Generally, the PV module shows a non-linear power characteristic, which varies as solar irradiation (in kW/m²) and ambient temperature vary. The model’s classes of grid systems with PV cells have been described in several pieces of the literature (Estévez-Bén et al., 2020). As a distributed intermittent power resource, a solar PV module is reflected as the significant importance in the HPGS. The operational framework of a solar PV cell formulates the key characteristic in building a solar PV module by Eq. 7 (Patel, 2006). The photovoltaic module is made up of various PV cells interconnected in sequences of the array. The PV module is developed by a series of PV cells that are associated in a particular order. The schematic circuit of the stand-alone PV cell with the diode is shown in Figure 2A.

This circuit consists of a PV cell that is composed of the power supply current ( I p h ) , a diode ( D j ), a parallel resistance   ( R p a r ) , and q   is the electron charge; and a series resistance ( R s e )   increasing the unit of a PV cell to maximize the PV generated power range is shown within (Nwaigwe et al., 2019), and the resulted current (in ampere) of the analyzed PV array range is indicated by Eq. 7. I P V = N i j I p h − N j k I N L { exp [ q ( V P V + R P 1 I P V ) k A T N s e N i j ] − 1 } − ( ( V P V + R P 2 I P V ) R P 2 ) . (7)

Here, V P V is the voltage of a PV module, PV range, N i j , and N j k   are typically the required PV units attached in parallel as distributed generation, individually (Zolfaghari et al., 2018), where N s e is the quantity of the sequence-linked cells in a PV module; the parallel resistances are R p 1 and R p 2 correspondingly, and I P h and I N L are a particular phase current (in amp) and no-load current (in amp), respectively, in Eqs 8, 9. I P H = [ I T , n + k i ( T p − T p n ) ] . ( G G n ) , (8) I N L = I T , n . ( T p T p n ) 3 . (9) Additionally, generated current ( I 0 , k ) is shown in Eq. 10. I 0 , k = I s c , k { exp [ ( q V o c , k ) / ( k A T k N i j ) ] − 1 } , (10) where T is the cell temperature (in ^o K) and G is the inversion (in K) and photovoltaic irradiance (in kW/m ² ). Individually, for instance, the electron energy is related within semiconductors ( e v ). Also, A   is the charge of the electron (in C), the Boltzmann constant (within J/K), in addition to the diode operation. Figure 2C displays the simplified analysis through the DC/DC converter, which is coupled with the PV module to the DC link connection. The switching behavior of a diode (D) as a switch (S) is demonstrated in Figure 2B. The chopper DC/DC converter is displayed in Figure 2C. Hence, the dynamic calculation is utilized to simulate the DC/DC converter. F c × P w × V P V = i P V − i L P , (11) ( F L ) . ( P w ) .   ( i L P ) = − R P i L P + V P V − ( 1 − K P ) . V D C , (12) i P V − D C = ( 1 − K p ) . i L P . (13)

Here F _c is typically the filter capacitance, i P V − D C   is the current of dc-link through the PV module, L C and R C are the certain values of converter inductance and resistance, respectively. In addition to this, K P will be the liable ratio from the current DC/DC converter. Also, i L P is the inductor current and V D C is the dc-link voltage.

The power and current characteristics with respect to the voltage of a PV module are demonstrated in Figure 3. The proposed solar PV module is simulated for average insolation (800 W/m²) with a temperature of 27 ℃ .

The main objective is to feed extracted HPGS power via the boost conversion stage into the distributed grid by distributed grid-side converters. This is accomplished by the control of DC-link voltage as mentioned in the study by Hoon et al. (2016). In this research work, a VSI was linked to the distributed grid through an L-type filter for better response. Similarly, boost converters or the PV inverters have been controlled by the controlling of the duty cycles, and hence, output voltage was regulated. Various techniques based on PWM can be employed to generate the duty cycle for the inverters. A bipolar modulation has been preferred due to its low leakage current. This is important for transformer-less PV systems. There are two switching states in bipolar modulation: 1). SW₁, SW₄ are on, and SW₂, SW₃ are off, and 2). SW₁, SW₄ are off, and SW₂, SW₃ is on. Required expressions are given by Eq. 8. { L . d i g r d t = v D C − R × i g r − v g r ,     0 ≤ t < d . T s w L . d i g r d t = − v D C − R × i g r − v g r ,     d . T s w ≤ t < T s w , (14) where v D C is an input DC voltage from the VSI by boost stage with voltage ( v g r ) and current ( i g r ) parameters of the HPGS grid. For the design of a controller associated with the PV inverter, the transfer function in the frequency domain is to be achieved. So forth, a small-signal model was introduced to make the dynamic equations linear. Eq. 9 has been used to calculate the average value of the inductor voltage. Finally, using Eq. 10, the transfer function for the system can be obtained and thus used to design the controller for the associated PV inverters. L d ( i g r ) d t = ( 2 d − 1 ) × ( v D C ) − R . ( i g r ) − v g r , (15) i ^ g r ( s ) = 2 D − 1 L s + R . v ^ D C ( s ) + 2 V D C L s + R . d ^ ( s ) − 1 L s + R × v ^ g r ( s ) . (16)

High-order filters have been used for the suppression of desired harmonics; it is a combination of L and C together. Therefore, higher frequency harmonics have been suppressed by the LCL filter. A damping-based resistor is generally employed in the design of an L-C-L filter to rectify the system resonance issues that may be responsible for instability in a hybrid grid system (Teodorescu, 2011).

The FFT-based THD analysis of voltage characteristics is shown in Figure 4. Initially, when the load is not increasing, the THD in the distributed voltage is found at 14.67% and the waveform has a nominal value of RMS voltage.

The design of the wind power generation control system is based on a mass-spring-damper control system (Kumar et al., 2021) that can be modeled as p ( z ) = U z , (17)       m t × p × u z = F w − K s − K D . u z − P f , (18) where the gear operator is symbolized by p, the floater wind speed is denoted by z , U z are the distances, m t is the overall size of the specific floater, F w is the certain force coming from the wind, and P f   is the pressure exerted on the floater. In Eq. 18,   K D along with K s are the dissipating ratio and spring constants, respectively. The LPMG’s representation based upon the d-q axis mechanism of WPG is expressed as (Penalba et al., 2017) L d L G × p × i d L G = ω z . L q L G . i q L G − R L G . i d L G − v d L G , (19) L q L G × p × i q L G = ω z . ψ P M − ω z . L d L G . i d L G − R L G . i q L G − v q L G . (20)

In Eq. 19, i d L G , i q L G , v d L G , and v q L G are both the d and q-axis voltages and current parameters, respectively. Also, R L G is the resistance of stator-winding. In Eq. 20, L d L G _G and L q L G include the d-q-axis inductance parameters individually. i q L G _ r e f = ( 2 τ K D U z ) ( 3 π ψ P M ) . (21)

The control mechanism of WPGS for VSC is designed based on d and q axis parameters related to the particular translation for the L_PMG, which is modified in the control operator associated with the VSC. The operation of the WPGS control operator is to get optimum energy coming from the wind energy resources and enhance the energy in the L_PMG. In research (Kulkarni et al., 2016), we can evaluate the energy loss in the particular L_PMG and the effective energy transformed from the wind resources to manage and simply by d and q axis parameters of the L_PMG ( i d L G and i q L G ), individually. As referred from Figure 5, the proposed control structure associated with the VSC to determine the d-q axis parameters of L_PMG as per reference indexes i d L G _ r e f and i q L G _ r e f , and optimal use of PI controller.

Solar and wind are both energy resources that are intermittent in nature. Thus, stability can be improved by the hybrid resources of these two systems to comply with the remote areas. These systems can be further employed for the application of storage systems through the admittable converter systems, that is, AC-DC converters, then DC-DC converters, and again DC-AC converters. The load demands can be consummated by these renewable sources combined with storage systems and various converter/inverter systems. This hybrid system has shown diverse dominance to supply the energy demand by converter topology. Eqs 21, 22 used for the DC-to-DC converter for boost operation are expressed as follows: [ p i E p v c ] = [ − R E + ( 1 − D ) ( R C / / R ) L E − ( 1 − D ) R L E ( R + R C ) ( 1 − D ) R ( R + R C ) C − 1 ( R + R C ) C ] . [ i E v C ] +   [ 1 L E 0 ] v g , (22) v E =   [ ( 1 − D ) ( R C / / R ) R ( R + R C ) ] .   [ i E v C ] . (23)

Eqs 21, 22 are representing the converter state vector model and its output, i E is the current through inductor L E , v c is the voltage within capacitor C, v E is the voltage across the output terminals of the converter, and D is the duty ratio of the switching device to transfigure input voltage to other higher-level voltage. DC-AC inverter system linearization modeling is shown by the following equations.

The output voltage on a per-unit basis across the VSI terminals is given as follows, in Eq. 23: V I =   V q I × cos ⁡ γ I − (   π 6   ) × X C I × I I . (24)

In Eq. 23, V I and I I are representing inverter input side voltage and current, respectively, γ I is the excitation angle, X C I depicts the commutation choke reactance of VSI, and V q I is a voltage of the inverter output side. The system linearization equations for the HPGS along with converters and inverter topology can be simplified in the most appropriate formats as follows: p [ X ] = f ( [ X ] , [ U ] ,  t ) , (25) where [X] is representing the state vector, [U] is the control input vector, and f   is a non-linear vector of the concerned hybrid system. Eq. 24 can be written in the linearized format as follows: p [ ΔX ] = [ A ] [ ΔX ] +   [ B ] [ ΔU ]   . (26)

Here, [ A ] and [ B ]   represent the HPGS and controller matrix, respectively, and d e t ( [ A ] −   γ [ I ] ) is the characteristic system linearization equation expressed as follows in Eq. 27. det ( [ A ] −  γ [ I ] ) = 0 , (27) where [ I ] is the identity matrix of a suitable rank, and γ is the eigenvalue of the proposed system matrix [ A ] . When any characteristic root is being on the right side of the principal complex Laplacian plane, then the HPGS may be subjected to disturbance, and it may lead toward its overall system instability.

The equivalent circuit of the supercapacitor (SC) is shown in Figure 6A. In addition to the capacitance (C _SC ) of the SC, the optimal placement of SC represents as well as holds into consideration reactive power compensation using SC. From Figure 6A, the dynamic computation of the SC is employed and represented by Eq. 31. S C c × P w = S C SCPS C SCVC , (31) V S C = V q s c s c ⁡ csc , (32) where SC is the voltage ( V S C )   and current ( i S C ) , respectively, and V C S C is the voltage across C S C .

The simplified layout of the control schemes is presented in Figure 6B. The DC-DC converter comprises two switches (SW₁ and SW₂), which are operated in a switch operation. The converter is enabled by the boost setting of the circulation current. Switch SW₂ works as a switch, whereas SW₁ works as a diode, and the energy is fed by the SC to the DC link. Furthermore, once operating the DC-DC converter in the supercharged condition, the transitioning approach is usually overlooked. Figures 6C,D describe the active mean value of bidirectional DC-DC topology, which is step-down and setting of desired voltage. The mathematical expression of the converter mode topology is given in the following text.

a) In the boost mode:

The control configuration of a typical bilateral DC-to-DC converter via DC link and the SC is capable of maintaining the active power flow, as presented in Figure 6E. Generally, a bidirectional DC-to-DC converter-based controller is employed, which is dependent on current control loops. The external current control loop measures the V D C   through DC link as per V D C _ r e f and the internal control loop for i S C control via SC. Once the DC voltage reaches the typical desired range, then SC is followed by i S C _ r e f , which is further controlled by the external loop via DC connection voltage.

The power generated by the solar PV module is fluctuating and changes with the solar irradiance and temperature. To operate the HPGS efficiently, the maximum power delivered by the PV module needs to monitor constantly. To achieve this objective, the step-up DC/DC converter associated with the PV module is regulated to control the V D C of the module to get its desired maximum value. Various algorithms of MPPT are proposed in several stability studies (Hlaili and Mechergui, 2016). Furthermore, this article used the extensive MPPT control technique; that is, P&O is applied for controlling the operation of step-up DC/DC converter.

The controller layout of voltage d-q axis parameters is represented in Figure 7A. The controller is associated with the VSI, which is utilized to manage the distributed energy transmitted via the VSI, and also to stabilize the voltage value of the PCC at the grid side. To get the optimal control associated with the active grid elements, the active power of the VSI is carried out within the d-q axis lined up at the PCC. As a result, the energy transmitted using the voltage source inverter associated with the PCC is managed by the i d and i q   d-q axis for VSI correspondingly. The controller associated with the VSI offers the rapid control system, as demonstrated in Figure 7A, where the external control loops deal with the effective energy flow from VSI at a different voltage.

The active power is controlled through the VSI via reference PCC of the HPGS (PV and wind power generating systems). Thus, the control mechanism allows the VSI to smoothly transmit the HPGS-based electricity to the grid. It can be achieved by applying the external control scheme as presented in Figure 7B. The developed HPGS power is transferred through the low-pass filter (LPF).

The time constant of low-pass filters is symbolized by T ₁ and T ₂ , respectively, as shown in Figure 7B. To minimize the power variation caused by wind, speed fluctuation (T ₁ ) should be chosen equivalent to the wind duration (T _w ). However, another constant can be chosen, that is, T ₁ to control the power developed by P _LG and the PV module.

The voltage controller associated with a grid-connected inverter has been designed in three parts: voltage regulator for V P V control, current controller for i P V control, and PI controller for V d c   synchronization. Grid synchronizer has been designed to generate reference grid current so that P and Q can be under a specified limit (Zhang et al., 2010).

It is observed from Figure 8 that v g β = grid voltage  ( v g ) , and v g is in a similar quadrant as virtual voltage v g α ; therefore, v m g = v g β 2 + v g α 2 , (37) where magnitude v m   is denoted by v m . g . A controller for current control, proportional-resonant (PR) has been proposed in the study by Ngo and Santoso (2016). This PI controller is accomplished of desired reference signals, and it does not require an error value (Ozdemir, 2016). This task is accompanied only due to the reason that the PR controller has a higher gain value even at the resonant frequency by utilizing the advantage of delay caused by the PWM generator (Kazimierczuk, 2000). The filter transfer function for the grid is obtained with the difference between the VSI output V and I. The current controller expression can be represented as F C c c ( s ) = F C P R ( s ) + F C R C ( s ) = k p r + k i r s 2 + w 0 2 + k r c Q ( s ) e − s T 0 1 − Q ( s ) e − s T 0 . G f ( s ) . (38)

In Eq. 38, PR-based controller is signified by F C P R ( s ) for fundamental grid frequency controller, where k p r is proportional control gain and k i r is the resonant control gain to develop PR-based controller. F F C P R ( s ) is represented through values of w 0 and RC controller ( k r c ) , and the fundamental time ( T 0 )  is  2 π ω 0 . Q ( s ) is an LPF and G f ( s ) is a phase-lead compensator which is equal to e s T c with compensation time ( T c ) (Jadoon et al., 2017).

In a grid-connected VSI, two control loops (external and internal) are typically employed (Hossain et al., 2017). The external control loop is employed to power control and control power fed into the distribution power grid through the DC link. However, the inner control loop is employed for current control and manages the power quality problems in faster response (Hossain et al., 2017). There are different types of proportional control schemes for the outer control power loop which depends on grid operation. But the most common control scheme is PI through the DC link voltage controller presented in Figure 9.

For the control of DC-link voltage, a closed-loop controller is needed to adjust within a specified voltage limit. The reference parameters of DC link voltages are adjusted by a voltage controller-based external loop, which controls the injected distributed grid current values. DC voltage was fed through a proposed PV module to regulate a reference current (i _ref) for injected grid current (i _grid ). The DC voltage control is usually controlled using a PI-based controller. A PI controller is employed for DC link voltage balance. This PI controller representation is shown in Figure 9.

In Figure 11, the inverter configuration has been shown, which is connected through an inductor–capacitor–inductor (LCL) filter scheme. Here DC side of the inverter is interconnected with a supercapacitor (SC). A buck-boost (DC-to-DC) converter has taken advantage of connecting SC, which employs an inductor L s c and bidirectional switches SW₃/SW₄. In this configuration, i s c is the current through the SC, d s c duty cycle, and VSC is the voltage across the SC terminals. The designed LCL filter embodies two inductors L 1 and L 2 , and one capacitor C f , as exhibited in Figure 11. At PCC, the grid has been connected to the hybrid system, where U S a b c is the rated grid voltage, Z g is the impedance of the grid, i g is a current injected into the grid, i C is the capacitor current, i i n v is the inverter current, and finally U g a b c is the voltage of the distributed grid at PCC, where abc is the reference frame (stationary) that was converted into a synchronous-based reference frame (d-q) by taking advantage of Park’s transformation. The power developed by the solar subsystem is expressed as P ∗ , P g is the injected power in the grid network, while K P is the droop gain for developing the controller loop, w is the system frequency, w ∗ is the designed system frequency for the hybrid system, and finally θ   corresponds to the synchronizing angle.

The specific range of solar irradiance of the solar PV module is shown in Figure 13A. The variation in irradiance was taken for 0.5 s in the MATLAB simulation tool. The fluctuation of the solar power irradiance is supposed to consist of substantial imbalances due to the intermittent nature of the local atmospheric circumstances.

The dynamic response of the voltage of the PV module is shown in Figure 13B. The PV generation characteristic is displayed in Figure 13B and the resulting distributed energy of the PV module through controlled VSI boosts output and reduction of unwanted harmonics in the load side. Furthermore, it is observed the distributed energy of the particular PV module is certainly impacted by the variation of solar irradiance and maximize their value up to 10 kW.

Figure 13C displays the relative response of the battery charging energy provided to the distribution grid via a dc-link connection. Active response of the analyzed method is obtained within the variations of the solar power. It is noticed that if the PI control scheme is investigated together with the battery, then the effective energy is given directly into the distributed main grid. Furthermore, when the SC is generally absent, the distributed generated energy into the main grid fluctuates substantially.

These subsections investigating the effect of different sizes of SC are used for stability evaluation. In Figure 14A, the relative dynamic response of the stored energy is given to distributed grid for three different ratings of SC. Three different supercapacitors with a capacity of 60/30/95 kW are typically used in the stability analysis of the HPGS.

From the simulation results, validation of response characteristics illustrated in Figures 14B,C, it has resulted that at the 50 Hz constant frequency, under RL and without load conditions, the proposed HPGS generates an overrated RMS voltage of 960 V or 1.02 kV (peak). Now, the simulated loads are added at PCC through a DC bus system.

Here in Figure 14C, we can examine that specific voltage drops in the HPGS demonstrated at 0.25–0.6 s. After 0.6 s duration, the load will be removed, and we have noticed that the reduced rated voltage is now stabilized.

From Figure 14D, the stability of a grid-connected HPGS has been successfully investigated at no load; also, during 3–4 s, I waveforms tend to stabilize. To reduce the fluctuation of a connected grid system, a supercapacitor-based energy storage technology was deployed. Furthermore, a d-q control technique is proposed for maintaining the power flow balance and obtaining maximum power from an HPGS coupled to the grid system. The d-q based control scheme is proposed to suppress power variation from the distributed energy resources to the distribution grid, while maximum energy extracts were obtained through a P&O-based MPPT controller. The results presented for the power quality are considerably higher by enabling the integration of an effective distributed HES approach operated by the PI controller.

To stabilize the current drop phenomena, the proposed RL load is presented to the HPGS referred from Figure 14E. It was observed that SC was integrated with DC bus along with RL load at 0.67 s, as depicted from Figure 14F. When SC is placed into the HPGS along with sudden RL load, the proposed system maintained their current and frequency values at rated values of 425 V (RMS value) or 590 V (peak value). Furthermore, these results show that the optimal placement of SC by VSI voltage is controlled, even though the DC bus voltage is controlled. It is depicted that, in the current response, roughly transients are there because of switching time in the SC. Thus, nevertheless, of these slight transients, the simulated results show a stability enhancement approach in the performance of the proposed HPGS.

The grid-side harmonics of the proposed HPGS model is reduced, that is, 1.06% THD. Figure 15A shows the output power was ensured comparable with IEEE standards 519–2014 (Kanjiya et al., 2015). Since the voltage of the DC buses was modified by modifying the resonance frequency using a PI controller, it should also be specified that the recommended converter has the highest converter efficiency for boost operation.

The FFT algorithm based THD of the V and I in the proposed simulated HPGS as shown in Figures 15B,C under the RL load scenario was examined as per the IEEE THD range (<5%) (Kanjiya et al., 2015).

Simulated results of the proposed PI control scheme conclude that the SC enables the HPGS to achieve the optimal implementation and limits reactive power at a steady state. The PI controller was utilized to achieve the active power flow through the DC link in the HPGS and stored SC energy to maintain the load level. The PI controller uses different modes to get optimal results from the HPGS.