High performance digital control of UPS inverters

HIGH-PERFORMANCE DIGITAL CONTROL OF UPS INVERTERS DENG HENG NATIONAL UNIVERSITY OF SINGAPORE 2007 HIGH-PERFORMANCE DIGITAL CONTROL OF UPS INVERTERS DENG HENG (M.Eng., Wuhan University, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledgement Acknowledgement I would like to express my sincere gratitude towards Prof Ramesh Oruganti, my chief advisor, for his inspiring guidance and constant support during my doctoral research His immense enthusiasm and encouragement made undertaking this study a pleasure I also would like to give my thanks to Prof Dipti Srinivasan, my secondary advisors, for her invaluable advice and help throughout my study I would like to thank Prof S.K.Panda and Prof Xu Jian-Xin for their valuable comments and suggestions I thank lab officers Mr Teo Thiam Teck, Mr Seow Hung Cheng, Mr Jessica, Mr Woo Ying Chee and Mr Chandra for their support for the many troubles I brought to them Without their help, the research project would have taken a longer time Special thanks go to Mr Abdul Jalil Bin Din for his prompt PCB fabrication services I am grateful to all my friends in the Centre for Power Electronics for their support In particular, I would like to thank Dr Kanakasabai Viswanathan, Dr Sahoo Sanjib Kumar and Mr Xu Xinyu for helpful discussions Deep in my heart are special thanks to my wife Her love has accompanied me through bad and good moments Finally, I want to thank my parents who made all this possible i Table of Contents Table of Contents Chapter Introduction……………………………………………… 1.1 Background………………………………………………………………………2 1.2 Research Objectives…………………………………………………………… 1.3 Thesis Contributions…………………………………………………………… 1.4 Thesis Organization…………………………………………………………… 10 Chapter Literature Survey on Control of UPS Inverters…………12 2.1 Dynamic Model and Output Impedance of UPS Inverters………………………12 2.1.1 State space Model…………………………………………………………………… 13 2.1.2 Transfer Function Model………………………………………………………………… 14 2.1.3 Output Impedance……………………………………………………………………… 15 2.2 Requirements of UPS Inverter Control………………………………………….20 2.3 Classification of Control Methods for UPS Inverters………………………… 22 2.4 Model base instantaneous feedback control of UPS inverters………………… 23 2.4.1 Multi-loop Control Schemes………………………………………………………………23 2.4.2 Other Linear Feedback Control Schemes………………………………………………26 2.4.3 Summary of Linear Feedback Control Schemes………………………………………27 2.5 Repetitive Control of UPS Inverters…………………………………………….28 2.6 Nonlinear Control of UPS inverters…………………………………………… 32 2.6.1 Sliding-mode Controllers………………………………………………………………….32 2.6.2 Neural Controller………………………………………………………………………… 35 2.7 Chapter Conclusions…………………………………………………………….37 Chapter Instantaneous Feedback Control of UPS Inverters…….40 3.1 Introduction…………………………………………………………………… 40 3.2 Generalized Minimum Variance Control of UPS Inverters…………………….41 3.2.1 Prediction of Output Voltage with Minimum Variance…………………………………42 3.2.2 Design of the GMV Controller……………………………………………………………44 3.2.3 Stability Analysis………………………………………………………………………… 46 ii Table of Contents 3.2.4 Design of λ ……………………………………………………………………………… 47 3.2.5 Robustness……………………………………………………………………………… 50 3.3 Pole-placement Control with Minimum Output Impedance……………………53 3.3.1 Design of D( z −1 ) and E ( z −1 ) ………………………………………………………….…55 3.3.2 Design of F ( z −1 ) ………………………………………………………………………… 56 3.3.3 Explanation of F ( z −1 ) Design in the Time Domain………………………………… 59 3.3.4 Verification of F ( z −1 ) Design…………………………………………………………… 61 3.3.5 A Simplified Expression for the Closed-loop Impedance………………………………62 3.3.6 Robustness…………………………………………………………………………………64 3.4 Comparison of Control Methods……………………………………………… 67 3.4.1 Benchmark Controller…………………………………………………………………… 67 3.4.2 Output Impedance…………………………………………………………………………68 3.4.2 Simulation and Experimental Results……………………………………………………68 3.5 Chapter Conclusions…………………………………………………………… 75 Chapter Iterative Learning Control of UPS Inverters……………78 4.1 Introduction………………………………………………………………………78 4.2 A Brief Overview of ILC…………………………………………………………81 4.3 Direct ILC Scheme for UPS Inverters…………………………………………….84 4.3.1 Dynamic Model of Single-phase Inverter…………………………………………………84 4.3.2 Proposed Direct ILC for Inverters………………………………………………………….85 4.3.3 Design of Φ( z ) …………………………………………………………………………… 88 4.3.4 Effect of Learning Gain γ ………………………………………………………………….90 4.3.5 Effect of Forgetting Factor α …………………………………………………………… 91 4.3.6 Design of L-C Filter………………………………………………………………………….92 4.4 Hybrid ILC Scheme for UPS Inverters………………………………………… 93 4.4.1 Modified PD Controller…………………………………………………………………… 94 4.4.2 Design of the ILC with the PD Controller……………………………………………… 96 4.5 Design Details of the ILC Methods……………………………………………………… 98 4.5.1 Design of Direct ILC……………………………………………………………………… 98 4.5.2 Design of the Hybrid ILC…………………………………………………………………100 4.6 Experimental Results…………………………………………………………….102 4.6.1 Forgetting Factor………………………………………………………………………… 102 4.6.2 Steady-state Performance……………………………………………………………… 105 iii Table of Contents 4.6.3 Error Convergence……………………………………………………………………… 105 4.6.4 Transient Performance…………………………………………………………………….107 4.6.5 Resetting of Memory………………………………………………………………………109 4.7 ILC with Inductor Voltage Compensation………………………………………111 4.7.1 Feedforward Compensation of Inductor Voltage Drop……………………………… 112 4.7.2 Design of the ILC with Inductor Voltage Compensation……………………………….114 4.7.3 Experimental Results of the ILC with Inductor Voltage Compensation………………117 4.8 Chapter Conclusions……………………………………………………………121 Chapter ANN Based Learning Control of UPS Inverters……….123 5.1 Introduction…………………………………………………………………… 123 5.2 Adaptive Linear Neural Controller for UPS Inverters………………………… 127 5.2.1 Dynamic Model of UPS Inverters……………………………………………………… 128 5.2.2 ADALINE Identifier……………………………………………………………………… 131 5.2.3 ADALINE Controller………………………………………………………………………133 5.2.4 Convergence Analysis……………………………………………………………………135 5.2.5 Hard Limitation of 1/ R ( k ) ……………………………………………………………… 136 5.2.6 Experimental Results…………………………………………………………………… 137 5.3 B-spline Network Controller for UPS Inverters…………………………………142 5.3.1 Proposed Controller………………………………………………………………………145 5.3.2 BSN Controller…………………………………………………………………………….147 5.3.3 Frequency Domain Analysis of the BSN Controller……………………………………152 5.3.4 Design of B-spline Support………………………………………………………………156 5.3.5 Choice of the Learning Gain 5.3.6 Effect of Forgetting Factor γ ………………………………………………………….161 α ……………………………………………………………162 5.3.7 Design Steps of the Proposed Control Scheme……………………………………….163 5.3.8 Comparison with the Hybrid ILC Scheme…………………………………………… 164 5.3.9 Comparison of Multi-layer NN Controller and the BSN Controller………………… 167 5.3.10 Experimental Results……………………………………………………………………168 5.4 Chapter Conclusions……………………………………………………………173 Chapter Design Guidelines of UPS Inverters…………………… 174 6.1 Introduction…………………………………………………………………… 174 6.2 Design Issues of UPS Inverters………………………………………………….175 6.2.1 Switching Frequency and PWM Methods………………………………………………175 iv Table of Contents 6.2.2 Cut-off Frequency of L-C Filter………………………………………………………… 178 6.2.3 Inductance and Capacitance…………………………………………………………… 181 6.3 UPS Loading……………………………………………………………………183 6.4 Design of DC Voltage………………………………………………………… 186 6.5 Suggested Design Guidelines……………………………………………………188 6.6 Design Examples……………………………………………………………… 189 6.7 Chapter Conclusions…………………………………………………………….190 Chapter Implementation Issues of Digital Controllers………… 191 7.1 Introduction…………………………………………………………………… 191 7.2 Configurations of Digital Controller for UPS Inverters…………………………192 7.3 Analog Pre-filtering…………………………………………………………… 194 7.4 Programming Aspects………………………………………………………… 194 7.5 Time Delay Due to Sampling/Calculation………………………………………195 7.5.1 Problems Due to Time Delay…………………………………………………………….196 7.5.2 Novel PWM Methods for Handling Time Delay……………………………………… 198 7.5.3 Verification of the proposed PWM methods……………………………………………205 7.5.4 Application of the Proposed PWM Methods for Inverters with Bipolar Switching… 209 7.6 Chapter Conclusions……………………………………………………………211 Chapter Conclusions and Future Work………………………… 212 8.1 Background…………………………………………………………………… 212 8.2 Feedback Control of UPS Inverters…………………………………………… 213 8.3 Learning Control of UPS Inverters…………………………………………… 214 8.3.1 Iterative Learning Control of UPS Inverters…………………………………………… 214 8.3.2 ANN Based Learning Control of UPS Inverters…………………………………………216 8.4 Comparison and Summary of the Proposed Control Schemes………………… 217 8.5 Design and Implementation Issues of UPS Inverters……………………………218 8.6 Future Work…………………………………………………………………… 219 v Table of Contents References…………………………………………………………… 223 Appendix…………………………………………………………… 228 A DS1104 Controller Board…………………………………………………………228 B Discrete Total Harmonic Distortion………………………………………………231 C Schematic Circuit Diagrams………………………………………………………233 D Flowchart of Control Program……………………………………………………236 E Photographs of Experimental Circuits…………………………………………….237 F Details of the Benchmark Cascade Control Methods…………………………… 239 G Derivation of Equation (4.10)…………………………………………………….243 H Derivation of Equation (4.15)…………………………………………………….245 vi Summary Summary The performance of an Uninterruptible Power Supply (UPS) is measured both in terms of steady-state and transient performances Most of the high-performance control techniques reported in literature require additional high-bandwidth current sensor for sensing inductor/capacitor current incurring extra cost The focus of this thesis is to develop advanced digital control techniques with potential for lower cost for UPS inverters capable of higher performance than those currently available The proposed control solutions fall under two categories: feedback control methods and learning control methods Under the feedback control methods, firstly, a method based on general minimum variance (GMV) prediction is proposed Then, a pole-placement controller is proposed which aims to reduce the output impedance through feedback of load current The design, stability and robustness analyses of both control methods are also presented Both proposed feedback control strategies are capable of achieving very good dynamic and steady-state responses with only output voltage and load current sensing The pole-placement controller is shown to have higher robustness than the GMV controller In order to achieve even better steady-state performance, two types of learning based controllers were then investigated: iterative learning based control (ILC) schemes and artificial neural network (ANN) based schemes Firstly, using a direct ILC method, the ILC is combined with the reference feedforward and excellent steady-state performance is achieved Next, a hybrid ILC, where the ILC is paralleled with a PD controller, for improved dynamic response is presented Lastly, an ILC with Inductor Voltage compensation (IVC ILC) is proposed vii Summary in which the dynamic performance is further improved by a feedforward of the inductor voltage The detailed design methods for the schemes to achieve rapid error convergence and robustness are also presented The ILC based schemes achieve almost near perfect steady-state performance and rapid error convergence Though ILC schemes give very good overall performance, the design is quite complex Hence, linear ANN based learning controllers capable of achieving similar performance but are easier to design and implement have been investigated The Adaptive Linear Neural (ADALINE) controller is a simple linear single neuron adaptive controller based on estimation of system parameters Experimental results show that the ADALINE controller can achieve satisfactory performance with only output voltage being sensed Though the scheme is simple, the performance is not as good as the ILC schemes Therefore, B-spline network (BSN) controller has been investigated A BSN controller that is easy to implement is proposed next for UPS inverters Detailed design formulas for the two parameters of B-spline network: the B-spline support width and the learning gain, are given based on stability analysis in frequency domain Compared with the ILC schemes, the proposed BSN controller is easier to design while achieving comparable performance because it has only two parameters to be tuned The performance of UPS inverters is determined not only by control methods but also by the design of the inverters The guidelines for determining switching frequency, PWM methods, DC voltage and parameters of L-C filter are also presented viii Appendix Appendix B Discrete Total Harmonic Distortion The total harmonic distortion (THD) of a voltage v is defined as V − V12 %THD = 100 × V1 Where V1 is the fundamental RMS value of the voltage v while V is the RMS value of the voltage v In the research, following discrete THD is used to evaluate the performance of the inverter The THD is still calculated by %THD = 100 × V − V12 V1 But the V1 and V are discrete RMS value achieved by following formulas: T / Ts Discrete RMS value of a voltage: V = ∑ v (k ) k =1 T / Ts Discrete fundamental RMS value a voltage: V1 = Re + Im T / Ts where Re = T / Ts k =1 k =1 ∑ sin(ωkTs )v(k ) T / Ts , Im = ∑ cos(ω kTs )v(k ) T / Ts , T is the period of one fundamental cycle and Ts is the sampling period 231 Appendix It may be noticed that the discrete THD value may be different with different sampling period even for a same waveform It is found that the discrete THD value is smaller with lower sampling frequency especially when the distortion is very low 232 Appendix Appendix C Schematic Circuit Diagrams Fig C.1 Schematic circuit diagram of voltage sensor Fig C.2 Schematic circuit diagram of current sensor 233 Fig C.3 Schematic circuit diagram of driver circuit 234 Appendix Appendix Fig C.4 Schematic circuit diagram of the circuit for step change resistive load 235 Appendix Appendix D Flowchart of Control Program Fig D.1 Flowchart of control program 236 Appendix Appendix E Photographs of Experimental Circuits Fig E.1 Photo of the voltage sensor board Fig E.2 Photo of the current sensor board 237 Appendix Fig E.3 Photo of the driver board Fig E.4 Photo of the board for testing step-change load 238 Appendix Appendix F Details of the Benchmark Cascade Control Methods For comparison purposes, experiments and simulations were also carried out with the cascade digital controller proposed in [5], which is a popular high-performance controller in industry The structure of this benchmark controller is shown in Fig 3.19 The scheme consists of two control loops, with the control objective of inner loop being the inductor current, while that of the outer loop being the output voltage A feedforward decoupling of output voltage is used for improving the performance of inner current loop Besides, a feedfroward decoupling of load current is adopted for the outer voltage loop The parameters K v and K c are designed to obtain deadbeat responses of voltage loop and current loop, respectively From stability considerations, the sampling frequency of the inner loop is kept double that of the sampling frequency of the outer loop The parameters of the gains for two control loops are based on following Fig 3.19 Cascade controller for comparison purpose 239 Appendix formulas Kv = Kc = C 2TS − CrC rL e − 1− e rL TS L − rL TS L According to the parameters of the inverter for experiments, K v = 0.1526 and K c = 3.9139 were achieved With the proposed control method, both output voltage and inductor current are controlled to achieve deadbeat response for fast dynamic performance It must also be pointed out that the benchmark scheme requires sensing of three variables, output voltage, load current and inductor current 240 Appendix Appendix G Derivation of Equation (4.10) The model of the inverter in iteration i is Vo (k ) = P(z)u(k ) + di(k ) i (AG1) The direct ILC scheme is u (k ) = Vref ( k ) + ui ( k ) , (AG2) with learning law ui (k ) = (1 − α )ui −1 (k ) + γΦ ( z )ei −1 (k ) (AG3) Assuming N is the number of samples in one iteration, according to (AG3), ui −1 (k ) = γΦ ( z )ei −1 (k ) z N −1 + α (AG4) Thus, ui (k ) = γΦ ( z )ei (k ) γΦ ( z ) z N ei −1 (k ) = z N −1+ α z N −1+ α (AG5) By substituting (AG5) into (AG2) γΦ ( z ) z N ei −1 (k ) (AG6) u (k ) = Vref (k ) + z N −1+ α By substituting (AG6) into (AG1), the output voltage is Voi (k ) = P ( z )Vref (k ) + γΦ ( z ) z N P( z )ei −1 (k ) + di (k ) z N −1+ α (AG7) By substituting z N ei −1 (k ) = ei (k ) in to (AG7), Voi (k ) = P ( z )Vref (k ) + γΦ ( z ) P( z )ei (k ) + di (k ) z N −1+ α (AG8) 241 Appendix According to (AG8), the tracking error in iteration i is ei (k ) = Vref (k ) − Voi (k ) = Vref (k ) − P( z )Vref (k ) − (AG9) γΦ ( z ) P( z )ei (k ) − di (k ) z N −1 + α Equation (AG9) can be changed to (1 + γΦ ( z ) P( z ) )ei (k ) = Vref (k )(1 − P( z )) − di (k ) z N −1+ α (AG10) Thus, ei (k ) = (1 − P( z ))( z N − + α ) z N − + α + γΦ ( z ) P( z ) Vref ( k ) + 1−α − z N z N − + α + γΦ ( z ) P( z ) di (k ) (AG11) In steady state, we have z N Vref (k ) = Vref (k ) and z N di (k ) = di (k ) By replacing z N with in (AG11), e( k ) = (1 − P( z ))α −α Vref (k ) + di (k ) α + γΦ ( z ) P( z ) α + γΦ ( z ) P ( z ) (AG12) By substituting z = e jωTs , the magnitude of the tracking error is e( k ) = α (1 − P( jωTs )) α di(k ) + V (k ) α + γΦ( jωTs ) P( jωTs ) α + γΦ( jωTs ) P( jωTs ) ref (AG13) 242 Appendix Appendix H Derivation of Equation (4.15) With Assumption F2 in Section 4.2, the initial state of the inverter is considered to be the same at the start of each iteration Therefore, this initial state need not be considered in the model With (4.6), the output voltage at iteration i is Voi (k ) = Pi ( z )ui (k ) + dii (k ) + dmi (k ) (AH1) where the dii (k ) is the disturbance of the plant at iteration i and dmi (k ) is the measurement noise at iteration i Because the disturbance of the plant is periodical, dii (k ) = dii −1 (k ) = di (k ) Due to Assumption F3, Pi ( z ) = P( z ) and due to Assumption F4, dmi (k ) = Therefore, the output voltage at iteration i is Voi (k ) = P ( z )ui (k ) + di (k ) (AH2) With Fig 4.3 and Assumption F1 Vrefi (k ) = Vrefi −1 (k ) = Vref (k ) , the input voltage of the inverter at iteration i is ui (k ) = u PDi (k ) + Vrefi (k ) + u ILCi (k ) = u PDi (k ) + Vref (k ) + uILCi (k ) , (AH3) where u PDi (k ) = GPD ( z )ei (k ) , (AH4) and u ILCi (k ) is the output from ILC at iteration i By substituting (AH3) and (AH4) into (AH2), the tracking error of iteration i can be presented as ei (k ) = Vref ( k ) − Voi (k ) = Vref (k ) − di (k ) − P ( z )Vref (k ) − P( z )GPD ( z )ei (k ) − P( z )uILCi (k ) (AH5) Obtaining the tracking error of iteration (i-1) in a similar fashion 243 Appendix ei −1 (k ) = Vref (k ) − di (k ) − P ( z )Vref (k ) − P( z )GPD ( z )ei −1 ( k ) − P( z )uILCi −1 (k ) (AH6) By using (4.8), (AH6) and (AH5), it can be shown that α P( z )ui −1 ( z ) − γΦ( z ) P( z ) + GPD ( z ) P( z ) ei −1 ( z ) + + GPD ( z ) P( z ) + GPD ( z ) P( z ) α P( z )ui −1 ( z ) P( z ) )e ( z ) + = (1 − γΦ( z ) + GPD ( z ) P( z ) i −1 + GPD ( z ) P( z ) ei ( z ) = (AH7) 244 Appendix Appendix I Derivation of Equation (5.31) For each b-spline, there are only 2m sampling points that keep µi (l ) ≠ Therefore, ∑ l =0 T / Ts µi (l ) = ∑ l =0 µi (l ) 2m (AI1) Because the b-spline is symmetric, it can be changed to m −1 ∑ l =0 µi (l ) = 2∑ l =0 µi (l ) + µ (m) 2m (AI2) i According to the feature of b-spline, it is easy to find µi (l ) = l m (AI3) Therefore, ∑ l =0 T / Ts m −1 µi (l ) = 2∑ l =0 µi (l ) + µi (m) m + + + + )+( + + + m m m m m m m (0 + m)(m + 1) (0 + m − 1)m = + 2m 2m m +1 m −1 = + 2 d =m= 2TS =( + m −1 ) m (AI4) 245 ... Background…………………………………………………………………… 212 8.2 Feedback Control of UPS Inverters? ??………………………………………… 213 8.3 Learning Control of UPS Inverters? ??………………………………………… 214 8.3.1 Iterative Learning Control of UPS Inverters? ??…………………………………………... compare control performance of UPS inverters with different parameters of L-C filter The digital implementation issues of control methods for UPS inverters are presented in Chapter The problem of. .. modeling of UPS inverters This could serve as the basis for developing novel control techniques of UPS inverters in the future * Classification and detailed review of control methods applied to UPS inverters