Analysis, Design and Control of DC-DC Converters

Transcription

1 TUM Jan 216 Analysis, Design and Control of DC-DC Converters Damian Giaouris BEng, BSc, PG Cert, MSc, PhD Senior Lecturer in Control of Electrical Systems Electrical Power Research Group School of Electrical and Electronic Engineering, Merz Court, Newcastle University, Newcastle Upon Tyne, NE1 7RU, United Kingdom

2 Outline

3 Power Converters A device that changes input voltage/current levels: i Input u Input DC-DC Converter i Output u Output iinput uinput ioutput uoutput In each clock cycle: S S u in S us out u in u out S: Opens (S=) and Closes (S=1) periodically with Period T 1 1 k T uout uoutdt T u in D u out C u out R u out - L i L + kt d t on Step Down or Buck Converter T t T 2T 3T 1 1 k d T k T uout uoutdt uoutdt T kt k d T k d T k 1T 1 uout uindt dt T kt k d T 1 uindt T ud in

4 Main Power Converters S L + u in D C R u out - i L D u d u out in duout uout i dil uout uin,, S On dt RC C dt L C duout uout i dil uout,, S Off dt RC C dt L u in L i L S C R Step Up or Boost Converter u out u out 1 1 d u in duout ilr uout dil uin uout,, S On dt RC dt L duout uout dil uin,, S Off dt RC dt L Aonx Bonu, S On x Aoff x Boff u, S Off fon x, u, t, S On x foff x, u, t, S Off Piecewise smooth systems

5 Simulink Model Vin=2; L=2e-3; C=47e-6; R=22; T=1/25; D=.5

6 OL Response

7 OL Response D= Δi L = or 22.78% Δv out = or 1.67%

8 Converter Design Basic Steps 1) Power level 2) Clock Frequency 3) Input/Output Voltage/Current (conversion Rate) 1) Standard: a) Minimise ripple b) Design a low pass filter 2) Advanced: a) Artificial Intelligence b) Optimisation methods Minimise Current/Voltage Ripple duout ilr uout dil uin uout,, S On dt RC dt L duout uout dil uin,, S Off dt RC dt L dil u u u DT dt L L i in in in il DT L L duout uout uoutdt C dt RC R u out 4 uin il DT 2% L.2 4 uoutdt uout 1.9% 4 RC

9 Control of Power Converters x ref (t) + - e(t) G c x con (t) G x real(t) x real (t) u ref (t) + - e(t) G c u con (t) G u real(t) u real (t) u ref (t) + - e(t) G c u con (t) PWM u G u real(t) u real (t)

10 On-Off Controllers u t sign e t u 1-1 e

11 On-OFF Controllers

12 On-Off Controllers

13 Hysteresis Controller u 1-1 e

14 Hysteresis Controller ε=+/-1 ε=+/-.1

15 Average Controllers Set the switch at t= ON and then compare u con (t) with a Ramp signal u Ramp (t): u Ramp u con uramp t VL VU VL T t u

16 Average Controllers

17 Similarly for current control: Average current controller

18 Average current controller

19 Change of ramp: +/-2, +/-1, +/-2

20 Change of k: 3 and.3 K=.3

21 Change of offset u con u Ramp u

22 Change of offset

23 Another point of view Hysteresis also called Peak Controller: P-Controlled averaged also called Peak Controller: x ref x x ref +x ramp x u t u t t t Switching occurs at: k x x x t x x p ref ramp x x t ramp ref x k p ref x ramp k p t x

24 PI Control Averaged model: x Aonx Bonu, S On Aoff x Boff u, S Off 1 x A x B u d A x B u d on on off off Linearise around the FP x A x B u Bode Plots, Root Locus Any other type of advanced controller!

25 Transfer Functions D u in L i L S C R u out G v out, d s 1 1 s LC RC 1 D RC LC 2 s s D 2 2 S L + u in D C R u out - i L G v out s, d 2 s b as 1 cs 1

26 Latch What if slope of v con >v ramp? u con u Ramp Switch On again! u Ramp u con u con u Ramp Sliding or Multiple switching. u u u Latch to avoid sliding S Q S T 2T 3T u R Q

27 Current/Voltage Control u ramp u con u ramp s closed u con i G C + - u ref (t) i con i ramp s closed i ramp i con G C + - i ref (t) S L + u in D C R u S L i + - u in D C R u - Voltage Controlled Converter Current Controlled Converter

28 Cascade Control i ramp u ramp u con u ramp s closed u con i G C + - u ref (t) i con i G Ci + - i ref (t) G C2 - + V ref(t) S L + S L + u in D C R u - u in D C R u - Voltage Control Current Control i ramp i con i ramp i con G C - i ref (t) s closed + i S L + u in D C R u -

29 Cascade Control

30 Cascade Control

31 Transfer Functions D u in L i L S C R u out G v out, d s 1 1 s LC RC 1 D RC LC 2 s s D 2 2 S L + u in D C R u out - i L G v out s, d 2 s b as 1 cs 1

32 Minimum Phase Systems u ramp u con u ramp s closed u con i G C + - u ref (t) S L + u in D C R u -

33 State space L i L D v out V in S C R State Space Controller PWM

34 State space d u Ramp 1 1 T 2T

35 State space d Ramp T 2T u 1 d T 2T d 1-d dt T 2T T T 2T 2T 1 T dt 2T

36 State space, T u k x xss uss k k1 k2 T

37 State space, T u k x xss uss k k1 k2 T k k 1 2 1A 2A,, k k a b x nt c k k a b x nt c 1B 2B

38 MPC Geyer, T.; Papafotiou, G.; Morari, M., "Hybrid Model Predictive Control of the Step-Down DC DC Converter," in Control Systems Technology, IEEE Transactions on, vol.16, no.6, pp , Nov. 28

39 Complete Modelling D L i L I ref i u in S C R u out u t t duout ilr uout dil uin uout,, S On dt RC dt L duout uout dil uin,, S Off dt RC dt L S On when i I S Off when i I h x t, t = i L I ref L L ref ref S On when h x t, t S Off when h x t, t

40 duout ilr uout dil uin uout,, S On dt RC dt L duout uout dil uin,, S Off dt RC dt L h x t, t = i L I ref Complete Modelling u out h< h> h x t, t = i L I ref x I Kxt Rt 2 ref 1, T, K x t u t i t R t I out L ref I ref i L x Aonx Bonu, S On Aoff x Boff u, S Off h: Defines a switching surface in the state space, h x t t Kx t R t

41 Complete Modelling - V tri + V con Controller DSP V out V ref + - S L Off On Off On Off On Off On Off On E D C R uramp t VL VU VL T t Cont p ref out u t K V u t, h x t t Kx t R t T t K K, xt u t i t, Rt V V V K V T p out L L U L p ref

42 State Space - h

43 State Space - h

44 Instabilities (Bifurcations) output voltage, V current, A Period Doubling Bifurcation time, s Period 2 Current ripple>.2a Current ripple<.1a Period 1 output voltage, V current, A time, s

45 Instabilities (Bifurcations) x T Saddle Node Bifurcation

46 Instabilities (Bifurcations) Slow Scale Bifurcation

47 PI Voltage Controlled Converter Bifurcation Diagram

48 Bifurcation Diagram PI Voltage Controlled Converter For V in =2.V, i L (n)=.443 For V in =2.1V, i L (n)=.442 For V in =2.2V, i L (n)=.438 For V in =2.3V, i L (n)=.436 For V in =2.4V, i L (n)=.433 For V in =2.5V, i L (n)=.431 For V in =2.6V, i L (n)=.43 For V in =2.7V, i L (n)=.426 For V in =2.8V, i L (n)=.425 For V in =2.9V, i L (n)=.422 For V in =21.V, i L (n)=.417 AND.422 For V in =21.5V, i L (n)=.418 AND.3863 i L (n) V in 2 21 V in

49 Bifurcation Diagram PI Voltage Controlled Converter i L (n) 2 21 V in

50 Digital State Feedback Bifurcation Diagram

51 Digital State Feedback Bifurcation Diagram

52 Smooth Limit Cycle x t, t, x t, x 1 t, t, x 1 t, t, t, x t, t x, t,t,x t,t,x t,t,x x t,t,x t,t,x t,t,x f x,t d t,t,x t,t,x dt x x x x t,t,x x t t,, x t, x 2 t, x Ax t,t,x Att x e t, x 1 t, x t, t, x t,t,x t,t t,t t,t,x x t t,, x x 1 t, t 1 1 t, t 1, x, x 1 x 2 t t 2 2, t, t 1, x, x 1

53 Smooth Limit Cycle Φx t x x pp t t xx t t x t x x p p t t t T,t,x eigs Unstable r=1 Stable 53

54 Nonsmooth Limit Cycle Δx t t t x p Δxt 1 t 1 t 4 Δxt 4 t 2 t 1 t 3 Δx t 2 t 2 Δxt 3 t 3 t 4

55 Saltation Matrix t 1 x x x ' S2 t 1 S x 1 x' t 3 t 4 x' t 2 x t 2 x' Saltation or jump matrix (Aizerman, Filippov, Muller) t 3 t 4 S lim f tt tt I T n lim f x tt xt lim f xt t h t T n Dynamics and Bifurcations in Non-Smooth Mechanical Systems R. I. Leine and H. Nijmeijer T Off Φ Φ T,dT S Φ dt, 1 Φ e S e T A d T A dt On

56 Application of Filippov s Theory Damian Giaouris, Soumitro Banerjee, Bashar Zahawi, and Volker Pickert: Stability Analysis of the Continuous Conduction Mode Buck Converter via Filippov's Method, IEEE Transactions of Circuits and Systems-I, vol. 55, no. 4, pp , May 28 current, A V f f n T f n T f voltage, V X(dT ) X() n Orbit V Imaginary part of the eigenvalues V in = 24V V in = 25V r = 1 r = Real part of the eigenvalues

57 Bifurcation Classification r=1 r=1 Period Doubling Slow Scale Saddle - Node Sampled State Variable Bifurcation Variable x2 x 1 Sampled State Variable Bifurcation Variable x2 x 1 Sampled State Variable Bifurcation Variable x2 x 1

58 High Frequency Injection L C R D S E V ref + - V out + - V con V tri Controller DSP Supervising Controller t a V V ref ref sin 1 A T V V t a V A T V V t t h L U ref L U / cos / ) (

59 Dual Input

60 Interleaved Operation

61 Buck Boost

62 Resonant

63 Boost Boost

64 Any questions?

65 Millennium Bridge Newcastle upon Tyne