Download Stability Analysis of a Microgrid System with a Constant Power Load

ANNUAL JOURNAL OF ELECTRONICS, 2011, ISSN 1313-1842
Stability Analysis of a Microgrid System with a
Constant Power Load
Joan Peuteman, Jean-Jacques Vandenbussche
Abstract – Microgrids not only incorporate renewable and
non-renewable distributed generation units, they also feed
linear and non linear loads. The present papers models a
microgrid and investigates stability properties of such a
microgrid in case the microgrid feeds a Constant Power Load
CPL. The impact of several parameters on the stability
properties is studied by linearizing the nonlinear differential
equations describing the microgrid system. Local stability
properties and the large scale behavior of the microgrid
system is studied.
Keywords – Microgrid, Constant Power Load, asymptotic
stability, differential and difference equations.
I. INTRODUCTION
In order to incorporate larger amounts of small scale
renewable and non-renewable distributed generation units,
the use of microgrids is an interesting challenge. These
microgrids – containing generation units, energy storage
devices, linear and nonlinear loads – are able to operate
connected with the public grid or in island mode. The
present paper studies stability properties in case such a
microgrid operates in island mode.
First, a simple model describing the behavior of the
microgrid is studied. When considering a linear load or a
Constant Power Load CPL, the static and the dynamic
resistances are determined. Such a CPL consumes a
constant power independent of the supply voltage. For
instance, some induction cookers, DC-DC converters with
a constant output voltage and a fixed resistive load, DC-AC
converters feeding a motor with a fixed mechanical load,
can behave as a CPL [3].
Using the static resistance, the steady state equilibrium
points of a DC microgrid are calculated allowing to study
the local asymptotic stability properties of these
equilibrium points. Due to the negative dynamic resistance
of the CPL, stability problems can occur.
The impact of the parameters in the model on the
stability properties are studied in case the microgrid feeds a
CPL. The stablizing effect of a linear load in parallel with
the CPL is also investigated. Using difference equations,
the large scale behavior of the microgrid system is
visualized by calculating the trajectories.
II. MODELING THE MICROGRID AND THE CPL
Joan Peuteman is with the Katholieke Hogeschool BruggeOostende, Departement IW&T, Zeedijk 101, B-8400
Oostende - Belgium (e-mail: [email protected]) and
with the Katholieke Universiteit Leuven, Department
Electrical Engineering (ESAT), Kasteelpark Arenberg 10,
B-3000 Leuven – Belgium.
Jean-Jacques Vandenbussche is with the Katholieke
Hogeschool Brugge-Oostende, Departement IW&T, Zeedijk
101, B-8400 Oostende – Belgium.
Figure 1 models the microgrid [2] feeding a CPL having
a positive static resistance RNL since it consumes a power
𝑃 > 0. Notice the voltage source E(t) in combination with
a resistance R and an inductor L giving a Thevenin
equivalent circuit of the microgrid. The capacitor C models
the capacitance of the feeding cable or models a discrete
capacitor in parallel with the CPL in order to improve the
stability properties.
Alternatively, the Thevenin equivalent circuit of the
microgrid can contain E(t) and the resistance R. The L and
the C can model a low pass filter filtering out kHz
components in case the CPL is a SMPS.
In case a CPL consumes a constant power P,
independent of the applied voltage U, the consumed current
𝐼 = 𝑃 ⁄𝑈 is inversely proportional with U. The nonlinear
static resistance RNL of the CPL increases as the applied
voltage increases. Indeed,
𝑅𝑁𝐿 =
𝑈2
𝑃
=
𝑃
𝐼2
.
(1)
FIGURE 1. MODEL OF THE MICROGRID FEEDING A CPL
In case the applied voltage U changes to 𝑈 + 𝑑𝑈, the
current changes to 𝐼 + 𝑑𝐼 with (𝑈 + 𝑑𝑈)(𝐼 + 𝑑𝐼) = 𝑈𝐼 =
𝑃. The dynamic resistance
𝑟𝑁𝐿 =
𝑑𝑈(𝐼)
𝑑𝐼
=
𝑑
𝑃
( )= −
𝑑𝐼 𝐼
𝑃
𝐼2
= −𝑅𝑁𝐿 < 0.
(2)
When considering a linear resistance RL, the static
resistance equals RL and the dynamic resistance 𝑟𝐿 = +𝑅𝐿 .
III. THE STEADY STATE BEHAVIOR
Considering Figure 1 with 𝐸(𝑡) = 𝐸 and (1), one obtains
𝐼=
𝐸
𝑅+𝑅𝑁𝐿
,
𝑈=
𝑅𝑁𝐿 𝐸
𝑅+𝑅𝑁𝐿
,
(3)
giving two equilibrium points with voltages 𝑈𝑒𝑞 and
currents 𝐼𝑒𝑞 which equal
TABLE 1. EQUILIBRIUM POINTS
ANNUAL JOURNAL OF ELECTRONICS, 2011
−
𝑅
1
𝑅
1 2
4
𝑅
+
± √( −
) −
(1 −
)
𝐿 𝐶 𝑅𝑁𝐿
𝐿 𝐶 𝑅𝑁𝐿
𝐿𝐶
𝑅𝑁𝐿
𝜆1,2 =
In case 𝐸 2 < 4𝑅𝑃 there exists no equilibrium point. In
case 𝐸 2 = 4𝑅𝑃 there exists one single equilibrium point
and in case 𝐸 2 > 4𝑅𝑃 there are two distinct equilibrium
points. Notice that a large 𝑈𝑒𝑞1 corresponds with a small
𝐼𝑒𝑞1 and that a small 𝑈𝑒𝑞2 corresponds with a large 𝐼𝑒𝑞2 as
defined in Table 1 implying 𝑈𝑒𝑞1 𝐼𝑒𝑞1 = 𝑈𝑒𝑞2 𝐼𝑒𝑞2 = 𝑃.
In the equilibrium point P1, the large static resistance
equals
𝑅𝑁𝐿1 =
𝑈𝑒𝑞1
𝐼𝑒𝑞1
=𝑅
𝐸+√𝐸 2 −4𝑅𝑃
(4)
𝐸−√𝐸 2 −4𝑅𝑃
and in the equilibrium point P2, the small static resistance
equals
𝑅𝑁𝐿2 =
𝑈𝑒𝑞2
𝐼𝑒𝑞2
=𝑅
𝐸−√𝐸 2 −4𝑅𝑃
𝐸+√𝐸 2 −4𝑅𝑃
.
(5)
Since the efficiency of the power transfer to the CPL
equals
𝑅
𝜂 = 𝑅+𝑅𝑁𝐿 ,
(6)
𝑁𝐿
the use of equilibrium point P1 is preferred. The next
sections will study the local asymptotic stability properties
of P1 and P2.
IV. ASYMPTOTIC STABILITY
The behavior of the microgrid loaded with a CPL is
determined by the set of nonlinear differential equations
𝑑 𝐼(𝑡)
[
𝑑𝑡
𝑅
1
1
= − 𝐿 𝐼(𝑡) − 𝐿 𝑈(𝑡) + 𝐿 𝐸(𝑡)
𝑑 𝑈(𝑡)
𝑑𝑡
1
𝑃
= 𝐶 𝐼(𝑡) − 𝐶 𝑈(𝑡)
(7)
having equilibrium points (𝐼𝑒𝑞1 , 𝑈𝑒𝑞1 ) and (𝐼𝑒𝑞2 , 𝑈𝑒𝑞2 ).
Linearizing (7) gives the set of linear differential equations
[5, p. 51] (𝑅𝑁𝐿 equals 𝑅𝑁𝐿1 or 𝑅𝑁𝐿2 )
𝑅
𝑑 𝑖(𝑡)
[𝑑 𝑑𝑡
]
𝑢(𝑡)
𝑑𝑡
=[
1
−𝐿
−𝐿
1
1
𝐶
𝐶 𝑅𝑁𝐿
1
𝑖(𝑡)
][
] + [ 𝐿 ] 𝑒(𝑡).
𝑢(𝑡)
0
2
.
Notice that 𝜆1,2 both depend on 𝑅𝑁𝐿 which depends on
the equilibrium point taken into consideration. Since
𝑅𝑁𝐿1 > 𝑅𝑁𝐿2, it is easier to obtain the desired stability
properties for equilibrium point P 1. Since the square root in
𝜆1,2 is either positive or purely imaginary, the asymptotic
stability property can only be obtained when
𝑅
1
−𝐿 +𝐶𝑅
𝑁𝐿
< 0.
(10)
This implies an increase of 𝑅 ⁄𝐿 improves the stability
properties and also an increase of C improves the stability
properties [1]. In case the power P of the CPL increases,
𝑅𝑁𝐿 decreases making it more difficult to satisfy (10).
In order to obtain eigenvalues 𝜆1,2 which are strictly
located into the left half plane, it is necessary but not
sufficient to satisfy (10). In order to prevent that the
eigenvalue with the largest real part (expression (9) with
the +) is positive, also the condition 𝑅 < 𝑅𝑁𝐿 must be
satisfied. Combining this condition with (10), R is not
allowed to be too large nor too small.
V. THE EIGENVALUES
Using MATLAB, it is possible to calculate and visualize
the location of the eigenvalues of the system matrix in case
the parameters RNL, C, R and L are changing [1]. Figure 2
visualizes these eigenvalues 𝜆1,2 in case 𝑅 = 3𝛺, 𝐿 =
10𝑚𝐻 and 𝐶 = 2200𝜇𝐹 (the horizontal axis visualizes the
real parts and the vertical axis visualizes the imaginary
parts of the eigenvalues).
The nonlinear load changes from 𝑅𝑁𝐿 = 1𝛺 to 𝑅𝑁𝐿 =
20𝛺. For small 𝑅𝑁𝐿 -values there is a positive and a
negative eigenvalue implying no asymptotic stability
property is obtained [4]. As 𝑅𝑁𝐿 increases, both real
eigenvalues become strictly negative and the asymptotic
stability property of the equilibrium point is obtained. In
case a further increase of 𝑅𝑁𝐿 occurs, two complex
conjugate eigenvalues appear with a strictly negative real
part implying the asymptotic stability property of the
equilibrium point is obtained.
(8)
Here, 𝑖(𝑡) = 𝐼(𝑡) − 𝐼𝑒𝑞 (consider 𝐼𝑒𝑞1 or 𝐼𝑒𝑞2 ), 𝑢(𝑡) =
𝑈(𝑡) − 𝑈𝑒𝑞 (consider 𝑈𝑒𝑞1 or 𝑈𝑒𝑞2 ) and 𝑒(𝑡) = 𝐸(𝑡) − 𝐸.
In case 𝑒(𝑡) = 0, (8) has an equilibrium point (0,0) which
is asymptotically stable when the two eigenvalues of the
system matrix are strictly located into the left half plane i.e.
these eigenvalues must have a strictly negative real part.
This implies local asymptotic stability of the equilibrium
point (𝐼𝑒𝑞 , 𝑈𝑒𝑞 ) of (7) [5, p. 139]. These two eigenvalues
equal
FIGURE 2. EIGENVALUES AS A FUNCTION OF RNL
ANNUAL JOURNAL OF ELECTRONICS, 2011
Figure 3 visualizes the eigenvalues 𝜆1,2 in case 𝑅 = 3𝛺,
𝐿 = 10𝑚𝐻 and 𝑅𝑁𝐿 = 10𝛺. The capacitor changes from
𝐶 = 100𝜇𝐹 to 𝐶 = 4400𝜇𝐹. For small C-values, the
eigenvalues are a complex conjugate pair with a positive
real part implying no stability properties are obtained. As C
increases, the real part decreases and finally becomes
strictly negative implying asymptotic stability of the
equilibrium point is obtained. A further increase of C
makes the eigenvalues real but strictly negative implying
the asymptotic stability property is maintained.
VI. COMBINING A CPL AND A LINEAR LOAD
Figure 5 visualizes the situation where the microgrid is
feeding a CPL with a static resistance RNL (consuming a
power P) and a linear load RL. The current 𝐼1 is the current
in the CPL, 𝐼1′ is the current in the linear load, 𝐼2 is the
current in the capacitor and 𝐼3 is the total current fed by the
voltage source 𝐸.
FIGURE 5. MICROGRID FEEDING A CPL AND A LINEAR RESISTOR
Based on Figure 5, the set of nonlinear differential
equations can be written as
𝑑 𝐼3 (𝑡)
[
FIGURE 3. EIGENVALUES AS A FUNCTION OF C
Figure 4 visualizes the eigenvalues 𝜆1,2 in case 𝐿 =
10𝑚𝐻, 𝑅𝑁𝐿 = 10𝛺 and 𝐶 = 2200𝜇𝐹. The resistance R
changes from 𝑅 = 0.01𝛺 to 𝑅 = 20𝛺. For small R-values,
the eigenvalues are a complex conjugate pair with a
positive real part implying no asymptotic stability is
obtained. As R increases, the real part of the complex
conjugate pair becomes strictly negative implying the
desired asymptotic stability property. Due to a further
increase of R, the eigenvalues become real and strictly
negative implying the asymptotic stability property is
maintained.
In case a further increase of R occurs, the stability
property is lost again since the condition 𝑅 < 𝑅𝑁𝐿 is no
longer satisfied. Indeed, a positive real eigenvalue appears
as visualized in Figure 4.
𝑅
1
1
= − 𝐿 𝐼3 (𝑡) − 𝐿 𝑈(𝑡) + 𝐿 𝐸(𝑡)
𝑑𝑡
𝑑 𝑈(𝑡)
𝑑𝑡
1
𝑃
= 𝐶 𝐼3 (𝑡) − 𝐶 𝑈(𝑡) −
𝑈(𝑡)
(11)
𝐶 𝑅𝐿
In case E is sufficiently large, there are two equilibrium
points (𝐼1𝑒𝑞1 , 𝑈𝑒𝑞1 ) and (𝐼1𝑒𝑞2 , 𝑈𝑒𝑞2 ) with 𝑈𝑒𝑞1 𝐼1𝑒𝑞1 =
𝑈𝑒𝑞2 𝐼1𝑒𝑞2 = 𝑃 given by
2
𝑅𝐿
𝑅
4𝑅𝑅𝐿 𝑃
)±√𝐸 2 ( 𝐿 ) −
𝑅+𝑅𝐿
𝑅+𝑅𝐿
𝑅+𝑅𝐿
𝐸(
𝑈𝑒𝑞 1,2 =
(12)
2
𝑅+𝑅𝐿
)
𝑅𝐿
𝐸 ∓√𝐸 2 −4𝑅𝑃(
𝐼1𝑒𝑞 1,2 =
2𝑅
.
(13)
By linearizing (11), obtain a linear differential equation
𝑑 𝑖3 (𝑡)
[ 𝑑 𝑑𝑡
]=[
𝑢(𝑡)
𝑑𝑡
𝑅
−𝐿
1
−𝐿
1
𝑅𝐿 −𝑅𝑁𝐿 ] [
𝐶
𝐶 𝑅𝐿 𝑅𝑁𝐿
1
𝑖3 (𝑡)
] + [ 𝐿 ] 𝑒(𝑡). (14)
𝑢(𝑡)
0
Notice that 𝑖3 (𝑡) = 𝐼3 (𝑡) − 𝐼3𝑒𝑞 , 𝑖1 (𝑡) = 𝐼1 (𝑡) − 𝐼1𝑒𝑞 ,
𝑢(𝑡) = 𝑈(𝑡) − 𝑈𝑒𝑞 and 𝑒(𝑡) = 𝐸(𝑡) − 𝐸. When 𝑒(𝑡) = 0,
(14) has an equilibrium point (0,0) which is asymptotically
stable when the two eigenvalues of the system matrix are
strictly located into the left half plane i.e. these eigenvalues
must have a strictly negative real part. This implies local
asymptotic stability of the equilibrium point (𝐼3𝑒𝑞1 , 𝑈𝑒𝑞1 )
of (11). These two eigenvalues 𝜆1,2 equal
(−
FIGURE 4. EIGENVALUES AS A FUNCTION OF R
Similar calculations show asymptotic stability of the
equilibrium point is lost in case L increases.
𝑅 𝑅𝐿 − 𝑅𝑁𝐿
𝑅 𝑅 − 𝑅𝑁𝐿 2
4
𝑅(𝑅𝑁𝐿 − 𝑅𝐿 )
+
) ± √(− + 𝐿
) −
(1 +
)
𝐿 𝐶 𝑅𝐿 𝑅𝑁𝐿
𝐿 𝐶 𝑅𝐿 𝑅𝑁𝐿
𝐿𝐶
𝑅𝐿 𝑅𝑁𝐿
.
2
VII. THE EIGENVALUES
ANNUAL JOURNAL OF ELECTRONICS, 2011
Using MATLAB, it is possible to calculate and visualize
the location of the eigenvalues of the system matrix of (14)
in case the parameters RL, RNL, C, R and L are changing.
Figure 6 visualizes these eigenvalues 𝜆1,2 in case 𝑅 = 1𝛺,
𝐿 = 10𝑚𝐻, 𝐶 = 2𝜇𝐹 and 𝑅𝑁𝐿 = 10𝛺. The linear load
changes from 𝑅𝐿 = 1𝛺 to 𝑅𝐿 = 20𝛺.
For small 𝑅𝐿 -values, the eigenvalues are both real and
strictly negative implying the asymptotic stability property
is obtained. As 𝑅𝐿 increases, the eigenvalues become a
complex conjugate pair having a strictly negative real part
implying the asymptotic stability property is maintained.
As 𝑅𝐿 further increases, the complex conjugate pair has a
positive real part implying the stability property is lost. A
further increase of 𝑅𝐿 gives real positive eigenvalues
implying no asymptotic stability is obtained.
This shows the linear load, with a 𝑅𝐿 -value which is
sufficiently small i.e. the power consumed by the linear
load is sufficienty high in comparison with the power P
consumed by the CPL, has a stabilizing effect on the
microgrid. Adding linear loads, in parallel with a CPL, has
a stabilizing effect.
The horizontal axis visualizes the current 𝐼(𝑡𝑘 ) and the
vertical axis visualizes the voltage 𝑈(𝑡𝑘 ).
FIGURE 7. TRAJECTORIES OF THE MICROGRID SYSTEM
Trajectories with an initial voltage which is sufficiently
high converge to the asymptotically stable equilibrium
point (𝐼𝑒𝑞1 , 𝑈𝑒𝑞1 ) and they do not converge to the unstable
equilibrium point (𝐼𝑒𝑞2 , 𝑈𝑒𝑞2 ) (with 𝑈𝑒𝑞1 > 𝑈𝑒𝑞2 ). When
the initial voltage is too low, the voltage collapses and no
convergence to equilibrium point (𝐼𝑒𝑞1 , 𝑈𝑒𝑞1 ) is obtained.
IX.CONCLUSION
Due to the negative dynamic resistance of a CPL, the
CPL can cause stability related problems when considering
a microgrid. The impact of several parameters on the
asymptotic stability property of the equilibrium points have
been studied. The positive effect of a capacitor or a linear
load in parallel with the CPL has been shown.
REFERENCES
FIGURE 6. EIGENVALUES AS A FUNCTION OF RL
VIII. TRAJECTORIES
Using linearization, only a local asymptotic stability
analysis is performed. This means the solutions of the
differential equations (7) and (11) converge to the stable
equilibrium point when their initial condition is sufficiently
close to the equilibrium point.
The behavior of the differential equation (7) can be
approximated by the difference equations
𝛥𝑇. 𝑅
𝛥𝑇
𝛥𝑇
𝑖(𝑡𝑘 ) −
𝑢(𝑡𝑘 ) +
𝐸(𝑡𝑘 )
𝐿
𝐿
𝐿
𝛥𝑇
𝛥𝑇. 𝑃
1
𝑢(𝑡𝑘+1 ) = 𝑢(𝑡𝑘 ) +
𝑖(𝑡𝑘 ) −
.
𝐶
𝐶 𝑈(𝑡𝑘 )
𝑖(𝑡𝑘+1 ) = 𝑖(𝑡𝑘 ) −
[
Here, for all integer k values 𝑡𝑘+1 = 𝑡𝑘 + 𝛥𝑇. Using
MATLAB, the difference equations allow to calculate the
trajectories i.e. the evolution of the voltage 𝑈(𝑡𝑘 ) and the
current 𝐼(𝑡𝑘 ) [6,7].
Figure 7 shows different trajectories starting with
different initial conditions (𝐸 = 220𝑉, 𝑅 = 1.5𝛺, 𝐿 =
10𝑚𝐻, 𝐶 = 2200𝜇𝐻, the CPL consumes a 𝑃 = 4420𝑊).
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