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An Optimal Virtual Inertia Controller to Support Frequency
Regulation inAutonomous Diesel Power Systems with High Penetration
of Renewables
Miguel Torres and Luiz A.C. Lopes
Power Electronics and Energy Research GroupDepartment of
Electrical and Computer Engineering
Concordia University, Montreal, Quebec, Canada H4G 2M1Phone:
+1-514-8482424 (ext. 3080), e-mail: {mi torre, lalopes
}@ece.concordia.ca
Abstract. This paper addresses the problem of frequencycontrol
in autonomous diesel-based power system with highpenetration of
renewables. Usually, small power systems withhigh penetration of
renewable energies are supplied by one ortwo small diesel
generators, resulting in a system with a rela-tively low moment of
inertia, and which can be susceptible tosignicant frequency
variations. However, frequency regulationcan be supported by
modifying the inertial response of the
system in an articial way, i.e., by adding a virtual inertia.
Thelatter can be performed by controlling the power
electronicsinterface of a distributed generator or an energy
storage unit. Inthis work, a controller is designed to provide the
optimal virtualinertia which minimizes, according to the proposed
performanceindex, variations in the fundamental frequency as well
as inthe power ow through the energy storage system. The
optimalcontroller is compared by simulations with other virtual
inertiacontrol strategies.
Keywords
frequency regulation, virtual inertia, optimal control,energy
storage, hybrid power system.
1. Introduction
Diesel generators (gensets) are very popular as mainpower source
in stand-alone power systems. For instance,in autonomous
wind-diesel power systems (AWDPS) [1],the power grid is established
by a diesel generatorwhichis a controllable source of energyand a
wind generator(WG) is used to complement power production when
thewind turbine reaches a specic speed. With an appro-priate
control strategy fuel consumption can be reduced,
lowering the cost of the energy produced. However, asidefrom
reducing fuel consumption, there also exists thenecessity of
controlling the frequency of the voltage beingsupplied to the load.
The latter is specially important insmall power systems (10-200kW)
with high penetrationof renewable energies [2], [3]. Usually, this
kind of gridsare supplied by one or two small gensets, resulting in
asystem with a relatively low moment of inertia, and whichcan be
susceptible to signicant frequency variations dueto sudden
variations in load demand and/or in the outputpower of a renewable
energy source.
Virtual synchronous machine/generator (VSM, VISMA,
VSG) is a novel concept and it has been proposed asa control
strategy to tackle stability issues in powersystems with a large
fraction of inertia-less distributedgenerators [ 4][9]. The main
idea is to control the power
electronics interface (PEI) of a distributed generator, orenergy
storage unit, in order to behave as a conventionalrotating
generator, which consists of a prime mover anda synchronous machine
[ 10] . Virtual inertia control is aparticular case of a VSM
implementation, where onlythe action of the prime mover is emulated
to supportfrequency control.
A. Interest of the work
Since the fundamental concept of VSMs was formallyintroduced, it
has been subject of several publications.Most of the research deals
with specic issues on theimplementation of VSMs such as: the
necessary require-ments of the PEI to perform a VSM [11], selection
of the storage media [ 12], and laboratory prototypes
forexperimental tests [ 13] . The most recent literature relatedto
VSMs found on the IEEE database has been reviewedand, to the best
of our knowledge, no research work hasbeen reported concerning
optimal control.
B. Objectives
1) Main objective. Design an optimal controller for theESS in
order to emulate a virtual inertia to supportfrequency control in a
diesel based power system.
2) Secondary objectives. i) Model main components of the system,
ii) Compare the performance of the optimalcontroller versus other
virtual inertia controllers, andiii) analyze the effects on the ESS
of using the optimalcontroller.
C. Main contribution
The optimization of the virtual inertia controller withthe
performance index being the integrated sum of theweighted quadratic
regulating errors (QRE) of the powerfrequency and the power ow
through the ESS.
2. System overview and modeling
The system of this study is an autonomous power sys-
tem based on a diesel generator and a wind generatorconnected in
parallel (see Fig. 1). It also has an energystorage system which is
interfaced to the ac-bus bya bidirectional power electronics
interface. The active
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Fig. 1. Power system.
power balance of the system is governed by the
followingequation:
pe + pw + ps = pl . (1)
A. Load
The load represents the uncontrolled end-user power
demand and it is considered as a disturbance signal pl .
B. Wind generator
The WG used in this work is a xed-speed xed-pitch(FSFP)
conguration. Main advantages of this congura-tion are its
simplicity and the robustness of the squirrelcage induction
generator (SCIG). On the other hand, themain disadvantages comes
from the fact that the SCIGoperates over a narrow range around the
synchronousspeed. More details about this conguration and
com-parisons with other conversion schemes can be found
on [14]. The modeling of the WG is based on equationspresented
in [ 3]. A conceptual block diagram of the modelis shown in Fig. 2
and each of its components are herewithexplained.
1) Wind speed. Wind speed, vw , is dened by fourcomponents:
vw = V wb + vwr + vwg + vwt (2)
where, V wb is the base wind component, vwr theramp component,
vwg the gust component, and vwt theturbulence component. The ramp
component is dened
as:
vwr (t) =V wr
t 2 r t 1 rt , t 1 r t t 2 r ,
0 otherwise.(3)
the gust component as:
vwg (t) =V wg
2 (1 cos) , t 1 g t t 2 g ,
0 otherwise.(4)
where = 2(t t1 g )/ (t 2 g t1 g ). Finally, theturbulence
component is dened in terms of its power
spectral density (PSD):
vwt = 2N
k =1
{S (k )}12 cos(k t + k ) (5)
Fig. 2. Wind generator conceptual block diagram.
where k = ( k 1/ 2) and k = rand(0 , 2) arethe frequency and
phase of the k-th cosinusoidal com-ponent, and S (k ) is the
spectral density functiondened as:
S (k ) = 2z0 F 2 |k |
2 1 + F kv2
43
(6)
2) Wind turbine. The model of the wind turbine consistsof two
elements: a rst order system ( 7) which repre-
sents the ltering of the high frequency components inthe wind
speed by the turbine, and the output powerequation of the wind
turbine ( 8).
w vwf = vwf + vw (7)
pwt = 12
R wt2 c p ()vwf
3 (8)
where, w is the time constant which depends on theturbine size,
vwf is the ltered wind speed, is theair density, Rwt is the radius
of the turbine rotor, andc p () is the performance factor of the
turbine givenby:
c p () = c1 c21
c5 c3 e c 4 (1
c 5 ) (9)
with being the tip speed ratio:
= Rwt wt
vwf (10)
where wt is the rotational speed of the WT. Coef-cients c1 .. 5
are dened later to t the characteristiccurve of the turbine of
interest.
3) Transmission system. The transmission system consistsof the
high speed shaft (IG side) and the low speedshaft (WT side)
connected together by a gear box. Themodel considered is the
following:
J w rig = 1N gb
T wt T ig (11)
where, J w is the lumped inertia of rotating parts, N gbis the
conversion ratio of the gear box, T wt the me-chanical torque of
the turbine, T ig the electromagnetictorque of the induction
generator, and rig is the rotorspeed of the induction generator.
Now, the rotationalspeed of the WT can be dened as
wt = rigN gb (12)
4) Induction generator. The proposed model for the IG isbased on
the fact that the generator operates in a small
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region around the synchronous speed sync [15], thenthe torque
can be approximated by:
T ig 3V t
2 (rig sync )sync 2 R 2
(13)
where, V t is the terminal voltage, R2 is the rotorresistance,
and sync is the synchronous speed.
C. Diesel generator
The diesel generator model consists of two main com-ponents
coupled by a common shaft: the synchronousgenerator (SG) and the
diesel prime mover. Due to thescope of this work, the electrical
dynamics of the SG areneglected. Also, in order to focus the study
on frequencyvariations, it is assumed that the amplitude of grid
volt-ages are kept constant by the automatic voltage regulator(AVR)
of the SG. Therefore, the proposed model for thediesel generator
and its governor can be represented bythe following equations:
r = (kf + kd )
J r +
1J
T m 1J
T e (14)
T m = 1 e
T m + ke e
ue (t td ) (15)
ue = kc r (16)
where r is the SG rotor speed, ue is the controlsignal from the
speed regulator (governor) to the fuelinjection system, T m is the
mechanical torque providedby the diesel engine at the common shaft,
and T e isthe electromechanical torque produced at the commonshaft
due to the active power balance of the grid (1).
The system has the following parameters: J the lumpedinertia of
rotating parts, kf the friction coefcient of theshaft, kd the SG
damping torque coefcient, e and kethe fuel injection system time
constant and gain, td theengine dead time which represents the
elapsed time untiltorque is produced at the engine shaft, and kc
the speedregulator gain (see block diagram in Fig. 3).
D. Energy storage system
The ESS consists of two main components: the PEI andthe storage
media. The PEI is a two-level VSC and
the storage media is considered as an ideal dc voltagesource, V
dc , connected to the dc-bus of the VSC. If theabc-dq
transformation is perfectly synchronized with gridvoltages, the
output active power ow can be dened as:
ps = V d i ds (17)
where V d and i ds are the d-axis component of the the
gridvoltages and VSC ac currents, respectively. From ( 17) itcan be
seen that the active power of the VSC can becontrolled by means of
i ds , whose dynamic is given by:
L
i ds = r i
ds + 2 f L i
qs + V dc m
d V d (18)
where i qs is the q-axis component of the VSC ac currents,m d is
the d-component of the modulation indexes, f is the grid frequency,
L and r are the inductance andresistance of the output lter of the
VSC. To decouple
and operate i ds in closed loop, the following control lawis
dened:
md = 1V dc
(ksi + r ) i ds ref
ksi i ds 2f L iqs + V d (19)
where, ksi is the current controller gain, and i ds ref is
the d-axis current reference signal (see block diagram inFig.
3). For more details on the modeling and control of a VSC, refer to
[16].
3. Optimal virtual inertia control
A. Virtual inertia concept
If the active power through the PEI of the ESS iscontrolled in
inverse proportion to the derivative of thegrid frequency, we are
emulating a virtual inertia in thepower system, thus enhancing the
inertial response to
changes in the power demand. Then, the control law forthe active
power of the VSC will be:
ps ref = kvi kr2 f o f (20)
where kr = 4/n p is the conversion factor between rotorspeed and
grid frequency with n p being the number of poles of the machine,
kvi is the virtual inertia added tothe system, and f o is the
nominal grid frequency (seeblock diagram in Fig. 3).
B. Optimal controller
Instead of being constant, the virtual inertia kvi can
bedesigned to be optimal for a given design specication.In order to
nd such optimal value, we will use thedeterministic linear
quadratic regulator (LQR) approach.For details on the theoretical
framework and mathematicalderivations related to the LQR problem
refer to [17][19] .For the design of the optimal controller,
consider thesystem in the form:
f = a f + b ps (21)
with a = (kd + kf )/J and b = 1 / (J ro kr ). The gen-eral
formulation of the LQR problem requires to de-ne a mathematical
performance criterion, often calledperformance index or function
cost. Then, the proposedperformance index is dened as the
integrated sum of theweighted quadratic regulating errors of the
grid frequencyand the ESS power (22). By doing this, we are
lookingfor minimizing the frequency variations, but since thisis
accomplished by the PEI of the ESS, at the sametime we penalizeby
means of the weighting factor the amount of power owing through the
ESS. Theexpression for the performance index is:
I = 0 f 2 (t) + ps 2 (t) dt (22)where is the weighting factor.
Therefore, the controlinput which minimizes I is dened as:
ps = 1
bF f (23)
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Fig. 7. ESS power demand for different values of .
inertia, kvi = kvi ; and iii) a modied virtual inertiadened
as:
kvi =kvi if f f 0,0 otherwise.
(27)
The system is simulated for a 100s wind speed prole anda
constant load demand. The wind prole consists of awind gust at t =
20s followed by a positive ramp. All theparameters used in
simulations are summarized in Tab. I,Tab. II, Tab. III, Tab. IV,
Tab. V, Tab. VI, and presented inthe appendix section. The
simulated wind speed is shownon Fig. 8. Fig. 9 shows the output
power of the windturbine, which is the result of its interaction
with the windand with the grid frequency, in this case. Fig. 10
shows thefrequency variation in the system, for different
conditionsof virtual inertia. Finally, Fig. 11 shows the variations
inthe power ow of the ESS, which is the one that emulatesthe
virtual inertia.
As can be seen from Fig. 10, when the system is operatedwith
constant inertia ( kvi = 0 and kvi = kvi ) the inertialresponse is
better at the beginning of the perturbation (lessreactive to
changes) but the oscillations last for longertime, since there is
no additional damping in the system.At the same time, as the
constant virtual inertia increases,the peak power through the ESS
increases as well. Thelatter has direct inuence on the necessary
power ratingsof the storage media and its power converter.
In the case when the system is operated with variable
virtual inertia ( kvi = kvi and kvi = kvi ) it is possi-ble to
say that the response is better than in the case
of constant inertia. However, since control law ( 27)
ispiecewise constant, each change of condition represents afast
commutation from one power level to another, whichbecomes into an
operational requirement for the storagedevice.
Compared with the proposed cases, the optimal controllerpresents
a good performance in terms of, oscillationmitigation, peak
frequency, and power consumption. Forinstance, consider Fig. 10 and
the case of the optimalcontroller vs. control law ( 27): both
operate within the
power range [ 7, 5]kW, however the optimal controllerachieves
almost 50% less in peak frequency deviationand it settles back at
least 3 times faster. It also has theadvantage of having just one
parameter for tuning, .
Fig. 8. Wind speed.
Fig. 9. Comparison of WG output power variations.
Fig. 10. Comparison of frequency response for different virtual
inertiacontrollers.
Fig. 11. Comparison of power usage for different virtual
inertiacontrollers.
5. Conclusion
In this work, we presented the design of an optimalcontroller
for the emulation of virtual inertia to support
frequency regulation in a diesel based power system.The model of
each main component of the system wasdeveloped. The performance of
the optimal controllerwas compared with other controllers by
simulation. The
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results show a good performance of the optimal controllerin
minimizing frequency variations, in exchange of
powerconsumption/injection from/to the ESS. By adjusting onlyone
parameter of the controller, it is possible to changeits control
effort, which has direct inuence on the ESSpower ow requirements.
The possibility to include con-straints in the formulation of the
optimization problem,
to satisfy physical limitations of the system, should
beconsidered as future work.
Appendix: simulation parameters
Table I. - Diesel generator model
Parameter description Symbol Value UnitNominal power P en 30
kWNumber of poles n p 2Moment of inertia J 0.9 kg m 2
Damping coefcient kD 0.4 kg m2 s 1
Fuel injection gain ke 1Fuel injection time constant e 0.2
sDiesel engine delay d 0.011 sSpeed controller gain kc 0.4
Table II. - Wind generator model
Parameter description Symbol Value UnitNominal power P wn 20
kWNumber of poles n p 2Moment of inertia J w 1.2 kg m 2
Terminal voltage V t 400 VRotor resistance R 2 0.221
Air density 1 .275 kgm 3
Rotor radius R wt 5 mGear box ratio N gb 40Wind speed lter time
constant w 1 s
Table III. - Performance factor coefcients
Wind generator type c 1 c 2 c 3 c 4 c 5Fixed-speed xed-pitch
0.76 125 6.94 16.5 -0.002
Table IV. - Wind speed model
Parameter description Symbol Value UnitBase wind speed V wb 8
m/s
Ramp amplitude V wr 4 m/sRamp start time t1 r 20 sRamp end time
t2 r 60 sGust amplitude V wg -3 m/sGust start time t1 g 20 sGust
end time t2 g 30 sPSD components N 100PSD frequency step 0.37
rad/sLandscape roughness coefcient z0 0.004Turbulence length scale
F 400 mMean wind speed at reference height v 8 m/s
Table V. - Virtual inertia controller
Parameter description Symbol ValueConstant virtual inertia kvi
3J Optimal virtual inertia gain 1
Table VI. - Power electronics interface
Parameter description Symbol Value UnitFilter inductance L 32
mHFilter resistance r 0.2
Current controller gain ksi 10
Acknowledgment
The authors would like to thank the nancial support of the
Chilean Government, through the National Commis-sion for Science
and Technology Research (CONICYT),and the Faculty of Electrical
Engineering and ComputerScience of Concordia University.
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