Keywords

1 Introduction

Generation of energy from RES has been exponential in current years and is essentially driven by factors like the escalation of greenhouse emissions with depleting in fossil fuel reservoirs [1,2,3]. Photovoltaic (PV) energy, a prominent RE source, produces electricity directly from sunlight. Its operation is noiseless and harmless to the environment. The PV system is interfaced with various essential power electronic components to achieve necessary efficiency in energy conversion to harness renewable energy.

Controlling the output voltage of PV cells varies when regulating the irradiation and environmental [4,5,6,7]. Hence, the necessity of MPPT technique and boost converter is considered to generate desired output voltage by PV cells. A variety of control techniques has been worn to discover the maximum power point (MPP). Perturbation and observation (P&O) technique is very easy and simplicity with simple implementation. P&O works based on iterative method, and it swings near the point dP/dV = 0, i.e., MPP. Incremental conductance (IC) technique gives the best dynamic performance and accuracy when fast change in atmospheric environment. In this paperwork, IC MPPT is worn with the adaptive distinction in size of steps. IC method used to solve the difficulty of conventional process when the irradiance changes. IC method evaluates with some further variable step size methods; the worn method is not complex instruction coefficient and estimates to progress the computing speed.

Voltage production by PV is not efficient for network rating and hence considers the DC-DC boost converters to boost the PV panels output. The boost output voltage in form of DC, but requirements in commercial purpose in form of AC seeing as most of the loads are AC loads.

Hence, MLI topologies are used to produce AC voltages to improve the voltage performance with less THD [8,9,10,11].

CHB MLI, diode-clamped inverter (DCI), and capacitor-clamped inverter (CCI) are very common topologies and fine expression in the literature [12,13,14,15]. CHB MLI has various advantages compared to the conventional MLIs but requirements of devices are more. Hence, RDC MLI configuration topology with GA is used in this paper to minimize the THD.

The number of devices in MLI increases, and then cost, circuit configuration, and reliability also increase. Hence, the configuration of MLIs with less number of devices is the key element [16,17,18,19,20,21,22,23,24,25,26,27,28,29]. To generate the same output voltage levels such as 7-level (7L) CHB MLI, a novel configuration is designed with 7L-RDC MLI devoid of raising the number of bridges.

In this manuscript, SHE technique is taken to solve the nonlinear equations and produced the best results of MLIs. In addition, this manuscript reports the on PV 7-level (7L) RDC MLI topology; this process requires less switches and devices. The voltage produced by 7L-RDC MLI similar voltage output generates via 7L-CHB MLI.

In this work, estimating the values of switching angles is through solving of transcendental nonlinear equations with using GA. Newton–Raphson (NR) and resultant theory (RT) method suffer from some disadvantages to calculate the optimal value of switches [15]. NA and RT methods are not suitable when it applied to higher level MLIs. Hence, in this work bio-inspired optimization method-based GA SHE is used to the development of the system robustness.

GA is useful to solve the constrained and unconstrained troubles through the suitable assortment of unique population and gradually regularly adjust the individual population. By GA, find out the optimal values of angles of PV-based 7-L CHB MLI and 7L-RDC MLI, this work is performed the minimum value of THD.

2 Modeling of PV Cell

The simple way to represent a photovoltaic cell is with one diode model [2]. It consists of parallel with the current to a diode and required parameters such as diode identity factor \(\left( a \right)\), \(\left( {V_{{\text{oc}}} } \right)\) open-circuit voltage, and \(\left( {I_{{\text{sc}}} } \right)\) short-circuit current.

The realized characteristics of PV cell by the fundamental equation of a one cell as the source of current are placed in parallel by diode. The current output Eq. 1 is as follows:

$$I = I_{{\text{PV}}} - I_{\text{o}} \left[ {e\frac{{q\left( {V + {\text{IR}}_{\text{s}} } \right)}}{akT} - 1} \right] - \left[ {\frac{{V + {\text{IR}}_{\text{s}} }}{R_\rho }} \right]$$
(1)

\(I_{{\text{PV}}}\) is the function of irradiance \(\left( G \right)\) and the mathematical calculation of PV current is given as in Eq. 2

$$I_{{\text{PV}}} = \left[ {I_{{\text{PV}}\_{\text{STC}}} + K_i \Delta T} \right]\frac{G}{{G_{{\text{STC}}} }}$$
(2)

The saturation current \(\left( {I_{\text{o}} } \right)\) of diode is formulated in Eq. 3 as follows:

$$I_{\text{o}} = I_{{\text{o}}\_{\text{STC}}} \left( {\frac{T}{{T_{{\text{STC}}} }}} \right)^3 \exp \left[ {\frac{qEg}{{ak}}\left( {\frac{1}{{T_{{\text{STC}}} }} - \frac{1}{T}} \right)} \right]$$
(3)

The flowchart representation of the PV current calculation is given in Fig. 1. The detailed outputs of PV model are shown in Fig. 2.

Fig. 1
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Flowchart of PV cell current calculation

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Fig. 2
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Output of the PV model

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3 Modeling of Boost Convertors

Boost convector is known as step-up convertor which converts DC to DC power [2]. This is similarly to switched mode power supply (SMPS) among a diode, a transistor with storage element of energy such as capacitor and inductor. The basic principle of boost convertor is that it can work in two modes. The first one is when the switch is working mode, then energy stores in inductor and energy releases by capacitor. In case when switch is not working, then energy stores in capacitor and released by the inductor. The voltage across the inductor is mathematically given as represented in Eq. 4.

$$\begin{aligned} v_{\text{L}} & = V_{\text{s}} = L\frac{{{\text{d}}i_{\text{L}} }}{{{\text{d}}t}} \\ \frac{{V_{\text{s}} }}{L} & = \frac{{{\text{d}}i_{\text{L}} }}{{{\text{d}}t}} \\ \end{aligned}$$
(4)

4 Incremental Conductance Method

IC is the method with the high performance, increased tracking speed better efficiency, and ease in implementation [2].

The basic idea in IC algorithm is that slop of the PV curve becomes zero on MPP. The slope of PV derivative power with respect to the voltage of PV has successive relation with MPP [1, 2].

\(\frac{\Delta P}{{\Delta V}} = 0\) at MPP, \(\frac{\Delta P}{{\Delta V}} > 0\) left side for MPP,

$$\frac{\Delta P}{{\Delta V}} < 0\,\,\,\,{\text{Right side for MPP}}$$
(5)
$${\text{At the MPP}}:\frac{{{\text{d}}I}}{{{\text{d}}V}} = - \frac{I}{V}$$
(6)

When working point left side for MPP:

$$\frac{{{\text{d}}I}}{{{\text{d}}V}} > - \frac{I}{V}$$
(7)

When working point right side for MPP:

$$\frac{{{\text{d}}I}}{{{\text{d}}V}} < - \frac{I}{V}$$
(8)

where dI/dV is IC and \(\frac{I}{V}\) is instantaneous conductance; MPP is found by evaluate instantaneous conductance to the IC. The pictorial form of the IC is shown in Fig. 3, and comparisons of different MPPT techniques are shown in Table 1.

Fig. 3
figure 3

Flowchart of IC method

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Table 1 Different of MPPT techniques
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MPPT methods are key enablers of an extra energy sustainable culture, due to their utilization ease, low cost, and malleable operation. The analysis is done on variable irradiance and temperature conditions. It is observed that IC has better performance than P&O techniques.

5 PV 7L-CHB MLI Topology

CHB MLI modular topology gives the preferred output voltage, power with higher stability. Modular topology feature enables each inverter to be a module with a comparable circuit topology, power arrangement, and modulation [9, 11]. Each inverter level gives three different output voltages, i.e., + Vdc, 0 or − Vdc [12]. Three-level structure investigated by [12], and total voltage positive half cycle is + 3Vdc, and in negative, half cycle is −3Vdc. The number of output phase voltage steps is calculated via n = (2 k + 1), and here k defines number of DC supply. The PV 7L-CHB MLI configuration scheme is shown in Fig. 4. This configuration produced 7L voltage steps with the use of 12 IGBTs electronics device and 3 DC supply. MLI switching function with 12 switches and 3 DC supply is shown in Table 2.

Fig. 4
figure 4

Configuration of 1-ɸ 7L-CHB MLI with PV scheme

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Table 2 Switching function of 7L-CHB MLI
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6 PV 7L-RDC MLI Topology

In MLIs, the number of switches represents the price, volume of circuit, reliability, and complexity. So the number of necessary switches to generate the essential voltage level is necessary to designing a MLI. To achieve the same output in 7L-CHB MLI, a novel structure is configured and less count of number of switches exclusive of rising the number of H-bridges. Figure 5a shows the proposed MLI structure, and Fig. 5b shows the novel MLI voltage waveform with consequent switching condition. The switches operation (or polarity generation switches) is working at low switching frequency while the steps formation switches T1–T3 working at high switching frequency. Consequently, a price efficient solution is to choose lower switching power equipments of the H-bridge and high switching power equipments for the steps formation.

Fig. 5
figure 5

Seven-level RSMLI and its output voltage a Structure of 7L-RDC MLI, b Output voltage waveform with corresponding switching condition

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In comparison with a 7L-CHB MLI, the required number of switches in 7L-RS MLI is very less. In 7L-CHB MLI, 12 switches are used, whereas only 7 switches with 2 discrete diodes are used in design of 7L-RDC MLI. The 7L-RDC MLI working principle is summarized in the following modes.

Mode (a): When T1 is ON, the current flows through two diodes and H-bridge switches S1 and S2. Output voltage at load is + Vdc. Similarly, S3 and S4 are switched ON, and voltage output is −Vdc.

Mode (b): In this mode, switch T2 is make ON and then path of current through diode D2, switches S1 and S2. Output voltage at load is + 2Vdc. Similarly, S3 and S4 are switched ON, and voltage output is −2Vdc.

Mode (c): T3 is ON, and this mode of operation starts. Path of current is switches S1 and S2. Output voltage at load is + 3Vdc. Diodes D1 and D2 are not performed operation. Similarly, S3 and S4 are switched ON, and voltage output is −3Vdc.

Mode (d): Any set of S1 and S3 or S2 and S4 are ON in this mode. Switches T1, T2, and T3 are in OFF condition. Obtained load across voltage is zero.

The RSC-MLI switching scheme is represented in Table 3. The voltage is obtained from a RSC-MLI with corresponding switching states is depicted in Fig. 5a, b.

Table 3 Switching scheme of 7L-RSC-MLI
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7 Calculation of THD Using Optimization Algorithm

As described earlier, the SHE technique significantly makes to produce the output voltage of MLIs. MLIs voltage output waveform is positive half is equal to the negative half, i.e., exhibits quarter symmetrical output. The formation of output voltage waveform is shown in Fig. 7b, and it can be defined in Fourier form as,

$$V{\text{(t)}} = \sum_{n = 1}^\infty {V_n } \sin (n\alpha_n ) + B_n \cos (n\alpha_n )$$
(9)

The even harmonics usual canceled because it is quarter-wave symmetry, i.e., \(B_n\) = 0 for all n. So the novel equation of output voltage becomes

$$V{\text{(t)}} = \sum_{n = 1}^\infty {V_n } \sin (n\alpha_n )$$
(10)

The amplitude \(V_n\) can be defined in terms of Fourier form with \(\alpha\) varying in the range 0 to π/2,

$$V_n = \frac{{4V_{{\text{dc}}} }}{n\pi }(\cos n\alpha_1 + \cos n\alpha_2 + \cos n\alpha_3 )$$
(11)

Three switching angles \(\alpha_1\),\(\alpha_2\), and \(\alpha_3\), the fundamental component of output voltage is adjust by the modulation index (M). The dominant lower order harmonics is necessary to eliminate from the output voltage. Accordingly, the three nonlinear equations for elucidation of the problem are used in this paperwork as:

$$\begin{aligned} V_1 & = \frac{{4V_{\text{dc}} }}{\pi }(\cos \alpha_1 + \cos \alpha_2 + \cos \alpha_3 ) = m \\ V_5 & = \frac{{4V_{\text{dc}} }}{5\pi }(\cos 5\alpha_1 + \cos 5\alpha_2 + \cos 5\alpha_3 ) = 0\; \\ V_7 & = \frac{{4V_{\text{dc}} }}{7\pi }(\cos 7\alpha_1 + \cos 7\alpha_2 + \cos 7\alpha_3 ) = 0 \\ \end{aligned}$$
(12)

where\(m = \frac{V_f }{{({{4V_{{\text{dc}}} } / \pi })}}\), modulation index = \(M = m/N_{{\text{DC}}}\) and NDC is the number of DC sources, \(V_f\) is the required value of fundamental voltage.

Fifth- and seventh-order harmonics are targeted to eliminate by solving the above nonlinear as Eq. 13. The solution of these equations leads to discontinuity for the certain modulation index. Conventional technique such as NR method is able to solve these problems involves much calculation time and difficult mathematical arrangement. In order to reduce this, GA is used here to solve the objective function to find the best value of switching angles. The optimal fitness function (FF) is taken to remove the 5th and 7th component harmonics at optimal values of switching angles. Fundamental component equation goes as under:

$${\text{FF}} = \frac{1}{h}*\left[ {\left| {M - \frac{{\left| {V_1 } \right|}}{{N_{{\text{DC}}} V_{{\text{DC}}} }}} \right| + \left( {\frac{{\left| {V_5 } \right| + \left| {V_7 } \right|}}{{N_{{\text{DC}}} V_{{\text{DC}}} }}} \right)} \right]$$
(13)

Subject to condition that,

$$0 < \alpha_1 < \alpha_2 < \alpha_3 < \frac{\pi }{2}\;$$
(14)

To obtain least THD with possible modulation index, the total set is multiplied with a factor \(\frac{1}{h}\). The target of this paper, 5th and 7th harmonics are set under the error limit 1% (i.e., h = 0.01). To analyze the superiority of voltage THD, it can express and calculated as under:

$${\text{THD}}\,(\% ) = \left[ {\frac{1}{V_1^2 }\sum_{k = 2}^\infty {(V_k )^2 } } \right]^\frac{1}{2} \times 100$$
(15)

where \(V_k\) is the voltage of exacting harmonics.

8 Genetic Algorithm (GA)

In this section, GA results obtained for the switching angles of 7L-CHB MLI and 7L-RDC MLI are described. GA [15] is the method of random search process used to find optimal solutions of optimization problems. This is a search technique inspired by biological evolution, such as selection, crossover, and mutation. Usually, evolution starts from a random initial population and repeatedly modifies the solution in each of generation. Multiple switching angles are selected over the successive generation from current population and are used in the next iteration of the algorithm. It is repeated till an optimal solution is reached with defined accuracy.

With continuous number of generations and a large population in each production, the algorithm finally finds for a set of solutions of the problem. It computes the different values of three switching angles to obtain the minimum FF keeping the harmonics within the limit. Flowchart collection of the GA shown in Fig. 6 represents the different steps for calculation of FF values.

Fig. 6
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Flowchart arrangement of the GA

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Fig. 7
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Output voltage (V) and current (A) of PV system

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The GA toolbox results obtained from GA are summarized in Table 4. To use GA toolbox, The following information are entered.

Table 4 Results obtained from GA FOR 7L-MLIS
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Fitness function: The FF as given entered in the form @FF, where FF.m is a program file that computes the FF.

No of variables: The length of input vector to the FF is entered. Three switching angles are considered as variables.

Constraint function for the problems is specified in constraints pane of toolbox. Boundary variable for the variable is also specified. Population size, initial population, and command are set in the options pane.

9 Results and Analysis

The configuration of PV, boost converter (DC-DC), and IC MPPT integrated with MLI is designed at MATLAB platform. The generated voltage and current output by PV without boost voltage are shown in Fig. 7.

The voltage should be increased for each PV panel as shown in Fig. 8. The PV output voltage of 7L-RDC MLI interfaces with boost converter is shown in Fig. 8.

Fig. 8
figure 8

Output of 7L-CHB MLI between boosted voltage (V) and time (s) of a PV panel1, b PV panel2, and c PV panel3

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The simulated output voltage is observed from 7L-CHB MLI and 7L-RDC MLI as shown in Fig. 9a, b.

Fig. 9
figure 9

a Overall output voltage of 7L-CHB MLI, b 7-L RDC MLI

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FFT analysis of THD present in 7L- CHB MLI output voltage is 15.50% and 7L-RDC MLI 10.36% at the fundamental frequency of 50 Hz which is as shown in Fig. 10. The comparisons of MLIs performance and different MPPT control schemes are shown in Table 5.

Fig. 10
figure 10

FFT analysis of THD present in the voltage output for a 7L-CHB MLI, b 7-L RDC MLI

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Table 5 A comparison of MLIS performance and MPPT
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Using a new 7L-RDC MLI topology reduced switch count was presented and also compared with the conventional 7L-CHB MLI. The total number of switches is almost halved as compared to the CHB MLI topology. So, switching loss is expected to be reduced significantly and cost is also less.

10 Conclusion

This paperwork describes 1-ɸ PV 7L-RDC MLI and 7L-CHB MLI with DC-DC converters, IC control technique and compared it. The feature of PV-based 7L-RDC MLI plays the important role for produce the sufficient power to fulfill the load demand with fewer number of semiconductor switches. The THD performance of output voltage is improved with less number of components reducing the volume, size, and price of MLIs. The MLIs have less the %THD without any filter circuit requirement. GA optimization technique is used to faster response with less time-consuming. SHE equation of MLIs is solved by GA toolbox, and simulation is done at MATLAB platform. From the calculated results by simulation, it clarifies GA-based SHE technique can be useful to MLI. When the level of MLI is increased, the output voltage waveform just goes to smooth sinusoidal appearance and also decreases the THD.