Output high order sliding mode control of unity-power-factor in three-phase AC/DC Boost Converter

Transcription

1 Output high order sliding mode control of unity-power-factor in three-phase AC/DC Boost Converter JianXing Liu, Salah Laghrouche, Maxim Wack Laboratoire Systèmes Et Transports (SET) Laboratoire SeT

2 Contents of the presentation Introduction Problem Formulation Second order sliding mode controller design Second order sliding mode observer design Simulation results with proposed controller Conclusions

3 AC-DC power conversion The electric utility grid has a sinusoidal waveform, and most electronic equipment needs a DC power supply. AC-DC power conversion is required by all electronic devices virtually MOSFET, IGBT, are commonly used for AC-DC Converters

4 AC-DC power conversion Problem:. The performance and efficiency of power converters with unknown varying load and internal uncertainties. Minimize the number of the sensors Power factor reflects the efficiency and quality of such process. Different control algorithms have been used to achieve unity power factor. An output second order sliding mode control is designed here.

5 Model of three phase AC/DC converter The circuit of three phase AC/DC boost converter i link Fig Three phase boost type AC/DC converter

6 Model Model in Phase Coordinate Frame: di r U U 0 = i ( u u u 3) + dt L 6L L di r U U 0 = i ( u u u ) + dt L 6L L di U 3 r U i 0 = 3 ( u 3 u u ) + dt L 6L L du U ( ) 0 0 = + ui + ui + ui 33 dt RlC C g g g3

7 Model in (d,q) Coordinate Frame It is convenient to design the control in the rotating reference frame synchronized with the supply frequency. Transformation Matrix: cos( ) cos( ) cos( ) x x ωt ωt π ωt+ π d 3 3 x x = q 3 sin( ωt) sin( ωt π) sin( ωt π) + x Define: C cos( ) cos( ) cos( ) ωt ωt π ωt+ π 3 3 = 3 sin( ωt) sin( ωt π) sin( ωt π) + 3 3

8 Model in (d,q) Coordinate Frame di r U U d 0 = i + ω i u + dt L d q L d L di r U U q = i ω i 0 u + dt L q d L q L du U 3 0 = 0 + ( iu + iu ) dt R C 4C d d q q l Where ω is the source frequency gd gq

9 Control objectives Unity power factor Ripple free output voltage with desired level i d i * * tracks id, q tracks i q Satisfy Power Balance Equation 3 ( U i + U i ) = U U * * 0 gd d gq q 0REF Rl Considering unity power factor, the desired currents are as follows: * d i = i = 0 U U * 0REF 0 q 3U R gq l

10 Control structure Observer-based control structure is shown is fig. Fig Observer-based control structure

11 Super-Twisting sliding mode x= f(,) xt + bxtu (,) y = Design sx ( ) where x, f, b! n Control Objective: Force sx ( ) to zero. The relative degree with respect to Advantages: Robustness property with respect to perturbations and parametric uncertainties. Smoother than the classic sliding mode control sx ( ) is equal to one. u = υ(), s υ() s = λs sign() s + αsign() s dt,( λα, > 0)

12 Step:Sliding manifolds design Design the switching functions: s = i * i d d d = * q q q s i i To find a domain in the system space from which any state trajectory converges to the sliding manifold(s d =0,s q =0). U * r gd u d d q s i i ωi d L L U u 0 d fd U0 = + C u = + U u * r gq L q fq L s q iq iq ωi + d u 3 L L

13 Step: Control design Super twisting sliding mode controller can be designed: u - υ(s ) u = - υ(s ) u 3 - υ(s 3) υ( e) = λe sign( e) + αsign( e) dt,( λα, > 0) The transformed vector s should be designed such that, under the above controller, s d and s q will vanish in finite time. s L 3 s T d s = C U0 s q s 3

14 Super-Twisting Observation Design Assuming that only the output voltage(u 0 ) is measured, a supertwisting sliding observer is constructed as a copy of the original system. di r d U U i i 0 gd = d+ ω q u d + k υ( e 3) dt L L L diq r U U i i 0 gq = u k q ω d q + υ( e 3) dt L L L du0 U0 3 = + ( iu d d + iu q q ) k3υ ( e3 ) dt RlC 4C Where Define observation error: e = id id e = iq iq e = U U υ( e ) = λe sign( e ) + αsign( e ) dt,( λα, > 0)

15 Super-Twisting Observation Design The estimation error dynamics are: de r ud = e + ω e e 3 k υ( e 3) dt L L u q de = r e ω e e 3 k υ( e 3) dt L L de3 3 = e3+ ( ue d + ue q ) k3υ ( e3 ) dt RlC 4C The sliding surface defined as s=e 3 =0. Choosing a large positive constant k 3 can assure the convergence of s(e 3 =0) in finite time.

16 Observer convergence proof Dynamics on the sliding manifold The equivalent switching function is: υ ( e3) = ( ue d + ue q ) kc 3 Substitute into the two equations of error dynamics, r r e ω ~ L e k k ud uq e ω L k k ud uq, A, A, U = = = dq = r e kc 3 k k 0 0 e r k k 0 0 e ω ω L e ψ ( e ) L As A is a Hurwitz matrix, there exists a unique positive definite symmetric matrix P that satisfies the equation with positive definite symmetric matrix Q. T PA + A P = Q The Lyapunov function is given as: V ( e ) = e Pe T

17 Observer convergence proof The derivative of V(e ) is V V V ( e ) = Ae + ( e ) e ψ e The first term: V Ae T T T = e ( PA + A P ) e = e Qe λmin ( Q ) e e The second term can be expressed as: V T ( e ) ep ( e ) P e ( e ) e ψ ψ ψ ~ η ψ( e ) A U dq e ρ e, ρ = kc kc 3 3 ~ ~ T ~ A λmax( A A) ( k k ) η = = = + If ρ < λmin ( Q) λ ( P) max V( e ) λ ( Q) e + ρλ ( P) e min max,the origin is globally exponentially stable.

18 Observation of Source Phase Voltage A link current sensor is used to estimate the source phase voltage. if u u = u link if u u = u3 Design the sliding mode observer: i3 if u3 u = u di r U0 i ( u u u 3) M ( e = + υ ), r Ug Mυ( e) dt L 6L L e= e+ L L υ( e) = λ e sign( e) + αsign( e) dt, e = ui link i Ug = Mυ( e) i 3 = i Choose sufficiently large observer gain M, the sliding mode will be enforced in finite time. i

19 Observation of Load Resistance du0 U0 = + ( ui + ui + ui 33 ) dt RlC C du0 U0 = + ( ui + ui + ui 33 ) + kυ ( e ) dt RC C l υ( e) = λ e sign( e) + αsign( e) dt, e = U U 0 0 Error dynamics: U U e= k e 0 0 ( ) υ( ) RC l RC l Choose sufficiently large observer gain k, sliding mode will be enforced. υ e U = Rl = kcυ ( e) + R U 0 ( ) ( ) kc R Rl l l 0

20 Power Factor Calculation I I PF = PFh PFd = cos( φ) I main current harmonic PFh harmonic distortion I total current RMS φ phase shift between input current and main voltage T RMS( i( t)) = i( τ) dτ T is the period of i(t) T 0 Remark: Using Fourier analysis, harmonic distortion measurement, trigonometric modules in matlab, the power factor of the system can easily be obtained.

21 Simulation Results Parameter Value Description R Load Resistance Ω r 0.0 Parasitic phase resistance Ω L e-3 Phase inductor H C 00e-6 Output capacitance F E 50 Amplitude of source voltage V f t=.5s t=.0s Source voltage frequency Hz U 0ref 650 Desired output DC voltage V; V(0)=V Load resistance and frequency are varied to test controller s ability to handle with varying load and frequency.

22 Input current in phase with source voltage Figure Input current and source voltage in case of super-twisting SMC Figure Input current and source voltage in case of traditional SMC Traditional SMC results in higher harmonics compared with super-twisting SMC.

23 Source Voltage Estimation Figure3 Source voltage U g estimation within the windows when i link is equal to i Source voltage can be reconstructed with super-twisting observer without designing a low-pass filter.

24 Load Resistance Estimation Figure4 Load resistance R estimator performance Figure4 shows the performance of load resistance estimation.

25 The performance of observer Figure5 Output voltage estimator performance U 0 Figure6 Current estimator performance i q

26 The performance of observer Figure7 Phase Current estimator performance Figure5-7 show the performance of the observer. i a

27 The performance of output voltage Figure8 Output voltage performance with super-twisting control Figure9 Output voltage performance with traditional sliding mode control A bit higher than the desired voltage level and more oscillating compared with super-twisting SMC.

28 Power Factor Figure0 Power factor with super-twisting control Figure Power factor with Traditional SMC Less value and more oscillations compared with supertwisting control. The super-twisting control was able to produce a power factor that was more than 99%, and can withstand the changing conditions.

29 Conclusions The proposed control method can achieve a power factor close to unity. Power Balance Condition is taken into account to achieve the desired performance of the system. The proposed super-twisting observer demonstrates its robustness to the change in operational conditions. Source Voltage Estimation is achieved with i link via super-twisting method without using low-pass filter.

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