Capacitor volume evaluation
based on ripple current
in Modular Multilevel Converter
Toshiki Nakanishi1, and Jun-ichi Itoh1
1

Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka, Niigata, Japan

Abstract-- This paper discusses the overall volume
evaluation of capacitors in Modular Multilevel
Converter (MMC). First, a theoretical formula of theripple voltage on each cell capacitor in the MMC isclarified. Second, a formula of the total electrostatic
energy is clarified. As a result, it is difficult to evaluate
the overall volume of the capacitors because the total
electrostatic energy is constant even the number ofcells changes. Against the problem, the overall volumeof the capacitors is designed by using the database ofcommonly-marketed electrolytic capacitors which are
generally implemented into many products. In
conclusion, the parallel connection of the capacitor
with the small ripple current is better than using one
capacitor with the large ripple current in order toachieve the size reduction of capacitors. Finally, it isnecessary to close the ratio of the charge voltage to thewithstand voltage to 1.0 in order to achieve thevolume minimization.
Index Terms—Modular Multilevel Converter, Step-down
rectifier, DC micro-grid, Capacitor volume evaluation.

I. INTRODUCTION
Recently, a micro-grid and a DC power grid have been
actively researched as next generation power supplies [1][3]. Specifically, the DC micro-grid is suitable to connect
a battery energy storage system and a renewable energy
source such as a photovoltaic generation system or a fuelcell to the DC grid because the output voltages of many
renewable energy sources are DC voltage [1]-[3]. From
the above advantages, the micro-grid has been applied tothe power grid in isolated islands as a stand-alone powersystem [3]. Moreover, an offshore wind power generation
system is also actively researched [3]. The wind power
generation is able to apply as the power source in isolatedislands because weather conditions around isolatedislands are suitable for the wind power generation.
Presently, the wind power generation system which isconnected to an AC grid has a transformer between thegenerator and the power converter in order to match thegenerator voltage and the AC grid voltage [4]. In the caseof the wind power generation system which is connectedto the DC micro-grid, the step-down rectification isrequired because the voltage rating of the wind power
generator with high power capacity is higher than the DC
bus voltage in the DC micro-grid. Specifically, thevoltage rating of the wind power generator is several
kilovolts [5]. On the other hand, the DC bus voltage inthe DC micro-grid is 400 V [2]. The step-down operation
is able to be achieved simply by the transformer.
However, the volume of the transformer is bulky. The
main reason why the transformer becomes bulky is thatthe operation frequency of the transformer is low,
because the transformer is required to operate insynchronization with the low generator frequency.
Moreover, the reasons why the transformer is bulky
includes that the high transform ratio is required.
As one of solutions, the employment of Modular
Multilevel Converter (MMC) to the wind power
generation system connected to the micro-grid is aneffective method. The MMC has been actively researched
as a next generation high power converter without thebulky transformer [6]-[7]. Therefore, the wind power
generation system achieves the size reduction by applying
the MMC into the system. However, in the general ACDC converter using the MMC which consists of chopper
cells, it is difficult to achieve the step-down rectification
because chopper cells cannot output the negative voltage.
In order to overcome the above problem, a step-down
converter using the MMC with H-bridge cells has been
proposed [6]-[7]. The MMC with the H-bridge cells
achieves the reduction of the bulky transformer because
the H-bridge cell has no limit of the output voltage. Thus,
the system with the H-bridge cells operates as a highpower step-down rectifier. On the other hand, the design
method of the capacitance in each cell and the overall
volume evaluation when the number of cells is changed,
have been reported scarcely [8]. So far, the volume of thecapacitor based on the electrostatic energy has beenreported [8]. However, it is seems that the relationshipbetween the capacitor volume and the number of cells has
not been reported. Specifically, it is important to consider
a ripple current of the capacitors in terms of lifetime of
the capacitor when the capacitor with the large capacity
such as an electrolytic capacitor is applied.
This paper discusses the overall volume evaluation of
the capacitors in the MMC. First, a formula of the totalelectrostatic energy is clarified by using a formula of thecapacitance, because the capacitance is designed in orderto suppress the ripple voltage. As a result, theelectrostatic energy does not change even the number ofcells changes because the formula of the total
electrostatic energy does not include the number of cells
as a variable number. Thus, it is necessary to evaluate thevolume of the capacitors by other methods. On the other
hand, the volume of the capacitors changes when thecapacitor parameters such as the voltage rating and thecapacitance changes. Moreover, the ripple current of thecapacitor is one of the important factors for the capacitor
design when the electrolytic capacitors are applied.
Therefore, it is necessary to evaluate the relationshipbetween the ripple current and the volume of thecapacitors as a first step of the capacitor design. In thispaper, the overall volume of the capacitors is evaluatedby applying the database of commonly-marketed
electrolytic capacitors which are generally implemented
into many products. Finally, the relationship between theoverall volume and the number of cells is shown.

vAr1

cell

vAr(n/2)

vcAr1

cell

Wind
power
generator

G
cell

S1

S3

S2

S4

iAr
ir
is
it

Lb

iBr

Lb

vBr1
C

H-bridge cellvcAr(n/2)

vBr(n/2)

DC bus of
DC distribution

Vmmc

Isolated
DC-DC
Converter

vcBr1

cell

vcBr(n/2)
cellFig. 1. Circuit configuration of the MMC for the wind power generation
system connected to the DC micro-grid. The MMC with the H-bridge
cells is able to operate as the step-down rectifier.

II. STEP-DOWN RECTIFIER USING MMC FOR WIND
POWER GENERATION SYSTEM CONNECTED TO DC MICROGRID

A. Main circuit configuration
Fig. 1 shows the main circuit configuration of the
proposed step-down rectifier using the MMC. Each leg
consists of two buffer reactors Lb and the H-bridge cells.
The converter outputs a multi-level voltage waveform inorder to reduce harmonic distortions of the input current.
Thus, many cascaded cells are used in practical because
the reduction effect of the harmonic distortion increases
by increasing the number of cells. Additionally, the
MMC is able to reduce the voltage rating of devices oneach cell because of cascade connections of cells.
Therefore, lower voltage rating devices are utilized to the
MMC. On the other hand, the output DC voltage of the
MMC depends on the sum of the cell’s output averagevoltage. Thus, the output voltage of the cell includes the
AC component and the DC component in order to control
the input current and the output voltage. Moreover, theMMC converts the high AC voltage to the DC voltage ofseveral hundreds volts. After this step, the DC voltage of
400 V is supplied to the DC bus of the DC distribution by
an isolated DC-DC converter. Thus, the input side and theoutput side are isolated by the DC-DC converter.
B. Relationship between cell configurations and output
voltageFig. 2 shows relationships between cell configurations
and cell output voltages. In the AC-DC converter using
the MMC, the maximum value of the generator output
voltage is applied to each arm of the MMC when thewind generator operates. Thus, the peak-to-peak value ofthe cell output voltage vp-p is given by (1) from thegenerator output voltage and the number of cells per leg.
v p− p = 4

2 E
3 n

(1)

(a) Chopper cell(b) H-bridge cellFig. 2. Circuit configuration and output voltage of each cell. The
chopper cell has the lower limit of the output voltage because thechopper cell cannot output the negative voltage. On the other hand, theH-bridge cell does not have the lower limit of the output voltage.

where E is an effective value of an output line-to-line
voltage in the wind power generator, and n is the numberof cells at each leg.
In addition, the output DC voltage of the AC-DC
converter using the MMC Vmmc equals to the sum of thecell’s average output voltage at one leg. Hence, Vmmc isgiven by (2).

Vmmc = nvcell_ ave(2)

where vcell_ave is an average output voltage of one cell.
Fig. 2(a) shows the relationship between the chopper
cell and its output voltage. In the case of applying the
chopper cell, the lower limit of the cell’s output averagevoltage is uniquely determined from the peak-to-peakvalue of the cell voltage vp-p and the number of cells at
each leg n, because the chopper cell cannot output thenegative voltage. Therefore, in the chopper cell, the lower
limit of the cell’s output average voltage is given by (3).

1vcell _ ave = v p− p2(3)

From (1), (2) and (3), the lower limit of the output DC
voltage Vmmc equals twice the value of a maximum
voltage of the wind generator phase voltage. From theabove principle, the MMC with chopper cells cannot
achieve the step-down rectification.
Fig. 2(b) shows the relationship between the H-bridge
cell and its output voltage. In the case of the H-bridge cell,
there is no limit of cell output average voltage because it

is possible for the H-bridge cell to output the negative
voltage. Hence, the H-bridge cell is able to control lower
voltage than that of (3). Therefore, the MMC with the Hbridge cells achieves the step-down rectification.
III. CONTROL STRATEGY
Fig. 3 shows the control block diagram of the proposed
step-down converter. The proposed control system isbased on the single-phase power factor correction (PFC)
converter because the H-bridge cell has same circuit
configuration as singe-phase PFC converter. Moreover,the PFC converter and the H-bridge cell have many
common points in the circuit operation and the control
system. For example, the capacitor voltage has to be
maintained constant in both circuits. Moreover, eachconverter has to control the input AC current in order tomaintain the capacitor voltage and reduce the total
harmonic distortion (THD). Differences between the PFC
converters and the H-bridge cells are; there is the DC
component in the output voltage of the H-bridge cell, anda voltage balancing control is necessary in order tocorrect unbalance voltages among all capacitors. Thus, inorder to deal with above problems, it is necessary to add
two control systems to control the DC component andachieve the voltage balancing into the conventional
control system of the PFC converters.
Specifically, in order to control each arm as shown in
Fig. 1, the control block diagram is separated to thecapacitor voltage control block and the input current
control (ACR) block. Moreover, the capacitor voltage
control block consists of an average voltage control(AVR) system and a voltage balancing control system.
First, the average voltage control system regulates theaverage value of the capacitor voltage in the arm into thecapacitor voltage command. On the other hand, thevoltage balancing control system corrects the unbalanceamong all capacitor voltages in the arm.
Moreover, the PI controller in AVR outputs the
positive phase component’s command of the arm current.
Therefore, the positive phase component of the arm
current is regulated in order to keep the capacitor voltage
constant. On the other hand, the zero phase component of
the arm current is controlled in order to supply the output
power to the DC micro-grid side.
A. Average Voltage Control
The average voltage control is applied in order tocontrol the average value of all capacitor voltage in thearm [6]. Therefore, the average value of all capacitor
voltages has to be calculated in each arm.
The error between the capacitor voltage command andthe average value of all capacitor voltages in the arm iscorrected by a PI controller. The output value of the PI
controller is given as the positive phase component’s
command of the arm current.
Moreover, the H-bridge cell outputs the DC voltage inorder to achieve the rectification. Therefore, the capacitor
voltage includes the ripple voltage whose cycle is same as
the input AC voltage. It should be cautious the control
system may become instability due to the ripple voltage.

Fig. 3. Control block diagram for step-down rectifier using the MMC.
The proposed control is based on a control system of a single-phasepower factor correction (PFC) converter. The proposed control system isapplied to each arm of the MMC in order to control each arm.
Moreover, the ACR controls only the AC component of the arm current.

Thus, in order to keep the stability of the control system,
a filter in order to damp the frequency component of the
input voltage cycle should be applied.
B. Arm Current Control
Each arm current includes the AC component and the
DC component. First, the AC component of the arm
current flows from the wind power generator. The ACcomponent is defined as the positive phase component.
Moreover, from the previous section, the positive phasecomponent’s command is generated by the averagevoltage control. In other words, the AC component of thearm current is controlled in order to maintain the averagevalue of all capacitor voltages in each arm constant.
The PFC operation in the input side of the MMC isachieved in the control system by regulating only the ACcomponent of the arm current. Furthermore, as thegenerator control system, it is necessary for the proposed
control system to achieve the high efficiency at same time.
In general, the id = 0 control is applied in order to achieve
the high efficiency in a lot of adjustable drive systems. In
the proposed converter, the control system achieves the id= 0 control by conforming the phase of the positive phasecurrent’s command to the back electromotive force (backEMF)’s one of the wind power generator.
On the other hand, the DC component of the arm
current flows in order to supply the DC power into theload. The DC current shunts into the each leg of the
MMC. Generally, the shunted DC current is constant andhas same value among arms when the output DC voltageof the MMC is divided equally among cells. The DC
current is considered as the zero phase component
because the DC current is constant at a specific load.
In the ACR, it is necessary to eliminate only the zero
phase component from the arm current in order to control
the capacitor voltage. However, it is difficult for a highpass filter to extract only the positive phase component
because the AC current frequency is very low. The
methods to extract the positive phase component havebeen proposed [6]. In this paper, the application of therotational coordinate transform is applied. This method
extracts the positive phase component by using therotational coordinate transform in each arm group; theupper side and the lower side. The zero phase component
is eliminated by the rotational coordinate transform andthe inverse transform.
C. Output Voltage Control System
The output DC power is controlled by the zero phasecomponent. The command of the output DC voltage isadded into the output value of the controller in the arm
current control system. Moreover, the output DC voltageof the MMC is divided on each leg. Thus, the output DC
voltage of one cell is fundamentally set as Vmmc*/n when
the command of the MMC output voltage control is
Vmmc*.
D. Voltage Balancing Control System
The average voltage control is used to maintain the
average value of all capacitor voltage. However, an
unbalance voltage which occurs among capacitors insame arm cannot be suppressed by only the averagevoltage control, because the average voltage control
corrects only the error between the voltage command andthe average value of all capacitor voltages in the arm. So
far, several methods to balance each capacitor voltage
have been reported [7], [9]. However, in most of theconventional control systems, it is difficult to understandthese control principle. Moreover, a design method ofcontrol parameters has not been reported. In the control
method of this paper, the positive phase component of thearm current is applied in order to maintain the averagevalue of all capacitor voltages in each arm. Therefore, thecommon positive phase component flows into all cells insame arm. Thus, it is difficult to adjust the positive phasecomponent of the arm current in order to balance thecapacitor voltage on each cell. In order to solve thisproblem, in the proposed control system, the output DC
voltage of the cell is varied depending on the capacitor
voltage. The control principle is given by (4).
*Vmmc _ mkj = Avd _ mkjVmmc*

m = A, B

k = r, s, t

(4)where Vmmc_mkj* is the output DC voltage command ofeach cell. i is the index of the cell number. Moreover, mshows an index which is the upper side A or the lower
side B. k is an index of each phase. m, k and j are matchedin both side of (4). Avd__mkj is a dividing ratio of each cellwhich is varied depending on the capacitor voltage.
The dividing ratio which is generated by the proposed
control system is given by (5).

Avd _ mkj =1 vcmkj
2 n/ 2∑ vcmkx

m = A, B

k = r, s, t

(5)

x =1

where Vcmkj is each capacitor voltage. The denominator of(5) is the summation of all capacitor voltages in each arm.
The dividing ratio Avd__mkj is set into 1/n when the all
capacitor voltages in the arm are same. On the other hand,the dividing ratio Avd__mkj is varied depending on thecapacitor voltage when each capacitor voltage fluctuates.
For example, the dividing ratio Avd__mkj of the cell is set
into high when its capacitor voltage is higher than thecommand of the capacitor voltage. Thus, the output
power of that cell becomes high because the output DC
voltage of the cell is high due to the high dividing ratio.
Therefore, the electric charge of the capacitor isdischarged. In contrast, the dividing ratio Avd__mkj of thecell is set into low when its capacitor voltage is lower
than the command. Thus, the electric charge of thecapacitor is charged because the output power of the cellbecomes low due to the low dividing ratio. The main
advantage of the proposed voltage balancing control
system is that it is not necessary to design control
parameters because the dividing ratio Avd__mkj is adjusted
automatically depending on the capacitor voltage. Thus,
the proposed voltage balancing control system is simpler
than the conventional control system.
IV. CLARIFICATION OF TOTAL ELECTROSTATIC ENERGYIN CELL CAPACITORS
Each cell of the MMC has the capacitor. However, thevolume of the capacitors has not been discussed
thoroughly. As one of reasons, the capacitor which isconsidered to apply to the cell has the special
specification such as the high voltage rating and the highcapacitance. The above capacitor has a low number of
product data due to these special specifications. On theother hand, as the common view of the capacitor volume,the volume of the capacitor becomes large when thevoltage rating is high and the capacitance is large. Hence,the volume of the capacitors varies based on theelectrostatic energy which is the function of the voltage
rating and the capacitance. Thus, the evaluation methodof the capacitor volume has been reported in terms of theelectrostatic energy of the capacitors [8]. However, in theabove evaluation, types of the capacitors have not been
discussed.
In general, an electrolytic capacitor is known as thecapacitor which has the high capacity. The ripple currentof the capacitor is one of the important factors for thedesign of the MMC when the electrolytic capacitors areapplied to the MMC, because the ripple current affects
the lifetime of the capacitors. Moreover, the voltage
rating of the capacitor drastically changes according to
the number of cells, whereas the ripple current alsochanges depending on the voltage rating and thecapacitance.
In this paper, the ripple current is applied as one ofevaluation indexes in order to evaluate the volume of thecapacitors. Additionally, the relationship between thecapacitor volume and the ripple current is evaluated froma database of commonly-marketed electrolytic capacitors
which are generally implemented into products.

A. Calculation of capacitor’s voltage rating
The output DC voltage’s command in the MMC Vmmc*
is added to the output block of the input current control.
In the MMC, the output DC voltage Vmmc is applied ineach leg because each leg is connected to the load inparallel. In addition, the maximum value of the input
phase voltage is applied to each arm. Therefore, eachcapacitor voltage depends on both the input voltage andoutput voltage. The capacitor voltage command vc* isgiven by (6). Note that the modulation index λ is set to
0.8 or less. Besides, the voltage drop of the buffer
inductor is ignored because the voltage drop of the buffer
inductor is sufficiently small compared with the input
phase voltage and the output voltage of the MMC.

*

vc ≥

⎞
1 ⎛⎜ 2*2 E + Vmmc ⎟⎟nλ ⎜⎝ 3⎠

(6)

B. Calculation of ripple voltage
In general, the capacitance of the converter is designed
based on the ripple voltage against the average value ofthe capacitor voltage. Additionally, it is important toclarify the formula of the capacitor voltage in order todesign the fluctuation of the capacitor voltage. On theother hand, in the MMC, the system has many capacitors
and the voltage rating of the capacitor changes dependingon the number of cells. Furthermore, the formula of thecapacitor voltage which includes the circuit parameters of
the MMC should be clarified.
First, the cell output voltage vcell is given by (7) inorder to control the input current and the output DC
voltage. The arm current iarm is given by (8). The arm
current iarm includes the AC component and the DC
component. The AC current flows from the generator inorder to maintain the capacitor voltage. On the other hand,
the DC current flows into the load in order to supply thepower. Moreover, the arm current iarm flows into each cellin same arm. Note that the sign of positive or negative isdecided based on the lower arm of Fig. 1.

vcell (t ) =⎞1 ⎛⎜ 22 E sinωt + Vmmc ⎟⎟n ⎜⎝ 3
⎠

(7)

1 2 S
P
sin(ωt − φ ) +
2 3 E
3Vmmc

(8)

iarm (t ) = −where S is an input apparent power. P is an input active
power. Q is an input reactive power. φ is an input phasedifference. Moreover, S is defined by (9).
2S = P +Q

2

(9)

From (7) and (8), the instantaneous power of the cell is
calculated. The energy which flows into the capacitor is
calculated by integration of the instantaneous power. Theenergy which flows into the capacitor is given by (10).

WC (t ) = ∫ vcelliarm dt=

1 2 Vmmc S
2 2 EPcosωtcos(ωt − φ ) −2 3 nωE
3 3 nωVmmc
+

(10)

S
sin(2ωt − φ ) + WC 06nω

where WC0 is the constant of integration.
WC0 does not change with time. Thus, WC0 is definedas the average value of the energy which flows into thecapacitor. Moreover, the relationship between thecapacitor voltage vc(t) and the energy which flows into
the capacitor WC(t) is given by (11) in terms of anelectrostatic energy.

WC (t ) =1
2
CvC (t )2(11)

From (10) and (11), the capacitor voltage vc(t) is givenby (12). Note that the theoretical formula is calculated in
the linear approximation by Taylor expansion.

vc (t ) = VC 0 +1 2 Vmmc S
cos(ωt − φ )2 3 nωCEVC 0

−

2 2
EP
cosωt
3 3 nωCVmmcVC 0+

S
sin(2ωt − φ )6nωCVC 0(12)where the constant value VC02 is defined by 2WC0/C.
The ripple voltage includes the fundamental frequency
component whose frequency is same as the generator
voltage’s frequency and the high frequency component
whose frequency is as twice as the frequency componentof the generator voltage’s frequency (second-orderfrequency component). Moreover, VC0 is defined as theaverage value of the capacitor voltage because WC0 does
not change with time.

C. Analysis of ripple voltage
From (12), it is understood that the ripple voltage withthe fundamental frequency is function of the input phasedifference φ because the input active power P changes
depending on the input phase difference φ . Therefore, therelationship between the input phase difference φ and theripple voltage has to be clarified in order to design thecapacitance for the worst case when a magnitude of theripple voltage is the largest. The ripple voltage with thefundamental frequency Δvc_1(t) is given by (13). Note that(13) is calculated by synthesizing fundamental frequency
components of (12). An angle β is given by (14).

Δvc _1 (t ) =−2⎧⎪⎛ E ⎞ 2 ⎛ E
Vmmc ⎞
S
2
⎜
⎟
⎜
⎟
+
−
⎨
3 nωCVC 0 ⎪⎜⎝ 3Vmmc ⎟⎠ ⎜⎝ 3Vmmc 2E ⎟⎠
⎩

(13)

1

⎫⎪ 2
⎛ E ⎞⎛ E
V ⎞
⎟⎟⎜⎜
+ 2⎜⎜
− mmc ⎟⎟ cos 2φ ⎬ cos(ωt + φ + β )⎪⎭⎝ 3Vmmc ⎠⎝ 3Vmmc 2E ⎠
⎡
⎤
⎛ E
V ⎞
− ⎜⎜
− mmc ⎟⎟ sin 2φ
⎢
⎥
3Vmmc
2E ⎠
⎝
−1 ⎢
⎥
β = tan ⎢⎥⎛ E ⎞ ⎛ E
Vmmc ⎞⎟⎟ + ⎜⎜⎟⎟ cos 2φ ⎥
⎢ ⎜⎜
−
2E ⎠
⎢⎣ ⎝ 3Vmmc ⎠ ⎝ 3Vmmc
⎥⎦

(14)

From (13), it is understood that the magnitude of theripple voltage with the fundamental frequency becomesmaximum when the input phase difference φ is 0 or π.
Fig. 4 shows the relationship between the input phasedifference φ and the ripple voltage. In Fig. 4, thefundamental frequency component is drawn by (13). The
second-order frequency component is drawn by the lastterm of (12). As a result, the second-order frequency
component of the ripple voltage does not changed against
the input phase difference because the input apparent
power S does not change. On the other hand, thefundamental frequency component becomes maximum
when the input phase difference φ is 0 or π. Thus, it isnecessary to design the capacitance only when the input
phase difference φ is 0.

D. Clarification of total static energy
In general, the ripple voltage is evaluated as the ripple
factor. Moreover, the ripple factor is defined by (15) [10].

Aripple =ΔvC _ peak−to− peak
2VC 0

⎛2 E
2
1 Vmmc ⎞
P
⎜
⎟
−
2 ⎜
3 nωCVC 0 ⎝ 3 Vmmc 2 E ⎟⎠Moreover, from (6) and (16), the capacitance C iscalculated by (17). Note that the capacitor voltage VC0
equals to the command of the capacitor voltage.

C=

⎛2 E
2 nλ2 P
1
1 Vmmc ⎞⎜⎟⎟ (17)−2 ⎜3 ωAripple ⎛ 2⎞ ⎝ 3 Vmmc 2 E ⎠⎜ 2 E + Vmmc ⎟
⎜ 3
⎟
⎝
⎠

The required capacitance C to suppress the ripple
voltage according to the ripple factor Aripple is calculatedby (17). Moreover, the required capacitance becomeslarge with increasing the number of cells n when the
ripple factor Aripple is not changed at any number of cells.
It means that the ripple voltage becomes large relatively
because the average value of the capacitor voltage
becomes small with increasing the number of cells.
Finally, the total electrostatic energy EC_sum is clarified.
The electrostatic energy is calculated by replacing theparameters in function (11) by (6) and (17). Additionally,
it is able to calculate the total electrostatic energy tomultiple total numbers of cells 3n. Thus, the total
electrostatic energy is given by (18).

3n
2
CVC 0
2
3 2 P ⎛2 E
1 Vmmc ⎞⎜⎜⎟=−2 3 ωAripple ⎝ 3 Vmmc 2 E ⎟⎠EC _ sum =(18)

The formula (18) implies that the total electrostatic
energy does not change when the ripple factor Aripple isconstant at any number of cells. Thus, it is difficult to

0.014

Secong-order frequency components

0.012

0.07
0.06

0.010

0.05

0.008

0.04

0.006

0.03

0.004

0.02

0.002

0.01

(16)

0.016

1 p.u. = Average value of capacitor voltage

0.08

(15)

where Aripple is the ripple factor and Δvc_peak-to-peak is thefluctuation range of the ripple voltage. Note that thefluctuation range equals to the peak-to-peak value of theripple voltage when amplitude of the ripple voltage issymmetric about the average voltage.
In addition, from the previous section, it is necessary
to design the capacitance when the input phase difference
φ is 0. Thus, the ripple factor is defined when the input
phase difference φ is 0. From (13) and (15), the ripple
factor of the fundamental frequency component is givenby (16). Note that φ in (13) is 0 and S is replaced by P
because Q is zero.

Aripple =

0.10
0.1

0.09

Fundamental frequency components

0
0

30

60

90

0.000
120

150

180

Input phase difference
Fig. 4. Relationship between the input phase difference φ and the ripple
voltage. The fundamental frequency component is calculated by (13)and the second-order frequency component is calculated by the last termof (12). The magnitude of the fundamental frequency components is the
maximum when the input phase difference φ is 0 or π.

evaluate the relationship between the total volume of thecapacitors and the number of cells in terms of the total
electrostatic energy when the ripple factor is constant.

E. Clarification of ripple current
In general, the ripple current value of the capacitor isshown in the datasheet of commonly-marketed
electrolytic capacitors which are generally implemented
into products. Moreover, the capacitor should be selected
in order that the ripple current value which flows to eachcapacitor is sufficiently small compared with thespecified value on each datasheet in terms of the lifetime.
Additionally, the number of the capacitors which areconnected in parallel on each cell should increase in order
to meet the specified value of the ripple current.
Therefore, this makes the capacitor volume change
drastically. Thus, the relationship between the ripple
current and the capacitor volume has to be clarified
before the relationship between the number of cells andthe capacitor volume is evaluated.
The relationship between the capacitor current ic andthe capacitor voltage vc is given by (19).

iC = C

dvC
dt

(19)

From (13) and (19), it is concluded that therelationship between the input phase difference φ and the
maximum value of the amplitude does not change after
the differential of the ripple voltage. Therefore, theamplitude is maximum when φ is 0. Thus, the maximum
effective value of the ripple current IC_1 is given by (20).

I C _1 =⎛2 E
λP
1
1 Vmmc ⎞
⎜
⎟
−
⎜
3⎛ 2
⎞ ⎝ 3 Vmmc 2 E ⎟⎠⎜ 2 E + Vmmc ⎟
⎜ 3
⎟
⎝
⎠

(20)

From (20), it is understood that the maximum effective
value of the ripple current does not change when thenumber of cells and the capacitance changes.

V. EXPERIMENTAL RESULTIn this section, experimental results by the miniaturemodel of the step-down rectifier using the MMC areshown. In the experiment, a utility power source of 200 V
is applied as the input voltage source.
Fig. 5 shows waveforms of the input phase voltage, theinput current and the output DC voltage. Firstly, from the
waveforms of the input phase voltage and the input
current, it is confirmed that the unity power factor is
obtained in the input side. Moreover, the total harmonic
distortion (THD) of the input current is 5.6%. Second, thewaveform of the output DC voltage in lower side of Fig.
5 shows that the miniature model converts from the input
voltage of 200 V into the output DC voltage of 65 V.
From this result, it is confirmed that the output voltage is
kept constant. Therefore, the miniature model of the
MMC achieves the step-down rectification.
Fig. 6 shows the waveforms of cell capacitor voltages
which are connected to the r-phase leg. The cell capacitor
voltage is controlled according to the capacitor voltage
command. As a result, the proposed system maintains thecapacitor voltage of each H-bridge cell to the voltage
command of 120 V. In addition, the maximum voltage
error between the voltage command of the cell capacitor
and the measured voltage is 5% or less.
VI. EVALUATION OF CAPACITOR VOLUME
The overall volume of the capacitors is evaluated byusing the database of commonly-marketed electrolytic
capacitors which are generally implemented into many
products. Conditions to make the database are as follows;a) Evaluated targets are the capacitors which eachmanufacturer (Nippon Chemi-Con, Nichicon and
Rubycon) recommends as “For inverters” and “For highpower application”.
b) Withstand voltages are from 100 V to 600 V.
c) Each capacitor volume is the average value in thecapacitor volumes of each ripple current value which isrounded the ripple current of the capacitor to unit.

A. Relationship between ripple current and capacitor
volumeFig. 7 shows the relationship between the ripple
current and the capacitor volume as a database example.
The figure in the graph shows the ripple current of one
capacitor. Therefore, the number of the capacitors which
are connected in parallel on one cell increases withincreasing the ripple current. Moreover, the start point ofeach line means the ripple current and the volume of one
capacitor. As a result, the overall volume of the capacitors
becomes small by connecting the capacitor with the small
ripple current in parallel, compared with using one
capacitor with the large ripple current. Thus, the parallel
connection of the capacitor with the small ripple current
is better than using one capacitor with the large ripple
current in order to achieve the size reduction of capacitors.
Moreover, the ripple current does not change against thecapacitance and the number of cells. Thus, the minimum
point of capacitor volume against the number of cells
achieves the minimum point of the overall volume.

Fig. 5. Waveforms of input voltage, input current and output voltage.
The unity power factor is obtained in the input stage. The THD of the
input current is 5.6% when the %Z of the inductor is 2.5%.

Fig. 6. Waveforms of the capacitor voltage in r-phase leg. The proposedsystem maintains the capacitor voltage of each H-bridge cell to thevoltage command of 120 V. In addition, the maximum error between
the capacitor voltage command and the measured voltage is 5% or less.

B. Relationship between number of cells and capacitor
volumeFig. 8 shows the relationship between the number ofcells and the capacitor volume. Note that the number ofseries connection capacitors increases when the charge
voltage which is calculated by (6) is larger than thewithstand voltage. Additionally, the charge voltage is set
30% more than the value which is calculated by (6). In
this evaluation, the capacitor volume is determined by themultiplied value of the number of series connection
capacitors and the number of cells. The capacitor volume
changes drastically when the number of cells is small,because both factors changes. On the other hand, thecapacitor volume increases linearly as the number of cells
increases when the number of series connection
capacitors is kept constant. Moreover, there are several
points which the capacitor volume is minimum.
Fig. 9 shows the ratio of the charge voltage to thewithstand voltage. The ratio is defined as a voltage
utilization factor. Fig. 9 shows that the number of cells
which the capacitor volume becomes minimum, and thenumber of cells which the voltage utilization factor
closest to 1.0 match. On other words, it is possible toachieve the volume minimization by closing the voltage
utilization factor to 1.0. As a conclusion, it is necessary tomeet following conditions in order to achieve the volume
minimization of the capacitors.
1) Applying the capacitor with the small ripple current
2) Maximizing the voltage utilization factor

Volume of capacitors [dm3](Only parallel connection of capacitors)

VII. CONCLUSION
This paper discussed the overall volume evaluation of
capacitors by applying a database of commonly-marketed
electrolytic capacitors which are generally implemented
into many products. Specifically, the relationship between
the capacitor volume and the number of cells are
considered. It is necessary to consider the ripple current
in terms of the lifetime when the electrolytic capacitor isapplied to the MMC. Thus, the number of capacitors
changes drastically due to the ripple current of one
capacitor. In this paper, the number of parallel connection
capacitors is evaluated in terms of the ripple current.
Moreover, the number of series connection capacitors isevaluated in the terms of the voltage utilization factor. As
a conclusion, it is necessary to meet following conditions
in order to achieve the volume minimization of capacitors.
1) Applying the capacitor with the small ripple current
2) Maximizing the voltage utilization factor
As a future work, the design method including thethermal design by the loss analysis in order to achieve thevolume minimization of the MMC will be considered.

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(a) Withstand voltage : 100 V, Endurance : 85Ԩ, 5000hour

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(a) Withstand voltage : 400 V, Endurance : 105Ԩ, 3000hour
Fig. 7. Relationship between the ripple current and the capacitor
volume. The parallel connection of the capacitor with the small ripple
current is better than using one capacitor with the large ripple current inorder to achieve the size reduction of capacitors.

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40

50

60

70

80

90

100

Fig. 8. Relationship between the number of cells and the capacitor
volume. The capacitor volume is determined by the multiplied value ofthe number of series connection capacitors and the number of cells. The
capacitor volume changes drastically when the number of cells is small,because both the number of series connection capacitors and the numberof cells changes. On the other hand, the capacitor volume increases
linearly with increasing the number of cells when the number of seriesconnection capacitors is kept constant.
1.0
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Fig. 9. Relationship between a voltage utilization factor and the numberof cells. The number of cells which the capacitor volume becomesminimum, and the number of cells which the voltage utilization factor
closest to 1.0 match.