IEEE PEDS 2011, Singapore, 5 - 8 December 2011
Small Signal Stability Analysis of Arctan Power
Frequency Droop
C.N. Rowe†12 , T.J. Summers‡1 , R.E. Betz∗1 and D.J. Cornforth∼2
1 School of Electrical Engineering and Computer Science
University of Newcastle, Australia, 2308
2
CSIRO Energy Centre
Steel River, Newcastle, NSW
email:† Christopher.Rowe@newcastle.edu.au; ‡ Terry.Summers@newcastle.edu.au;
∗ Robert.Betz@newcastle.edu.au; ∼ David.Cornforth@csiro.au.
Abstract—
The microgrid is an environment constructed with the aim
of controlling local distributed resources. The most common
power flow control technique utilised in a standalone microgrid
is- a technique known as power frequency droop. An arctan
droop controller was previously presented in [1]. This paper
analyses the stability of this arctan power frequency droop.
SABER simulations are performed to obtain the operating points
about which the system is linearised. Considering a resistive
coupling the arctan droop can delay the onset of oscillatory
modes. Given an inductive coupling under high output power the
arctan gradient is able to maintain larger stability margins in
the microgrid, for the discrete operating points analysed. Limited
hardware results are provided to confirm the operating points
are achievable.
Index Terms—
Distributed Generation, Microgrids, Power Electronics, Power
Frequency Droop.
I. I NTRODUCTION
Existing problems with Distributed Generation (DG) highlight the need for an appropriate environment in which to
develop and control these relatively immature technologies.
By characterising the microgrid as a small autonomous unit
within a power system, we create an environment where
solutions can be formed on a microgrid level whilst mitigating
the need for major augmentations to the distribution network.
The most widely accepted method to ensure real and reactive
power are shared correctly in microgrids involves drooping
the frequency and voltage of supply relative to the real and
reactive power supplied by each DG unit.
This paper begins by introducing the existing theory on
power - frequency and reactive power - voltage droop. After
exploring the existing theory the arctan algorithm is introduced and the stability of this control scheme compared to
that of normal droop. Both systems are linearised and the
eigen values of the systems plotted at various operating points.
Conclusions are drawn about the benefits of the arctan droop
based control scheme.
II. F IXED G RADIENT AND A RCTAN D ROOP
A. Fixed Gradient Droop
This paper is based on the implementation of the traditional
power frequency droop method discussed in [2] and [3]. This
general approach is derived from the power flow across a
single impedance.
The general accepted form of the P-F and Q-V droop
equations is given in (1) and (2). The frequency is decreased
relative to the real power supplied and the voltage increased
relative to the reactive power flow to the load.
f − f0 = −mP (P − P0 )
(1)
U − U0 = −mQ (Q − Q0 )
(2)
where f is the operating frequency of the inverter, fo is the
frequency set point, mP is the frequency droop coefficient,
P is the real power of the inverter and Po is the real power
set point. In (2), U is the operating voltage of the inverter,
U0 is the voltage set point, mQ is the voltage droop coefficient,
Q is the reactive power of the inverter and Qo is the reactive
power set point
B. The Linear Rotational Transform
The physical equations that govern the power flow across a
single inductance change dependent on the inductance and resistance of the line (X/R ratio). To consider this phenomenon,
De Brabandere [3] introduces a linear rotational transform.
The transform is provided in (3) and explained fully in [3].
′ X
−R
P
P
Z
Z
= R
(3)
X
Q′
Q
Z
Z
where P and Q are the real and reactive powers. Z, X
and R are the coupling impedance, inductance and reactance,
respectively. P ′ and Q′ are the elements of real and reactive
power directly related to frequency and voltage.
The traditional droop equations are altered to consider the
transform, with P and P0 becoming P ′ and P0′ . Further
Q and Q0 become Q′ and Q′0 .
In a two inverter microgrid with a known X/R ratio we can
utilise this transform to create a P ′ entirely coupled to the
frequency. To simplify the terminology the original droop term
978-1-4577-0001-9/11/$26.00 ©2011 IEEE
787
III. S TABILITY A NALYSIS
Figure 1.
The effect of variations in ρ
It has been observed in [5] that with increasing output
power the traditional fixed gradient droop becomes increasingly unstable. The corollary is true – that decreasing the
droop gradient causes the dominant poles of the system to
become more stable. As arctan droop has a decreased tangential gradient at high output powers it essentially performs
this operation. Further it was shown that increasing the droop
gradient causes a two inverter microgrid to gain oscillating
poles [6]. The stability analysis provided below considers
the impact of the X/R ratio on the prevalence of these two
different phenomenon.
A. System Linearisation
Figure 2.
Control Block Diagram
’Power-frequency droop’ is maintained throughout this paper.
It refers to the use of any droop, utilised for power sharing,
irrespective of the X/R ratio of the coupling impedance.
The system configuration in this case is shown in Fig. 3.
The system linearisation in this case is similar to that presented in [7].
The inverter voltages and currents are related by
~1
I~1
E
Z1 + ZL
ZL
(5)
~2 =
ZL
Z2 + ZL
I~2
E
Converting into the alpha beta frame to utilise only real
quantities, we obtain


ed1
 eq1 

 e  = [Zs ] 
d2
eq2

C. Arctan Droop
The arctan droop scheme proposed in [1] removes the
constant frequency droop slope and replaces it with an arctan
based algorithm. By implementing this arctan based power
profile the microgrid operator can ensure that the operation
frequency of the microgrid is always within preset bounds.
Dynamic droop adjustment is the subject of [4] with the aim to
gain better control whilst implementing frequency and voltage
bounding.
The arctan function provides adequate control over the
gradient of droop about the power set point, has desirable
horizontal asymptotes and existing function libraries in most
coding languages. The droop equation from [1] is characterised as shown in (4). Where f is the operating frequency of
the inverter, fo is the frequency set point, ρ is the arctan droop
coefficient, P ′ is the output power of the inverter coupled to
the frequency and Po′ is the dash-power set point.
1
(arctan(ρ(P ′ − Po′ )))
(4)
π
By characterising the function in this way it is naturally
bounded in the frequency domain from (fo + 0.5) Hz to
(fo − 0.5) Hz. The control over the gradient (and inherently
concavity) is exacted by changing ρ. It is worthy of note that
under the application of the small angle criteria, the arctan
algorithm reduces to the same direct δ ∝ P relationship as the
general form of droop. The effect of variations in ρ are plotted
in Fig. 1. The control scheme for arctan droop is depicted in
Fig. 2.
f = fo −

id1
iq1 
id2 
iq2
(6)
where
R1 + RL
 X1 + XL
[Zs ] = 
RL
XL

−(X1 + XL )
R1 + R2
−XL
RL
RL
XL
R2 + RL
X2 + XL

−XL
RL

(7)
−(X2 + XL ) 
R2 + RL
and edi = Ei cos(δi ), eqi = Ei sin(δi ), idi = Ii cos(δi +
θi ), iqi = Ii sin(δi + θi ).
The linearisation of the system is given by
[△i] = [Zs ]
−1
[△e]
(8)
Linearisation of the fixed gradient droop control scheme
has been previously presented in the s-domain in [8] and as
a state space system in [7]. This state space system is further
validated by its use in [6]. The state space system, as in [7],
is described by








˙ 1
△ω
△e˙ d1
△e˙ q1
˙ 2
△ω
△e˙ d2
△e˙ q2


 
 = Ma

Mc


Mb
Md






△ω1
△ed1
△eq1
△ω2
△ed2
△eq2

 
+ C a

Cc

Cb
Cd


△P1
 △Q1 
 △P 
2
△Q2
(9)
where Mb , Mc , Cb , and Cc are all null matrices.
Considering any X/R ratio makes the computations somewhat more complex, the linearisation of the control equations
given in [7] becomes
788
△ω(s) =
−kp ωf ′
(P (s))
s + ωf
(10)
Thus
=
△ω̇
−
=
−
=
−
=
ωf △ω + −kp ωf △P ′
X
R
ωf △ω − kp ωf
△P − △Q
Z
Z
kp ω f X
kp ω f R
ωf △ω −
△P +
△Q
Z
Z
(13)
−kq ωf ′
(Q (s))
s + ωf
(15)
(11)
(12)
(14)
Similarly
△E(s) =
gives
=
△Ė
−
=
−
=
−
=
ωf △E + −kq ωf △Q′
R
X
ωf △E − kq ωf
△P + △Q
Z
Z
kq ω f R
kq ω f X
ωf △E −
△P −
△Q
Z
Z
System Configuration
(18)
(19)
The Ma, d matrices are unaffected by this change yet the
Ca, d matrices gain three additional cross coupling terms
which are evident in (14) and (19).
The manipulation of the state space equation (9) eventually
gives rise to a system of the form
h
i
△Ẋ = [A] [△X]
(20)
where the eigen values of [A] can be used to determine
the entire systems stability. Full details of the intermediate
steps can be gleaned by first gaining familiarity with the
linearisation in [7].
The arctan droop is considered to be a fixed gradient near
the points of linearisation, the gradient is thus given by the
differential of the arctan function
d
−2ρ
(ωo − 2 arctan(ρ(P ′ − Po′ ))) =
(21)
dP ′
1 + ρ2 (P ′ − Po′ )2
Hence the linearisation term becomes
dω
−2ρ
=
△P ′
′
2
dP
1 + ρ (P ′ − Po′ )2
Figure 3.
(16)
(17)
(22)
This function provides the tangential gradient of the arctan
droop at the point about which it is linearised. The pertinent
point in this calculation is that the linearisation is heavily
dependent on the operating power of the inverter as it is a
nonlinear function.
IV. E XPERIMENTAL S YSTEM
The control scheme was implemented in a two-inverter
microgrid. The microgrid system is shown in Fig. 3. A
hardware system was constructed and a simulation model of
Figure 4.
Overview of dSPACE Hardware System
the hardware system developed. The nominal system voltage
and frequency are 230 V (RMS, line to neutral) and 50 Hz
respectively.
SABERr Simulation
The simulation was performed in the SABERr simulation
package. The new control algorithm was implemented in a
dynamic link library written in the ‘C’ language. The dynamic
link library is executed once every 250 µs, which is equivalent
to a main control loop frequency of 4.0 kHz. In this way the
SABERr simulation mimics the single task implementation
of the dSPACEr hardware.
The model of the passive load contains only resistive and
inductive elements. The implementation of load variations in
this way models the possible load variations achievable in the
hardware system.
dSPACEr Hardware System
The hardware system consists of two inverters in parallel, connected to a passive resistive and inductive load.
The inverters are SEMITEACH three leg IGBT stacks from
Semikron. Control of each stack is achieved via dSPACEr
DS1103 controllers (which utilise high performance Power
PC processors). The hardware system is shown in Fig. 5.
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Table I
F IXED G RADIENT D ROOP - O PERATING P OINTS
Equilibria
Figure 5.
Figure 6.
dSPACEr and SEMITEACH Hardware System
A
(0.098 s)
B
(0.395 s)
C
(0.695 s)
D
(0.995 s)
E
(1.28 s)
ed1
261.84
-3.1659
-2.5247
-1.3104
321.05
eq1
-198.86
-323.84
-323.65
-323.07
1.0867
ed2
261.89
-3.2049
-2.572
-1.377
321.12
eq2
-198.89
-323.9
-323.71
-323.13
0.96499
id1
0.45865
-0.75757
-0.92672
-1.2291
0.72658
iq1
-0.37012
-0.6074
-0.61047
-0.63357
-2.0308
id2
0.46895
-0.68565
-0.83714
-1.1293
0.6935
iq2
-0.33438
-0.5847
-0.58122
-0.60068
-1.9237
mp
5 e−6
5 e−6
5 e−6
5 e−6
5 e−6
mq
5 e−6
5 e−6
5 e−6
5 e−6
5 e−6
R1 +
jX1
0.148
+j1.759 m
0.148
+j1.759 m
0.148
+j1.759 m
0.148
+j1.759 m
0.148
+j1.759 m
R2 +
jX2
0.220
+j0.848 m
0.220
+j0.848 m
0.220
+j0.848 m
0.220
+j0.848 m
0.220
+j0.848 m
RL +
jXL
283
+j0.0
111.99
+j133.42
87.71
+j126.15
58.16
+j107.79
26.26
+j72.64
P1 /ph
94.954
85.208
85.547
85.59
86.269
Q1 /ph
18.946
121.09
146.99
195.87
333.03
P2 /ph
92.964
82.412
81.884
81.572
82.376
Q2 /ph
12.306
108.94
131.9
179.05
316.1
Table II
A RCTAN D ROOP - O PERATING P OINTS
Arctan Operating Points derived from SABER Simulation
One custom PCB provides firing signals from the
dSPACEr system to the stacks and another provides voltage
and current measurements to the dSPACEr analogue to digital
converters. The sensing devices used are closed loop hall
effect devices due to the fast response time that these devices
exhibit. An overall diagram of the system is provided in Fig. 4.
V. R ESULTS
Simulation Results
Considering a Resistive Coupling: The SABERr simulation has been described in Section IV. This simulation was
utilised to obtain a number of different steady state operating
points for both the fixed gradient and arctan droop. Fig. 6
shows the output power of each inverter increasing under
stepped changes in load. The increase provided is purely
in reactive power. This allows us to observe the effect of
increased reactive power on the eigenvalues of the system.
Remark 1. Note that the ringing in Fig. 6 during transients is a
manifestation of the current unbalance caused by contacting in
a star connected inductive load without zero voltage switching.
Table I shows the operating points (A→E) for the fixed
gradient droop, about which the system has been linearised.
Equilibria
A
(0.098 s)
B
(0.395 s)
C
(0.695 s)
D
(0.995 s)
E
(1.28 s)
ed1
261.83
-2.229
0.4314
3.9427
320.92
eq1
-198.85
-323.79
-323.65
-323.11
9.6971
ed2
261.88
-2.2711
0.37906
3.8704
320.98
eq2
-198.88
-323.84
-323.69
-323.16
9.5698
id1
0.46061
-0.74723
-0.90716
-1.2021
0.80645
iq1
-0.37118
-0.62682
-0.6419
-0.67808
-1.9928
id2
0.46695
-0.69389
-0.84815
-1.1391
0.71831
iq2
-0.33332
-0.57055
-0.56591
-0.59418
-1.9258
ρ
1 e−7
1 e−7
1 e−7
1 e−7
1 e−7
mq
5 e−6
5 e−6
5 e−6
5 e−6
5 e−6
R1 +
0.148
0.148
0.148
0.148
0.148
jX1
+j1.759 m
+j1.759 m
+j1.759 m
+j1.759 m
+j1.759 m
R2 +
jX2
0.220
+j0.848 m
0.220
+j0.848 m
0.220
+j0.848 m
0.220
+j0.848 m
0.220
+j0.848 m
RL +
jXL
283
+j0.0
111.99
+j133.42
87.71
+j126.15
58.16
+j107.79
26.26
+j72.64
P1 /ph
95.266
88.255
89.468
89.882
90.576
Q1 /ph
18.827
119.79
145.52
194.14
330.55
P2 /ph
92.334
79.536
77.837
77.33
77.845
Q2 /ph
12.275
110.67
135.44
183.28
317.33
790
Power (W), (VARs) : time (s)
Power (W), (VARs)
1.0k
Real Power − Inverter 1
Real Power − Inverter 2
Reactive Power − Inverter 1
500.0
Reactive Power − Inverter 2
0.0
Current (A)
5.0
: time (s)
Id1
Current (A)
Iq1
0.0
−5.0
0.0
25.0m
50.0m
75.0m
0.1
0.125
0.15
0.175
0.2
0.225
0.25
0.275
0.3
0.325
0.35
0.375
0.4
0.425
0.45
0.475
0.5
time (s)
Figure 11.
Simulation
Figure 7.
Figure 8.
Figure 10.
Eigenvalue plot for Arctan with ρ = 1e−7
Eigenvalue plot for Fixed Gradient with mp = 5e−6
Figure 9.
Inductive coupling equalibria, derived from alternate SABER
Eigenvalue plot for Arctan with ρ = 1e−6
Eigenvalue plot for Fixed Gradient with mp = 5e−5
Similarly Table II shows the operating points about which the
arctan droop has been linearised.
The fixed gradient droop under matched power conditions
has eigen values that are real and stable. Null vectors are of no
interest in the stability analysis [7]. In the case of the arctan
droop, the points of linearisation are very similar and the two
inverters output power matched. In this case the eigen values
of the system are also real and stable.
As the gradient of droop is increased the poles of the
system become complex. This decreases the system damping
and leads to deleterious oscillating components in the control
variables (system frequency - in this case). These complex
poles are depicted in Fig. 8. Similar results are observed in [6].
The arctan function is able to delay the onset of these modes
as shown in Fig. 7. The droop coefficients (ρ and mp ) are
increased by a factor of ten for each case in Fig. 8 and 9, large
oscillatory modes are present in the fixed gradient approach
whilst the arctan function has delayed the onset of these
modes.
Considering an Inductive Coupling: In the case of an
inductive coupling it is possible to perform a similar linearisation. Fig. 11 shows the output power of each inverter increasing under stepped changes in load. The increase provided is
purely real power to observe the effect of this on the dominant
low frequency modes of the system.
The fixed gradient and arctan droop controls have eigenvalues that are real and stable. As the output power of the
system is increased the system tends towards instability. This
has previously been observed in [5].
Fig. 12 shows the tendency of the low frequency dominant
poles under a fixed gradient whilst Fig. 13 shows the tendency
for arctan droop. The fixed gradient and arctan gradients
are mp = 1 e−5 and ρ = 5 e−7 respectively.
In the case of the arctan droop, as the output power of
the system is increased the eigenvalues are larger (negative)
and the rate at which the eigen values tend to instability
decreases. That is for equal step increases in real power
there is a decrease in the d(ℜ(λi )). This indicates, for the
operating points investigated, relative stability margin of the
arctan droop is superior.
Hardware Results
The hardware system was used to confirm the operating
points used for the linearisation were achievable. The two
791
Figure 12. Inductive Coupling - Fixed Gradient Dominant Low Frequency
Eigenvalues
Figure 13.
values
Inductive Coupling - Arctan Dominant Low Frequency Eigen-
inverter microgrid was operated under various steps in reactive
load described in Table I and II. Due to hardware limitations
the microgrid was operated at 200 V RMS and thus the
reactive power points are slightly lower than that expected.
A change from ZL = 283+j0.0 Ω to ZL = 26.26+j72.64 Ω
was implemented, the reactive power increase of inverter one
and two are shown in Fig. 14. The currents and voltages of
inverter one during this transient are given in Fig. 15. These
preliminary hardware results validate the operating points used
in the linearisation.
VI. C ONCLUSIONS
This paper contributes a two inverter microgrid control
system linearisation that is able to consider a various X/R ratios. This provides basis for the stability analysis of arctan
Figure 14.
Figure 15.
Hardware Inverter 1 Currents and Voltages
droop control. The control system was simulated in the
SABERr simulation package to obtain operating points about
which to linearise. System linearisation was performed and
the behavior of the eigen values investigated. Considering a
resistive coupling the arctan droop can delay the onset of
oscillatory modes. Given an inductive coupling under high
output power the arctan gradient is able to maintain larger
stability margins in the microgrid, for the discrete operating
points analysed. For the investigated operating points the
arctan droop was found to be more stable. Also increasing
output power was found to cause the arctan droop to tend
towards tends towards instability at a slower rate than the fixed
gradient droop. Limited hardware results from a 200 V RMS
two inverter microgrid were provided to confirm the operating
points simulated were achievable.
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Hardware Reactive Power Transient
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