Stability and Control of dc
Micro-grids
Alexis Kwasinski
Thank you to Mr. Chimaobi N. Onwuchekwa
(who has been working on boundary controllers)
May, 2011
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© Alexis Kwasinski, 2011
Overview
•Introduction
•Constant-power-load (CPL) characteristics and
effects
•Dynamical characteristics of DC micro-grids
•Conclusions/future work
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© Alexis Kwasinski, 2011
Micro-grids
• What is a micro (or nano) grid?
• Micro-grids are independently controlled (small) electric
networks, powered by local units (distributed generation).
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© Alexis Kwasinski, 2011
Introduction
• ac vs. dc micro-grids
• 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?
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© Alexis Kwasinski, 2011
Introduction
• ac vs. dc micro-grids
•DC is better suited for energy storage, renewable and
alternative power sources
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© Alexis Kwasinski, 2011
Constant-power loads
• Characteristics
•DC micro-grids comprise cascade distributed power architectures
– converters act as interfaces
•Point-of-load converters present constant-power-load (CPL)
characteristics
0

i (t )   PL
 v(t )
if v(t )  Vlim
if v(t )  Vlim
dvB
PL
z 
 2
diB
iB
•CPLs introduce a destabilizing effect
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© Alexis Kwasinski, 2011
Constant-power loads
• Characteristics
Simplified cascade distributed power architecture with a buck LRC.
 diL
 L dt  q (t )( E  RS iL )  (1  q (t ))(VD  iL RD )  iL RL  vC
 dv
P v
C C  iL  L  C
vC Ro
 dt
with iL  0, vC  
• Constraints on state variables makes it extremely difficult to find a closed form
solution, but they are essential to yield the limit cycle behavior.
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© Alexis Kwasinski, 2011
Constant-power loads
• Characteristics
• The steady state fast average model
yields some insights:
d 2 x2 LPL dx2
LC 2  2
 x2  DE
dt
x2 dt
dx
P
with x2   and x1  C 2  L  0
dt x2
• 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: d (t ) E  vC  1  PL  iL  PL
2
L
• Note: x1 = iL and x2 = vC
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© Alexis Kwasinski, 2011
C  vC
 vC
Constant-power loads
• Characteristics
• 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.
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© Alexis Kwasinski, 2011
Controls
• Characteristics
E1  400V , E 2  450V , PL1  5 kW , PL 2  10 kW ,
LLINE  5 H , RLINE  10m, CDCPL  1mF
LRC parameters:
L  0.5mH , C  1mF , D1  0.5, D2  0.54, RL  0.8
• Large oscillations and/or voltage collapse are observed due to
constant-power loads in micro-grids without proper controls
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© Alexis Kwasinski, 2011
Stabilization
• Passive methods – added resistive loads
• Linearized equation:
 Ri
P
1  Ri
 L2 

L
R
C
CV
o
o

 L
2   
 1
PL  1


0

2 
R
C
LC
CV
o 
 o
• Conditions:
C
 1
1 
1 
PL  Vo2  Ri   , PL  Vo2   
Ro 
L
 Ri Ro 
where Ri  RS D  RD (1  D )  RL
• Issue: Inefficient solution
1Ω resistor Ro in
parallel to CDCPL
E  12.5V , L  480  H , C  480  F , Ro  2, D  0.9, PL  43.8W
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© Alexis Kwasinski, 2011
Stabilization
• Passive methods – added capacitance
• Condition: C  PL2 L
Vo Ri
• Issues: Bulky, expensive and may reduce reliability. But may
improve fault detection and clearance
E  12.5V , L  480  H , C  480  F  200mF ,
D  0.9, PL  35W
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60 mF added in parallel to CDCPL
© Alexis Kwasinski, 2011
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.
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© Alexis Kwasinski, 2011
Stabilization
• Passive methods – load shedding
• It is also based on the condition that: C  PL2 L
Vo Ri
• Issues: Not practical for critical loads
Load reduced from 49 W to 35 W
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Load dropped from 10 to 2.5 kW at
0.25 seconds
© Alexis Kwasinski, 2011
Stabilization
• Linear controllers – Passivity based analysis
• Initial notions
A system
 x  f ( x, u ),
:
 y  h( x, u ),
x(0)  x0  R n , f : R n  R m  R n
h : Rn  Rm  R p
with
f is locally Lipschitz
and
f(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
dH
H ( x) 
f ( x, u )  u T y
dx
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( x, u )  R n  R m
© Alexis Kwasinski, 2011
Stabilization
• Linear controllers – Passivity based analysis
• Initial notions
• Σ is output strictly passive if:
dH
2
H ( x ) 
f ( x, u )  u T y   o y
( x, u )  R n  R m , 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 y (t )  0  x (t )  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.
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© Alexis Kwasinski, 2011
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
Mx  [ J  R ( x)] x  dE
where
L 0
M

0
C


 0 1
J 


1
0


0
R ( x)  
0

0
PL 

x22 
E
E 
0
 x1 
x 
 x2 
PL

 I L   DE
Equilibrium point: xe     

V
DE
 o 

Mx  [ J  R ( x)] x  dE
x  x  xe (Coordinate change)
Mx  [ J  R ( x )] x  dE  [ J  R ( x )] xe
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© Alexis Kwasinski, 2011
(1)
Stabilization
• Linear controllers – Passivity based analysis
•Define the positive definite damping injection matrix Ri as
 Ri1
Ri  
 0



1 PL 
 2
Ri 2 x2 
0
 Ri1
Then, R t  R  R i  
 0


From (1), add R i ( x ) x on both sides:
Mx  [ J  R ] x  dE  [J  R ( x )] x  R ( x ) x
PL
1

Ri is positive definite if
.
Ri 2 x22
t
e
:

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0
1
Ri 2
i
  0 (Equivalent free
E
evolving system)
Mx  [J  R t ] x  0
© Alexis Kwasinski, 2011





Stabilization
• Linear controllers – Passivity based analysis
• Consider the storage function
 ( x)  1 x T Mx
H
2
Its time derivative is:

H
( x )  x T Mx   x T [ J  R t ] x   x T R t x  0
if y  x
 ( x ) =  y T R y   y 2 ,
H
t
o

1 
where  o  max  Ri1 ,

R
i2 

 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.
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© Alexis Kwasinski, 2011
Stabilization
• Linear controllers – Passivity based analysis
• Since E  dE  [J  R ( x )]xe  R i ( x ) x  0
then,  dE  Vo  Ri1 x1  0

PL x2 Vo

I

 L x  R  R 0
2
i2
i2

Hence,
1
Vo  Ri1 ( x1  I L ) 
E
PL
Vo PL x2

x

Cx

I

 
and since 1
and
2
L
x2
Ri 2 x2 Ri 2
d
Thus,

R
R
1
d    Ri1Cx2  i1 x2  Vo  i1 Vo 
E
Ri 2
Ri 2 
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© Alexis Kwasinski, 2011
This is a PD controller
(e  VO  x2 )
Stabilization
• Linear controllers – Passivity based analysis
• 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 not
sufficient condition for stability is
P
dE  x2 1  PL
   x1  L2
L
C  x2
 x2
• Line and load regulation can be
achieved by adding an integral
term but stability is not ensured
d  D  k p e  kd e  ki  edt
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© Alexis Kwasinski, 2011
Stabilization
• Linear controllers
x1
• Experimental results (buck converter)
x2
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© Alexis Kwasinski, 2011
Stabilization
• Linear controllers
• Experimental results (buck converter)
Line
regulation
Load
regulation
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© Alexis Kwasinski, 2011
Stabilization
• Linear controllers – Passivity based analysis
• The same analysis can be performed for boost and buck-boost
converters yielding, respectively
D ' D '2  4
d '  1 d 
Ri1 
1 

Ce
e


V0 
Ri 2 
,
2
2
 E 
Ri1 
x2  V0 
E

4
 

Cx


 2

(V0  E )
(V0  E ) 
Ri 2 
 V0  E 
d'
2
• Engineering criteria dictate that the non-linear PD controller can
be translated into an equivalent linear PD controller of the form:
d '  kd e  k p e  D '
• Formal analytical solution:
min
e emax , e emax
 d ' d '(k p , kd ) 
subject to
0  1  d  d max
0  1  d  d max
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© Alexis Kwasinski, 2011
Stabilization
• Linear controllers – Passivity based analysis
• Perturbation theory can formalize the analysis (e.g. boost conv.)
• Consider
:

x  f (t , x )  M 1[J  R t ]x
Unperturbed system with nonlinear PD controller, with
• And
 0 d '
J 


d
'
0


 :

p
x  f (t , x )  g (t , x )  M 1[ J  R t ]x
Perturbed system with linear PD controller, with
 0 d '
J 

d
'
0



•The perturbation is
g (t , x )  M 1[J  J ]x
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© Alexis Kwasinski, 2011
Stabilization
• Linear controllers – Passivity based analysis
• Lemma 9.1 in Khalil’s: Let x  0 be an exponentially stable equilibrium
 . Let V (t , x ) be a Lyapunov function of the
point of the nominal system 
nominal system which satisfies
2
c1 x  V (t , x )  c2 x
2
V (t , x ) V (t , x )

f (t , x )  c3 x
t
x
2
V (t , x )
 c4 x
x
in [0,∞) X D with c1 to c4 being some positive constants. Suppose the
perturbation term g (t , x ) satisfies
g (t , x )   x
t  0, x   where  :
x   and  
Then, the origin is an exponentially stable equilibrium point of the
 .
perturbed system 
p
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© Alexis Kwasinski, 2011
c3
c4
Stabilization
• Linear controllers – Passivity based analysis
•Taking V (t , x )  H ( x )
•It can be shown that
c1 = λmin(M)
c2 = λmax(M)
c3  min (2J  3R t )
c4  M  max (MT M)  max(L2 , C 2 )  max(L, C )
•Also, x  0 is an exponentially stable equilibrium point of  ,
g (t ,0)  0
and
g (t , x )   x
t  0, x   where  :
with
x  
 ( d ' d ') (d ' d ') 
  G  max 
,

L
C


min (2J  3R t )
'
'
d

d

•Thus, stability is ensured if
C L 
max  , 
L C
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© Alexis Kwasinski, 2011
Stabilization
• Linear controllers
• Experimental results boost converter
Load Regulation
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Line Regulation
© Alexis Kwasinski, 2011
Stabilization
• Linear controllers
• Experimental results voltage step-down buck-boost converter
Load Regulation
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Line Regulation
© Alexis Kwasinski, 2011
Stabilization
• Linear controllers
• Experimental results voltage step-up buck-boost converter
Load Regulation
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Line Regulation
© Alexis Kwasinski, 2011
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
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© Alexis Kwasinski, 2011
Stabilization
• Boundary controllers
• 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.
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© Alexis Kwasinski, 2011
Stabilization
• Geometric controllers – 1st order boundary
• Linear switching surface with a negative slope:
x1  k ( x2  x2OP )  x1OP
Switch is on below the boundary and off above the boundary
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© Alexis Kwasinski, 2011
Stabilization
• 1st order boundary controller (buck converter)
• Switching behavior regions are found considering that trajectories
are tangential at the regions boundaries.
fx
x x
dx
1
 1  k  1 1OP
f x2 dx2
x2  x2OP
• For ON trajectories:
x x
C ( E  x2 )
 1 1OP
L( x1  PL / x2 ) x2  x2OP
ON ( x ) : L[ x12 x2  x1OP x1 x2  PL x1  PL x1OP ]
 C[ Ex2OP x2  ( E  x2OP ) x22  x23 ]  0
• For OFF trajectories:
x1  x1OP
C ( x2 )

L( x1  PL / x2 ) x2  x2OP
OFF ( x ) : L[ x12 x2  x1OP x1 x2  PL x1  PL x1OP ]  C[ x2OP x22  x23 ]  0
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© Alexis Kwasinski, 2011
Stabilization
• 1st 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) 
x  xOP
2
35
2
0

P 
V ( x)  (1  k 2 )( x2  x2OP )  x1  L 
x2 

© Alexis Kwasinski, 2011
Stabilization
• 1st order boundary controller (buck converter)
• Simulated and experimental verification
L = 480 µH, C = 480 µF, E = 17.5 V, PL = 60 W, xOP = [4.8
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© Alexis Kwasinski, 2011
12.5] T
Stabilization
• 1st order boundary controller (buck converter)
• 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
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© Alexis Kwasinski, 2011
Stabilization
• 1st order boundary controller (buck converter)
• Line regulation is unnecessary. Load regulation based on moving boundary
Line regulation: ∆E = +10V (57%)
Load regulation: ∆PL = +20W (+29.3%)
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Line regulation: ∆E = +10V (57%)
No regulation: ∆PL = +45W (+75%)
© Alexis Kwasinski, 2011
Load regulation: ∆PL = +45W (+75%)
Stabilization
• 1st order boundary controller (boost and buck-boost)
• Same analysis steps and results than for the buck converter.
Buck-Boost
(k<0)
Boost (k<0)
Buck-Boost
(k>0)
Boost (k>0)
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© Alexis Kwasinski, 2011
Stabilization
• 1st order boundary controller (boost and buck-boost)
• Experimental results.
Buck-Boost
(k<0)
Boost (k<0)
Buck-Boost
(k>0)
Boost (k>0)
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© Alexis Kwasinski, 2011
Stabilization
• 1st order boundary controller (boost and buck-boost)
• Experimental results for line and load regulation
Buck-Boost:
Line regulation
Boost: Line
regulation
Buck-Boost:
Load regulation
Boost: Load
regulation
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© Alexis Kwasinski, 2011
Stabilization
• Geometric controllers
• First order boundary with a negative slope is valid for all types of
basic converter topologies.
• Advantages: Robust, fast dynamic response, easy to implement .
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© Alexis Kwasinski, 2011
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.
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© Alexis Kwasinski, 2011
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 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.
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© Alexis Kwasinski, 2011