Stability and Control of dc Micro-gridsusers.ece.utexas.edu/~kwasinski/dc microgrid control v3 96 dpi.pdf · • Incompatibility with induction (AC) motors • Power electronics help

1 © Alexis Kwasinski, 2011

Stability and Control of dcMicro-grids

Alexis Kwasinski

Thank you to Mr. Chimaobi N. Onwuchekwa(who has been working on boundary controllers)

May, 2011


•Introduction

•Constant-power-load (CPL) characteristics and effects

•Dynamical characteristics of DC micro-grids

•Conclusions/future work

OverviewOverview


• Micro-grids are independently controlled (small) electric networks, powered by local units (distributed generation).

MicroMicro--gridsgrids• What is a micro (or nano) grid?


• 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?

IntroductionIntroduction• ac vs. dc micro-grids


•DC is better suited for energy storage, renewable and alternative power sources

• ac vs. dc micro-gridsIntroductionIntroduction


•DC micro-grids comprise cascade distributed power architectures – converters act as interfaces

•Point-of-load converters present constant-power-load (CPL) characteristics

•CPLs introduce a destabilizing effect

lim

lim

0 if ( )( ) if ( )

( )L

v t VPi t v t V

v t

2B L

B B

dv Pz

di i

ConstantConstant--power loadspower loads

• Characteristics


Simplified cascade distributed power architecture with a buck LRC.

• Characteristics


( )( ) (1 ( ))( )

with 0,

LS L D L D L L C

C CLL

C o

L C

diL q t E R i q t V i R i R v

dtdv vP

C idt v R

i v

• Constraints on state variables makes it extremely difficult to find a closed form solution, but they are essential to yield the limit cycle behavior.


• The steady state fast average modelyields some insights:

• 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: x1 = iL and x2 = vC

22 2

22 22

22 1

2

with and 0

L

L

d x LP dxLC x DE

dtdt xdx P

x x Cdt x

• Characteristics


2

( ) 1C L LL

C C

d t E v P Pi

L C v v



• 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 2.25 Ω resistor (dashed trace); E = 24 V, L = 0.2 mH,

PL = 100 W, C = 470 μF.

• Characteristics


1 2 1 2400 , 450 , 5 , 10 ,L LE V E V P kW P kW

LRC parameters: 5 , 10 , 1LINE LINE DCPLL H R m C mF

1 20.5 , 1 , 0.5, 0.54, 0.8LL mH C mF D D R

• Large oscillations and/or voltage collapse are observed due to constant-power loads in micro-grids without proper controls

ControlsControls• Characteristics


12.5 , 480 , 480 , 2 , 0.9, 43.8o LE V L H C F R D P W

StabilizationStabilization• Passive methods – added resistive loads

• Linearized equation:

• Conditions:

• Issue: Inefficient solution

22 2

1 1 1 0i iL L

o oo o

R RP PL R C L R C LCCV CV

2 21 1 1,

where (1 )

L o i L oo i o

i S D L

CP V R P VL R R R

R R D R D R

1Ω resistor Ro inparallel to CDCPL


12.5 , 480 , 480 200 , 0.9, 35L

E V L H C F mFD P W

StabilizationStabilization• Passive methods – added capacitance

• Condition:

• Issues: Bulky, expensive and may reduce reliability. But may improve fault detection and clearance

60 mF added in parallel to CDCPL

2L

o i

P LCV R


StabilizationStabilization• Passive methods – added bulk energy storage

• It can be considered an extension of the previous approach.

• Energy storage needs to be directlyconnected to the main bus withoutintermediate power conversioninterfaces.

• Issues: Expensive, it usually requiresa power electronic interface, batteriesand ultracapacitors may have cellvoltage equalization problems, andreliability, operation and safety maybe compromised.


StabilizationStabilization

• Passive methods – load shedding

• It is also based on the condition that:

• Issues: Not practical for critical loads

2L

o i

P LCV R

Load dropped from 10 to 2.5 kW at 0.25 seconds

Load reduced from 49 W to 35 W


A system

withf is locally Lipschitz

andf(0,0) = h(0,0) = 0

is passive if there exists a continuously differentiable positive definite function H(x) (called the storage function) such that

0( , ), (0) , : R R R:

( , ), : R R R

n n m n

n m p

x f x u x x R f

y h x u h

( ) ( , ) ( , ) R RT n mdHH x f x u u y x udx

• Linear controllers – Passivity based analysis


• Initial notions


• Σ is output strictly passive if:

• A state-space system is zero-state observable from the output y=h(x), if for all initial conditions we have

• 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.

2( ) ( , ) y ( , ) R R , and 0T n mo o

dHH x f x u u y x udx

( ), R nx f x x (0) R nx ( ) 0 ( ) 0y t x t



• Initial notions


[ ( )]x x x d M J R E

22

0 0( ) 0 LP

x

x

R00L

C

M0 11 0

J0E

E 1

2

xx

x



• Consider a buck converter with ideal components and in continuous conduction mode. In an average sense and steady stateit can be represented by

where

[ ( )] [ ( )] ex x x d x x M J R E J R

ex x x

LPL DE

eo

Ix

V DE

Equilibrium point:

(Coordinate change)

(1)

[ ( )]x x x d M J R E


22 2

1 L

i

PR x

•Define the positive definite damping injection matrix Ri as1

22 2

010

i

i L

i

RP

R x

R

Ri is positive definite if . Then,



: [ ] 0tx x M J R

[ ] [ ( )] ( )t e ix x d x x x x M J R E J R R From (1), add on both sides:

0E

1

2

010

i

t i

i

R

R

R R R

(Equivalent free evolving system)

( )i x xR


1( ) 2

Tx x xH M

• Consider the storage function



• is a free-evolving output strictly passive and zero-state observablesystem. Therefore, is an asymptotically stable equilibrium point ofthe closed-loop system.

( ) [ ] 0T T Tt tx x x x x x x H M J R R

Its time derivative is:

y x

2 1

2

1( ) = , where max ,Tt o o i

i

x y y y RR

H R

0x

if


[ ( )] ( ) 0e id x x x x E E J R R

1 1

2

2 2 2

0

0

o i

oLL

i i

dE V R xVP xI

x R R

• Since

then,



Hence,

1 22

LPx Cxx

1 11 ( )o i Ld V R x IE

2( )Oe V x

and since and

1 11 2 2

2 2

1 i ii o o

i i

R Rd R Cx x V VE R R

Thus,This is a PD controller

2

2 2 2

o LL

i i

V P xIR x R


• Remarks for the buck converter:• xe 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 notsufficient condition for stability is

• Line and load regulation can beachieved by adding an integralterm but stability is not ensured



21 2

2 2

1 L LdE x P Px

L C x x

p d id D k e k e k edt


x2

x1


• Experimental results (buck converter)

• Linear controllers



• Experimental results (buck converter)


Line regulation

Load regulation


2 1

0 2

1' ' 4' 1 ,

2

i

i

RD D Ce eV R

d d



• The same analysis can be performed for boost and buck-boost converters yielding, respectively

2

1 2 02

0 0 0 2

4( ) ( )

'2

i

i

R x VE E CxV E V E V E R

d

• Engineering criteria dictate that the non-linear PD controller can be translated into an equivalent linear PD controller of the form:

' 'd pd k e k e D

max max,

max

max

min ' '( , )

subject to 0 1 0 1

p de e e e

d d k k

d dd d

• Formal analytical solution:


• Consider

• And

•The perturbation is

1: ( , ) [ ]tx f t x x M J R

1: ( , ) ( , ) [ ]p tx f t x g t x x M J R

1( , ) [ ]g t x x M J J

0 '' 0

dd

J

0 '' 0

dd

J

Unperturbed system with nonlinear PD controller, with

Perturbed system with linear PD controller, with



• Perturbation theory can formalize the analysis (e.g. boost conv.)


• Lemma 9.1 in Khalil’s: Let be an exponentially stable equilibrium point of the nominal system . Let be a Lyapunov function of the nominal system which satisfies

in [0,∞) X D with c1 to c4 being some positive constants. Suppose the perturbation term satisfies

Then, the origin is an exponentially stable equilibrium point of the perturbed system .

0x ( , )V t x

2 21 2( , )c x V t x c x

23

( , ) ( , ) ( , )V t x V t x f t x c xt x

4( , )V t x c xx

( , )g t x

3

4

( , ) 0, where : and cg t x x t x xc

p




•Taking•It can be shown that

•Also, is an exponentially stable equilibrium point of ,

and

with

•Thus, stability is ensured if

( , ) ( )V t x H x

c1 = λmin(M)c2 = λmax(M)

3 min (2 3 )tc J R

( ' ') ( ' '),d d d dmaxL C

G

(2 3 )' ',

min td dC LmaxL C

J R

24 ( ) , ) , )T 2

maxc max(L C max(L C M M M

( , ) 0, where : g t x x t x x

0x ( ,0) 0g t




Line RegulationLoad Regulation


• Experimental results boost converter





• Experimental results voltage step-down buck-boost converter





• Experimental results voltage step-up buck-boost converter• Linear controllers


• All converters with CPLs can be stabilized with PD controllers (adds virtual damping resitances).

• An integral term can be added for lineand load regulation

• Issues: Noise sensitivity and slow

StabilizationStabilization• Linear Controllers - passivity-based analysis


• Boundary control: state-dependent switching (q = q(x)).

• Stable reflective behavior is desired.

• At the boundaries between differentbehavior regions trajectories are tangential to the boundary

• An hysteresis band is added toavoid chattering. This band contains the boundary.

StabilizationStabilization• Boundary controllers


• Linear switching surface with a negative slope:

Switch is on below the boundary and off above the boundary1 2 2 1( )OP OPx k x x x

StabilizationStabilization• Geometric controllers – 1st order boundary


• Switching behavior regions are found considering that trajectories are tangential at the regions boundaries.

• For ON trajectories:

• For OFF trajectories:

• 1st order boundary controller (buck converter)

1

2

1 11

2 2 2

x OP

x OP

f x xdxk

f dx x x

1 12

1 2 2 2

( )( / )

OP

L OP

x xC E xL x P x x x

21 2 1 1 2 1 1

2 32 2 2 2 2

( ) : [ ] [ ( ) ] 0

ON OP L L OP

OP OP

x L x x x x x P x P xC Ex x E x x x

1 12

1 2 2 2

( )( / )

OP

L OP

x xC xL x P x x x

2 2 31 2 1 1 2 1 1 2 2 2( ) : [ ] [ ] 0OFF OP L L OP OPx L x x x x x P x P x C x x x



2( ) 02 OPCV x x x 2

2 2 12

( ) (1 )( )OP

LPV x k x x xx

• Lyapunov is used to determine stable and unstable reflective regions. This analysis identifies the need for k < 0





• Simulated and experimental verification


L = 480 µH, C = 480 µF, E = 17.5 V, PL = 60 W, xOP = [4.8 12.5] T



• Simulated and experimental verification


Buck converter with L = 500 μH, C = 1 mF, E = 22.2 V, PL = 108 W, k = –1, xOP = [6

18] T


Line regulation: ∆E = +10V (57%)

Load regulation: ∆PL = +20W (+29.3%) No regulation: ∆PL = +45W (+75%) Load regulation: ∆PL = +45W (+75%)

Line regulation: ∆E = +10V (57%)


• Line regulation is unnecessary. Load regulation based on moving boundary• 1st order boundary controller (buck converter)



• Same analysis steps and results than for the buck converter.

• 1st order boundary controller (boost and buck-boost)

Boost (k<0)

Boost (k>0)

Buck-Boost(k<0)

Buck-Boost(k>0)



• Experimental results.


Boost (k<0)

Boost (k>0)

Buck-Boost(k<0)

Buck-Boost(k>0)



• Experimental results for line and load regulation


Boost: Load regulation

Boost: Line regulation

Buck-Boost: Load regulation

Buck-Boost: Line regulation


• First order boundary with a negative slope is valid for all types of basic converter topologies.

• Advantages: Robust, fast dynamic response, easy to implement .

• Geometric controllers



• 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.

ConclusionsConclusions


• 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 2011 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. 25, no. 8, pp. 2018-2032, August 2010.• 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) 2011, Fort Worth, Texas, March, 2011.• A. Kwasinski and P. T. Krein, “Stabilization of Constant Power Loads in Dc-Dc Converters Using Passivity-Based Control,” in Rec 2007 International Telecommunications Energy Conference (INTELEC), pp. 867-874.• A. Kwasinski and P. Krein, “Passivity-based control of buck converters with constant-power loads,” in Rec. PESC, 2007, pp. 259-265.

References for Additional DetailsReferences for Additional Details

Documents

Stability and Control of dc Micro-gridsusers.ece.utexas.edu/~kwasinski/dc microgrid control v3 96 dpi.pdf · • Incompatibility with induction (AC) motors • Power electronics help