# A Data-Driven Condition Monitoring Method for Capacitor in Modular
Multilevel Converter (MMC)

Shuyu Ou∗, Mahyar Hassanifar†, Martin Votava‡, Marius Langwasser†, Marco
Liserre†‡,
Ariya Sangwongwanich∗, Subham Sahoo∗, and Frede Blaabjerg∗
∗ Department of Energy, Aalborg University, Aalborg, Denmark
† Chair of Power Electronics, Kiel University, Kiel, Germany
‡ Electronic Energy Systems, Fraunhofer Institute for Silicon Technology ISIT,
Kiel, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>0000-0002-6339-6984 0000-0003-1506-6312 0000-0001-5762-7646
0000-0001-9909-7516 0000-0002-0818-2684 0000-0002-2587-0024
0000-0002-7916-028X 0000-0001-8311-7412

###### Abstract

The modular multilevel converter (MMC) is a topology that consists of a high
number of capacitors, and degradation of capacitors can lead to converter
malfunction, limiting the overall system lifetime. Condition monitoring
methods can be applied to assess the health status of capacitors and realize
predictive maintenance to improve reliability. Current research works for
condition monitoring of capacitors in an MMC mainly monitor either capacitance
or equivalent series resistance (ESR), while these two health indicators can
shift at different speeds and lead to different end-of-life times. Hence,
monitoring only one of these parameters may lead to unreliable health status
evaluation. This paper proposes a data-driven method to estimate capacitance
and ESR at the same time, in which particle swarm optimization (PSO) is
leveraged to update the obtained estimations. Then, the results of the
estimations are used to predict the sub-module voltage, which is based on a
capacitor voltage equation. Furthermore, minimizing the mean square error
between the predicted and actual measured voltage makes the estimations closer
to the actual values. The effectiveness and feasibility of the proposed method
are validated through simulations and experiments.

###### Index Terms:

Condition monitoring, capacitor, data-driven, modular multilevel converter.

##  I Introduction

Modular multilevel converter (MMC) attracts wide research attention in medium
voltage and high voltage applications such as HVDC and STATCOM due to its high
modularity, low harmonic component, high efficiency, etc [1, 2, 3, 4]. Since
MMC consists of a large number of components, the reliability of critical
components in the MMC should be evaluated carefully. Capacitors are one of the
critical components in the MMC, as they can stabilize the submodule voltage
and are key elements in voltage level generation. Similar to any other
component, capacitors are subjected to aging over time as they suffer from
electrical and mechanical stresses during operation. The degradation can be
monitored and maintenance plan can be updated accordingly. The method of
scheduling the maintenance intervention based on the health status is known as
predictive maintenance, which can mitigate the loss of unplanned downtime and
additional maintenance costs [5]. The key to do predictive maintenance is
getting a reliable and accurate health status of the critical components,
where condition monitoring methods can help.

There are different approaches to monitoring the health status of capacitors
in the MMC, and the main research direction is monitoring the capacitance.
This is due to the fact that the voltage and current harmonics in MMC are
mainly located in the low-frequency region (below a few kHz) and the
capacitance dominates the equivalent circuit model in this region, so that the
other components in the equivalent circuit model, namely equivalent series
resistance (ESR) and equivalent series inductance (ESL), are neglected in the
capacitance calculation process [6]. The capacitance can be calculated with
voltage and current harmonics, and these harmonics can come from either normal
operation [7, 8, 9, 10] or injected harmonics [11]. The capacitance can also
be calculated with a capacitance-voltage relationship based on a reference
submodule, where the submodule voltage ripple is inversely proportional to the
capacitance when the same switching signals are given to the submodule under
test and the reference submodule [12, 13, 14]. It is possible that the
capacitance-voltage relationship is built based on the discharging voltage
curve, however, with the expenses of the exclusion of a submodule [15]. Using
data-driven methods is another way to monitor the capacitance, e.g., the
Kalman filter [16] and recursive least square [6], which needs a more
complicated structure than model-based methods.

Besides the capacitance, ESR in the equivalent circuit model can also be used
to monitor the health status since ESR can deteriorate faster than the
capacitance and reaches the end-of-life criteria much earlier than the
capacitance, as reported in [17, 18, 19, 20]. In these cases, if the condition
monitoring method only monitors capacitance, the converter might operate with
a degraded capacitor that has a high ESR, and the end-of-life warning is not
triggered until the capacitance reaches the end-of-life criteria which makes
the estimated health status unreliable. Considering the importance of
monitoring ESR in reflecting the capacitor degradation, recently published
research has started to monitor the ESR for capacitors in an MMC, e.g., by
using the wavelet decomposition to analyze the turn-on transient voltage step
which requires extra components to measure the fast and small voltage step
[21] and the harmonic-based method [22].

There are certain limitations in the existing methods: 1) only monitoring a
single health indicator, e.g., [12, 13, 14, 11, 7, 8, 9, 10]. Considering the
capacitance and ESR can shift at different speeds, the end-of-life time
indicated by them can be different. For instance, if ESR degrades faster than
the capacitance, the end-of-life time related to ESR will be shorter than the
one related to capacitance. Between these two end-of-life times, the end-of-
life warning is not triggered if only monitoring the capacitance even though
the actual ESR already exceeds the end-of-life criteria. 2) the normal
operation of the converter is affected during the condition monitoring due to
changes in the control behaviors, e.g., [12, 13, 14].

To solve these problems, this paper applies a data-driven condition monitoring
method to monitor capacitance and ESR together, so the health status can be
derived no matter which health indicator (capacitance or ESR) degrades faster.
The reason for using capacitance and ESR as health indicators is that these
two health indicators cover the main failure mechanism, i.e., electrolyte
evaporation of the aluminum electrolytic capacitor which is suitable for MMC
applications [23]. Adding more health indicators makes the condition
monitoring method more complicated without covering more failure mechanisms.
For instance, the weight of the capacitor decreases as the electrolyte
evaporates. However, measuring the weight requires disassembling the capacitor
from the print circuit board, and the weight is less sensitive than the
capacitance and ESR [24, 25].

Specifically, the proposed method uses particle swarm optimization (PSO) to
update the position of particles, where the positions are estimated
capacitance and ESR. Then the estimations are used to predict the submodule
voltage in a sampling window based on a capacitor voltage equation. The mean
square error between the predicted and measured submodule voltage is used as
the objective function. The update process reduces the voltage prediction
errors, and in the meantime, the estimations of health indicators are closer
to the actual values.

Compared with the existing methods, the proposed method has the advantages of:

  * •

Monitoring both capacitance and ESR so the condition monitoring result is more
reliable compared with methods based on one of the health indicators.

  * •

The method runs offline so the normal operation of the converter is not
affected, and it requires no extra components.

The system description is given in Section II; the proposed method is
introduced in Section III; and validations are provided in Sections IV-V.

##  II System Description

This section introduces the MMC topology and operation, as well as the failure
mechanism and health indicators of capacitors.

Figure 1: A modular multilevel converter, (a) Converter topology, (b) half-
bridge submodule, and (c) model of the submodule capacitor with parasitic
parameters.

###  II-A Modular Multilevel Converter (MMC)

Three-phase MMC with one phase in detail is depicted in Fig. 1(a). As
demonstrated, three phases are connected with a medium voltage ac grid
($v_{a}$, $v_{b}$, and $v_{c}$) with a filter inductor $L_{f}$ and a filter
resistor $R_{f}$. Each phase of MMC is composed of two arms, known as the
upper and lower arms. A series of connections of submodules with an arm
inductor $L_{arm}$ and equivalent arm resistor $R_{arm}$ forms an arm. The
topology of the half-bridge submodule is shown in Fig. 1(a), which consists of
a capacitor and two IGBTs with their antiparallel diode, namely $C$, $S_{1}$,
$S_{2}$, $D_{1}$, and $D_{2}$, respectively.

The MMC can be used as an inverter or an active front-end rectifier, where the
outer control loop is responsible for it. In a typical MMC controller, the
generated current references are considered as inputs of the inner control
loop. Then, the reference of the arm voltages is produced and used in the
modulation step. Apart from the mentioned controllers, MMC exploits some other
controllers to improve the performance of the submodules, such as circulating
current control, energy balancing control, etc. [26, 27].

###  II-B Failure Mechanism of Capacitors

The capacitor being analyzed in this section is limited to the aluminum
electrolytic capacitor (AEC), which is suitable for MMC applications because
of its high energy density and relatively low cost [23].

The capacitor can experience electrical stress (e.g., voltage and current
ripple) and thermal stress in the operation stage. For AECs, these stresses
can lead to electrolyte evaporation, oxide film degradation, and anode foil
degradations [19].

The degradation mechanisms can be monitored with electrical or non-electrical
health indicators. The electrical indicators, e.g., capacitance and ESR, can
be measured online and they have a clear relation with the end-of-life
criteria. While the non-electrical indicators, e.g., weight and pressure, are
difficult to measure online and require a conversion stage to estimate the
remaining useful lifetime. Therefore, using electrical indicators to estimate
the health status is main research direction in capacitor condition
monitoring.

For the AEC, the capacitor reach the end-of-life time if one of these criteria
is satisfied [23]:

  * •

Capacitance reduces to $80\text{\,}\mathrm{\char 37\relax}$ of the initial
value;

  * •

Equivalent series resistance increases to $200\text{\,}\mathrm{\char
37\relax}$ of the initial value.

The equivalent circuit of the capacitor is given in Fig. 1(c), consisting of a
main capacitance $C$, an equivalent series resistance ESR, and an equivalent
series inductance ESL. The capacitance and ESR affect the impedance when the
frequency is below a few $\text{\,}\mathrm{MHz}$ so the ESL is negligible. The
degradation process and the shifting of these two parameters are shown in Fig.
2(a)-(b), where $C$ reduces gradually from the initial value $C_{0}$ to
$80\text{\,}\mathrm{\char 37\relax}$ of $C_{0}$ at time $t_{C-EOL}$, and the
ESR increases from the initial value $ESR_{0}$ to $200\text{\,}\mathrm{\char
37\relax}$ $ESR_{0}$ at $t_{R-EOL}$.

The figure shows also that these two indicators can vary at different speeds
so they reach the end-of-life criteria at different times, i.e.,
$t_{C-EOL}\neq t_{R-EOL}$.

Fig. 2(a) shows the ESR degrades faster than $C$ so $t_{R-EOL}<t_{C-EOL}$ and
Fig. 2(b) shows the opposite trend so that $t_{R-EOL}>t_{C-EOL}$. The worst
case is that the condition monitoring method only monitors the health
indicators that degrade slower, e.g., only monitor $C$ in Fig. 2(a) so the
end-of-life condition can not be detected at time $t_{R-EOL}$. From
$t_{R-EOL}$ to $t_{C-EOL}$, ESR exceeds the end-of-life criteria, which can
generate higher losses and lead to capacitor failure, but the end-of-life time
warning can only be triggered at $t_{C-EOL}$.

Figure 2: Health indicators of capacitors changing gradually with operation
time. When the capacitance reduces to 0.8 of the initial value $C_{0}$ or the
ESR increases to two times the original $ESR_{0}$, the capacitor reaches the
end-of-life time $t_{C-EOL}$ or $t_{R-EOL}$, respectively. (a) ESR reaches the
end-of-life criteria earlier than the capacitance, and (b) Capacitance reaches
the end-of-life criteria earlier than the ESR.

##  III Condition Monitoring Method

Figure 3: A system diagram of the proposed condition monitoring method: (a) a
flowchart of the condition monitoring method; (b) a sampling window of three
input signals: the submodule voltage $v_{SM}(t_{k})$, the switching state
$v_{sw}(t_{k})$, and the arm current $i_{arm}(t_{k})$, where time
$t_{k}=t_{1},t_{2},...,t_{N}$ consists of $N$ points; (c) the particle swarm
optimization (PSO) method updates the positions of each particle to have a low
voltage error; (d) the voltage prediction module predicts the submodule
voltage $\hat{v}_{SM}(t_{k})$ in the sampling window, where the prediction
with 1 p.u. estimations is closer to the measured voltage; (e) the mean square
error between predicted and measured voltage error is calculated, and it is
minimized when both C and ESR are 1 p.u.

To overcome the limitations in the previous capacitor condition monitoring
methods, this paper proposes a method to estimate the two health indicators
(being capacitance and ESR) together, while not affecting the converter
operation.

The system diagram of the proposed method is illustrated in Fig. 3(a). The
proposed method starts with taking input signals and initializing capacitance
and ESR which are randomly distributed in the solution space. Then the
capacitance and ESR are updated with particle swarm optimization, and the
updated estimations are used to predict the submodule voltage within a
sampling window based on the capacitor voltage equation. The mean square error
between the predicted and measured voltage is calculated and compared with an
error limit to determine if the update should continue. The condition
monitoring method is repeated multiple times ($R_{m}$) and the median values
of these estimations are taken as the final estimations to avoid using an
outlier as the final estimations. The key modules in the flowchart are
introduced in the following sections.

###  III-A Input Signal

Three signals related to one submodule being monitored are measured, including
the submodule voltage $v_{SM}(t_{k})$, the switching state of the submodule
$v_{sw}(t_{k})$, and the arm current $i_{arm}(t_{k})$.

An example waveform is shown in Fig. 3(b). These three signals are time series
in a sampling window, $t_{k}=t_{1},t_{2},...,t_{N}$. The sampling window is
selected as half of the fundamental cycle within which there are some
switching transients because it is important for ESR estimation [21], while
the window is shorter than a fundamental cycle to mitigate the error in the
voltage prediction process. The sampling frequency is
$100\text{\,}\mathrm{kHz}$, which is much higher than the switching frequency
of an MMC (usually lower than several $\text{\,}\mathrm{kHz}$ [28]) to ensure
the prediction accuracy.

###  III-B Update C and ESR with Particle Swarm Optimization

Particle swarm optimization (PSO) is a classical metaheuristic search method.
It searches the solution space with a group of particles and the positions of
the particles are updated gradually with shared information to find the
optimal solution.

In the proposed method, each particle has a 2-dimensional position: estimated
capacitance and ESR. These two values are assigned randomly within the
solution space during the initialization stage, marked as green circles in
Fig. 3(c). Each particle also has a randomly given initial velocity. The
position of each particle leads to an individual voltage error. In each
iteration, particles update their velocities and positions [29]:

$\begin{split}v_{j}=\omega_{j}v_{j-1}+c_{1}r_{1j}(x_{p}-&x_{j-1})+c_{2}r_{2j}(x_{g}-x_{j-1})\\\
x_{j}=&x_{j-1}+v_{j}\end{split}$ (1)

where $v_{j}$ and $v_{j-1}$ are velocities, $x_{j}$ and $x_{j-1}$ are
positions of the $j^{th}$ and $(j-1)^{th}$ iteration, respectively; $x_{p}$
and $x_{g}$ are individual and global optimal positions; $\omega_{j}$,
$c_{1}$, and $c_{2}$ are inertia weight, cognitive weight and social weight,
respectively; $r_{1j}$ and $r_{2j}$ are two random numbers between 0 and 1,
varying in each iteration. The inertia weight $\omega_{j}$ decreases gradually
so that as the iteration increases, the velocity reduces and the searching
step becomes more refined [29]. The reducing velocity can be seen on the
trajectory, which moves in a wider range at the beginning and becomes more
stable at the end. Three simplified trajectories are plotted in Fig. 3(c) as
green dashed lines.

Eq. 1 shows that the positions are updated according to the global and
individual optimal solution, so the voltage errors of all particles reduce
gradually and the positions of particles move to the optimal solution
gradually. In other words, the update process uses particles to approach the
actual values.

###  III-C Voltage Prediction and Error Calculation

The voltage prediction block takes the measured signals (i.e.,
$i_{arm}(t_{k})$ and $v_{sw}(t_{k})$) as well as the estimated parameters
(i.e., $\hat{C}$ and $\hat{R}_{C}$) to predict a series of submodule voltage
$\hat{v}_{SM}(t_{k})$, where $t_{k}=t_{1},t_{2},...,t_{N}$ covering the entire
sampling window. At the first time step $t_{1}$,
$\hat{v}_{SM}(t_{1})=v_{SM}(t_{1})$. The prediction starts from the second
time step $t_{2}$ to the last time step $t_{N}$ in the sampling window, and
the prediction is based on a capacitor voltage equation:

$\begin{split}\hat{v}_{SM}(t_{k})=&\hat{v}_{SM}(t_{k-1})+\frac{i_{c}(t_{k})T_{s}}{\hat{C}}+{\hat{R}_{C}}[i_{c}(t_{k})-i_{c}(t_{k-1})]\end{split}$
(2)

where $T_{s}$ is the sampling time; $i_{c}(t_{k})$ is the capacitor current
and it is the product of the arm current and the switching state:

$\begin{split}i_{c}(t_{k})=i_{arm}(t_{k})v_{sw}(t_{k})\end{split}$ (3)

where the switching state $v_{sw}(t_{k})$ is one when the upper switch of the
submodule is turned on and the capacitor is in the charging/discharging state;
while $v_{sw}(t_{k})$ is zero when the lower switch in the submodule is turned
on and the capacitor is bypassed.

The prediction is evaluated with the mean square error between the predicted
submodule voltage $\hat{v}_{SM}(t_{k})$ and the measured voltage
$v_{SM}(t_{k})$.

$V_{err}=\frac{1}{{V_{m}^{2}}N}\sum_{t_{k}=t_{1}}^{t_{N}}(v_{SM}(t_{k})-\hat{v}_{SM}(t)_{k})^{2}$
(4)

where ${V_{m}}$ is the maximum measured submodule voltage within the sampling
window and $N$ is the number of time steps in the sampling window.

The submodule voltage error is compared with an error limit to determine if
the capacitance and ESR estimations should be updated. A large error indicates
that the estimations deviate significantly from the actual value while a small
error reveals that the estimations are close to the actual values. Fig.3(d)
shows the voltage prediction waveforms. When the estimated capacitance and ESR
are different from 1 p.u., the prediction waveforms in dashed lines
significantly deviate from the measured voltage (a solid blue line). The
voltage error against the normalized capacitance and ESR when their
counterparts are 1 p.u. are given in Fig. 3(e), which shows that the voltage
error is minimized when both estimations achieve 1 p.u.

##  IV Simulation Study

This section validates the design, which is the selection of swarm size and
error limit for the data-driven condition monitoring method, and the result
which includes voltage predictions and health indicator estimations.

###  IV-A Simulation Setup

The condition monitoring method is validated with a three-phase MMC simulation
model built in PLECS. The circuit model and parameters are listed in Table I
and the MMC converter is operating in the rectifier mode at full load, i.e.,
$P_{in}=$3\text{\,}\mathrm{MW}$$. The key waveforms are illustrated in Fig. 4.
The current is regulated by a current controller where it can be seen that the
grid current ($i_{a}$, $i_{b}$, and $i_{c}$) is in phase with the phase
voltage ($v_{a}$, $v_{b}$, and $v_{c}$). The upper and lower arm currents
($i_{armu}$ and $i_{arml}$) are balanced by a circulating current controller.
The submodule voltage is controlled with a sorting method to balance the
submodule voltages of 20 submodules in one arm ($v_{SMu1-20}$ and
$v_{SMl1-20}$).

After running the simulation model, the signals of the submodule voltages, arm
current, and switching signals are recorded for condition monitoring purposes.

TABLE I: Simulation and experimental setup parameters. Parameter | Var. | Simulation | Experimental
---|---|---|---
Input power | $P_{in}$ | $3\text{\,}\mathrm{MW}$ | N/A
DC voltage | $V_{DC}$ | $40\text{\,}\mathrm{kV}$ | $200\text{\,}\mathrm{V}$
AC voltage | $V_{AC}$ | $16.5\text{\,}\mathrm{kV}$ | N/A
AC frequency | $f_{g}$ | $50\text{\,}\mathrm{Hz}$ | $50\text{\,}\mathrm{Hz}$
SM switching frequency | $f_{sw}$ | $3\text{\,}\mathrm{kHz}$ | $1\text{\,}\mathrm{kHz}$
Full bridge switching frequency | $f_{cg}$ | N/A | $10\text{\,}\mathrm{kHz}$
Filter inductance | $L_{f}$ | $40\text{\,}\mH$ | $5.4\text{\,}\mH$
Arm inductance | $L_{arm}$ | $10\text{\,}\mH$ | N/A
Filter resistance | $R_{f}$ | $100\text{\,}\mathrm{m\SIUnitSymbolOhm}$ | $1\text{\,}\mathrm{\SIUnitSymbolOhm}$
Arm resistance | $R_{arm}$ | $100\text{\,}\mathrm{m\SIUnitSymbolOhm}$ | N/A
Number of SM per arm | - | 20 | 1
SM capacitance | $C$ | $2.2\text{\,}\mF$ | $2.26\text{\,}\mF$
SM capacitance ESR | ESR | $40\text{\,}\mathrm{m\SIUnitSymbolOhm}$ | $44.12\text{\,}\mathrm{m\SIUnitSymbolOhm}$
Sampling window | - | $10\text{\,}\mathrm{ms}$ | $10\text{\,}\mathrm{ms}$
Sampling frequency | $f_{sa}$ | $100\text{\,}\mathrm{kHz}$ | $100\text{\,}\mathrm{kHz}$
Voltage loop bandwidth | $f_{v}$ | N/A | $20\text{\,}\mathrm{Hz}$
Current loop bandwidth | $f_{i}$ | $800\text{\,}\mathrm{Hz}$ | $800\text{\,}\mathrm{Hz}$
Repeat times | $R_{m}$ | 15 | 15
Swarm size | - | 10 | 10
Error limit | - | $10^{-6}$ | $10^{-6}$
Maximum iteration | - | $100$ | $100$
Cognitive weight | $c_{1}$ | 1.49 | 1.49
Social weight | $c_{2}$ | 1.49 | 1.49
Boundary of $C$ estimation | - |  $[1.1,6.6]$$\text{\,}\mF$ |  $[1.1,6.6]$$\text{\,}\mF$
Boundary of $ESR$ estimation | - |  $[20,120]$$\text{\,}\mathrm{m\SIUnitSymbolOhm}$ |  $[20,120]$$\text{\,}\mathrm{m\SIUnitSymbolOhm}$
Figure 4: MMC simulation waveforms of (a)three-phase grid voltage, (b) three-
phase arm current, (c) arm current in phase A, (d) submodule voltage of the
upper and lower arm in the phase a.

###  IV-B Parameter Design of Data-driven Condition Monitoring

Figure 5: The effect of the swarm size on estimation results distribution.
Figure 6: The effect of the error limit on estimation results distribution.

To tailor the optimization method for condition monitoring, the effect of two
parameters is studied: the number of swarms (also named the swarm size), and
the error limit.

The swarm size is usually between 20 and 30 [29] therefore 30 is the maximum
swarm size to be studied. The test conditions are listed in Table I, and the
capacitor has capacitance and ESR at their initial values, 1 p.u. The
condition monitoring method repeats for a hundred times to visualize the
distribution of estimations, as shown in Fig. 5.

The scatter plot and the boxplot illustrate the distribution of estimations.
With an increasing swarm size, the estimations (blue dots) concentrate
gradually. The box plots also illustrate the concentration of estimations; the
height of the box (named the interquartile range or IQR) covers most of the
estimations and reduces gradually with increased swarm sizes. For instance, if
the swarm size is 5, the interquartile ranges of capacitance and ESR
estimations are $0.21\text{\,}\mathrm{\char 37\relax}$ and
$14.47\text{\,}\mathrm{\char 37\relax}$ of 1 p.u., respectively; while if the
swarm size is 10, these values are $0.11\text{\,}\mathrm{\char 37\relax}$ and
$2.95\text{\,}\mathrm{\char 37\relax}$, respectively. Considering the
acceptable error ranges for $C$ and ESR are $\pm$1\text{\,}\mathrm{\char
37\relax}$$ and $\pm$10\text{\,}\mathrm{\char 37\relax}$$, respectively [22],
the IQR of ESR when the swarm size is 5 is $14.47\text{\,}\mathrm{\char
37\relax}$ which is higher than the acceptable error range
$\pm$10\text{\,}\mathrm{\char 37\relax}$$. Therefore, it is better to have a
larger swarm size, and the swarm size is selected as ten, which gives stable
estimations in the acceptable error range.

The second parameter that needs to be considered is the error limit. The error
limit is usually in the range of $10^{-3}$ and $10^{-7}$, and its effect on
estimations and iterations are shown in Fig. 6. When the error limit reaches
$10^{-6}$, both capacitance and ESR estimations concentrate around the median
value, with interquartile ranges of $0.19\text{\,}\mathrm{\char 37\relax}$ and
$3.86\text{\,}\mathrm{\char 37\relax}$, respectively. Therefore, the error
limit is selected as $10^{-6}$.

###  IV-C Voltage Prediction

The voltage prediction function is validated in the full load condition with
two capacitors: a healthy capacitor ($C$ = ESR = 1 p.u.) and a fully degraded
capacitor ($C$ = 0.8 p.u. and ESR = 2.0 p.u.).

The sensed voltage is compared with the predicted voltage in Fig. 7(a). The
voltage prediction in dashed lines can follow the voltage measurement in solid
lines with an instantaneous voltage error below $2\text{\,}\mathrm{V}$, which
is $0.1\text{\,}\mathrm{\char 37\relax}$ of the
$2\text{\,}\mathrm{k}\mathrm{V}$ average submodule voltage, as seen in Fig.
7(b). In the zoomed-in waveform on the right side, voltage steps caused by
current steps during the switching transient can be seen. The magnitude of the
voltage step is proportional to the ESR (Eq. (2)) so the degraded capacitor
has a higher voltage step because of the higher ESR.

Considering the voltage error is below $2\text{\,}\mathrm{V}$ or
$0.1\text{\,}\mathrm{\char 37\relax}$ of the $2\text{\,}\mathrm{kV}$, the
estimation error is acceptable and should only affect the capacitance and ESR
estimations in a limited range.

Figure 7: Simulation results: (a) waveforms of the sensed submodule voltage
$v_{SM}$ (solid lines) and predicted submodule voltage $\hat{v}_{SM}$ with
different capacitances and ESRs (dashed lines); (b) the instantaneous error
between $v_{SM}$ and $\hat{v}_{SM}$; (c) and (d) are boxplots of the estimated
capacitances and ESRs.

###  IV-D C and ESR Estimation

This section validates the function of capacitance and ESR estimation with
different health statuses. The test condition is that $C$ and ESR vary
linearly between the healthy condition and the end-of-life criteria:

  * •

$C$: 1, 0.99, …, 0.8 p.u.

  * •

ESR: 1, 1.05, …, 2 p.u.

The solution space is half and three times the nominal value (i.e., $\hat{C}$
$\in$ [0.5,3] p.u. and $\hat{R_{C}}$ $\in$ [0.5,3] p.u.) to cover both the
end-of-life criteria and a $\pm$20\text{\,}\mathrm{\char 37\relax}$$
tolerance.

The condition monitoring method repeats 15 times for 20 submodules in the
upper arm of phase A, and all normalized estimations are illustrated with
boxplots in Figs. 7(c) and (d). The acceptable estimation ranges,
$1\pm$1\text{\,}\mathrm{\char 37\relax}$$ and $1\pm$10\text{\,}\mathrm{\char
37\relax}$$ for capacitances and ESRs, respectively, are colored with a white
background, while the other ranges are marked with a light red background. The
median values of estimations are red lines, and a green line at 1 p.u. is
added as a reference. The median values for the 20 monitored capacitors are in
the acceptable range. In total, the average estimation errors of capacitance
and ESR are $0.18\text{\,}\mathrm{\char 37\relax}$ and
$5.47\text{\,}\mathrm{\char 37\relax}$.

The boxplot can also show the distribution of all estimations; the blue box
represents the $\pm$25\text{\,}\mathrm{\char 37\relax}$$ range around the
median value. For the capacitance and the ESR estimations, most of the blue
boxes are in the ranges of $\pm$1\text{\,}\mathrm{\char 37\relax}$$ and
$\pm$10\text{\,}\mathrm{\char 37\relax}$$, which shows that the condition
monitoring method can have acceptable estimations during most of the
iterations.

##  V Experimental Result

###  V-A Test Setup

Experiments are carried out with a mission profile emulator (MPE). The MPE is
a simplified reliability test setup; it emulates a mission profile for the
submodule under test similar to the full MMC operation. It can reduce the
required testing time and cost for building and testing an entire MMC
converter because an MMC usually has a large number of components and
complicated control schemes.

The mission profile emulator MPE parameters are listed in the experimental
column of Table I and the control diagram is illustrated in Fig. 8(a). The MPE
includes a DC voltage supply, a full-bridge current generator, a filter
inductor $L_{f}$, a filter resistor $R_{f}$, and a submodule for testing (also
named the device under test or DUT). There are two control loops in the
control diagram, a current control loop and a voltage control loop. The
current controller gets the reference current from the simulation and
regulates the filter current $i_{arm}$ by controlling the full-bridge current
generator; while the voltage controller maintains the DC component in the
submodule voltage $v_{SM}$ by adding an offset $v_{SMDC}^{*}$ to the voltage
reference. Therefore, the filter current $i_{arm}$ includes a DC component and
an AC component.

The MPE test setup is illustrated in Fig. 8(b). The submodule voltage $v_{SM}$
and the filter current $i_{f}$ are sensed with a voltage divider and a current
sensor LEM CKSR 15NP, respectively. These signals are sent to the controller
implemented on an FPGA and an ARM processor on a ZedBoard for control
purposes. They are also sent to the oscilloscope and saved for condition
monitoring validation. The DUT is connected to a group of three aluminum
electrolytic capacitors. The overall capacitance is $2.26\text{\,}\mF$ and the
ESR is $44.12\text{\,}\mathrm{m\SIUnitSymbolOhm}$ which are measured with an
impedance analyzer Keysight E4990A.

Figure 8: Mission profile emulator test setup: (a) Topology and control
diagram of the mission profile emulator used to validate the method. (b) Test
setup.

###  V-B Experimental Result and Discussion

Figure 9: Experimental waveforms: (a) test scenario 1 and (b) test scenario 3. TABLE II: Experimental test conditions and estimation errors of capacitance and ESR. Test scenario | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
$V_{SMDC}^{*}$ (V) | 50 V | 30 V
${I}_{AC}^{*}$ (A) | 9 | 6 | 3 | 9 | 6 | 3
${C}_{err}$ ($\%$) | -0.65 | -0.63 | 1.84 | -0.43 | 1.02 | 2.39
${ESR}_{err}$ ($\%$) | 2.28 | -4.46 | -4.45 | 5.01 | 8.15 | 5.85
Figure 10: Comparison between the measured submodule voltage and the predicted
submodule voltage. (a) The measured and estimated submodule voltage waveforms
for scenario 1 and scenario 3 in $0.05\text{\,}\mathrm{s}$; (b) The voltage
error between the measurement and estimation; (c) and (d) are boxplots of
capacitance and ESR estimations in six scenarios. respectively.

The experiments are designed to validate that the proposed method can provide
accurate estimations under different load conditions. The testing scenarios
are listed in Table II, showing the DC component of the submodule voltage
varies from $50\text{\,}\mathrm{V}$ to $30\text{\,}\mathrm{V}$, and the
magnitude of the AC reference current varies between $9\text{\,}\mathrm{A}$,
$6\text{\,}\mathrm{A}$, and $3\text{\,}\mathrm{A}$. The operating waveforms
for the first and the third test scenarios, including the submodule voltage
$v_{SM}$, the filter current $i_{f}$, and the switching signal of the upper
switch $v_{sw1}$ are shown in Fig. 9.

To verify the function of the voltage prediction block, the predicted
submodule voltage $\hat{v}_{SM}$ is compared with the voltage measurement
$v_{SM}$, as shown in Fig. 10. The voltage error $v_{SMerr}$ is the
instantaneous difference between the prediction $\hat{v}_{SM}$ and the
measurement $v_{SM}$. As seen in Fig. 10(c), $v_{SMerr}$ increases as the
prediction time increases, but the error is lower than
$0.5\text{\,}\mathrm{V}$ within a $0.05\text{\,}\mathrm{s}$ period.

The normalized capacitance $C$ and ESR estimations are given in Figs. 9(c) and
(d), wherein the green dashed line is the reference value; the red line is the
median estimation; the blue box represents the $\pm$25\text{\,}\mathrm{\char
37\relax}$$ estimations; the whisker above and under the box means outliers of
estimations; the red areas are the range above $\pm$1\text{\,}\mathrm{\char
37\relax}$$ and $\pm$10\text{\,}\mathrm{\char 37\relax}$$ for the $C$ and ESR,
respectively. Except for test scenario 3 and test scenario 6, the median
values of estimated capacitance and ESR are around
$\pm$1\text{\,}\mathrm{\char 37\relax}$$ and $\pm$10\text{\,}\mathrm{\char
37\relax}$$, respectively. The higher error of $C$ for test scenarios 3 and 6
can come from the small voltage ripple as shown in Fig. 9 (b). These
estimation errors are listed in Table II. Compared with simulation results in
Figs. 7(c) and (d), the experimental results in Fig. 9 have a wider blue box,
meaning that the results have higher oscillations.

##  VI Conclusion

Based on the fact that health indicators can change at different speeds and
lead to different end-of-life times, this paper proposes a condition
monitoring method that can monitor both capacitance and ESR, which makes the
condition monitoring results more reliable compared with other methods that
monitor only one of them. The method combines the particle swarm optimization
and the capacitor voltage prediction functions. The design of key parameters
(swarm size and error limit) is studied to guarantee that the estimation
errors of capacitance and ESR are in the expected ranges. The effectiveness of
the method is validated in the simulations and experimental stages. Except for
light-load cases, the estimation errors of the capacitance and ESRs are around
$1\text{\,}\mathrm{\char 37\relax}$ and $10\text{\,}\mathrm{\char 37\relax}$,
respectively.

## Acknowledgment

This project is supported by the European Union’s Horizon 2020 research and
innovation program under the Marie Skłodowska-Curie grant agreement No.
955614.

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