1 Introduction

The rising need for electricity has led to a widespread adoption of renewable energy sources(RESs). This means we need to do something to extend the usefulness of our most basic forms of energy. To extend the availability of fossil fuels, various distinct methods have been developed. Efficient use of existing power sources is one option, as is expanding the use of renewable energy technologies like solar panels, wind turbines, thermal energy conversion from the ocean, wave energy converters, and so on. As a matter of procedure, power electronic converters are liable for meeting those commitments by using an inverter (DC to-AC converter) that produces an asymmetrical square wave. Low-pass filters are employed to eliminate the high and low harmonics of a waveform, improving its overall quality. From the studies of [1,2,3,4], it can be seen that, two level inverters were firstly conceived to convert AC into DC. However, these converters had a number of drawbacks, including excessive switching losses, high switching losses, high THD, high voltage stress (dv/dt), low efficiency, and the requirement for filters, among others. This has led to the development of multilevel inverters (MLI), which are ideal for medium-voltage and high-power applications. MLI produces a frequency response with a better emission than two level inverters. [5, 6], find out by comparing multilevel inverters to two-level inverters, there are various advantages such as less voltage stress on switches, less harmonic distortion on voltage waveforms, and less dv/dt on switching devices. Its benefits are mostly concentrated on enhancing the output signal quality and increasing the nominal power of the inverters. [7,8,9,10,11,12,13,14], suggests that researchers and scholars may now concentrate exploring emerging innovations that must be linked to renewable power facilities due to the increasing attention on sustainable power resources, particularly solar photovoltaic (SPV) power. Multilevel inverters/converters (MLIS), particularly, the field which has drawn the interest of many scholars [15]. shows that the usage of renewable energy is now more advantageous from a political and financial standpoint. Due to contemporary technology, ambiently renewable energies, encompass energy from the sun panels along with energy from the wind turbines, there are plenty different categories in energy and have grown incredibly common and challenging. In reality, PV solar sources have a variety of uses, such as pollution-free sources. [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32], through their analysis shows cascaded H Bridge Multilevel (CHB) Inverters are typically utilised in static var uses, like as battery-based and renewable energy applications. The wye form or the delta form can be used with CHB multilevel inverters. From [30] it can be seen that modules can also be used to create multilayer converters, giving rise to multi-modular converters (MMCs). A certain multilevel converter can produce a levels using a DC source and two switches, hence it may be regarded as a module. More levels can be attained if these modules are joined in succession. Through the study by [33] it is a common fact that problems for grid-tied users significant concern with voltage stability, that can be brought on by frequency inverters in photovoltaic systems and affects utility firms as well as power variance. From the analysis of [34] non-linear electric and electronic loads cause harmonics, which are multiples of the fundamental frequency. From studies of [35] multilevel converters with lower power switches & very few circuit components are necessary to construct in order to decrease the price, and inefficiencies in the multilevel inverter and satisfy the power needs. But at the other end, the symmetrical MLI has the ability to be a more effective PV converter because it can raise the inverter's voltage levels despite running at a lower switching frequency. [36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58], states the distortion of the harmonic components also with the resultant spectrum are also among the key benefits of using the SHE methodology over alternative methods. Since the total eradication of a specific set of harmonics, being the primary function of such filters, is guaranteed, it is essential to take into account this modulating technique in order to enhance the performance of passive active filters.

From Table 1 Based on the research findings the objectives to design the proposed topology has the following benefit.

  • Reduces the number of Power switch components usage.

  • Reduces the usage of DC sources, Capacitors & gate drive circuits.

  • Increases the number of output voltage level with lesser number of components.

  • Reduces the total harmonic distortion as per the IEEE519 standard.

  • Implementation of SHE based on soft-computing technique with PI controller improves voltage regulation.

  • Minimizes the losses which leads to the improvement of efficiency.

  • Better the reactive power compensation

Table 1 Conventional literature survey on 27 -Level reduced components count Multilevel inverter
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The organisation of this article is as follows: The overview of reduced multilevel connections is covered in Sect. 2. The converter's operating modes are included in the design. The state space analysis of the converter is covered in Sect. 3. The modelling of solar PV Module is covered in Sect. 4. The Theory of SHE is covered in Sect.5. The switching process of the proposed converter is covered in Sect. 4. The loss analysis of the proposed RMC is shown in Sect. 5. The system's reactive power compensation is described in Sect. 6. The cost calculation is presented in Sect. 7. The result and discussion findings are presented in Sect. 8. The comparative analysis of an asymmetrical 27-level inverter is covered in Sect. 9 The ultimate conclusions are presented in Sect. 10.

2 Proposed multilevel connections

The proposed approach only needs twelve switches to build the power circuit, while the conventional and trinary structures each need a different control circuit in order to reach level 27. Figs.1 and 2a depicts the structure of a simple, single-phase RMC. The switching sequence indicates in order to achieve 27 levels, Fig. 2b the polarity generation switches that generate positive half-cycles (T1,T2) and negative half-cycles (T3,T4) must be set to the ON position, while the switches responsible for producing level generation (S1–S4, B1–B4) as shown in Fig. 2c must be adjusted so that intermediate levels are produced. The PV system supplies the DC input voltages. Most likely, PV1, PV2, PV3& PV4 are the inputs. PV1 is the only source of energy for the first tier while the other PV sources are running. As with the previous levels, Level 27 is unlocked by toggling switches. According to the PV inputs, the RMC has the potential to produce 11 different levels. The range of possible output levels from the RMC given the PV inputs. The level 27 is generated by a combination of input DC sources fed by the PV array in the following ratio: 1:2:3:7.

Fig. 1
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Proposed Multilevel Connections

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Fig. 2
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a Reduced Multilevel Connections, b Level Generation Circuit, c Polarity Generation Circuit

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Table 2 shows the elements of the proposed inverter based on number of levels and number of units on symmetric & Asymmetric Configuration.

Table 2 Elements based on proposed system
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When the developed structure is connected in cascaded mode Fig. 3 the amount of voltage levels obtained is 2NL i.e., it will be equal to 54-Voltage Levels for the above topology.

Fig. 3
figure 3

Cascaded Mode Connection

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The proposed design has various modes of operation which are shown in Fig. 4a–n. During the positive half cycle the voltages + 1Vdc, + 2Vdc, + 3Vdc, + 4Vdc, + 5Vdc, + 6Vdc, + 7Vdc, + 8Vdc, + 9Vdc, + 10Vdc, + 11Vdc, + 12Vdc, + 13Vdc are obtained. For the negative obtained. e half cycle the output voltages obtained are– 1Vdc, – 2Vdc, – 3Vdc, – 4Vdc, – 5Vdc, – 6Vdc, 7Vdc, 8Vdc, – 9Vdc, – 10Vdc, – 11Vdc, – 12Vdc, – 13Vdc are. Table 3 shows the current path, number of switches in conduction and the developed voltage level corresponding to the switching strategy. The Positive voltage levels V0 =  + 1Vdc, + 2Vdc, + 3Vdc, + 4Vdc, + 5Vdc, + 6Vdc, + 7Vdc, + 8Vdc, + 9Vdc, + 10Vdc, + 11Vdc, + 12Vdc, + 13Vdc are obtained by using the same switching strategies at positive half cycle. In the negative half cycle, the polarity generation switches T1 and T2 are switched on and Stages (I-XIV) are achieved. Tables 4 and 5 show the voltage & current stress of the switches at each level which is significantly reduced from the proposed topology which leads to the reduction of switching & conduction loss. By the reduction of switching & conduction loss the overall efficiency is improved. Figure 5a, b shows the switching and conduction loss of the proposed topology when compared to the existing topology.

Table 3 Current Path of the Proposed system
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Table 4 Voltage stress on the switches for each level
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Table 5 Current stress on the switches for each level
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Fig. 4
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a–n: Modes of operation of the RMC positive half cycle

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Fig. 5
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a Switching loss, b Conduction loss

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3 State space modeling of proposed RMC

A system is represented in state space model by taking the systems input, output parameters and state variables.

General state space model is given by

$$ \dot{X} = AX + BU\;\left( {\text{State Space Equation}} \right) $$
(1)
$$ Y = CX + DU\;\left( {\text{Output Equation}} \right) $$
(2)

Here X = State Variable, U = Input Variable, Y = Output Variable & \(\dot{X}\) = State Derivative The state space model of the RMC is shown in Fig. 6.

Fig. 6
figure 6

a Basic structure of RMC, b Simplified RMC circuit for modelling

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Let,

$$ {\text{Hb = }}\left\langle {\begin{array}{*{20}c} {1} & {{\text{for }} + {\text{ve half cycle}}} \\ { - 1} & {{\text{for }} - {\text{ve half cycle }}} \\ \end{array} } \right\rangle $$
(3)

The variable Hb is denoted for the H-Bridge in the proposed topology. Hence, the net output VL for the inverter is given by:

$$ {\text{V}}_{{\text{L}}} { = }\left\| {1 1 1 1} \right\|\left[ {\begin{array}{*{20}c} {\vartheta {\text{1 Hb}}} \\ {\vartheta {\text{2 Hb}}} \\ {\vartheta {\text{3 Hb}}} \\ {\vartheta {\text{4 Hb}}} \\ \end{array} } \right]\left\| {{\text{Vdc}}} \right\| $$
(4)

where, Vdc is the sum of each dc source, and \(\vartheta 1\) to \(\vartheta 4\) is the comparator output signals.

By applying KVL to the circuit:

$$ {\text{V}}_{{\text{L}}} { = }R_{{\text{switch }}} {\text{i}}_{{\text{L}}} {\text{(t) + }}R_{{\text{o}}} {\text{i}}_{{\text{L}}} {(}t{) + }L\frac{{{\text{d}}i_{{\text{L}}} {\text{(t)}}}}{{{\text{d}}t}} $$
(5)

where, R0=Re+ Rinductor , and Re, Rinductor, Rswitch, and L are the total load resistance, internal resistance of inductor, Resistance of all conducting IGBTs in the ON state and load inductance in mH at each level. In this RMC the total conducting component in each level is same therefore in equation (15) substitute the value of VL from equation (14).

$$ L\frac{{{\text{d}}i_{L} (t)}}{{{\text{d}}t}} = - \left( {R_{{{\text{switch}} }} + R_{o} } \right)i_{L} \left( t \right) + \left\| {1111} \right\|\left[ {\begin{array}{*{20}c} {\vartheta 1{\text{Hb}}} \\ {\vartheta 2{\text{Hb}}} \\ {\vartheta 3{\text{Hb}}} \\ {\vartheta 4{\text{Hb}}} \\ \end{array} } \right]\left\| {{\text{V}}_{{{\text{dc}}}} } \right\| $$
(6)

Assuming the state variable as \(X={i}_{L(t)}\) (current through the inductor) and the input variable \(U ={\mathrm{V}}_{\mathrm{dc}}\)

$$ \frac{{{\text{d}}i_{L} (t)}}{{{\text{d}}t}} = \frac{{ - \left( {R_{{{\text{switch}} }} + R_{0} } \right) i_{L} \left( t \right) + \left\| {1111} \right\|\left[ {\begin{array}{*{20}c} {\vartheta 1{\text{Hb}}} \\ {\vartheta 2{\text{Hb}}} \\ {\theta 3{\text{Hb}}} \\ {\vartheta 4{\text{Hb}}} \\ \end{array} } \right]\left\| {{\text{V}}_{{{\text{dc}}}} } \right\|}}{L} $$
(7)

where,

$$ \dot{X} = \frac{{{\text{d}}iL\left( t \right)}}{{{\text{d}}t}}\left( {{\text{State}} {\text{Derivative}}} \right); $$
(8)

\(A = - \left( {R_{{{\text{Switch}}}} + R_{O} } \right)\); \({\text{B}} = [1111]\left[ {\vartheta 1{\text{Hb}}\vartheta 2{\text{Hb}}\vartheta 3{\text{Hb}}\vartheta 4{\text{Hb}}} \right]^{{\text{T}}}\)

The output voltage across the load

$$ V_{{{\text{out}}}} = R_{O} i_{L} (t) + L\frac{{{\text{d}}i_{L} (t)}}{{{\text{d}}t}} $$
(9)

where \(Y = V_{{{\text{out}}}}\); \(C = \left( {R_{0} + L\frac{{{\text{d}}i_{L} (t)}}{{{\text{d}}t}}} \right)\)

Therefore, the generalized (State space & Output) equation for proposed topology is given by:

$$ \frac{{{\text{d}}i_{L} \left( t \right)}}{{{\text{d}}t}} = \frac{{ - \left( {R_{{{\text{switch}} }} + R_{0} } \right)i_{L} \left( t \right) + [\vartheta 1{\text{Hb }}\vartheta 2{\text{Hb }}\vartheta 3{\text{Hb }}\vartheta 4{\text{Hb}}]^{T} \left[ {{\text{V}}_{{{\text{dc}}}} } \right]}}{L} $$
(10)
$$ {\text{V}}_{{{\text{out}} }} = R_{o} i_{L} (t) + L\frac{{{\text{d}}i_{L} \left( t \right)}}{{{\text{d}}t}} $$
(11)

where Vdc is the magnitude of each dc source fed from solar PV and iL(t) is the current through the inductor.

4 Modelling of solar PV module

Photo Voltaic system consist of.

  • Photo voltaic modules

  • Boost converter (DC/DC Converter).

  • Maximum Power Point Tracking (MPPT) controller.

Solar PV cells are contained in Solar PV modules (SPV), which are connected in series. By interconnecting the PV segments in parallel and series, a solar PV array is created. The voltage–current (V–I) characteristics of individual solar cells vary. In order to simulate the MPPT boost converter, various elements are used, including the PV model.The major purpose of the PV model is to generate the Current–Voltage I–V characteristic curve of PV module. A reverse biased diode that is shunted across a resistor Rsh is used in the equivalent circuit of a solar panel, as shown in Fig. 7.

Fig. 7
figure 7

Equivalent circuit of Solar PV Panel

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4.1 Design of solar PV panel

Each solar cell has an open circuit voltage of 0.5 V and a short circuit current of 7 A. There are 24 solar cells in a single module. At typical test circumstances (1000 W/m2 and 25 °C), these cells are wired in series to provide 12 V, 7A. A 24 V, 7 A solar PV array is created by connecting two single modules in series. [54, 55],. The fixed interconnection design is put into place, and the ratio of DC sources is kept at 1: 2: 3: 7. The designed inverter produces 27 levels of output voltage from four DC sources.

$$ \begin{aligned} {\text{V}}_{{{\text{peak}}}} = & {\text{24V}} + {\text{48V}} + {\text{72V}} + {\text{168V}} \\ { = } & {\text{312V}} \\ \end{aligned} $$

4.2 DC–DC (BOOST) converter with MPPT control

To deal with the issue of the PV module being partially shaded, DC-DC converters are used. The Converter used in the proposed system is a boost converter. [48, 49]. A boost converter is a most efficient type of converter module that has been incorporated into this system. The boost converter is essentially a DC–DC converter that generates an output voltage that is higher than the input voltage.

As shown in Fig. 8, where the inducer, diode, capacitor and switches are the components of the boost converter which provides there required voltage level with the MPPT controller, the MPPT technique provides the extracted voltage and current from the PV source to the boost converter.

Fig. 8
figure 8

Circuit Diagram of DC–DC converter

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4.2.1 Maximum power point tracking (MPPT) method

To achieve 27 Levels, the proposed inverter uses various DC Sources. So, a discrete boost converter integrated with MPPT is used to obtain the desired output voltage [48, 49],. The power output by PV panels can be increased by using the MPPT method as shown Fig. 9 which employs an Adaptive modified perturb and observe algorithm [59]

Fig. 9
figure 9

Flow chart of Adaptive modified P & O Method

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5 Theory on selective harmonic elimination (SHE)

Among the various modulation techniques (Sinusoidal PWM, Multi-Carrier PWM, SHE PWM & Space-Vector PWM) the most efficient way to eliminate specific lower order harmonics is done by SHE-PWM[55],.It is derived from the Fourier transform, which contains a variety of triggering states over its complete cycle. These degrees of freedom can be calculated to verify that it is possible to generate a fundamental voltage component without (S-1) harmonics. By developing a set of equations to indicate when the triggering of switches should occur, these degrees of freedom are employed. In this work it is mainly focused to cancel the odd order harmonics.

In order to lower the 'S ang-1' harmonics for an N-level inverter, 'S ang' triggering angles must be determined. Therefore, in order to get the triggering angles for the 27-level RMC, the following Equation must be utilized.

$$ S_{{{\text{ang}}}} \,=\, \frac{{27 - 1}}{{2}} = 13$$
(12)

The switching angles must adhere to the following requirements:

$$ \beta_{1} < \beta_{2} < \beta_{3} < \ldots \beta_{13} < \prod /{2} $$

Equations below presents the nonlinear equations required to obtain the triggering angles.

$$ B_{1} = \frac{{4{\text{V}}_{{{\text{PV}}}} }}{1\pi }\left[ {{\text{cos}}\;\beta_{1} + {\text{cos}}\;\beta_{2} + \ldots + {\text{cos}}\;\beta_{13} } \right] $$
(13)
$$ B_{5} = \frac{{4{\text{V}}_{{{\text{PV}}}} }}{5\pi }\left[ {{\text{cos}}\;5\beta_{1} + {\text{cos}}\;5\beta_{2} + \ldots + {\text{cos}}\;5\beta_{13} } \right] $$
(14)
$$ B_{7} = \frac{{4{\text{V}}_{{{\text{PV}}}} }}{7\pi }\left[ {{\text{cos}}\,7\beta_{1} + {\text{cos}}\;7\beta_{2} + \ldots + {\text{cos}}\,7\beta_{13} } \right] $$
(15)
$$ B_{{{11}}} { = }\frac{{{\text{4V}}_{{{\text{PV}}}} }}{{{11}\pi }}\left[ {{\text{cos}}\,11\beta_{1} + {\text{cos}}\;11\beta_{2} + \ldots+ {\text{cos}}\;11\beta_{13} } \right] $$
(16)
$$ B_{{13}} = \frac{{4V_{{PV}} }}{{13\pi }}\left[ {\cos \;13\beta _{1} {\text{ }} + \cos \;13\beta _{2} + \ldots + \cos \;13\beta _{{13}} } \right] $$
(17)
$$ B_{17} = \frac{{4{\text{V}}_{{{\text{PV}}}} }}{17\pi }\left[ {{\text{cos}}\;{17}\beta_{{1}} + {\text{cos}}\;{17}\beta_{{2}} + \ldots + {\text{cos}}\;{17}\beta_{{{13}}} } \right]{,} $$
(18)
$$ B_{{{19}}} = \frac{{{\text{4V}}_{{{\text{PV}}}} }}{{{19}\pi }}\left[ {{\text{cos}}\;{19}\beta_{{1}} {\text{ + cos}}\;{19}\beta_{{2}} + \ldots + {\text{ cos}}\;{19}\beta_{{{13}}} } \right]{,} $$
(19)
$$ B_{23} = \frac{{4{\text{V}}_{{{\text{PV}}}} }}{23\pi }\left[ {\cos \,23\beta_{1} + \cos \;23\beta_{2} + \ldots + \cos \;23\beta_{13} } \right] $$
(20)
$$ B_{{{25}}} = \frac{{{\text{4V}}_{{{\text{PV}}}} }}{{{25}\pi }}\left[ {{\text{cos}}\;{25}\beta_{{1}} + {\text{cos}}\;{25}\beta_{{2}} + \ldots + {\text{ cos}}\;{25}\beta_{{{13}}} } \right]{,} $$
(21)
$$ B_{{{29}}} = \frac{{{\text{4V}}_{{{\text{PV}}}} }}{{{29}\pi }}\left[ {{\text{cos}}\;{29}\beta_{{1}} {\text{ + cos}}\;{29}\beta_{{2}} + \ldots + {\text{ cos29}}\;\beta_{{{13}}} } \right]{,} $$
(22)
$$ B_{31} = \frac{{4{\text{V}}_{{{\text{PV}}}} }}{31\pi }\left[ {\cos 31\beta_{1} + \cos 31\beta_{2} + \ldots + \cos 31\beta_{13} } \right] $$
(23)
$$ B_{{{35}}} = \frac{{{\text{4V}}_{{{\text{PV}}}} }}{{{35}\pi }}\left[ {{\text{cos}}\;{35}\beta_{{1}} + {\text{cos}}\;{35}\beta_{{2}} + \ldots + {\text{ cos}}\;{35}\beta_{{{13}}} } \right], $$
(24)
$$ B_{37} = \frac{{4{\text{V}}_{{{\text{PV}}}} }}{37\pi }\left[ {\cos \;37\beta_{1} + \cos \;37\beta_{2} + \ldots + \cos \;37\beta_{13} } \right] $$
(25)

The resultant equations, however, must be solved using quantitative tools because they are nonlinear and transcendental in nature. Due of limitations in computing the optimal switching angles mathematically, optimization techniques like Newton Rampson (NR), Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) are employed to compute the triggering angles. The PSO approach is superior among other methods [59]. in terms of switching angles prediction accuracy, fundamental output voltage generation and total harmonic distortion reduction in the case of different level CHB inverters. So, in this research work, the switching angles determined for the asymmetric 27-level RMC using the PSO approach is shown in Fig. 10.

Fig. 10
figure 10

Flow chart of PSO Algorithm for calculating Switching angles

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By solving the preceding equations, we obtain 13 triggering angles from that the harmonics of orders 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, and 34 are cancels each other. The SHE approach was applied for the 27-level RMC.For RMC with Mi = 0.95, the corresponding switching angles are listed in Table 6 which is shown in Fig. 11.

Table 6 Switching angles of the proposed system for different inverter levels (Mi = 0.95)
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Fig. 11
figure 11

Modulation Index Vs Switching Angles for 27-Level RMC

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6 Reactive power compensation in power system

Generally Reactive power can be compensated using solar power systems as per the IEEE1547a standard (Fig. 12). The designed controller injects reactive power into the photovoltaic inverter and smooths its apparent power transition. To reduce voltage, drop, reactive power is fed into the distribution line, when real power drops unexpectedly. However, frequent reactive power from the PV fed inverter generates reactive current, which increases power system losses. When the PV system's power reaches the proportional gain Kp maximum, the slope of the apparent power of inverter (Sin v) is fixed with a constant integral gain Ks value.

Fig. 12
figure 12

Reactive Power Compensation Method

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The MPPT controller receives PV panel voltage and current. The proposed system tracks solar panel power using the Adaptive Improved P and O method [58]. Cascaded control loops use the MPP block output signal as a voltage reference. A PLL determines grid phase and angle. The PI controller detects the DC error. The PLL angle and PI controller current signal form the reference active current component, \({{i}_{\mathrm{a}}}^{\mathrm{ref}}\). Q produces reactive current, \({{i}_{\mathrm{a}}}^{\mathrm{ref}}\). The inner current controller of the cascaded control system tracks the sum of the actual and reactive current references The PI controller receives the error from the inverter current and the reference currents (\({{i}_{\mathrm{a}}}^{\mathrm{ref}}\) + \({{i}_{\mathrm{r}}}^{\mathrm{ref}}\)). The SHE block receives the output from PI. The proposed inverter's switches are triggered by pulses from the SHE block, compensating reactive power.

The reactive power Q can be calculated by performing the steps.

Step 1: Sense the Grid voltage and Inverter current.

Step 2: Determine the active power, the active power's rate of change over time, and the apparent power.

Step 3: After determining \(\frac{\mathrm{d}{P}_{\mathrm{inv}}}{\mathrm{d}t}\) value, equate the obtained \(\frac{\mathrm{d}{P}_{\mathrm{inv}}}{\mathrm{d}t}\) value with the proportional gain \((-{K}_{\mathrm{p}})\).This stage decides the amount of reactive power needed for compensation by the proposed inverter.

Step 4: This stage helps to determine, At this point, we determine how much reactive power the proposed inverter needs to compensate for.

$$ {\text{If}}\frac{{{\text{d}}P_{{{\text{inv}}}} }}{{{\text{d}}t}} < - K_{p} ;\;{\text{than}}\;{ }\frac{{{\text{d}}S_{{{\text{inv}}}} }}{{{\text{d}}t}}{\text{is}}\;{\text{fixed}}\;{\text{to}} - K_{{\text{s}}} $$
$$ {\text{If}}\;\frac{{{\text{d}}P_{inv} }}{{{\text{d}}t}}{ \nless } - K_{p} ;\;{\text{then check}} $$
$$ {\text{If}}\;S_{{{\text{inv}}}} > P_{{{\text{inv}}}} \;{\text{than; }} \frac{{{\text{d}}S_{{{\text{inv}}}} }}{{{\text{d}}t}}\,{\text{is}}\;{\text{fixed}}\;{\text{to}}\; - K_{{\text{s}}} $$
$$ {\text{If}}\;S_{{{\text{inv}}}}{ \ngtr }P_{{{\text{inv}}}} ;\;{\text{than}}\;\frac{{{\text{d}}S_{{{\text{inv}}}} }}{{{\text{d}}t}} = \frac{{{\text{d}}P_{{{\text{inv}}}} }}{{{\text{d}}t}}\;{\text{no reactive power injection}} $$

Step 5: The apparent power \({S}_{\mathrm{inv}}\) is updated by using \(\frac{\mathrm{d}{S}_{\mathrm{inv}}}{\mathrm{d}t}\).The updated \({S}_{\mathrm{inv}}\) and the \({P}_{\mathrm{inv}}\) values are used to determine reactive power \((Q)\). From this \(Q\) value, the reactive reference current is achieved. The Q value is updated from time to time to successfully fix the reactive power compensation (Fig. 13)

Fig. 13
figure 13

Flow chart to determine reactive power Q

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.

7 Cost calculation

The total cost of the RMC depends on the voltage and current rating of the switches. It also depends on the voltage blocking capacity of switces if it doubles it cost of RMC will be more.The cost of RMC depends on the number of IGBT’s, number of driver circuits, number of different DC sources and standing voltage rating of switches.The quantity of IGBTs NIGBT, the quantity of driver circuits (Ndriver), the variety of DC sources (Ndiff DC), and the standing voltage rating of the switches (VTSV) all affect the price of RMC.

$$ {\text{RMC cost }} = \, N_{{{\text{IGBT}}}} + \, N_{{{\text{driver}}}} + {\text{ Nd}}_{{\text{iff DC}}} + \, \zeta V_{{{\text{TSV}}}} $$
(26)

The weight coefficient of total standing voltage on switches is ζ which is inverse to total power switches. Greater total standing voltage requires higher ζ. The proposed RMC uses fewer NIGBT, Ndriver, and lesser VTSV than the conventional configuration.

8 Results and discussion

8.1 Simulation results

Utilizing MATLAB/Simulink, the proposed PV fed RMC is simulated. A Diode rectifier in combination with R- L Load is used as Non-Linear Load. The simulation results is obtained for the duration 0.01–0.2 s for the proposed shunt active filter. To achieve a steady, constant DC voltage A series connection is made among 2,4,6,14 batteries of 12 V range. This RMC generates a peak voltage of 312 V. The DC inputs of 24 V, 48 V, 72 V and 168 V as in Fig. 14 The switching angles produced using PSO techniques are stored in the lookup table for switching sequences generation. The output waveform of the 27-level RMC is shown in Figs. 15 and 16 along with an analysis using the Fast Fourier Transform (FFT). The system's measured Total Harmonic Distortion (THD), which is does not include RMC-based APF, is shown in Fig. 16. The proposed active power filter is changed from 0.05 to 0.2 s in order to acquire the simulation results in the simulation model. Figure 16 displays the measured THD of the RMC-based APF.The measured voltage THD value for 0 ≤ t ≤ 0.05 s in Fig. 16c is non-sinusoidal in character and has a measured THD value of 30.43%.(not including APF). The measured voltage THD value drops to 3.11% when the APF is turned on for 0.05 ≤ t ≤ 0.2 s, and the current THD value is obtained as 0.81%, which is within IEEE -519 standards and suggests the waveform obtained is sinusoidal in nature. The measured current and voltage THDs after compensation are shown in Fig. 16a, b. After compensation, there is some short-term fluctuation in the source voltage, which has a harmonics of 3.11% The system's source voltage and source current are shown in Fig. 17. The system's load voltage and load current are shown in Fig. 18. The source current is initially not sinusoidal due to the nonlinear load, as shown in the Fig. 16 However, after the SAPF is switched from 0.05 to 0.2 s, it is discovered that the source current more closely resembles a sinusoidal waveform. When nonlinear loads are used in the load side of the SAPF, the RMC generates compensation currents that act as right angles to the harmonic currents generated by those nonlinear loads. Harmonic currents and compensation currents cancel each other at the Point of Common Coupling (PCC), which frees from the source current harmonics (Figs. 19, 20 and 21).

Fig. 14
figure 14

Solar input to the RMC

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Fig. 15
figure 15

27-level output of RMC

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Fig. 16
figure 16

a 27-level output of RMC, a Voltage THD, b Current THD, c Voltage THD without APF

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Fig. 17
figure 17

FFT analysis of a Voltage THD with SAPF, b Current THD with SAPF

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Fig. 18
figure 18

Source Voltage and Source Current

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Fig. 19
figure 19

Load Voltage and Load Current

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Fig. 20
figure 20

Compensation currents of each phase

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Fig. 21
figure 21

Active Power & Reactive Power

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Table 7 Components list with their respective rating
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8.2 Experimental validation

The prototype of the proposed RMC-based SAPF is developed. The developed prototype's is shown in Fig. 22. Each leg of the voltage source inverter consists of 12 switches FGA25N120 with a gate driver HCPL316J running at a switching frequency of 1 kHz to test the performance of the proposed system. To reduce current harmonics and balance reactive power, coupled inductors (1.9mh) are connected in series with the RMC (Fig. 23). As the load, a diode rectifier is used in. The Components list with their respective rating are shown in Table 7. The voltage and current sensors connected to the Controller board's analogue to digital converters generate the reference currents for the SAPF. Harmonics are introduced into the system through the introduction of a nonlinear load on the DC side. The Solar PV Systems parameters for hardware design are shown in Table 8. The hardware output of the 27-level RMC is shown in Fig. 24. The hardware output of the three phases' source voltage is shown in Fig. 22. The compensating currents generated by the RMC-based SAPF to reduce the current harmonics in the source side are shown in Fig. 23. The hardware output of the load current is shown in Fig. 24. The hardware output of the source current is shown in Figs. 25 and 26. Due to the injection of compensating currents at PCC, it can be shown in Figs. 27 and 28 that the source current has a sinusoidal nature.

Fig. 22
figure 22

Prototype of the Proposed system

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Fig. 23
figure 23

Solar Plant of rating 3kWp

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Table 8 Solar PV Systems Parameters
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Fig. 24
figure 24

a 27-level Output of RMC, b Magnified view of 27-levels

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Fig. 25
figure 25

Experimental output source Voltage

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Fig. 26
figure 26

Experimental output source Current

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Fig. 27
figure 27

Experimental output Load Current

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Fig. 28
figure 28

Experimental output Compensation Currents

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The active and reactive power of the system are represented in Fig. 21. The proposed PV fed RMC-based APF corrects the source side's power factor almost to unity by injecting harmonic currents. Only active power is provided from the source side, as can be shown in Fig. 21. The figure also shows that source-side reactive power delivered is almost negligible, resulting in a power factor of unity there. As a result, the RMC-based APF injects harmonic currents that are similar in size but in phase opposition at PCC to make the distorted source current approximately sinusoidal. Consequently, the proposed technology improves the quality of the power.

Figure 29 presents the change in output current when the resistive load is switched from 0 to 100 Ω. Figure 27 illustrates the change of output current when the RL load is changed from 50Ω + 100mh to 50Ω + 50mh (Fig. 30).

Fig.29
figure 29

Experimental output voltage & output current when the R load is switched from 0 to \(100\Omega \)

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Fig.30
figure 30

Experimental output voltage and output current when the RL load is changed from \(50\Omega +100\mathrm{mh}\) to \(50\Omega +50\mathrm{mh}\)

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Figure 31 illustrates the switching reaction of the system with and without SAPF. From Fig. 26, it is clearly seen that the source current before switching on the ASAPF is non-sinusoidal in nature and after switching exhibits there is a sinusoidal current wave of the injection of the compensation current.

Fig. 31
figure 31

Switching reaction of RMC based SAPF

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As a result, RMC topology improves power quality. The frequency spectrum of the proposed topology is acquired using a power quality analyzer and is shown in Fig. 32. The proposed topology, when used with SAPF, produces measured current THD of 2.33% and measured voltage THD of 2.61%, as can be shown in Fig. 32. The permitted IEEE standard limitations are not exceeded by this measured THD %. It has been demonstrated through simulation and experimentation that the suggested RMC decreases THD, raises voltage levels with fewer components, and compensates reactive power (Fig. 33).

Fig. 32
figure 32

Voltage & Current THD of the proposed system

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Fig. 33
figure 33

Comparative Analysis of the Proposed system THD with other 27-Level Asymmetric MLI

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9 Comparative analysis

Table 9 shows the comparison of the proposed work with other existing work, From the table, it is clear that the proposed 27 Level Multilevel inverter topology reduce the THD by 2.61 which is lower than 5%,satisfying the standard of IEEE519.

Table 9 Comparison of the Proposed work with other existing work
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10 Conclusions

In this research paper, Novel multilevel connections have been Modelled & Developed, when compared with conventional topologies it is more superior in terms of number of components, number of levels, simple switching pattern, reduction in total losses with improved efficiency.Interms of switch count, proposed topology uses only 12 IGBT’s & only 4 DC sources with various configuration for achieving 27-Levels.And switch stress is reduced to nearly 50% which leads to the improvement of efficiency. Modelling of PV Panels is carried out with integrated Step-up Converter(Boost converter) which uses Adaptive Modified P&O (MPPT) method for voltage regulation. In terms of harmonic mitigation, the SHE method is implemented with Particle swam optimization technique for the generation of 13 switching angles apart for this, the cost calculation is also done for the proposed method. Simulation analysis is carried out with the help of MATLAB-SIMULINK (R2020A) which proves that the total harmonic distortion is reduced to 3.62% for voltage and 3.85% for current. The Proposed RMC is experimentally investigated & Votlage THD is around 3.11% & Current THD is around 0.81% which is less as per IEEE 519 standard. Further, the developed inverter acts as shunt active filter is for reactive power compensation in power system application.Finally, this paper is concluded with the comparative analysis with other recent work.