Dynamic Phasors in Modeling, Analysis and Control of Energy Processing Systems

Transcription

1 Dynamic Phasors in Modeling, Analysis and Control of Energy Processing Systems 6/20/02 Alex M. Stanković Northeastern University, Boston, MA 1

2 Research Program Overview My research program focuses on the interface of control and energy processing. Emphasis on: 1. dynamical modeling and 2. experimental verification. Emerging energy conversion technologies: 1. efficiency driven and 2. require closed loop control - nonlinearity and uncertainty. NEU Energy Processing Laboratory (1994) is a confluence of research and educational efforts: 1. Areas: power electronics, electric drives and power systems, 2. Graduate students, 3. Sponsors in government and industry, 4. Technical collaborations. 2

3 Research Program Overview Control Modeling (with Experiments) Power Electronics Randomized Modulation (NSF, RIA) Resonant Converters I&I Control (ONR/VTB) Adaptive Control of Drives (ARO) Electric Drives Phasor Dynamics (ONR YIP) ANN in ID (NSF) Hilbert Space for Q comp. (NSF) IM Control Oscillation GM Hughes Suppression (NSF, Career) Naval Drives (ONR, PEBB) Quiet PMSM (Draper, Satcon) Power Systems Probabilistic Load Modeling CAD Methods (EPRI) 3

4 Presentation Overview Background and definitions, POWER SYSTEMS - Flexible AC Transmission Systems. ELECTRIC DRIVES - AC machine modeling, An interlude - extension to polyphase systems, POWER ELECTRONICS - active filters and switched-mode DC/DC converters. 4

5 Background New challenges in energy processing (power electronics, electric drives, power systems): reliance on switching operation for efficiency, new dynamic couplings, increased performance specifications, new problems (e.g., active filtering, pulsed power). Analytical tools for addressing ( close-to periodic) system operation: sinusoidal quasi-steady-state approximation in drives and power systems, time-domain simulations (in power electronics and drives), systematic exploration of the middle ground is largely missing. 5

6 Background Main features of dynamic phasors: large signal models, nonlinear, physical intuition used for simplifications, appealing mathematical structure. Origins and ideas related to our work: power electronics, power engineering ( space vectors, spiral vectors, polyphasors), nonlinear oscillations (classical averaging and recent variants), signal processing. 6

7 Definitions A (possibly complex) waveform using (short-time) Fourier series: can be represented on the interval Our dynamical models describe evolution of where are the complex, slowly time-varying Fourier coefficients, or dynamic phasors. ; for real we have 7

8 Definitions Two useful facts: Derivative of the -th dynamic phasor: Multiplication in time domain: 8

9 A Simple Example Consider a simple RL circuit (R=1, L=0.1) with a and some initial condition ( A): excitation (V=10), at 60Hz Current (A) Time(s) 9

10 A Simple Example, cont. 1 The dynamic phasors (according to our definition) Fundamental real part DC component Fundamental imag. part

11 Presentation Map Definitions, Power systems - Flexible AC Transmission Systems. Electric drives - AC machine modeling, An interlude - extension to polyphase systems, Power electronics - active filters and switched-mode DC/DC converters. 11

12 Flexible AC Transmission Thyristor Controlled Series Capacitors (TCSCs) are finding increasing application in power systems. L + _ C 1 φ α τ σ [s] 12

13 Flexible AC Transmission Fast and accurate models are needed for simulation and control. Analytical difficulties stem from the nature of TCSC that amalgamates continuoustime dynamics with discrete events (thyristor firings). Sampled-data models were the first to offer the needed accuracy, but model structure has no clear relation to the system configuration, hard to interface with the rest of the system which is usually described with phasor-based continuous-time models. 13

14 Flexible AC Transmission A state-space model for basic TCSC configuration: where is a switching function. is sinu- Evaluating the 1-phasor on both sides of each equation (and assuming soidal) we obtain a 2nd-order (complex) phasor model: where is: 14

15 Flexible AC Transmission has fast dynamics compared to, so we assume yielding where is the prevailing conduction angle and is the reference. is computed from steady-state assuming sinusoidal line current to the conventional approach which assumesto be sinusoidal). (in contrast 15

16 Flexible AC Transmission Line current for step changes of the firing angle: [s] [s] 16

17 Subsynchronous Resonance First IEEE benchmark test with TCSC: A L TCSC B L s1 R s C Ls2 Generator Infinite Bus 17

18 Subsynchronous Resonance 1 Modal damping (per second) TM0 TM2 TM3 TM1 TM Conduction angle [degrees] 18

19 Subsynchronous Resonance 15 Shaft Torque LPB GEN Torque deviation [pu] time [s] 19

20 Subsynchronous Resonance Expanded view (between 80s and 81s) of torque variations HP IP IP LPA LPA LPB LPB GEN GEN EXC 5 Shaft torque at different shaft positions time [s] 20

21 Presentation Map Definitions, Power systems - Flexible AC Transmission Systems. Electric drives - AC machine modeling, An interlude - extension to polyphase systems, Power electronics - active filters and switched-mode DC/DC converters. 21

22 Dynamic Phasors and Space Vectors Assuming now that all phase quantities are real; let / : This complex (scalar) quantity can encode two-dimensional information; for example,. 22

23 Electrical Machines Space vector model of a three-phase induction machine: 23

24 Electrical Machines Dynamic phasor model of a three-phase induction machine: 24

25 Electrical Machines Experimental evaluation: 40 v ca 30 v bc v bc and v ca (V) t (s) 25

26 Electrical Machines Experimental evaluation, cont. 2 i bs i as i as and i bs (A) t (s) 26

27 Electrical Machines Steady-state equivalent circuit ( and ): I p r I n r r s I p, s L l r + r r s r s I n s L l r + r r 2-s + V - p L Vpt l s ( ω s ) V ( ) L m - ω s L l s L m - V n t m * p V n t n m n Vpt * 27

28 Presentation Map Definitions, Power systems - Flexible AC Transmission Systems. Electric drives - AC machine modeling, An interlude - extension to polyphase systems, Power electronics - active filters and switched-mode DC/DC converters. 28

29 Extension to Polyphase Systems Dynamical symmetric components - recall /, / / 29

30 Properties of Dynamic Symmetric Components For real waveforms: Connection with space vectors: 30

31 Asymmetric Faults in Power Systems G F (a) Infinite bus 31

32 Asymmetric Faults in Power Systems Line voltages v as v bs v cs v abcs (p.u.) t (s) 32

33 Asymmetric Faults in Power Systems Field current 3 a i fd (p.u.) 1.5 b t (s) 33

34 Presentation Map Definitions, Power systems - Flexible AC Transmission Systems. Electric drives - AC machine modeling, An interlude - extension to polyphase systems, Power electronics - model reduction for switched-mode DC/DC converters. 34

35 Model Reduction - DC/DC Converters v (t) in L D + i S v C R Model in continuous conduction ( when S closed): 35

36 Model Reduction - DC/DC Converters 36

37 Model Reduction - DC/DC Converters 14 switched index-0 + index-1 (MFA) SSA Capacitor Voltage (V) index-0 (MFA) time (ms) 37

38 Model Reduction - DC/DC Converters 14 switched index-0 (CMFA) SSA Capacitor Voltage (V) index-0 (MFA) time (ms) 38

39 Estimation for our Simple Example Again we consider a simple RL circuit with excitation: / / / / 39

40 Estimation for our Simple Example, cont. 1 True (dash-dotted) and measured (solid line) current: True current (noise free) Current (A) Measured (noisy) current Time(s) 40

41 Estimation for our Simple Example, cont. 2 Convergence of estimates: Dash dotted estimates Solid true (noise free)

42 Estimation for our Simple Example, cont. 3 Another view of convergence of the fundamental phasor estimate: Estimated(:) and true ( ) phasors trace Imag Real x

43 Estimation for our Simple Example, cont. 4 Contrast with a model-free ( running FFT ) estimate: Measured(:) and true( ) phasors trace

44 Summary Dynamic phasors yield simple, but powerful large-signal dynamic models. A unified approach with additional applications in power electronics (resonant converters), electric drives (torque ripple minimization) and power systems (protection). Both refinements and simplifications are possible for various accuracy requirements. Models are modular, and compatible with engineering experience and intuition, A timely re-examination of analytical tools in energy processing addresses challenges posed by advances in semiconductor and computer technology. 44