Stability and Control of dc Micro-grids

Transcription

1 Stability and Control of dc Micro-grids Alexis Kwasinski Thank you to Mr. Chimaobi N. Onwuchekwa (who has been working on boundary controllers) May, Alexis Kwasinski, 011

2 Overview Introduction Constant-power-load (CPL) characteristics and effects Dynamical characteristics of DC micro-grids Conclusions/future work Alexis Kwasinski, 011

3 Micro-grids What is a micro (or nano) grid? Micro-grids are independently controlled (small) electric networks, powered by local units (distributed generation). 3 Alexis Kwasinski, 011

4 ac vs. dc micro-grids Introduction Some of the issues with Edison s dc system: Voltage-transformation complexities Incompatibility with induction (AC) motors Power electronics help to overcome difficulties Also introduces other benefits DC micro-grids DC micro-grids Help eliminate long AC transmission and distribution paths Most modern loads are DC modernized conventional loads too! No need for frequency and phase control stability issues? 4 Alexis Kwasinski, 011

5 ac vs. dc micro-grids Introduction DC is better suited for energy storage, renewable and alternative power sources 5 Alexis Kwasinski, 011

6 Characteristics Constant-power loads DC micro-grids comprise cascade distributed power architectures converters act as interfaces Point-of-load converters present constant-power-load (CPL) characteristics 0 if vt ( ) V it () PL if vt ( ) V vt () lim lim z dv B di B P i L B CPLs introduce a destabilizing effect 6 Alexis Kwasinski, 011

7 Characteristics Constant-power loads Simplified cascade distributed power architecture with a buck LRC. dil L qt ()( ERi) (1 qt ())( V ir ) ir v dt dvc PL vc C il dt vc Ro with i 0, v S L D L D L L C L Constraints on state variables makes it extremely difficult to find a closed form solution, but they are essential to yield the limit cycle behavior. C 7 Alexis Kwasinski, 011

8 Characteristics The steady state fast average model yields some insights: d x LPL dx dt x dt LC x DE dx P x x C dt x L with and 1 0 Lack of resistive coefficient in first-order term Unwanted dynamics introduced by the second-order term can not be damped. Necessary condition for limit cycle behavior: Note: x 1 = i L and x = v C Constant-power loads dte () v 1 P P i L Cv v C L L L C C 8 Alexis Kwasinski, 011

9 Characteristics Constant-power loads Large oscillations may be observed not only when operating converters in open loop but also when they are regulated with most conventional controllers, such as PI controllers. Simulation results for an ideal buck converter with a PI controller both for a 100 W CPL (continuous trace) and a.5 Ω resistor (dashed trace); E = 4 V, L = 0. mh, PL = 100 W, C = 470 μf. 9 Alexis Kwasinski, 011

10 Controls Characteristics E 400 V, E 450 V, P 5 kw, P 10 kw, 1 L1 L L 5 H, R 10 m, C 1mF LINE LINE DCPL LRC parameters: L 0.5 mh, C 1 mf, D1 0.5, D 0.54, R L 0.8 Large oscillations and/or voltage collapse are observed due to constant-power loads in micro-grids without proper controls 10 Alexis Kwasinski, 011

11 Stabilization Passive methods added resistive loads Linearized equation: Conditions: Issue: Inefficient solution Ri PL Ri PL CVo o o CVo C PL Vo Ri, PL Vo L Ro Ri Ro where R R DR (1 D) R L R C L R C LC i S D L 1Ω resistor R o in parallel to C DCPL E 1.5 V, L480 H, C 480 F, R, D0.9, P 43.8W o L 11 Alexis Kwasinski, 011

12 Stabilization Passive methods added capacitance Condition: P L C V R L o i Issues: Bulky, expensive and may reduce reliability. But may improve fault detection and clearance E 1.5 V, L480 H, C 480F 00 mf, D0.9, P 35W L 60 mf added in parallel to C DCPL 1 Alexis Kwasinski, 011

13 Stabilization Passive methods added bulk energy storage It can be considered an extension of the previous approach. Energy storage needs to be directly connected to the main bus without intermediate power conversion interfaces. Issues: Expensive, it usually requires a power electronic interface, batteries and ultracapacitors may have cell voltage equalization problems, and reliability, operation and safety may be compromised. 13 Alexis Kwasinski, 011

14 Stabilization Passive methods load shedding It is also based on the condition that: Issues: Not practical for critical loads C P L V R L o i Load reduced from 49 W to 35 W Load dropped from 10 to.5 kw at 0.5 seconds 14 Alexis Kwasinski, 011

15 Linear controllers Passivity based analysis Initial notions A system with and x f( x, u), x(0) x R, f : R R R n n m n 0 : n m p y h( x, u), h: R R R f is locally Lipschitz f(0,0) = h(0,0) = 0 Stabilization is passive if there exists a continuously differentiable positive definite function H(x) (called the storage function) such that dh H ( x) f( x, u) u y ( x, u) R R dx T n m 15 Alexis Kwasinski, 011

16 Stabilization Linear controllers Passivity based analysis Initial notions Σ is output strictly passive if: dh T n m H ( x) f( x, u) u yo y ( x, u) R R, and o 0 dx A state-space system x f( x), x R n is zero-state observable from the output y=h(x), if for all initial conditions x(0) R n we have yt () 0 xt () 0 Consider the system Σ. The origin of f(x,0) is asymptotically stable (A.S.) if the system is - strictly passive, or - output strictly passive and zero-state observable. - If H(x) is radially unbounded the origin of f(x,0) is globally asymptotically stable (A.S.) - In some problems H(x) can be associated with the Lyapunov function. 16 Alexis Kwasinski, 011

17 Stabilization Linear controllers Passivity based analysis Consider a buck converter with ideal components and in continuous conduction mode. In an average sense and steady state it can be represented by where L M 0 0 C J Mx [ JR( x)] xde 0 0 R( x) PL 0 x E E 0 x x x 1 Equilibrium point: x e I P L L DE V o DE Mx [ JR( x)] xde x x x e (Coordinate change) Mx[ JR( x)] x de[ JR( x)] x e (1) 17 Alexis Kwasinski, 011

18 Stabilization Linear controllers Passivity based analysis Define the positive definite damping injection matrix R i as Ri1 0 Ri 1 P L 0 Ri x 1 P Ri1 0 L R i is positive definite if. Then, Ri x Rt RRi 1 0 R i From (1), add R on both sides: i ( xx ) Mx [ JR ] x de[ JR( x)] x R ( x) x t e i E 0 (Equivalent free evolving system) : Mx[ JR ] x 0 t 18 Alexis Kwasinski, 011

19 Stabilization Linear controllers Passivity based analysis Consider the storage function Its time derivative is: 1 T H ( x) x Mx T T T H ( x) x Mx x [ JR ] x x R x 0 T 1 H ( x) = y Rty o y, where o max Ri1, Ri is a free-evolving output strictly passive and zero-state observable system. Therefore, x 0 is an asymptotically stable equilibrium point of the closed-loop system. t if y x t 19 Alexis Kwasinski, 011

20 Stabilization Linear controllers Passivity based analysis Since E de[ JR( x)] x R ( x) x 0 e i then, Hence, de Vo Ri 1x 10 P x V I L x R R L o i i 1 d Vo Ri1( x1il) E 0 Alexis Kwasinski, 011 P L and since x1 Cx and x Thus, d R Cx R x V R V 1 i1 i1 i1 o o E Ri Ri 0 I V P x o L L Ri x Ri This is a PD controller ( e V x ) O

21 Stabilization Linear controllers Passivity based analysis Remarks for the buck converter: x e is not A.S. because the duty cycle must be between 0 and 1 Trajectories to the left of γ need to have d >1 to maintain stability Using this property as the basis for the analysis it can be obtained that a necessary but not sufficient condition for stability is de x P P L C x x 1 L L x1 Line and load regulation can be achieved by adding an integral term but stability is not ensured p d i d D k e k e k edt 1 Alexis Kwasinski, 011

22 Stabilization Linear controllers Experimental results (buck converter) x 1 x Alexis Kwasinski, 011

23 Stabilization Linear controllers Experimental results (buck converter) Load regulation Line regulation 3 Alexis Kwasinski, 011

24 Stabilization Linear controllers Passivity based analysis The same analysis can be performed for boost and buck-boost converters yielding, respectively d R 1 1 i D' D' 4 Ce e V0 Ri ' 1 d, d ' E E Ri 1 x V0 4 Cx i ( V E) V E ( V E) R Engineering criteria dictate that the non-linear PD controller can be translated into an equivalent linear PD controller of the form: Formal analytical solution: d ' k e k ed' 4 Alexis Kwasinski, 011 d p e e, e e max max min d' d '( k, k ) subject to 0 1d d 0 1d d p max max d

25 Consider And The perturbation is Stabilization Linear controllers Passivity based analysis Perturbation theory can formalize the analysis (e.g. boost conv.) x f t x x 1 : (, ) M [ J R t ] Unperturbed system with nonlinear PD controller, with x f t x g t x x 1 p: (, ) (, ) M [ J Rt] Perturbed system with linear PD controller, with gtx M J J 1 (, ) [ ] x 0 d ' J d ' 0 0 d ' J d ' 0 5 Alexis Kwasinski, 011

26 Lemma 9.1 in Khalil s: Let x 0 be an exponentially stable equilibrium point of the nominal system. Let V(, t x ) be a Lyapunov function of the nominal system which satisfies 1 (, ) c x V t x c x Vtx (, ) Vtx (, ) f (, t x ) c 3 x t x Vtx (, ) x c in [0, ) X D with c 1 to c 4 being some positive constants. Suppose the perturbation term gtx (, ) satisfies Then, the origin is an exponentially stable equilibrium point of the perturbed system. 6 Alexis Kwasinski, x c gtx (, ) x t0, x where : x and c p Stabilization Linear controllers Passivity based analysis 3 4

27 Linear controllers Passivity based analysis Taking V(, t x) H ( x) It can be shown that T c4 M max ( M M) max(l, C ) max(l, C) Also, x 0 is an exponentially stable equilibrium point of, gt (,0) 0 and with Stabilization c 1 = λ min (M) c = λ max (M) c ( 3 ) 3 min J Rt gtx (, ) x t0, x where : x G ( d ' d') ( d' d ') max, L C d ' d' Thus, stability is ensured if 7 Alexis Kwasinski, 011 min(j 3 Rt ) C L max, L C

28 Linear controllers Stabilization Experimental results boost converter Load Regulation Line Regulation 8 Alexis Kwasinski, 011

29 Linear controllers Stabilization Experimental results voltage step-down buck-boost converter Load Regulation Line Regulation 9 Alexis Kwasinski, 011

30 Linear controllers Stabilization Experimental results voltage step-up buck-boost converter Load Regulation Line Regulation 30 Alexis Kwasinski, 011

31 Stabilization Linear Controllers - passivity-based analysis All converters with CPLs can be stabilized with PD controllers (adds virtual damping resitances). An integral term can be added for line and load regulation Issues: Noise sensitivity and slow 31 Alexis Kwasinski, 011

32 Boundary controllers Stabilization Boundary control: state-dependent switching (q = q(x)). Stable reflective behavior is desired. At the boundaries between different behavior regions trajectories are tangential to the boundary An hysteresis band is added to avoid chattering. This band contains the boundary. 3 Alexis Kwasinski, 011

33 Stabilization Geometric controllers 1 st order boundary Linear switching surface with a negative slope: x kx ( x ) x 1 OP 1OP Switch is on below the boundary and off above the boundary 33 Alexis Kwasinski, 011

34 1 st order boundary controller (buck converter) Switching behavior regions are found considering that trajectories are tangential at the regions boundaries. fx dx 1 1 x1 x1 OP k f dx x x For ON trajectories: For OFF trajectories: x Stabilization OP CE ( x ) x x L( x P / x ) x x 1 1OP 1 L OP ON x L x1x x1 OP x1x PL x1 PL x1 OP 3 CExOP x E xop x x ( ): [ ] [ ( ) ] 0 C( x ) x x Lx ( P/ x) x x 1 1OP 1 L OP ( x): L[ x x x x x P x P x ] C[ x x x ] 0 3 OFF 1 1OP 1 L 1 L 1OP OP 34 Alexis Kwasinski, 011

35 Stabilization 1 st order boundary controller (buck converter) Lyapunov is used to determine stable and unstable reflective regions. This analysis identifies the need for k < 0 C V( x) xxop 0 V ( x) (1 k )( x x ) x OP 1 P L x 35 Alexis Kwasinski, 011

36 Stabilization 1 st order boundary controller (buck converter) Simulated and experimental verification L = 480 µh, C = 480 µf, E = 17.5 V, P L = 60 W, x OP = [ ] T 36 Alexis Kwasinski, 011

37 Stabilization 1 st order boundary controller (buck converter) Simulated and experimental verification Buck converter with L = 500 μh, C =1mF, E =. V, P L = 108 W, k = 1, x OP = [6 18] T 37 Alexis Kwasinski, 011

38 Stabilization 1 st order boundary controller (buck converter) Line regulation is unnecessary. Load regulation based on moving boundary Line regulation: E = +10V (57%) Line regulation: E = +10V (57%) Load regulation: PL = +0W (+9.3%) No regulation: PL = +45W (+75%) Load regulation: PL = +45W (+75%) 38 Alexis Kwasinski, 011

39 Stabilization 1 st order boundary controller (boost and buck-boost) Same analysis steps and results than for the buck converter. Boost (k<0) Buck-Boost (k<0) Boost (k>0) Buck-Boost (k>0) 39 Alexis Kwasinski, 011

40 Stabilization 1 st order boundary controller (boost and buck-boost) Experimental results. Boost (k<0) Buck-Boost (k<0) Boost (k>0) Buck-Boost (k>0) 40 Alexis Kwasinski, 011

41 Stabilization 1 st order boundary controller (boost and buck-boost) Experimental results for line and load regulation Boost: Line regulation Buck-Boost: Line regulation Boost: Load regulation Buck-Boost: Load regulation 41 Alexis Kwasinski, 011

42 Geometric controllers Stabilization First order boundary with a negative slope is valid for all types of basic converter topologies. Advantages: Robust, fast dynamic response, easy to implement. 4 Alexis Kwasinski, 011

43 Conclusions Most renewable and alternative sources, energy storage, and modern loads are dc. Integration can be achieved through power electronics, but other stability issues are introduced due to CPLs. Control-related methods appear to be a more practical solution for CPL stabilization without reducing system efficiency. Nonlinear analysis is essential due to nonlinear CPL behavior. Boundary control offers more advantages than linear controllers and are equally simple to implement. Extended work focusing on rectifiers and multiple-input converters. 43 Alexis Kwasinski, 011

44 References for Additional Details A. Kwasinski and C. N. Onwuchekwa, Dynamic Behavior and Stabilization of dc Micro-grids with Instantaneous Constant-Power Loads. IEEE Transactions on Power Electronics, in print, appearing in the March 011 issue. C. N. Onwuchekwa and A. Kwasinski, Analysis of Boundary Control for Buck Converters with Instantaneous Constant-Power Loads. IEEE Transactions on Power Electronics, vol. 5, no. 8, pp , August 010. C. Onwchekwa and A. Kwasinski, Analysis of Boundary Control for Boost and Buck-Boost Converters in Distributed Power Architectures with Constant-Power Loads, in Proc. IEEE Applied Power Electronics Conference (APEC) 011, Fort Worth, Texas, March, 011. A. Kwasinski and P. T. Krein, Stabilization of Constant Power Loads in Dc-Dc Converters Using Passivity-Based Control, in Rec 007 International Telecommunications Energy Conference (INTELEC), pp A. Kwasinski and P. Krein, Passivity-based control of buck converters with constant-power loads, in Rec. PESC, 007, pp Alexis Kwasinski, 011