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MANUSCRIPT SUBMITTED TO IEEE TRANSACTIONS ON ENERGY CONVERSION 1
A Model Predictive Control Approach to Operation
Optimization of an Ultracapacitor Bank for
Frequency Control
Mateo Beus, Student Member, IEEE, Matej Krpan, Student Member, IEEE, Igor Kuzle, Senior Member, IEEE,
Hrvoje Pandˇ
zi´
c, Senior Member, IEEE, and Alessandra Parisio, Senior Member, IEEE
Abstract—This paper presents a nonlinear dynamic simulation
model of an ultracapacitor (UC) bank and the associated control
system. The control system at hand consists of two levels: the
lower level controls the inverter of the UC bank, while the upper
control level is responsible for providing charging/discharging
active power set points to be followed by the lower control level.
This paper focuses on the development of the upper control level
for frequency control. Specifically, two simulation case studies
are developed so as to assess the performance of the proposed
control framework. In the first case study the upper control level
is developed using a classical Proportional-Integral-Derivative
(PID) controller. In the second case study the upper control level
is devised using a Model Predictive Control (MPC) algorithm
based on internal linear prediction model of a nonlinear UC
bank. In both cases, a nonlinear UC bank simulation model
is used. The simulation case studies are modelled and tested
in Matlab/Simulink. The response of the MPC-controlled UC
bank is compared to the 3 existing PID-control algorithms for
frequency control. The simulation results show that the MPC
algorithm outperforms the conventional PID controllers.
Index Terms—frequency control, model predictive control,
ultracapacitor, supercapacitor, power system dynamics
I. INTRODUCTION
THE trend of increasing the share of inverter-interfaced
generation (IIG) in power systems throughout the world
and the subsequent reduction of synchronous inertia have
motivated many research efforts to understand the stability
of low-inertia systems as well as to develop new algorithms
for enabling the IIG participation in system frequency control
and other ancillary services [1]. Currently, the most prominent
technologies for tackling the issues (e.g. voltage, frequency
and angle stability, provision of balancing services [2]) brought
upon by an increased share of inverter-interfaced generation
are energy storage technologies such as batteries, flywheels,
supercapacitors/ultracapacitors (SC/UC) [1]. Until recently,
most attention has been paid to batteries as they are the most
M. Beus, M. Krpan, I. Kuzle and H. Pandˇ
zi´
c are with the Depart-
ment of Energy and Power Systems, Faculty of Electrical Engineering
and Computing, University of Zagreb, 10000 Zagreb, Croatia, e-mail:
{mateo.beus}{matej.krpan}{igor.kuzle}{hrvoje.pandzic}@fer.hr
A. Parisio is with the School of Electrical and Electronic Engi-
neering, University of Manchester, Manchester, UK, e-mail: alessan-
dra.parisio@manchester.ac.uk
This work was supported in part by the EU Horizon 2020 project “CROSS-
BOW”, grant agreement No. 773430, as well as by the Croatian Science
Foundation and European Union through the European Social Fund under the
project Flexibility of Converter-based Microgrids – FLEXIBASE (PZS-2019-
02-7747).
commercially mature and most versatile of all inverter-based
fast storage technologies with a good trade-off between power
density and energy density [3]. UC has been mostly used
in applications such as electric vehicles and uninterruptible
power supplies [3]. Only relatively recently UC storage has
been considered for bulk power system applications [4], [5].
Ultracapacitor (UC) energy storage system (ESS) can be
used both as a standalone ESS for grid support applications
or in combination with other storage systems or IIG as a part
of a hybrid energy (storage) system. Its high power density,
in particular, as well as the capability to perform hundreds of
thousands of charging/discharging cycles and fast discharge,
make it the most applicable technology during grid frequency
excursions when a fast injection or absorption of active power
is required. Similarly, it can be used to quickly stabilize inter-
mittent output of solar and wind generation. There are several
reasons for using UC systems for fast injection of high power
instead of other storage devices, e.g. batteries or flywheels [3],
[4]: i) UC bank can be fully charged or discharged in the time
scale of several tens of seconds or faster while the rated power
can be reached within a few milliseconds; ii) UCs have higher
power density than batteries and flywheels, i.e. a UC system
of the same power rating is much smaller than an equivalent
battery or flywheel system; iii) UCs can withstand significantly
more (hundreds of thousands) charging/discharging cycles;
iv) UCs have smaller operation and maintenance costs than
batteries and flywheels.
This paper presents a model predictive control (MPC)
algorithm for frequency control by a standalone UC bank.
Automatic control of active power and frequency in power sys-
tems is almost exclusively achieved by standard Proportional-
Integral-Derivative (PID) control, which is well-known and re-
liable [6]. Conventional power plants and large-scale batteries
have more than enough energy in normal operation in the time
scale of inertial response (IR) and primary frequency control
(PFC, tens of seconds) so that PID control can most of the time
give adequate performance. However, this is not the case when
UCs are utilized. UCs are energy limited so they can sustain an
output power profile only for a limited time (rated power can
be sustained for tens of seconds up to a minute, which is the
time scale of IR and PFC). The power output of a UC depends
on the PID controller parameters as well as on the size of
the frequency disturbance. Therefore, PID-controlled UC can
likely discharge (i.e. reach minimum voltage limit) while still
providing some amount of power. This can cause additional
MANUSCRIPT SUBMITTED TO IEEE TRANSACTIONS ON ENERGY CONVERSION 2
disturbance to the grid and a larger frequency nadir [7] as well
as overvoltages in the UC system. Since disturbances and their
size cannot be predicted, PID parameters cannot be optimally
chosen or tuned in real-time to provide maximum active power
support while avoiding abrupt discontinuation of power for all
possible operating scenarios. Thus, in order to address these
issues, we adopt an MPC approach to inherently consider
UC system constraints in the control system formulation and
to obtain a smooth power response to a disturbance of any
size. The proposed MPC approach is compared to a standard
PID controller for various disturbance sizes to demonstrate the
benefits of MPC over PID controllers.
A. Literature Review
In large-scale power applications, standalone UC banks
have been used for frequency control [5], [8]. In [5], an
electromagnetic transient model of a UC is developed using
ideal capacitor model, while in both [5], [8] the focus is
on developing a coordinated control between a UC, a gas
turbine and an adjustable-speed hydro turbine for IR and
(dynamic) droop control using standard PID formulation. It
is shown that the use of a UC in coordination with other units
can significantly arrest the frequency nadir and steady-state
deviation, however the issue of the sudden UC discharge when
using standard PID control is clearly visible. The negative
impact on system frequency is avoided because the UC bank is
coordinated with variable-speed pumped-hydro. Nevertheless,
the UC-PHS combination will not be case for most future
applications because availability of PHS is limited by location.
The aim of this paper is to design a more advanced control
strategy for the UC itself in order to improve its operation in a
power system with a high share of RES, regardless of whether
the UCs will be used standalone or in combination with some
other units.
The authors in [9] propose a simplified UC model for
inclusion in large power system simulation. Standard iner-
tial and droop controllers, i.e. proportional-derivative (PD)
controller, are used for controlling the UC output power.
Although sudden power reduction is avoided by using a power
ramp-down based on estimated energy, this ramp is fixed
so the expected performance in terms of extracted energy
and frequency response is not optimal for all possible cases.
Furthermore, only one case study is conducted, in which the
size of the disturbance is smaller than the size of the UC.
The authors have not investigated performance for disturbance
sizes larger than the size of UC which can have significant
impact as illustrated later in section IV. Similarly, authors in
[10] used a supercapacitor energy storage to support frequency
of an island system. However, an ideal supercapacitor model
is also used and there were no comprehensive studies on the
impact of disturbance size on the performance of the proposed
control.
Authors in [7], [11]–[14] focus on optimal sizing and
PID control system tuning of UCs and/or hybrid energy
systems with UCs for enhancement of system frequency. The
sizing and tuning are based on simulated system frequency
metrics, measured power fluctuations for various scenarios or
metaheuristic optimization. However, the approaches proposed
in the aforementioned papers use simplified or ideal UC
cell models. This has a significant impact on the control
performance, since the capacitance of the UC varies with
voltage, as shown in section II. Furthermore, it is concluded
in [7] that the control parameters have to be carefully chosen
as it is possible to aggravate the frequency even more when
the energy is depleted. The size of the disturbance in the case
studies is sufficiently small compared to the size of the UC or
a hybrid storage system, i.e. the performance is not tested for
different sizes of a disturbance.
The authors in [15] investigate the coordination of a UC
bank (with a realistic UC cell model) with a wind turbine in
order to improve the virtual IR of wind turbines. Standard
derivative control for virtual inertia is used, in which the
parameters are tuned to perform adequately for only one
disturbance size. Likewise, authors in [16], [17] investigate the
capabilities of an UC in the DC link of a back-to-back power
converter to provide IR using PID control. Ideal capacitor
model is used and the UC capacity is large enough compared
to the disturbance size so that the response is always adequate.
In the context of MPC, UC is usually part of a battery-
ultracapacitor hybrid system, where the control is either fo-
cused on automotive applications [18], [19] or on general
energy management [20], [21]. In [18] and [19] it is shown that
MPC of a battery-UC hybrid system can be used to improve
battery lifespan. More importantly, in [19] it is shown that the
proposed nonlinear MPC algorithm is implementable in real
time for under 10 ms required sampling time. Similarly, in
[20]–[22] a real-time implementation of MPC is successfully
tested. However, the focus of MPC was on general energy
management and on improvement of the hybrid energy system
by exploiting the strengths of each part (battery, fuel cell, UC).
These papers do not focus on power system frequency control
and ideal capacitor model is used in all papers above but [18].
Furthermore, the capacity of the UC/hybrid system is large
enough and charging/discharging power small enough at high
frequency so that the depletion of energy is not an issue for
the considered applications.
The authors in [23] implement a supervisory adaptive
generalized predictive control for utilizing a UC in reducing
frequency and tie-line deviations, and for damping of interarea
oscillations in a three-area power system. However, the UC is
modeled as an ideal capacitor and MPC requires an online
model estimation of the complete power system in order
to generate the UC power setpoints. Furthermore, since the
model is adaptive, it also needs to compute the UC controller
parameters in real-time. Although this may be feasible for a
simple dynamic representations of a power system, real-world
power systems are high-order nonlinear systems. Accurately
estimating the most important parameters of such complex
dynamic systems, as well as computing optimal controller pa-
rameters, is computationally intensive for real-time application
using standard industrial controllers. Moreover, the relevant
(reduced) grid topology may not be known to the UC owner.
To advance the body of knowledge in this area, we focus only
on the UC model and measurements that are directly available
without the need to know the grid topology or to estimate
MANUSCRIPT SUBMITTED TO IEEE TRANSACTIONS ON ENERGY CONVERSION 3
parameters/signals online in real-time.
The authors in [24], [25] developed an MPC strategy for
frequency control by a standalone supercapacitor using an
ideal model. However, the authors used MPC for predicting
the behaviour of microgrid frequency. This assumption may
hold in very small microgrids in which all the different
entities and parameters that impact the frequency response are
known, but this assumption does not hold for a bulk power
system whose parameters and topology change over time. Our
approach focuses only on directly available measurements and
predicting the behaviour of a supercapacitor using a realistic
model that can be readily estimated. The authors of [24],
[25], however, have shown that real-time utilization of MPC
is feasible.
Similarly, authors in [26] utilized functional MPC to predict
the inner states of a rotor-side converter of a DFIG wind
turbine with a supercapacitor in the DC link in order to damp
inter-area oscillations in coordination with a static synchronous
series compensator. The drawback of this approach is that
using MPC for inner converter control requires high compu-
tational power. MPC is used for power smoothing of a hybrid
PV-battery-supercapacitor system in [27] in which the authors
predict the power outputs of a hybrid system on a longer
time scale meaning the supercapacitor dynamics are neglected,
while authors in [28] proposed an operation framework for a
hybrid battery-supercapacitor system in a market environment
in which real-time dynamics are neglected as well. Moreover,
the optimization problem in [27] is nonlinear and takes too
long to solve to be practical for real-time applications.
B. Contribution
Based on the conducted literature survey, the following
drawbacks and shortcomings have been identified:
1) There are only a handful of papers to the best of
our knowledge that utilize a standalone ultracapaci-
tor/supercapacitor bank and MPC for fast frequency con-
trol, e.g. [23]–[25]. We elaborate on the shortcomings of
these papers in the introduction section in the literature
review. In summary, these three papers besides using
a nonrealistic supercapacitor model also use MPC for
predicting the behaviour of system frequency which is
not feasible because the parameters and the topology
of the grid are not always known and are continuously
changing, so continuous identification of prediction
model is necessary. On the other hand, we use MPC
for predicting the behaviour of supercapacitor voltage /
state-of-charge so that we can always smoothly extract
maximum amount of energy without causing additional
severe disturbances to the grid once the supercapacitor
is discharged.
2) There is not any paper that considers a realistic super-
capacitor model for system identification in MPC.
3) All other papers that utilize a supercapaci-
tor/ultracapacitor bank for system frequency control
have one or more of the following aspects in common:
•they use a non-realistic representation of a su-
percapacitor/ultracapacitor (it does not take into
account voltage-dependent capacitance and other
losses), which is not appropriate for frequency con-
trol applications and introduces an error in the state
estimation [29];
•they use PID control of which we explain the short-
comings for frequency control applications using a
supercapacitor (and compare MPC and PID in the
study cases)—these shortcomings are also reported
in the literature review and contribution sections of
the paper;
•they use a supercapacitor as part of a hybrid system
and/or carefully selected use cases which mask the
inherent shortcomings of utilizing a supercapacitor
for frequency control (low energy density making
smooth charge/discharge for all operating conditions
difficult to achieve by classical control).
4) Papers that utilize MPC for supercapacitor control are
mostly concerned with automotive applications or some
general energy management of hybrid systems. Applica-
tion of standalone UC banks for frequency control which
exhibits a different behaviour are missing. These papers
also use idealized models of supercapacitors.
To the best of our knowledge there are no papers using MPC
approach to fast frequency control by a standalone UC bank
that also consider a realistic UC model. Hence, this paper aims
to bridge this gap. The main argument against using MPC for
fast frequency control can be the computational cost and thus
the possibility that the solution cannot be always found within
reasonable time. In this paper, an implicit MPC framework is
used, which always calculates feasible solutions with solver’s
average solving time under 100 ms.
The advantages of using MPC for UC banks providing
frequency control is that the disturbance uncertainty of any
size can be mitigated by inherently incorporating UC system
constraints and predictive capabilities into the control design,
which is particularly relevant in today’s context, where power
systems have increasing shares of stochastic renewable energy
sources. By doing this, the control action is more robust
against the uncertainty coming from the renewable power gen-
eration, hence an improved frequency response to disturbances
can be obtained. UC only has energy available for tens of
seconds, which is also the timescale of frequency containment
control. On the one hand, increasing the contribution of UCs to
frequency response through PID gain tuning is risky because
the system frequency can be aggravated even more if the
stored energy is depleted while still providing a significant
power output (as shown in the simulation studies). On the
other hand, if the PID gains are tuned more conservatively,
the advantage of a fast and high power output are lost. In
both cases, the PID controller cannot be tuned to optimize the
contribution of a UC to frequency response for all possible
disturbance sizes. On the contrary, MPC is able to optimize
the behaviour of the control system against a desired criteria,
thus making the control system more flexible and responsive to
the time-varying power system conditions. Additionally, one
of the benefits of MPC is that system limitations are inherently
considered in the MPC formulation, and issues such as integral
MANUSCRIPT SUBMITTED TO IEEE TRANSACTIONS ON ENERGY CONVERSION 4
wind-up, which are present in PID controllers, can be avoided.
In this paper, an MPC control scheme is devised to always
guarantee that the UC bank discharges smoothly and that the
maximum energy is extracted while meeting the constraints of
the UC system.
The main contributions of this paper are:
1) the development of a realistic UC bank dynamic model;
2) the design of an MPC framework based on internal linear
prediction model for active power control by UC bank
3) the design of an autonomous frequency controller based
on voltage reference trajectory and MPC framework
4) the performance comparison between a classic PID
control and an MPC for different sizes of disturbances
to provide fast frequency control by a UC bank.
II. ULTRAC APACITO R BANK MOD EL
The core of the UC bank model is an ultracapacitor cell.
Detailed description of the equivalent RC circuit (experimen-
tally identified) of an UC cell can be found in [30]–[32]. For
short-term studies (tens of seconds up to a few minutes), it
is enough to observe the behaviour of the RC circuit shown
in Fig. 1, where Cuc is the UC capacitance. Finite series of
parallel RC groups R1C1–RNCNalongside Cuc model the
porosity of the UC electrodes. Experimental research [32] has
shown that 5 RC groups are enough to obtain an accurate
model. Capacitance Cuc as well as RC groups Rn,Cnare
dependent on the ultracapacitor voltage uC(t). This model
is nonlinear with time-varying parameters, which is why the
ideal capacitor representation is not always accurate enough
for frequency control applications [29].
Rs
+−
uRs
iuc
Cuc(uC)
+−
uC
C1(uC)
+−
uC
1
R1(uC)
CN(uC)
+−
uC
N
RN(uC)
+−
uuc(t)
Fig. 1: Ultracapacitor cell equivalent RC circuit.
The parameters Cuc,Rn,Cn(n∈ {1...N }) are calculated
according to (1)–(3):
Cuc(uC) = C0+kvuC(t),(1)
Cn=1
2Cuc, n ∈ {1...N},(2)
Rn=2τ(uC)
n2π2Cuc , n ∈ {1...N}.(3)
C0is the UC capacitance at 0 V and kvis a constant expressed
in F/V. τ(uC)is another experimentally determined parameter
(it has dimension of time), which can also be approximated
by (4) [32]:
τ(uC)≈3Cuc(Rdc −Rs),(4)
where Rdc is the equivalent series resistance experimentally
obtained at very low frequencies (DC), while Rsis the series
resistance obtained at very high frequencies.
UC bank is created by connecting a certain number of cells
Nsin series to form a string (to increase voltage rating) and
a certain number of strings Npin parallel (to increase current
rating). The commonly made assumption here is that all the
cells are identical, symmetrically loaded and no balancing
is needed, which is an assumption that does not always
hold. However, commercial UC solutions always have a sort
of balancing system integrated [33], so this assumption is
valid. Unbalanced cells are an electronic design issue and
are beyond the scope of this paper. Nonetheless, the main
issue of unbalanced cells is the risk that some cells exceed
rated voltage, which can lead to damage and a fault in the
string. However, for power system dynamics applications the
assumption of balanced cells is realistic and greatly simplifies
the aggregation of individual cells to form a bank with a higher
power rating.
The dynamic model of this UC bank can be easily derived
from the equivalent circuit shown in Fig. 1, by taking the UC
current iuc as an input, the UC voltage uuc as an output and
the capacitor voltages uCand uC
nas state variables. The actual
requested input current is calculated from the actual inverter
output power and the UC voltage. A complete nonlinear model
of the UC bank is analytically described by (5)–(10), where
Rs
nand Cs
nare defined by (2) and (3), respectively.
uuc(t) = iuc (t)Rs+uC(t) +
N
X
n=1
uC
n=y(t),(5)
iuc(t) = u(t) = Pinv(t)
uuc(t),(6)
uuc,s(t) = Nsuuc (t) = Nsy(t),(7)
iuc,m(t) = Npiuc (t) = Npu(t),(8)
duC
dt =iuc(t)
C0+kvuC(t),(9)
duC
n
dt =−uC
n
RnCn
+iuc(t)
Cn
.(10)
III. ULTRA CA PACITOR CO NT ROL DESIGN
The proposed hierarchical control architecture applied to the
UC bank consists of two control levels defined as follows:
•upper control level—controller at this level calculates the
active and reactive power set-points which are passed
on to the lower control level. Reactive power control by
UC is out of the scope of this paper because the focus
is on frequency control (thus, this set-point is always
0). Furthermore, reactive power control is related to the
inverter and is independent of active power control. The
main purpose of this level is to provide a viable solution
for power system frequency control using a UC bank.
•lower control level—controllers at the inverter level are
used for tracking set points received from the upper
control level.
The lower level controls the inverter and is not analysed in
detail since numerous relevant examples can be found in the
literature. In this paper, the UC inverter is modeled as a grid-
following current source where the inverter current references
MANUSCRIPT SUBMITTED TO IEEE TRANSACTIONS ON ENERGY CONVERSION 5
are directly determined by fast PI control loops based on P,Q
control errors [34]. The upper control level is in charge of the
UC bank frequency control. Fig. 2 illustrates the introduced
hierarchical control structure. The overall UC control structure,
the PID frequency controller and the MPC based frequency
controller are defined in the following subsections.
Fig. 2: Hierarchical control structure.
A. Overall Control System
An ultracapacitor bank system (UCBS) with the control
systems is shown in Fig. 3. Pand Qare active and reactive
power injected or absorbed by the inverter to or from the grid,
while the asterisk (*) denotes a set-point value (Q∗= 0 in
this paper). Vac,grid is the AC voltage of the bus the inverter
is connected to, while idand iqare the direct and quadrature
axis currents of the inverter. The inverter is controlled in the
grid voltage reference frame (standard PQ control block with
PI current control), where the grid voltage phase θis estimated
by the phase lock loop (PLL) [34]. The PLL also estimates
the system frequency for the frequency control block. The
DC current calculation block calculates the UC current for
charging or discharging by dividing the output power by the
UC DC voltage. Charging and discharging is controlled by
measuring the state-of-voltage (SoV, analogous to state-of-
charge–SOC in batteries): when a UC is charged to its nominal
voltage, the charging process stops and when its voltage falls
below the lower voltage limit, the discharging process stops.
This is achieved by controlling the maximum charging and
discharging currents (ich,max and idch,max, respectively,) through
the charge control block.
B. PID Frequency Controller
The grid frequency control block from Fig. 3 is expanded
in Fig. 4. The input to this block is a grid frequency signal
estimated by the PLL and the output is the requested change
in power. The type of implemented control algorithm for
frequency response can be arbitrary. It is assumed that the
Grid frequency
control (Fig. 4)
PQ
control
P∗;P
Q∗;Q
f
i∗
d
V∗;Vac,grid
δP Charge control
&
LVRT
i0
d
i0
q
DC current
calculation
Ultracapacitor/
supercapacitor
model (5)–(10)
iuc,m uuc;uuc,s
Inverter
i∗
d
i∗
q
ich,max
idch,max
PPLL
cos θ∗;
sin θ∗
Vac,grid
f
P
Q
Fig. 3: Ultracapacitor bank system with controls.
PLL exhibits strong frequency tracking behaviour and that the
conditions in the grid are balanced and not distorted, however
frequency measurement is delayed by τdf = 100 ms [35] to
emulate the performance of a nonideal system. Although the
PLL can cause instabilities and errors in frequency tracking
especially in low-inertia systems [36], these issues are not in
the scope of this paper. In the literature, frequency controller
is usually a proportional or a derivative controller, or a
combination of both (PD). However, the UC does not have a
large amount of stored energy available. If the standard droop
control (pure proportional controller) is employed, the UC
output power is proportional to the frequency deviation both
during the transient state as well as in the steady-state. Once
the UC reaches the lower voltage limit during discharging,
the output power will instantaneously drop to zero, which
will cause a higher secondary frequency drop. This can be
mitigated to a certain extent by only allowing the UC to
provide support during the transient state. This is achieved
through a washout filter, which is more akin to a derivative
controller.
Therefore, two control loops are used for the grid frequency
control block of Fig. 3. They both use a washout filter to
diminish the contribution of the UC bank in the steady-state.
The inertia controller has a lower washout time constant and
a bigger gain. Inertia controller will have a stronger response
during the frequency transient, but will diminish more quickly.
The transient droop controller is also proportional to the rate of
change of frequency, however the washout filter time constant
is much higher (τd
wτi
w) and the gain is much smaller,
which means that the UC output power will be lower, but it
will last longer while conventional units pick up. Hence, in the
PID formulation, this controller has the form of a derivative
controller with a low-pass filter to smooth out high-frequency
noise, since differentiating in time is a process that inherently
amplifies the noise.
Although not always guaranteed, the UC will not be com-
pletely discharged this way and the additional frequency drop
will not always occur. This depends on the required energy
during the transient process, which not only depends on the
parameters of the frequency controller, but also on the size of
the disturbance, which is something that cannot be controlled.
This issue will be tackled using MPC, which will always
guarantee that the UC bank discharges smoothly and that the
maximum energy is extracted while satisfying the constraints
of the UC system. The described frequency controller is
replaced by the proposed MPC controller in the following
subsections.
MANUSCRIPT SUBMITTED TO IEEE TRANSACTIONS ON ENERGY CONVERSION 6
Transient droop controller
Inertia controller
∆fe−sτdf
Time
delay
Kd
Gain
sτd
w
sτd
w+ 1
Washout filter
Ki
sτi
w
sτi
w+ 1
+δP
Fig. 4: Ultracapacitor bank grid frequency control module.
C. Predictive Model of a UC Bank
The predictive model of a UC bank in the form of transfer
function that relates the UC bank output voltage Vwith the
charging/discharging power Pis expressed as:
V(z)
P(z)=G(z) = a3z+a4
z2+a1z+a2
.(11)
The transfer function coefficients a1,a2,a3and a4are identi-
fied by conducting simulations on the nonlinear simulation
model (5)–(10) of the UC bank (nonlinearities are due to
(9)–(10)). The identification procedure has been conducted
using Matlab System Identification Toolbox. Prediction model
coefficients a1–a4are given in the Appendix. The sampling
time Tsused to discretize the identified linear UC bank model
is set to 50 ms (the root mean square error over 1500 s,
hence a validation data-set of 30000 samples, was 30.21).
Fig. 5 and 6 show the training and validation data sets used
to conduct the linear model identification, while Fig. 7 shows
the response of the nonlinear UC model (black line) compared
with the identified linear model (blue line). Identified linear
model response has fitness of 82.59 %. In terms of the
prediction model quality the identified linear model fitness
level is satisfying.
0 500 1000 1500
800
900
1000
UC Voltage [V]
Input and output signals
0 500 1000 1500
Time [s]
-4
-2
0
2
4
Power [MW]
Fig. 5: Training set.
Furthermore, the discrete-time predictive model used within
the MPC controller in this paper is derived by using the gen-
eralized formulation of Controlled Auto Regressive Integrated
Moving Average (CARIMA) model expressed as:
a(z)∆yk=b(z)∆uk+T(z)(z).(12)
Since the output, i.e. voltage, can be directly measured,
the prediction model uses variables of the output and input
increment, and assumes the best estimate of future random
term T(z)(z)=0.
0 500 1000 1500
400
600
800
1000
UC Voltage [V]
Input and output signals
0 500 1000 1500
Time [s]
-5
0
5
Power [MW]
Fig. 6: Validation set.
0 500 1000 1500
Time [s]
400
600
800
1000
UC Voltage [V]
Identified linear model
Simulated non-linear model (validation set)
Fig. 7: Simulated non-linear (validation set) and identified
linear model output.
In (12), a(z)and b(z)are polynomials that represent de-
nominator and numerator of the transfer function, respectively.
These polynomials are expressed as
a(z) = 1 + a1z−1+· · · +anz−n, a(z)∆ = A(z),(13)
b(z) = b1z−1+· · · +bmz−m.(14)
where ∆is the (1 −z−1)operator. Since disturbance
estimate is implicit within the use of increments there is no
need for disturbance estimate in this prediction model.
The UC bank output voltage predictions can be found by
using the following compact matrix/vector form:
Vk+1
−−→
=H∆Pk
−→ +K∆Pk−1
←−−
+QVk
←−,(15)
where H,Kand Qmatrices are expressed as:
H=C−1
ACb,K=C−1
AHb,Q=C−1
A,(16)
in which matrices CA,Cb,HAand Hbare given in the
Appendix.
Furthermore, in (15), the vector Vk+1
−−→
represents predicted
output voltage of an UC bank, while the vector ∆Pk
−→ repre-
sents control sequence in a form of active power increments
of the converter’s active power set point. These vectors are
expressed as follows:
Vk+1
−−→
= [V(k+ 1) V(k+ 2) ... V (k+N)]T,(17)
∆Pk
−→ = [∆P(k) ∆P(k+ 1) ... ∆P(k+N−1)]T.(18)
Since the MPC formulation in this paper is set up to control
the signal increment, i.e. power increment calculated at each
MANUSCRIPT SUBMITTED TO IEEE TRANSACTIONS ON ENERGY CONVERSION 7
time step, the control signal produced by the controller at time
instant kis calculated as follows:
P(k) = P(k−1) + ∆P(k).(19)
D. Novel Autonomous Frequency Controller Based on Voltage
Reference Trajectory and MPC
In this section, we introduce a novel autonomous frequency
controller which is shown in Fig. 8. If either frequency
deviation or rate-of-change-of-frequency (ROCOF) are outside
user-defined limits (modelled by deadzones in Fig. 8), the
UC system switches to fast discharge mode in which the
DC voltage trajectory is equal to Vs(∆f) = V0+KV∆f,
where V0is the initial voltage reference (SoC) of the UC and
KVis the DC voltage – frequency droop gain in VDC/Hz.
The change in voltage due to change in frequency KV∆fis
limited to only negative gradients and to some desired negative
maximum change ∆Vmin which corresponds to some user-
defined maximum change of SoC for a change in frequency.
On the other hand, once the frequency is stabilized, i.e. both
the frequency deviation and the ROCOF are inside the user-
defined limits, the timer is triggered. If frequency deviation
and ROCOF are inside the user-defined limits for a user-
defined period of time Tsteady, UC system switches to the slow
charging mode in which the voltage reference trajectory is set
to the user-specified value Vs=Vref, e.g. maximum voltage
for a full charge. Simultaneously, a flag is triggered which
induces a change of MPC constraints for the control signal:
pmax,pmin ,∆pmax,∆pmin . These constraints are set conserva-
tively which will reduce the charging power so that it doesn’t
cause additional grid frequency excursions. The slow-charging
∆f=f−fn
d
dt == 0
== 0
Timer
≥Tsteady
Trigger on
rising edge
Vref
Flag to set MPC constraints
for slow charging mode
pmax,pmin, ∆pmax, ∆pmin
KVΣ
+
V0
+
d∆V
dt |min =−∞
d∆V
dt |max = 0
∆Vmin
∆Vmax = 0
Vs(∆f)
Fig. 8: Novel autonomous frequency controller based on
voltage reference trajectory and MPC.
mode is not enabled before the frequency is stable for some
time Tsteady because both the frequency deviation and ROCOF
can temporarily pass through the defined deadbands during
frequency containment process. Without this time delay, an
oscillatory behaviour between charging and discharging modes
can be triggered which will be reflected in grid frequency
oscillations. This time delay can be set from 30 seconds up
to around tens of minutes which is usually the time range in
which the frequency is theoretically stabilized and returned to
nominal value.
The presented control is superior to classic active power –
frequency droop control because the latter can cause a sudden
reduction of output power if UC energy is depleted, which
will be shown in the study cases. On the other hand, the
proposed approach directly determines how much energy will
be used for a certain frequency deviation, while the MPC
framework calculates the required power set-points to the
lower level converter control such that all system constraints
are satisfied. This results in a smoother discharge curve and
better performance in arresting the frequency excursions.
In this paper, the parameters of the proposed controller are
set as follows: frequency deviation deadband is set to ±50
mHz, ROCOF deadband is set to ±100 mHz/s, Tsteady = 30
s, KV= 1050 Vdc /Hz, ∆Vmin =−600 V, V0=Vref = 1000
V.
E. MPC Problem Formulation
The objective function to be minimized at each current point
in time kis defined as:
J= [Vs1−Vk+1
−−→
]TQy[Vs−Vk+1
−−→
] + ∆PT
k
−→
Qu∆Pk
−→ +STRS,
(20)
where 1is a column vector of Nones, Vsis the UC bank
output voltage reference trajectory defined in Section III-D.
The objective function (20) has a quadratic form that
consists of two terms: i) the first term is used to express
the error between the output voltage reference trajectory Vs
and the predicted UC bank voltage output V;ii) the second
term models the control effort. The optimal control sequence
resulting from the MPC algorithm consists of active power
increments. At each time step optimization problem is solved
and the first element of the resulting control sequence is then
sent to the local controller that controls the converter. The
matrices Qyand Quin (20) contain the weighting factors:
qypenalizes the predicted UC bank voltage output deviations
from the reference trajectory, while qupenalizes the change of
the control signals. The weighting matrices Qyand Quover
the prediction horizon Ncan be built as follows:
Qy=
qy. . . 0
.
.
.....
.
.
0. . . qy
,Qu=
qu. . . 0
.
.
.....
.
.
0. . . qu
.(21)
One of the main advantages of MPC schemes is their ability
to incorporate different types of constraints on control and
output signals into the control design. By doing so, the optimal
control sequence is calculated taking into account relevant sys-
tem constraints. However, one should be very careful in con-
straint formulation since they can lead to infeasibility issues.
In order to ensure the feasibility of the optimization problem
within an MPC scheme, its constraints can be formulated as
’hard’ and ’soft’ constraints. Hard constraints must be always
satisfied, while soft constraints can be violated to a certain ex-
tent so as it is possible to find a feasible solution at each MPC
iteration. In the proposed MPC algorithm hard constraints are
considered for the control signals (charging/discharging power
and the incremental charging/discharging power), whilst the
output constraints on the UC bank terminal DC voltage can
be softened since UC bank voltage violations are permissible
(in an allowed range specified by the manufacturer). These
constraints are softened by introducing a vector of N slack
variables, S, which is penalized in the objective function (20)
by adding the term STRS, with Rbeing a diagonal matrix of
MANUSCRIPT SUBMITTED TO IEEE TRANSACTIONS ON ENERGY CONVERSION 8
penalties. The weighting matrix Rover the prediction horizon
Ncan be built as follows:
R=
r . . . 0
.
.
.....
.
.
0. . . r
(22)
These constraints are expressed as
Pmin ≤P≤Pmax,(23)
−∆Pmin ≤∆P≤∆Pmax,(24)
Vmin −S≤V≤Vmax +S. (25)
where Pmin,Pmax ,∆Pmin,∆Pmax ,Vmin and Vmax
are column vectors with Nelements of pmin,pmax ,∆pmin,
∆pmax,vmin and vmax .
IV. RES ULT S AN D DISCUSSION
Effectiveness of the proposed hierarchical control structure
is validated on a nonlinear simulation model of the UC bank.
The quality of the upper-level controller based on the MPC
algorithm is compared with the response of a 3 standard PID
control approaches: i) virtual inertia + transient droop (Fig.
4, Kd>0;Ki>0;τd
wτi
w>0); ii) virtual inertia only
(Kd= 0;Ki>0;τi
w>0); iii) virtual inertia and droop
with dynamic power limitation based on SoC estimation from
[9] (Kd>0;Ki>0;τd
w=τi
w= 0). Four study cases
are analyzed: three different disturbance sizes with automatic
generation control (AGC) enabled in the power system and one
study case for the largest disturbance without AGC enabled.
MPC and PID controller settings used in all simulation cases
are given in Table I.
The PID parameters were chosen such that the desired
response to changes in system frequency is obtained while hav-
ing gain and phase margins of an open-loop system (frequency
input to converter power output). MPC parameters were cho-
sen to obtain smooth and fast discharging profile. Closed-loop
stability can be guaranteed by adopting standard approaches
in the literature, i.e. by adding a terminal cost to the cost
function or a terminal set [37]. It should be emphasized that
the controller settings do not represent optimal settings for
PID nor MPC controllers. Nonetheless, different tuning of PID
controller does not lead to significant differences in control
performance. The main MPC controller tuning parameters are
the prediction horizon Nand the weighting factors qy,qu
and r. Constraints are defined based on system limitations
in the simulation setup. Namely, maximum permissible UC
bank voltage is 1000 V, while minimum permissible UC
bank voltage is 400 V. In addition, it is also necessary to
include constraints on the control signals due to the physical
limitations of the converter used to couple the UC bank with
the grid.
The power system is modeled as an equivalent single
machine described by an inertia (H), a damping (D) and
a steam turbine with governor to simulate a power system
dominated by thermal units (Fig. 9). The turbine model in-
cludes frequency deadband, ramp rate and active power limits
[38], [39] which are provided in the Appendix. Active power
disturbance is represented by the input δpd.R,Tg,Fhand
Trare permanent droop, governor time constant, fraction of
the power generated by the high-pressure turbine and reheater
time constant. This low-order model is adequate since we are
only interested in the average frequency dynamics to test the
performance of the UC [7]. Pr/Sbis a factor for conversion
from the UC bank p.u. base to the system p.u. base. The
parameters of the UC bank (values based on standard realistic
ranges [30]–[32], [40], [41]) and power system (standard
values for power systems based on [6]) are given in the
Appendix.
∆f1
2Hs +D
−1/R +1
sTg+ 1
FhTrs+ 1
sTr+ 1
∆pmin
∆pmax
dp
dt|min
dp
dt|max
Σ
−0.1
s
∆pmT +δpd
−
Ultracapacitor model
(Fig. 2) Pr/Sb∆pUC
Fig. 9: Power system model.
TABLE I: Controller settings.
Controller Settings
PID1Kd= 20/0/20 Ki= 150/150/150
τd
w= 30/0/0τi
w= 1/1/0
MPC2
N=50
pmax = 10/0.5MW pmin =−10/−0.5MW
∆pmax = 0.1/0.1MW ∆pmin =−0.1/−0.1MW
vmax = 1000 Vvmin = 400 V
qy = 1000 qu = 150
r= 1500
A. Case 1—Disturbance Exceeds the Size of the UC Bank
A 15 MW disturbance is applied at t= 5 s causing
a frequency drop. Results are shown in Fig. 10. It can be
seen that both PID and MPC controllers satisfy the physical
constraints in terms of the UC bank permissible voltage
range (Fig. 10b) and power (i.e. current, Fig. 10d, Fig. 10c)
limitations of the converter. However, the MPC controller
has a significantly stronger power injection and discharges
completely in a smooth manner around 25 s. Consequently,
the frequency nadir is significantly reduced compared to PID
controllers (Fig. 10e): MPC-based control results in 0.07 Hz
(−10%) smaller nadir compared to virtual inertia + (transient)
droop controllers and approximately 0.17 Hz (−23%) smaller
nadir compared to only virtual inertia control. Moreover, MPC
controller visibly reduces the drop of frequency and it delays
1Parameters for 3 different PID layouts used for comparison are delimited
with a slash: e.g. for virtual inertia control [layout ii)]: Kd= 0,Ki= 150,
τd
w= 0,τi
w= 1
2Control signal constraints for fast discharge / slow charge are delimited
with a slash, e.g. for fast discharge the constraints are: pmax = 10 MW,
pmin =−10 MW, ∆pmax = 0.1MW, ∆pmin =−0.1MW
MANUSCRIPT SUBMITTED TO IEEE TRANSACTIONS ON ENERGY CONVERSION 9
the occurrence of nadir. On the other hand, the initial ROCOF
is worse for the MPC control (around 0.03 Hz/s higher initial
ROCOF or +25%, Fig. 10f). However, this is due to the fact
that all other PID controllers are sensitive to the frequency
time derivative, while MPC operates only on the frequency
deviation. This can potentially be mitigated by introducing a
d/dt term in the proposed frequency controller. Nevertheless,
around 7 s mark the ROCOF is quickly reduced by the
MPC controller and after a few oscillations the frequency is
stabilized. In terms of ROCOF, the behaviour between the
PID controllers is similar, while the MPC controller shows
somewhat more oscillatory behaviour in this case. It can be
seen that once the frequency is stabilized around 80 s mark,
the MPC controller switches to slow charging mode and the
SoC is slowly brought back to full charge, ready for the next
disturbance (Fig. 10a).
B. Case 2—Disturbance is Equal to the Size of UC Bank
A 10 MW disturbance is applied at t= 5 s. Results are
shown in Fig. 11. Both controllers satisfy physical constraints
in terms of permissible voltage range (Fig. 11b) and power (i.e.
current, Fig. 11d, Fig. 11c) limitations of the converter. MPC
controller shows superior performance in terms of frequency
nadir (Fig. 11e) resulting in 0.1 Hz (−22%) lower nadir
compared to virtual inertia + (transient) droop controllers and
in 0.17 Hz (−32%) lower nadir compared to virtual inertia
only. In this case, performance of MPC controller in terms
of ROCOF is similar to conventional PID controllers (Fig.
11f). Initial RoCoF is marginally worse in the case of MPC
controller (0.085 Hz/s compared to 0.08 Hz/s, or +6%). On
the other hand, MPC controller strongly reduces the ROCOF
around 7 s mark delaying the nadir and resulting in a generally
smaller ROCOF afterwards. Slow charging starts around 80 s
mark and finishes around 220 s mark which doesn’t violate
the defined frequency controller limits and the UC is slowly
charged to 100% ready for the next disturbance (Fig. 11a).
It is interesting to note that both virtual inertia and virtual
inertia + transient droop controllers charge back to 100% fairly
quickly. This is due to the df /dt action with a low pass filter
in their respective controllers which will charge the UC up to
a certain point on positive frequency gradient. On the other
hand, the purely proportional part (droop) of virtual inertia +
droop controller will discharge the UC to a certain point, but it
will not autonomously recharge the UC bank back to nominal
voltage, a problem solved by the proposed MPC controller.
C. Case 3—Disturbance is Smaller than the Size of UC Bank
A 5 MW disturbance is applied at t= 5 s. Results are com-
pared in Fig. 12. Once again, all controllers satisfy physical
constraints in terms of permissible voltage range (Fig. 12b) and
power limitations of the converter (Fig. 12d, Fig. 12c). As in
the previous cases, the MPC controller has a stronger response
to system frequency excursion resulting in significantly lower
frequency nadir: 0.19 Hz compared to 0.24 Hz (−20%) by
virtual inertia + (transient) droop controllers and 0.19 Hz
compared to 0.27 Hz by virtual inertia controller (−30%).
Initial ROCOF is marginally higher compared to virtual inertia
and virtual inertia + transient droop controllers (+7%) and
19% higher than virtual inertia + droop controller, however
MPC controller damps the ROCOF much more quickly than
the conventional controllers.
D. Case 4—No Automatic Generation Control
In this study case, AGC is disabled (integral term in the
power system model is zero) to illustrate the benefits of the
proposed MPC controller even further. In previous study cases,
AGC will pickup the generation-load mismatch to drive the
frequency error to zero thus decreasing the contribution of
the ultracapacitor. Additionally, all PID controllers were tuned
more aggressively so that the power injection to grid frequency
change is greater (all gains were scaled by a factor of 4).
A 15 MW disturbance is applied at t= 5 s. The MPC
controller shows a better performance compared to traditional
PID approaches (Fig. 13e and Fig. 13f). Virtual inertia + droop
controller exhibits a oscillatory behaviour due to high gains
(Fig. 13d), while the problem of sudden power loss once
the energy is depleted is visible in virtual inertia + transient
droop and pure virtual inertia controllers which causes an
additional disturbance. In terms of grid frequency response
(Fig. 13e and Fig. 13f) it can be seen that initially the PID
controllers have a stronger response which results in smaller
frequency deviation compared to MPC controller up to 15 s
mark. However, once the energy of the UC bank controlled
by PID controller is depleted, a secondary frequency drop
occurs which reduces the frequency even further. On the other
hand, MPC controller brings the frequency to steady-state
in a more controlled manner. Around 28 s mark, frequency
drops to 49.0 Hz in the case with virtual inertia + (transient)
droop controllers while in the case with MPC controller the
frequency at this instant is 0.2 Hz higher, or 49.2 Hz which is
significant. Similarly, the initial ROCOF is higher for the MPC
controller but is initially damped more quickly. On the other
hand, secondary frequency drop around 20 s mark in cases
with virtual inertia + (transient) droop controllers causes 50%
to 100% higher ROCOF than with MPC controller at the same
time instant.
V. CONCLUSION
The main goal of this paper is to show the potential benefits
of applying an MPC algorithm to UC bank operation opti-
mization for frequency control. The applied MPC algorithm
is based on a linear prediction model of a UC bank, while
a nonlinear model of the UC bank is used as a simulation
model to validate the algorithm. An autonomous frequency
controller based on voltage reference trajectory and MPC
has been proposed and compared against three classical PID
control layouts for different sizes of disturbances and with or
without automatic generation control in the power system. The
proposed controller autonomously and slowly recharges the
UC bank in the steady-state by changing the MPC formulation
constraints, while providing a strong and smooth response
during disturbances in the grid. The main conclusion is that
the MPC provides smoother power decrease and improves the
system frequency response (in terms of frequency deviations
MANUSCRIPT SUBMITTED TO IEEE TRANSACTIONS ON ENERGY CONVERSION 10
and ROCOF), since it inherently considers the UC system
constraints. For the simulated study cases, the proposed MPC
controller reduces the frequency nadir between 10%–30%. On
the other hand, although the initial ROCOF for the MPC
controller is higher 6% to 25%, it is damped more quickly.
Moreover, this somewhat inferior behaviour is due to the fact
that used PID controllers all have d/dt term which reacts more
strongly to ROCOF. Finally, aggressive tuning of PID con-
trollers can lead to unwanted oscillatory behaviour and sudden
loss of power once the energy is depleted causing another
frequency disturbance and 50% to 100% higher ROCOF at
that point. This behaviour has been mitigated with the use
of the proposed MPC controller which behaves consistently
across all simulated study cases. In addition, a smooth power
decrease also prevents over-voltage issues, which can occur
in the converter due to instantaneous current reduction. An
additional strength of the introduced MPC algorithm is that
it is based only on direct measurements (i.e. ultracapacitor
terminal DC voltage and power) and therefore state estimation
is not necessary, making it less computationally intensive.
Future research includes implementation and experimental
validation of the proposed algorithm on a real UC system.
APPENDIX
Prediction model coefficients: a1=−1.9921.a2= 0.9921.a3=
−0.4106.a4= 0.4107.
Ultracapacitor bank parameters: Rs= 0.25·10−3Ω,Rdc = 0.5·10−3
Ω,kv= 90 F/V, C0= 750 F, V0= 2.5V, Np= 100.Ns= 400.
Pr= 10 MW. converter ramp rate: ±10 p.u./s, power limit: ±1
p.u.
Power system parameters: H= 5 s, D= 1 p.u., R= 0.05 p.u.,
Tg= 0.2s, Tr= 8 s, Fh= 0.3p.u., Sb= 400 MVA., deadband:
±10 mHz, ramp rate: ±0.5 p.u./s, power limit: ±0.15 p.u.
Matrices CA,Cb,HAand Hb:
CA=
1. . . 0 0
A11 0 0
.
.
.....
.
.0
AN−1AN−2. . . 1
,
HA=
A1A2· · · An−4An−3· · · An−1An
A2A3· · · An−3An−2· · · An0
.
.
.· · · · · · An−2An−1· · · 0 0
ANAN+1 · · · An−1An· · · 0 0
,
Cb=
b10 0 0
b2b10 0
.
.
.....
.
.0
bNbN−1. . . b1
,
Hb=
b2b3· · · bm−4bm−3· · · bm−1bm
b3b4· · · bm−3bm−2· · · bm0
.
.
.· · · · · · bm−2bm−1· · · 0 0
bN+1 bN+2 · · · bm−1bm· · · 0 0
.
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Mateo Beus (S’17) received his M. Sc. degree in
electric power engineering from the University of
Zagreb Faculty of Electrical Engineering and Com-
puting, Croatia, in 2014. His employment experience
includes working as an engineer at Brodarski Insti-
tute Ltd and Eccos Engineering Ltd. Currently, he
works as a senior researcher and teaching assistant
at the University of Zagreb Faculty of Electrical
Engineering and Computing. His research interests
include distribution grid automation, integration and
management of distributed energy resources, control
of hydro power plants and control & protection of microgrids.
Matej Krpan (S’17) received the bachelor’s and
master’s degrees in electrical engineering from Uni-
versity of Zagreb Faculty of electrical engineering
and computing, Croatia, in 2014 and 2016, respec-
tively. He is currently pursuing a PhD degree in
electrical engineering at the Department of Energy
and Power Systems of the University of Zagreb
Faculty of electrical engineering and computing. His
research interests include power system dynamics,
stability and control of low-inertia power systems
with focus on fast frequency control.
Igor Kuzle (S’94–M’97–SM’04) is a Full Professor
at the University of Zagreb Faculty of Electrical
Engineering and Computing, Croatia. His scientific
interests include electric power system dynamics
and control, maintenance of electrical equipment,
smart grids, and the integration of renewable energy
sources. He was awarded the highest Croatian Na-
tional Science Award for 2018 for his outstanding
contribution in the field of smart grid applications
in the transmission system. In 2019, he was given
the award for excellence in the technical field by the
Croatian Academy of Sciences and Arts. The Award recognizes the work in
the field of application of different control concepts to increase power system
flexibility and enable further integration of renewable energy sources. He is
a member of the editorial boards of 8 journals and he served as chairman of
several international conferences. He is a member of the IEEE PES Governing
Board.
Hrvoje Pandˇ
zi´
c(SM’2017) received the M.E.E. and
Ph.D. degrees from the University of Zagreb Faculty
of Electrical Engineering and Computing, Croatia, in
2007 and 2011, respectively. From 2012 to 2014, he
was a Postdoctoral Researcher with the University
of Washington, Seattle, WA, USA. He is currently
an Associate Professor and Head of the Department
of Energy and Power Systems with the University
of Zagreb Faculty of Electrical Engineering and
Computing. He has coordinated multiple European
and national research projects as well as commercial
projects for industrial partners. He received numerous awards for his research
work, including the award for the highest scientific and artistic achievements
in Croatia from the Croatian Academy of Science and Arts. His research
interests include power system operation and planning.
Alessandra Parisio (Senior Member, IEEE) is a
Senior Lecturer in the Department of Electrical
and Electronic Engineering at The University of
Manchester, UK, where she is/has been principal
or co-investigator on research projects supported by
EPSRC, Innovate UK, EC H2020 and industrial
partners in the areas of building energy management
and distributed control for flexibility service and
grid support provision. Dr Parisio is IEEE senior
member, vice-Chair for Education of the IFAC Tech-
nical Committee 9.3. Control for Smart Cities, and
member of IEEE Technical Committees on Smart Grids and Smart Cities.
She has been in the program committees of several international conferences
and serves as editor of the Elsevier journal Sustainable Energy, Grids and
Networks (SEGAN), Results in Control and Optimisation (RICO) and the
IEEE Transactions on Automation and Science Engineering. Dr Parisio
received the IEEE PES Outstanding Engineer Award in January 2021 and the
Energy and Buildings Best Paper Award for (for a ten-year period between
2008-2017) in January 2019. Her main research interests include the areas
of energy management systems under uncertainty, model predictive control,
stochastic constrained control and distributed optimisation for power systems.
MANUSCRIPT SUBMITTED TO IEEE TRANSACTIONS ON ENERGY CONVERSION 12
SoC [%]
(a) SoC
Voltage [V]
(b) DC voltage
Current [kA]
(c) DC current
Power [MW]
(d) AC Power
Frequency [Hz]
(e) Frequency
ROCOF [Hz/s]
(f) ROCOF
Fig. 10: Case 1 results.
SoC [%]
(a) SoC
Voltage [V]
(b) DC voltage
Current [kA]
(c) DC current
Power [MW]
(d) AC Power
Frequency [Hz]
(e) Frequency
ROCOF [Hz/s]
(f) ROCOF
Fig. 11: Case 2 results.
MANUSCRIPT SUBMITTED TO IEEE TRANSACTIONS ON ENERGY CONVERSION 13
SoC [%]
(a) SoC
Voltage [V]
(b) DC voltage
Current [kA]
(c) DC current
Power [MW]
(d) AC Power
Frequency [Hz]
(e) Frequency
ROCOF [Hz/s]
(f) ROCOF
Fig. 12: Case 3 results.
SoC [%]
(a) SoC
Voltage [V]
(b) DC voltage
Current [kA]
(c) DC current
Power [MW]
(d) AC Power
Frequency [Hz]
(e) Frequency
ROCOF [Hz/s]
(f) ROCOF
Fig. 13: Case 4 results.