Multilevel Synchronous Optimal PulsewidthModulation Generalized Formulation

Jackson Lago and Marcelo Lobo HeldweinFederal University of Santa Catarina (UFSC) – Electrical Engineering Department (EEL)

Power Electronics Institute (INEP) – www.inep.ufsc.br - Phone:+55(48)3721-9204

88040-970 — PO box: 5119 — Florianopolis, SC, BRAZIL

E-mail: jacksonl@inep.ufsc.br ; heldwein@inep.ufsc.br

Abstract—With the growing interest of the industry in highpower medium voltage multilevel inverters and the technologicallimitation in high voltage power semiconductors switches, thesynchronous optimal pulsewidth modulation technique, originallydeveloped for two/three–level inverters, became once again atopic of interest, now to optimize multilevel waveforms. Thiswork proposes a new formulation for the problem of optimizingthe modulation pattern of multilevel converters, including in asingle optimization problem the decision of the directions foreach step transition in addition to the switching angles and, thus,completely defining the optimized multilevel waveform for a givenmodulation index.

I. INTRODUCTION

The basic idea to perform synchronous modulation schemes

based on a set of predefined (off–line evaluated) optimized

switching angles emerged in the 1960s in order to make

good use of the low switching frequency allowed for the

semiconductor technology of the time. Initially, the technique

was used to completely eliminate a group of harmonics com-

ponents in two-level inverters (SHE — Selective Harmonic

Elimination) [1]. Soon later, [2] proposed to do not completely

eliminate a limited group of harmonic components, but to

control a whole spectrum to minimize the harmonic distortion

of the current of a motor. This was done considering the motor

mathematical model and waveform performance indexes such

as THD (Total Harmonic Distortion) and WTHD (Weighted

Total Harmonic Distortion). This technique, also developed to

two– and three–level inverters, became known as synchronous

optimal pulsewidth modulation (SOP), sometimes also referred

as optimized pulse pattern (OPP).

In the following decades, with the evolution of semiconduc-

tor technology that allowed higher switching frequencies,

the interest for this technique has been lost in low-voltage/-

power applications. But recently, with the advent of multilevel

converters for powering very high power medium voltage

systems, which are also subject to restrictions of low switching

frequency, this technique again became a topic of interest.

The mathematical formulation for the optimization of two–

level and three–level inverters waveforms is well establish

and extensively studied [3]–[9]. In this cases, the output

waveforms can assume only two levels within a semi–cycle,

and consequently, consecutive switchings just toggle the level

of the output waveform such that the direction of each step

transition is imposed and well defined. For multilevel inverters

with number of levels L grater than three, each step transition

from intermediary levels can assume two directions that results

in additional degrees of freedom in the modulation. Therefore,

for the modulation of a multilevel inverter it is not enough to

define the switching angles of the output waveform. To define

the direction of each step transition is also needed.

The original formulation for the two/three–level waveform

optimization problem does not include the decision on the

direction of each transition and, thus, the technique cannot be

directly used for optimizing multilevel waveforms.

The harmonic content of a multilevel waveform with a

quarter–wave symmetry as a function of both the switching

angles (θ) and step transitions directions (δ) is given by,

lh (θ, δ) =2

πh

N∑k=1

δk cos(hθk), (1)

where lh is the amplitude of the harmonic component of

order h of the inverter output waveform, N is the number

of switchings in a quarter of the waveform, and θk and δk are

respectively the k–th elements of the set of switching angles

θ and the set of the signals that represent the direction of each

step transition δ that define the waveform,

θ={θ1, θ2, θ3, . . . , θN} , 0 < θ1 < ... < θN < π/2δ={δ1, δ2, δ3, . . . , δN} , δk ∈ {−1,+1} (2)

With (1) it is possible to evaluate the WTHD (3), or

another similar performance index, of the inverter output

voltage waveform and use it as the objective function for the

optimization problem. However, a minimization problem that

must return the two sets of angles and directions is a mixed

integer nonlinear problem. There is no efficient optimization

algorithm for this type of problems, and nor it is possible to

ensure a solution for this kind of formulation.

σwthd (θ, δ) =H∑

h=6n∓1n∈N

∗

[lh (θ, δ)

h

]2

(3)

To get around this complexity [10] introduced the concept

of switching structures, also known as switching patterns. A

switching pattern is a valid sequence of the direction of the

step transitions along the waveform. Each switching pattern

completely defines the δ set in (1) and (2). For instance, a

Paper O1-2 Workshop on Control and Modeling for Power Electronics (COMPEL) 1

Fig. 1. The four possible switching patterns for a waveform with L=5 andN=5 and quarter–wave symmetry.

five–level inverter (L = 5) with five switching angles in a

quarter of the output waveform (N = 5) has four different

valid switching patterns as shown in Fig. 1.

Solve the optimization problem for a single switch pattern,

imposing δ, is mathematically similar to solve the problem

for a two–/three–level waveform. Since the switching angles

for a given modulation index M can be obtained by solving

the optimization problem for a defined switching pattern, the

remaining problem is to define the ideal switching pattern

for each value of modulation index M . Some authors [11]–

[14] just define a pattern and optimize the switching angles

assuming the chosen pattern gives good/acceptable results. In

fact, the ideal pattern is dependent of the modulation index M .

It is part of one of the constraints of the optimization problem

and an optimal solution for the modulator will change the

pattern for different ranges of M .

In order to get an optimum set of switching angles and the

ideal pattern for each modulation index M value reference

[10] proposes to evaluate the optimization problem for the

entire range of interest of the modulation index, for each

valid switching pattern and then select the pattern and its

corresponding switching angles that give the best solution (the

solution that provides the lower value for the objective func-

tion). This technique allows to find the optimal solution, both,

for the switching angles and switching pattern, completely

defining θ and δ. However, for an inverter with a relatively

high number of commutations per cycle or for a high number

of levels the number of switching patterns and consequently

the scale of the optimization problem that must be solved can

Fig. 2. Number of possible switching patterns for a L–level multilevelconverter with N commutations in a quarter of the fundamental period.

be impractically large. The increase of the number of different

valid patterns for inverters with three–, five–, seven– and nine–

level waveforms for up to 20 commutations in a quarter of the

fundamental period is shown in Fig. 2. Note that this graph is

in log scale.

Other attempt to adapt the two–level optimized modulation

to multilevel converters appear in the literature [10]–[19]. Most

of them use the concept of switching pattern and/or impose

unnecessary restrictions in order to simplify the optimization

problem.

This paper proposes a generalized adaptation of the opti-

mization problem formulation for multilevel waveforms that

does not use the concept of switching patterns and includes

the decision of the direction of each step transition within the

optimization problem as an expansion of the search space of

the optimization variables.

II. OPTIMAL SYNCHRONOUS PULSEWIDTH MODULATION

GENERALIZATION FOR MULTILEVEL CONVERTERS

The main complexity of optimizing generic multilevel wave-

forms is due to the necessity of finding, in addition to the

switching angles, their directions. Since the direction of the

steps are modeled as a set of binary variables (δ), it turns the

optimization problem into a mixed integer non–linear one. The

adaptation for the optimization problem formulation presented

in this section reduces the mixed integer non–linear to a non–

linear problem without imposing any simplification nor losing

generality. It simply explores the intrinsic symmetries of the

problem.

Just to emphasize, an optimization algorithm to minimize

(3), or a similar index, must return the optimized values for

both θ and δ that represent respectively the set of angles

(relative to the fundamental) where will be step transitions

in the inverter output waveform and their directions.

A generalized expression to evaluate the amplitude of any

harmonic component of a multilevel waveform was presented

in (1). This expression is the central point for the opti-

mization problems based on harmonic components control

and is the one that holds the relationship between the two

sets of variables to be found. The cosine term in (1) is a

direct consequence of the choice of having the fundamental

component of the output waveform in phase with a sine signal

Paper O1-2 Workshop on Control and Modeling for Power Electronics (COMPEL) 2

with the same frequency. Shifting the entire waveform by π/2rad, the fundamental component of the waveform becomes in

phase with a cosine signal, and the expression for the harmonic

content of the waveform is now given by

lh (θ, δ) =2

πh

N∑k=1

δk sin(hθk) =2

πh

N∑k=1

sin(hδkθk). (4)

With this shift in the reference point adopted in the formu-

lation, the cosine term in (1) becomes a sine term in (4) and

the odd symmetry of the function sine can be used to join the

two set of variables θ and δ as a new set of variable γ:

lh (θ, δ) =2

πh

N∑k=1

sin(hδkθk) = lh (γ) =2

πh

N∑k=1

sin(hγk),

(5)

where

γk = δkθk ,−π/2 < γk < π/2, (6)

and since θk is always positive, and δk has only a signal

information with fixed amplitude, both θk and δk can be

completely reconstituted from γk using (7), without losing any

information, which ensures that this variable integration does

not affect the validity or generality of the optimization problem

formulation.

θk = abs(γk) , 0 < θk < π/2δk = sign(γk) , δk ∈ {−1,+1} (7)

Exemplarily, the complete optimization problem that mini-

mizes the WTHD of the a generic multilevel waveform with

this new formulation can be expressed as:

minγ

H∑h=6n∓1n∈N

∗

4

π2h4

[N∑

k=1

sin(hγk)

]2

subject to

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−π/2 ≤ γk ≤ π/2

4

π

N∑k=1

sin(γk) = M

0 ≤ lk(γ) ≤ (L− 1)/2

(8)

where H is the highest harmonic order taken into account,

h is the index representing the harmonic order, M is the

modulation index, L is the maximum number of levels al-

lowed by the converter power structure, N is the number of

switchings performed in a quarter–wave and lk is the level

of the waveform at the angle θk. The first constraint of the

problem (8) ensures the validity of the range of the found

angles, the second one sets the amplitude of the fundamental

component to achieve the desired modulation index and the

third one ensures the solution does not use more levels that

the inverter can physically synthesize. In fact, the third one

can be relaxed for a modulation index lower than one.

To validate this unified formulation of the optimization

problem for a multilevel waveform, the solution found with

this formulation can be compared against the results found

with the conventional methodology, that optimizes the switch-

Fig. 3. Comparison between the minimum value for the objective function forall possible switching patterns (p1...p4) and the ones found by the proposedformulation (pγ ).

ing angles individually for all possibles switching patterns.

Fig. 3 shows the values of the objective functions obtained by

numerically solving the optimization problem (8) (identified in

the graph as pγ) and its equivalent formulations for the four

switching patterns (identified as p1...p4) for a waveform with

N =5 and L=5 (same condition of Fig. 1). The non–linear

numeric solver KNITRO was used in both formulations.

This comparison shows that the proposed formulation is

able to find the optimal solution minimizing, in this case

the WTHD of the waveform, for the entire valid range of

modulation index M , obtaining both the optimal angles and

the optimal switching pattern through the set of variables γ,

as a solution of an unified optimization problem.

III. SIMULATION RESULTS

Simulations of a more complex optimized modulator for a

five–level NPC H–bridge inverter with up to N = 13 (with

64 valid switching patterns) were carried out to demonstrate

the effectiveness of the proposed optimization technique. The

simulated system is shown in Fig. 4 and it consists of an entire

inverter/motor/load system (based on a 2 MW drive system).

The simulation includes the dynamic model of a thee–phase

induction motor coupled to the mechanical model of a pump,

the passive rectifiers for the three isolated dc–links of the NPC

H–bridge and the power electronic structure of a three-phase

NPC H–bridge.

The voltages vs1...vs6 from Fig. 4 in this simulation are

ideal sinusoidal voltage sources of 2.25 kV and 60 Hz prop-

erly phase–shifted for the 12–pulse rectifier operations which

powers each dc link segment with approximately E=2.9 kV.

The dc–links capacitances and the rectifier input inductance

are respectively C = 10 mF and Ls = 1.6 mH and the main

parameters of the inductor motor model used are resumed in

Table I.

In order to evaluate the effectiveness of the synchronous

optimal pulsewidth modulation a comparison against conven-

tional modulation techniques like SVM (space vector modula-

tion) and carrier–based PWM modulation schemes were car-

ried out. The same system was simulated in similar conditions,

with same switching frequencies and modulation index, for

Paper O1-2 Workshop on Control and Modeling for Power Electronics (COMPEL) 3

Fig. 4. Simulated system.

TABLE IINDUCTION MOTOR PARAMETERS.

Pn 2 kW rated power

Vn,rms 6.6 kV rated line–to–line rms voltage

In,rms 238 kV rated rms current

fn 74 Hz rated frequency

tree different modulation schemes. The first one named mIPD

(modified in–phase disposition) is carrier–based and consists

of an adaptation of conventional multilevel IPD (in–phase dis-

position) for the NPC H-bridge in order to balance the voltages

and losses between the two NPC legs that compose one phase

of the NPC H–bridge. The second one is a conventional 5–

level seven–segments regularly sampled SVM. The third one

is the synchronous optimal pulsewidth modulation presented

in this paper.

The OPP modulation scheme is a synchronous modulation

based on a look–up table with the optimum commutation

angles and directions obtained from solving the previously

presented optimization problem (8). The reconstruction of the

phase voltages from a table of γ and the gate signals for the

eight power electronic switches of each phase of the inverter is

done as shown in Fig. 5. The gate signals generation includes

an algorithm to balance the central point of each dc–link

through the intra–phase redundant states similar to the way

it is done in space vector modulation schemes.

Fig. 5. Output phase voltage signal reconstruction from γ and gate signalgeneration.

At first, this three systems with different modulation

schemes were simulated with N =13 (with carrier frequency

and sample sampling frequency that results in the same amount

of commutations per cycle). This is a relatively high switching

frequency for synchronous optimal pulsewith modulation or

other similar techniques like harmonic elimination, but it is

a low frequency for conventional modulation techniques like

mIPD and SVM though these ones still produce reasonable

performances. The graphs in Fig. 6 show a comparison be-

tween the voltage waveforms at the output of an NPC H-bridge

inverter and the currents of a motor powered by this inverter

with these three different modulations schemes. On Fig. 6(a)

and (d) the mIPD PWM was used, on Fig. 6(b) and (e) a

conventional seven–segments SVM was used and on Fig. 6(c)

and (f) the synchronous optimized modulation presented in

this parer was used. The optimization problem was modeled

to minimize the inverter output voltage WTHD to indirectly

minimize the motor current THD. From Fig. 6 it can be noticed

the great improvement with respect to the current distortions

that the optimized modulation provide over the carrier–based

ones even for a relatively high switching frequency.

For high power systems even N = 13 is a switching fre-

quency too high and several applications requires much lower

switching frequency. Another simulations were performed for

the tree modulation schemes previously analyzed, but this time

with N = 4. The graphs in the Fig. 7 show a comparison

between the voltage waveforms at the output of a NPC H-

bridge inverter and the currents of a motor powered by this

inverter with three different modulations schemes On Fig. 7(a)

and (d) was used the mIPD PWM, on Fig. 7(b) and (e) was

used a conventional seven–segments SVM and on Fig. 7(c) and

(f) was used the synchronous optimized modulation presented

in this paper.

This level for switching frequency (relative to the funda-

mental) is too low for the carrier–based techniques and the

results only testify that both mIPD and SVM were not suitable

to operate in this conditions, but the synchronous optimal

pulsewidth modulation presents very good results even with

such low switching frequency.

Fig. 8(a) shows the equivalent voltage output vectors, the

vector path on the αβ plane and the sampling times (propor-

tional with the diameter of the circumferences that represents

each sampled vector) for the OPP modulation with M=0.93,

N=4 shown in Fig. 7(c) and (f). The wide disparity between

Paper O1-2 Workshop on Control and Modeling for Power Electronics (COMPEL) 4

Fig. 6. Simulation results for the inverter output voltages and motor current with mIPD (a) and (d), SVM (b) and (e) and the optimized modulation (c) and(f) with N=13 and M=0.93.

Fig. 7. Simulation results for the inverter output voltages and motor current with mIPD (a) and (d), SVM (b) and (e) and the optimized modulation (c) and(f) with N=4 and M=0.93.

Paper O1-2 Workshop on Control and Modeling for Power Electronics (COMPEL) 5

Fig. 8. Path taken by the (a) voltage and (b) current in the αβ plane for OPPmodulation with N=4 and M=0.93 .

the sampling times make it clear that the switching angles

obtained from the optimization can not be generate by regular

sampled modulation techniques. Fig. 8(b) shows the path taken

by the motor current vector on the αβ plane showing the low

current distortion with respect to the fundamental component.

An inverter feeding a motor must be able to impose/control

its voltage output frequency in a wide range, as well as its

voltage. In the synchronous optimized modulation, unlike con-

ventional carrier–based modulation, the switching frequency

is dependent of the output voltage fundamental frequency

which means that during the acceleration of the motor the

switching frequency is constantly changing. Thus, for low

fundamental output frequency the number of commutation per

cycle N can be increased in order to ensure low distortion

without damaging the semiconductor switches due to high

switching losses. Since a motor operates with a nearly constant

V/f (voltage per frequency) relation, the switching frequency

becomes proportional to the modulation index M , and a table

with the values of the optimal γ with variable discrete values

for N can be generate in order to provide the optimal mod-

ulation with maximum frequency limitation for all the range

of voltage/fundamental frequency operation. Fig. 9 shows the

Fig. 9. Optimal values of γ from N=9 down to N=4 for a constant V/frelation.

optimal γ for modulation indexes from 0.4 to 0.98 varying

the number of commutations per cycle from N = 9 down to

N=4.

For very low output frequencies, this technique of variable

N is not appropriate since N should be changed very often,

for small changes of the modulation index M and fundamental

output frequency, which can result in several N changes within

the period of the fundamental depending on the increasing

ratio of the output frequency during the acceleration of the

motor. To overcome this problem a different modulation

technique like a carrier–based one can be used during the

initial stage of the motor acceleration, for very low output

fundamental frequency, and then, the modulation technique

can be switched to the OPP when the output fundamental

frequency becomes greater than a predefined minimum value.

Fig. 10 shows the results of a simulation of the start of

an induction motor through a carrier–based mIPD modulation

scheme subsequently switched to a variable N OPP modulator.

During the motor acceleration the OPP modulator change the

number of commutations per cycle from N=9 down to N=4.

On Fig. 10(a) the motor induced electrical torque and its ripple

are shown. It can be noted that the torque ripple increases

on the instant of N transitions due to the fact that at higher

number of commutations the harmonic distortions is smaller.

On Fig. 10(b) the motor current is shown and Fig. 10(c) shows

the line–to–line voltage imposed by the inverter. Fig. 10(d)

shows the mechanical angular velocity.

IV. CONCLUSIONS

This work proposed a way to overcome the additional

complexities for optimizing the modulation of multilevel con-

verters. The presented formulation does not use the concept

of switching patterns, that is the conventional way to solve

the problem of defining the direction of each step transition

for multilevel waveforms. It includes the decision of these

directions into the optimization problem as an expansion of the

search space of optimization variables. A comparison of this

technique with the conventional one was performed to validate

the formulation, showing that it found not only the switching

angles but also the optimum directions of each transition. To

Paper O1-2 Workshop on Control and Modeling for Power Electronics (COMPEL) 6

Fig. 10. Simulation results for a motor start with OPP modulation.

demonstrate the effectiveness of the proposed optimization

technique detailed simulations of an inverter feeding an in-

duction motor were performed. These simulations used at first

a relatively high number of commutation per cycle to demon-

strate that the optimization formulation proposed can deal with

generic waveform, with a high number of switching patterns.

Another set of simulations with very low commutation per

cycle shows the effectiveness of the optimized modulation

contrasting with the deficiencies of conventional modulation

techniques with severe switching frequency limitations.

ACKNOWLEDGEMENT

The authors would like to thank Petrobras for the prof-

itable technical discussions under the project process number

201200003-7.

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