In this paper, the Optimal Harmonic Stepped Waveform (OHSW) method is proposed in order to eliminate the selective harmonic orders available at the output of cascaded multilevel inverter (CMLI) fed by solar photovoltaic (SPV). This technique is used to solve the harmonic elimination equations based on stepped waveform analysis in order to obtain the optimal switching angles which in turn reduce the Total Harmonic Distortion (THD). The OHSW method considers the output voltage waveform as four equal symmetries in each half cycle. In the proposed method, a solar fed fifteen level cascaded multilevel is considered where the magnitude of six numbers of harmonic orders is reduced. A programmable pulse generator is developed to carry the switching angles directly to the semiconductor switches obtained as a result of OHSW analysis. Simulations are carried out in MATLAB/Simulink in which a separate model is developed for solar photovoltaic which serves as the input for cascaded multilevel inverter. A 3kWp solar plant with multilevel inverter system is implemented in hardware to show the effectiveness of the proposed system. Based on the observation the OHSW method provides the reduced THD thereby improving power quality in renewable energy applications.
1. Introduction
Harmonics is an important concern in all utility sectors. However, the electric utilities are recently designed to monitor and analyze the existence and effects of distortion on system and customer devices. The factors contribute the harmonic distortion on distribution systems are increased application of capacitors and non linear devices
[1]
. In order to improve the power quality and to maintain stable power supply performance, an inverter control strategy with harmonic reduction techniques can be employed
[2]
. Unlike conventional inverters, the multilevel inverters (MLI) can be employed because these converters are based on the series connection of single phase inverters and widely applied to interface to renewable energy sources, flexible AC transmission systems (FACTS), electrical drives and powerquality devices
[3

5]
. There are three types of MLI topologies such as diode clamped, flying capacitor and cascaded H Bridge. Amongst the three topologies, diode clamped multilevel inverters are widely used in industrial drive applications but it requires the number of diodes when the levels are increasing
[6

7]
whereas in flying capacitor, the number of capacitors are required when the level exceeds three
[8]
. Hence the cascaded multilevel inverter (CMLI) is an ideal choice to synthesize the desired output voltage from several separate DC sources (SDCs) like solar photovoltaic’s with reduced number of components when compared to other topologies
[9]
. In addition CMLI has no unbalance problem since each DC link is fed by an isolated DC source
[10]
.
The performance of multilevel converter (MLC) can be improved by Harmonic elimination, control strategies and new multilevel converter topologies
[11]
. Several methods are put forth for the harmonic elimination in literature. The methods proceeds from the basic sinusoidal pulse width modulation (SPWM) which are not able to eliminate lower harmonics completely
[12]
and space vector modulation (SVM) where the application of SVM in cascaded topologies is usually limited to a small number of levels due to the large number of switching vectors
[13]
. Selective harmonic elimination pulse width modulation (SHE PWM) is employed in multilevel inverters which require solving the non linear harmonic equations in order to eliminate the certain harmonic orders by the generation of switching angles corresponding to harmonic elimination.
Theory of resultant is proposed in
[14]
to derive the mathematical model, but this method is complicated, time consuming and not suitable for the varying input sources. Moreover the resultant theory is limited to find up to six switching angles for equal DC voltages and up to three switching angles for non equal DC voltage cases
[15]
. Homotophy algorithm provides only one set of solution
[16]
. Evolutionary concepts such as Bee algorithm (BA), Genetic algorithm (GA) and particle swarm optimization (PSO) methods are used to obtain the optimal switching angles in which additional to objective function the variables such as pheromone formation and its trail function in BA, crossover and mutation factors in GA, weighting function and random factors in PSO are required which makes the system complex
[17

18]
.
Waveform synthesis is used to suppress the harmonics and also to reduce the filtering requirement. Use of stepped waveform technique provides trade off between harmonic cancellation and equipment complexity
[19]
. Calculating switching angles which minimize the total harmonic distortion (THD) using step modulation and stair case modulation in a seven level inverter is proposed in
[20

21]
. Half wave symmetry for SHE is proposed for a five level inverter in [
22]
where 20 switching angles are considered in a half cycle incorporating its rising and falling edges, so 20 equations can be formulated. The number of switching angles for halfwave symmetry SHEPWM is twice that of quarter wave symmetry SHEPWM
[23]
.
Damoun et al., developed an equal area criteria based four equation method for five level inverter with unbalanced dc sources [
24]
. The line search trust region interior point algorithm to calculate the optimal switching angles in offline for a five level inverter is presented in
[25]
. All the methods referred deals with the minimum number of levels and the input DC other than solar PV is considered as the source for the multilevel converter.
A single phase grid connected photovoltaic inverter topology with boost converter, a low voltage three level inverter with an inductive filter and a step up transformer for interfacing the grid is considered in
[26]
. SHE for an eleven level inverter with Artificial Neural Network (ANN) based technique is proposed in [
27]
which requires an extensive data set required under various conditions which possess combinational problem.
In this paper, the optimized harmonic stepped waveform technique (OHSW) is applied to minimize loworder harmonics in a solar photovoltaic fed fifteen level CMLI, as well as to satisfy the desired fundamental component. Experimental results are presented to confirm the simulation results.
2. Problem Formulations
The cascaded multilevel inverter consists of a series connection of single phase full bridge inverter units as shown in
Fig. 1
. Each full bridge units known as ‘stages’ can generate three different voltages output: +Vpv, 0, Vpv. For the seven stage inverter the output waveform is illustrated in
Fig. 2
. The level and stage relation is given by the Eq. (1).
Fifteen level solar fed CMLI
Seven stage fifteen level output voltage waveform
where m is the number of levels and N
_{s}
is the number of inverter stages. Hence for a fifteen level output, seven inverter stages are required.
By using Fourier series the staircase output of multilevel inverter with non equal input sources is described in Eq.(2).
where k
_{i}
Vpv is the ith DC voltage, Vpv is the nominal DC voltage and the switching angles α
_{1}
to α
_{m}
must satisfy the following condition.
α
_{1}
≤ α
_{2}
≤ α
_{3}
≤……….≤ α
_{m}
≤
The number of harmonics which can be eliminated by the multilevel inverter is (m1). For a seven stage inverter, six numbers of harmonic orders can be minimized or eliminated. The equation governing the SHE for a fifteen level inverter is given in Eq. (3). Here the objective is to determine the switching angles α
_{1}
to α
_{7}
. The elimination of triple harmonics for the threephase power system applications is not necessary, because these harmonics are automatically eliminated from the lineline voltage
[17]
.
In CMLI based photovoltaic systems, the operating DC voltages of standard PV cells range from 15 to 35 V and from the CMLI based energystorage system with batteries, the voltages of batteries also change due to their states of charge
[21]
. Hence, the different voltages arrive at the inverter stages will cause unregulated fundamental with higher magnitude of low order harmonics which also to be considered.
3. Optimized Harmonic Stepped Waveform
The harmonic stepped waveform technique employing the quarter wave symmetry concept is much suitable for the multilevel inverter to determine the optimal switching angles inorder to provide the output voltage waveform with reduced total harmonic distortion (THD). Switching losses can also be minimized by turning the switch on and off only once per cycle.
There are three possible optimization techniques for reducing the loworder harmonics: 1) step heights are optimized with equally spaced steps, 2) step spaces are optimized with the steps of equal height and 3) optimizing both heights and spaces. In the variableinput scheme, the third optimization technique is more feasible than the other two techniques. The Fourier series expression for this method is given in Eq. (3). OHSW technique with equal input sources is given in
[28]
.
The OSHW is depicted as quarter wave symmetry where the first half cycle of the output waveform is given in
Fig. 3
. Considering the switching angles for the complete cycle as α
_{1}
to α
_{28}
, in which each set of α can be obtained by sampling the given waveform into four set of quarter. In the first quarter, from the interval zero to α1 the output voltage is zero and at α
_{1}
, the output voltage approaches to V
_{pv1}
. At α
_{2}
, the output voltage changes from V
_{pv1}
to (V
_{pv1}
+V
_{pv2}
) and the process continues upto
π
/2 where the voltage becomes (V
_{pv1}
+V
_{pv2}
+ V
_{pv3}
+V
_{pv4}
+ V
_{pv5}
+V
_{pv6}
+ V
_{pv7}
). In the second quarter the level of output voltage decreases to (V
_{pv1}
+V
_{pv2}
+ V
_{pv3}
+V
_{pv4}
+ V
_{pv5}
+V
_{pv6}
) at
π
α
_{7}
. The process will be repeated upto
π
α
_{1}
where the output voltage again approaches to zero to complete a half cycle. In the next half cycle the same process will be continued in the same manner other than the amplitude of the input voltage changes from positive to negative.
Quarter wave symmetry waveform for 15 level
First quarter:
Considering the switching angles α
_{1}
to α
_{7}
during the interval 0 to
π
/2.
Second quarter:
Third quarter:
Fourth quarter:
While considering the harmonic contents in the output waveform, the amplitude of DC components and all even order harmonics approaches near zero. Hence the odd harmonic in the quarter wave symmetric CMLI to be estimated. The set of non linear equations to obtain optimal switching angles is given in the Eqs. (4) to (10).
where
h
_{1}
is the amplitude of the fundamental component. As the proposed method involves the variable input sources, the V
_{pv}
is taken at the left hand side in the above expressions.
From Eq. (4), varying the modulation index value can control the amplitude of the fundamental component. The other nonlinear equations listed from (5) to (10) are the undesirable harmonic components can be eliminated which are set to be zero. Therefore, in an m level inverter, the lowest m1 odd harmonics can be removed.
The NewtonRaphson (NR) method is used to solve the nonlinear equation systems using successive approximation procedure which is suitable for implementing computer program. The following algorithm describes the steps involved to calculate switching angle.
Step 1
: Formation of switching angle matrix is given in the equation (11) where the unknown switching angles α
_{1}
to α
_{7}
to be determined.
Step 2
: Formation of nonlinear system matrix is given in the Eqs. (12) and (13).
Step 3
: Formation of harmonic amplitude matrix as per the Eq. (14).
Eq. (4) and (14) can be rewritten as per the following Eq. (15).
Using the matrices given in the Eqs. (11)  (15) and NR method the statement of algorithms is as follows to incorporate in the program using MATLAB software.
Step 1
: Guess for initial values of α as per Eq. (16). The process of finding the initial guess through equal area criterion is given in
[29]
.
Step 2
: This step includes the Eqs. (17)(20).
Linearizing the Eq. (15) at α
_{0}
we get
and
Solving the Eq. (19) using the inverse of the matrix given in the Eq. (20).
Step 3
: Updation of initial values is made as per the Eq. (21).
The steps (2)  (3) is repeated until
dα^{j}
is fulfilled and the condition to be satisfied is α
_{1}
, α
_{2}
, α
_{3}
………α
_{7}
<
π
/2.
4. Simulation and Results
Simulations are carried out in MATLAB / Simulink. In order to provide the source for a fifteen level inverter, a solar photovoltaic source model according to the specification of the commercial PV panel is used. The solar PV model comprises of many individual solar cells connected in series and parallel to obtain the required rating. The solar cell model consists of parallel combination of a current source, two exponential diodes, series (R
_{s}
) and parallel (R
_{p}
) resistances. The output current equation is given in Eqs. (22)(24). The PV array input to the each inverter stage is 48V, 7A. A single solar cell of rating 0.5V and 7A with standard test conditions of 1000W/mm
^{2}
and 25℃ of 96 numbers in series connection is chosen.
where
I
_{s}
and I
_{s2}
are the diode saturation currents, Vt is the thermal voltage, N and N
_{2}
are the quality factors (diode emission coefficients) and Iph is the solargenerated current. The quality factor is in the range of 1 to 2. The solar generated current Iph is given in the Eq. (25).
where I
_{ph0}
is the measured solargenerated current for irradiance I
_{r0}
. In order to verify the model, the PV and VI characteristics is plotted and shown in
Fig. 4
and
5
. The design is to generate the output peak voltage of 336V to deliver power to consumer loads as given in Eqs. (26)(27).
PV Characteristics of the solar panel
IV Characteristics of the solar panel
The simulation and analysis of optimal harmonic stepped waveform technique for solar fed CMLI is developed. The set of harmonic orders considered are 3, 5, 7, 9, 11 and 13. The initial guess values utilized are {0.01; 0.2989; 0.4387; 0.6950; 0.9240; 1.170; 1.52} and the process is repeated until all elements in
df^{j}
matrix are less than 0.0000001. A programmable pulse generator is developed which will carry the switching angles directly to the semiconductor devices which summarily reduces the complexity of the system.
Fig. 6
shows the output voltage waveform of OHSW based solar fed fifteen level inverter and
Fig. 7
shows the Fast Fourier transform (FFT) analysis of the waveform which shows the THD value of 8.55.
Table 1
shows the various sets of switching angles obtained for the modulation index M = 0.36 and M = 1.
Fifteen level output voltage waveform with OHSW
FFT Analysis for the output waveform with OHSW
Switching angles determination
Switching angles determination
The OHSW is compared with the conventional pulse width modulation (PWM) technique.
Fig. 8

9
shows the corresponding output voltage waveform and FFT analysis for a solar fed fifteen level multilevel inverter with PWM.
Fig. 10
shows the comparison of harmonic orders magnitude obtained using OHSW and PWM techniques. Based on the result it is observed that the OHSW technique improves the output quality of the waveform with reduced THD and magnitude of harmonic orders.
Fifteen level output voltage waveform with PWM
FFT Analysis for the output waveform with PWM
Comparison graph for harmonic order magnitudes
5. Experimental Results
A 3kWp solar PV power supply unit is designed and implemented for the seven stage fifteen level solar fed CMLI with the above mentioned algorithm. In India, much of the domestic loads operate from single phase 230V, 50Hz AC system. Hence the output voltage of the system is designed for the 230V AC supply. In order to achieve this, the input supply used in the setup is 48V.
Table II
shows the rating of the complete hardware setup. Each photovoltaic panel has a rated power of 115W with voltage variation of 16 to 21 volts (nominal 12V) depending on the operating conditions such as light intensity and so on. Four photovoltaic panels are connected in series to get a nominal voltage of 48V which constitutes as single individual source for CMLI. Four numbers of 12V, 100Ah battery packs are connected in series to get a nominal DC bus of 48 volts. These batteries are charged from the photovoltaic unit through a controlled charging circuit. There are seven sets of individual PV supply source used for the hardware implementation as they produce 48 x 7=336Vp (230Vrms).
Experimental setup specifications
Experimental setup specifications
Each input source to the CMLI provides the variation upto ±10%, the voltage sensors are provided at each source level. These inputs are given to the FPGA processor SPARTAN3E. Based on the OHSW algorithm incorporated in the processor, it generates the switching angles to the inverter switches MOSFET IRF840 through a driver circuit considering the sequences at each four quarters in the complete cycle.
Fig.11
shows the 3 kWp Solar PV plant and in
Fig.12
, the complete hardware setup comprising of charge controllers, battery and the proposed inverter is given.
Fig.13
shows the power and control circuit of inverter and
Fig.14
shows the FFT analysis using the Power Quality Analyser (PQA) WT3000 where the THD obtained in this method is about 9% with the possible minimisation of low order harmonics considered in this work. The variation of input PV voltage is measured at every instant which is used for solving OHSW equations. The comparison of the proposed method with the existing works is reported in
Fig.15
and
Fig.16
which shows that the proposed method provides higher voltage levels with V fundamental by considering more number of harmonic orders and results in lesser THD.
Solar plant of rating 3kWp
Hardware set up
Power circuit and control circuit of MLI
Output voltage waveform and FFT analysis
Comparison graph for performance parameters
Comparison graph for Vfundamental considered
In addition this method is implemented with solar photovoltaic’s as the input and also considered the variations of the solar panels with respect to temperature and irradiance. Based on the variations in the input V
_{pv1}
to V
_{pv7}
the necessary switching angles will be generated based on the equations given in (4)  (10). The advantage of the proposed method needs to satisfy the condition α
_{1}
, α
_{2}
, α
_{3}
…… α
_{7}
<
π
/2 rather than the condition α
_{1}
≤ α
_{2}
≤ α
_{3}
≤………≤ α
_{m}
≤
Adopting OHSW technique in
[30]
, the THD obtained is about 32.78 for a seven level inverter whereas for a fifteen level solar fed inverter the THD obtained in the proposed paper is 9.5. In both simulation and hardware, a solar fed fifteen level inverter of same specification and rating is taken into consideration. The correlation between the simulation and hardware results are made based on the THD analysis illustrated in
Fig. 17
which is related to the
Fig. 7
obtained through simulation.
Power Quality Analysis with WT3000
6. Conclusion
A fifteen level solar fed cascaded multilevel inverter for the elimination of certain harmonic orders is developed for the power quality improvement. The method is compared with the conventional PWM approach, which shows that OHSW based method is much suitable and an ideal choice for implementing selective harmonic elimination cases. The advantages of this method include simple computational algorithm and no requirement of filters, detailed look up tables and output transformers. Moreover, this technique can be implemented in both standalone and grid interacted PV systems.
Acknowledgements
The authors acknowledge and thank the Department of Science and Technology (Government of India) for sanctioning the research grant for the project titled, “Design and Development of Multilevel Inverters for Power Quality Improvement in Renewable Energy Sources” (Ref.No.DST/TSG/NTS/2009/98) under Technology Systems Development Scheme for completing this work.
BIO
S.Albert Alexander He received B.E. degree in Electrical Engineering from Bharathiar University, India. He received M.E. degree (Power Electronics and Drives) and Ph.D. degree from Anna University, India in 2007 and 2014 respectively. His research interests are multilevel inverters, renewable energy systems and soft computing techniques. He is the principal investigator for the major research projects of worth Rs.0.62 crores sponsored by Government of India.
Manigandan Thathan He received B.E. degree in Electrical Engineering from Government College of Technology, Coimbatore in 1996. He received M.E. degree in control systems from PSG College of Technology, Coimbatore, India in 1998 and Ph.D. degree from Anna University, India in 2005. His research interests are control systems, power electronics, Image processing and soft computing techniques.
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