an additional dimension in the parameter space. A distinct pattern in the stability margins due to inductance variation and line impedance is identified and stability margins at the vertices or along the edges of the parameter space are shown to be good enough for predicting overall stability.
The proposed design, control and stability analysis methods have been verified through simulation using MATLAB Simulink and through experimental results measured from a 5 kW single-phase grid-connected inverter with various LCL filters. The effectiveness of the proposed design method over conventional design methods has been verified for various design scenarios. All the proposed control modifications have also been validated through experimental results for various scenarios of line impedance and grid-voltage harmonics. The analytical results have also been shown to be in good agreement with experimental results, confirming the applicability of parametric approach to conventional stability analysis methods under filter inductance variation.
Keywords – LCL Filter, Design, Resonant Frequency, Damping, Harmonics, Line Impedance, Inductance Variation, Magnetic Permeability, Stability, D-Σ Digital Control, Virtual Impedance, Grid-Connected, Inverter.

List Of Figures VIII
List Of Tables XIII
CHAPTER 1. Introduction 1
1.1 Motivation 2
1.2 Why Single-Phase And Why LCL Filter? 3
1.3 A Brief Literature Review 5
1.3.1 Design of LCL filters 5
1.3.2. Control of the Inverter 6
1.3.3 Stability of the system 7
1.4 Dissertation Outline 8
CHAPTER 2. Direct Digital Control 11
2.1 Where does it fit in? 12
2.2 A Generalized Direct Digital Control Law 12
2.2.1 Control law for a single-phase grid-connected inverter with an LC filter 14
2.2.2 Control law for a single-phase grid-connected inverter with an LCL Filter 16
2.3 Direct Digital Control – Just Playing Around With Circuit Theory 23
CHAPTER 3. Filter Inductance 27
3.1 What Is Its Role? 28
3.2 The Variable Inductance Conundrum 30
3.3 Imagining Magnetic Hysteresis 35
3.4 Variable Structure Inductance Estimation Model (VSIEM) 37
3.5 Filter Inductance – Gain Or Pain For Direct Digital Control? 40
CHAPTER 4. Filter Design 41
4.1 The Natural Transition from L→LC→LXL 42
4.2 Why Resonance Matters? 42
4.3 The Resonant Frequency Band 43
4.4 The Varying Resonant Frequency 45
4.4.1 Filter Parameter Tolerance 45
4.4.2 Grid Strength 46
4.5 Band-Fit Resonant Frequency 48
4.6 Just Band-Fit Is Not Enough 52
4.6.1 Power Factor Consideration 53
4.6.2. Switching Ripple Consideration 53
4.7 Design Flow – Keeping It Simple 58
4.8 Design Validation – Integrity Check 59
4.8.1 Check#1: Does it really work? 59
4.8.2 Check#2: Does it perform better than a filter designed using conventional method? 61
4.9 Filter Design – Just Avoiding Resonance 63
CHAPTER 5. Compensated Control 65
5.1 Why Compensate? 66
5.2 Direct Digital Compensation – Is It Possible? 68
5.3 Not So Direct Digital Compensation 70
5.4 Will It Affect Steady-State Behavior? 73
5.4 Validation – Does It Actually Work? 73
5.5 Compensated Control – Its Circuit Theory Again, With A Twist 76
CHAPTER 6. Stability Analysis 77
6.1 Stability Analysis – The Complexity Involved 78
6.1.1 Changing Operating Point 78
6.1.2 Changing Filter Inductance and Line Impedance 78
6.2 Let’s see How It Works – Parameter Space 79
6.3 The Dynamic Model In The Uncertain Model Space 81
6.4 System Transfer Functions In The Uncertain Model Space 83
6.4.1 Basic Control Law (6.4) 84
6.4.2 Compensated Control Law (6.5) 85
6.5 Time For Plots 86
6.5.1 Loop Gain With 0.2≤k_Lx≤1.0 and k_ex=1 86
6.5.2 Loop Gain With 0.8≤k_ex≤1.0 and k_Lx=1 87
6.5.3 Impedance Plots With k_ex=1,k_Lx∈{0.2,1.0} and L_L∈{50 μH,2.6 mH} 88
6.6 Region Of Stability 92
6.6.1 Basic Control Law (6.4) 93
6.6.2 Compensated Control Law (6.5) 99
6.7 A Special Note On Compensation 103
6.8 Stability Analysis – Walking Through A Multidimensional Mesh 106
CHAPTER 7. Hardware Implementation 109
7.1 The Experimental Hardware 110
7.2 Validation Of Proposed Design Method 113
7.2.1 Test#1: Performance Comparison with Conventional Design Method 113
7.2.2 Test#2: Design Validation For Operation Under Extreme Conditions 115
7.2.3 Test#3: Importance of Considering Filter Inductance Variation In Design 117
7.3 Validation Of Proposed Compensation Method 118
7.3.1 Test#1: Steady-State Performance Comparison with Basic Control Law 118
7.3.2 Test#2: Stability Comparison with Basic Control Law 121
7.3.3 Test#3: Validating Change In Sign For Compensation 122
7.4 Validation Of Variable Structure Inductance Estimation Model 123
7.5 Hardware Implementation – Is It The End Or Its Just The Beginning? 124
CHAPTER 8. Conclusions & Future Work 125
8.1 Conclusions 126
8.2 Future Work 127
8.2.1. Optimization Algorithm Based Filter Design Application Software 127
8.2.2 A Direct Digital Nonlinear Control Law 130
8.2.3 A Hardware Trick For Robust Compensation 134
8.3 The Last Note 141
References 143
Publications 151

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