Active Disturbance Rejection Control of Photovoltaic Three-Phase Grid Following Inverters Under Uncertainty and Grid Voltage Variations

A PV three-phase grid following inverter (GFI) with LCL filters can reduce current harmonics and deliver active power to the grid. Controlling such higher-order systems is challenging due to system uncertainties, grid voltage variations, and filter resonance. Active disturbance rejection control (ADRC) provides resilience to these uncertainties and a prompt response to disturbances. A linear extended state observer and a phase-locked loop observer are integrated into the ADRC structure to optimize tracking performance. The proposed controller uses a structured inner and outer closed-loop ADRC that takes into account process delays and extended state observer dynamics. The GFI can operate under variable grid voltage frequencies or even deviations in frequency caused by grid failure, and is able to meet these challenges as part of the design process and the synchronization approach. Both simulation and experimental results are given to validate the dual ADRC-based methods for an 8 kW laboratory prototype PV-GFI system.

These energy sources connect to the grid at the point of common coupling using grid following inverter (GFI) and LCL-type input filters.While LCL filters improve grid power quality by reducing high-frequency switching components [1], they introduce control complexity and resonance issues.Ensuring high-quality grid-injected currents requires careful design of both the GFI and its associated controller to maintain reliable operation under various disturbance scenarios [2], [3].
A variety of approaches have been proposed to estimate and mitigate disturbance effects on physical GFI systems [4], [5], with active disturbance rejection control (ADRC) being a suitable solution.ADRC's high noise attenuation capability impacts controller dynamics during parameter uncertainties [4], and when based on an extended state observer (ESO), it can estimate uncertainties and external disturbances, enhancing GFI system dynamics [5].ESO and third-order ADRC demonstrate strong robustness against internal and external disturbances.Control methodologies in [6] and [7] effectively address imbalanced grid voltage and filter parameters using a combination of disturbance observers and a current controller.A higher-order linear ESO (LESO), called generalized ESO (GESO), compensates for total disturbance with minimal tracking error, improving system dynamics without relying on an accurate system model [8]; however, stability margin decreases as ESO order increases [8].
To enhance GFI stability, voltage sensorless control coupled with a grid-side current controller offers optimal performance while reducing hardware complexities [9].Despite these techniques achieving sensorless grid voltage operation, output currents remain highly sensitive to system parameter variations.Employing only grid-side current sensor installation, these systems utilize robust current management algorithms and disturbance observers (DOBs) to estimate grid voltage and state variables, successfully stabilizing LCL filtered GFIs under internal and external disturbances.However, none of the previously proposed robust control methods have considered grid frequency variation as a significant disturbance source during grid faults.Furthermore, a high disturbance rejection capability diminishes the observer's bandwidth, subsequently degrading system dynamics.The phase-locked-loop (PLL) dynamics, used for grid synchronization, impair system stability in weak grid This work is licensed under a Creative Commons Attribution 4.0 License.For more information, see https://creativecommons.org/licenses/by/4.0/Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
conditions [10], and PLL-less current control discussed in [11] is more suitable for ideal grid conditions while remaining sensitive to a wide range of grid impedance variations and parametric perturbations.The paper's objective is to investigate appropriate grid frequency and phase angle estimation under disturbance scenarios to increase GFI system stability.An ADRC-based controller effectively mitigates power fluctuations from maximum power point tracking (MPPT) and improves system dynamics under grid disturbances [12].Three-phase single-stage PV integrated systems produce low-frequency ripple at the DC-link and variable instantaneous active power, resulting in distorted and unbalanced grid current.Maintaining a stable DC-link under internal and external disturbances is essential for reliable PV integrated GFIs [13].PV-generated current often surpasses grid-injected current, causing energy accumulation in the DC-link capacitor.Feedback linearization with sliding mode control limits DC-link voltage under grid disturbances [13].During grid voltage sag, a robust DC-link controller is needed for stable DC-link voltage under MPP deviation-induced disturbances.Conventional ADRC-applied PV-GFI systems suffer from parameter uncertainties and external disturbances due to PV intermittency.LESO-based ADRC, as discussed in [14], improves noise filtering capability but may compromise stability with large gain and increased controller order.An enhanced ESO featuring a simple proportional gain that reduces the controller's order can boost stability with less complexity [15].
Inverter-side current control using reduced-order modelbased linear ADRC with capacitor current active damping is presented in [16], requiring extra current and voltage sensors for LCL filter resonance damping.The weighted average current strategy in [17] improves system robustness under weak grid conditions, but still necessitates additional sensors.An ESObased ADRC in [18] aims to mitigate resonance phenomena by estimating output and lumped disturbances with online disturbance rejection.However, handling unexpected grid disturbances and PV intermittency remains a challenge.Considering the above objectives, the original contributions of the article can be summarized as follows.
r Both DC-side and AC-side modelling of the PV-GFI are analyzed in dq-domain by considering the system parameter uncertainties and unknown grid disturbances.r A utilization of LESO and PLL based observer (PLLO) with the proposed outer ADRC is investigated to control the DC-link voltage, considering the uncertainties due to the photovoltaics and the grid voltage.r A proposed inner ADRC employs reduced-order ESO (RESO) and PLLO to control grid current injection at PCC by estimating grid voltage disturbances.This controller enables the estimation of the grid voltage phase angle and frequency without external PLL, using the normalized value of internally generated grid disturbance.Section II delves into the modeling of the PV-GFI system.The proposed inner and outer ADRCs, as well as stability analysis, are detailed in Section III.Section IV presents pertinent results and discussions, with conclusions provided in Section V.

A. System Description
Fig. 1 illustrates a three-phase GFI configuration comprising the dc-link capacitor (C dc ), the VSI, and the LCL filter.The gridside inductor (L g ) and the inverter-side inductor (L i ) include the parasitic resistances r g and r i , respectively, though their effects can be disregarded in worst-case scenarios.The capacitor of (C f ) the LCL filter plays a crucial role in harmonic attenuation, resonance frequency determination, damping, reactive power compensation, and voltage stability.v x cf and i x cf are the voltage across and current through the filter capacitor of LCL filter.The currents through these inductors are represented as i x g and i x i , while the potentials at the PCC and GFI terminals are denoted as v x g and v x i , respectively.The GFI terminal voltage depends on the DC-link voltage (v dc ) and the switching function (s x i ) generated by the proposed controller: v x i = s x i v dc .This switching function s x i governs the switching pulse for each GFI switch S a , S a , S b , S b , S c , S c , as depicted in Fig. 1.
where X denotes the variables {v : voltage, i : current, s : switching function}.j is considered a complex operator, and the superscript [ ] dq corresponds to a complex variable [ ] d + j[ ] q .
x ∈ {a, b, c} represent the subscripts for the three phases, and k ∈ {1, 2, 3} represents the subsequent phase number.The phase angle ( θg ) and the amplitude (v gm ) of the PCC voltage (v x g ) can be accurately estimated using the appropriate PLL mentioned in [19].Now, the ac side dynamics of GFI with state variables i x i , i x g , v x cf and v dc can be represented as (2).
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where τ i (= L i /r i ) and τ g (= L g /r g ) represent the time constants.As τ i and τ g are significantly higher with low parasitic resistance value r g and r i , respectively, the corresponding term in (2) can be neglected and the GFI model can be further modified as (3).
Eq. ( 3) represents the state-space representation of the GFI system in the abc-domain.In the dq-domain, the state-space model can be obtained as (4).
where ωg is the estimated angular grid frequency.It is obvious that s dq i denotes the control input and i dq g is the output of the overall PV-GFI system.The effect due to the grid voltage is completely considered a disturbance in the further problem formulation.In further modeling, the LCL filter parameters are considered with uncertainties, i.e., L i = L i0 + ΔL i , L g = L g0 + ΔL g and C f = C f 0 + ΔC f .Here, L i0 , L g0 and C f 0 denote the nominal parameter values and Δ indicate the parameter uncertainties.Implementing these modifications in (4), the modeling of the GFI system can be further represented as (5).
where ψ dq i , ψ dq g and ψ dq cf denote the uncertainties due to variations in the GFI system parameters.Internal and external disturbances are treated as a single lumped disturbance in ADRC, which is then evaluated by an observer and eliminated in real time via the control input.The uncertain GFI model is first converted into an integral chain form, as given in ( 6)- (11). where are the resonant angular frequency and the antiresonant angular frequency.b 0 are the combined effect of all uncertainties and unknown disturbances.The disturbance ψ dq cannot be measurable in real time and requires an implementation of ESO.As discussed in Section III-A, inner ADRC introduces ESO to estimate ψdq to nullify the impact of ψ dq .As shown in ( 9)-( 10), s dq i and y dq o can be the input and output of the ESO with the assumption that the extended state x dq 4 = ψ dq is differentiable and h dq = ψdq is bounded.Now, the state-space model can be represented as (12) which can be used in further detail for the inner ADRC design in Section III-A. where ) and Y dq are the internal state and output of the GFI system in dq-frame.A, B, C and D are the system matrix, the input matrix, the output matrix and the disturbance matrix, respectively, and can be defined as follows.
2) GFI DC-Side Modeling: The ac-side GFI current can be seen as ( 13) at the DC-link of the GFI.
where i dc is the current on the DC side of the GFI.The DC-side dynamics of the GFI can be represented as (14).
where and ¯denote the notation of the real component and conjugate of a complex variable.The power rating of C f in an LCL-type GFI system is supposed to be 5% of the rated power of GFI [20].Under steady-state conditions, i dq i is approximately equal to i dq g .Taking into account the effect of grid current (i dq g ) on Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.i dq i (i.e., χ f i dq g ) during steady state, ( 15) can be further simplified as (16).
where χ f is considered the short circuit reverse current gain during the steady state period and can be derived as (17).
Now, ( 16) can be simplified to (18) to establish the relation for the development of the outer ADRC.
) represent unmodeled dynamics due to solar photovoltaics, filter parameter variations and grid uncertainty.

III. PROPOSED CONTROL METHODOLOGY
Instead of conventional proportional integral (PI)-based controllers, this paper uses one first-order ADRC and one reducedorder ADRC.The structural block diagram of the overall control structure along with these ADRCs are provided in Fig. 2 and is described with the following methodology.
Step 1: Measure v pv (i.e.DC-link voltage v dc ), i pv and incremental change in PV power (ΔP pv ) and apply the appropriate incremental conductance MPPT control mentioned in [21] to generate the reference DC link voltage (v * dc ).
Step 2: Utilize First-order ADRC in the outer loop to control v dc with respect to v * dc against sudden disturbances, either by photovoltaics or by grid voltage.
Step 3: Measure the grid current i x g , x ∈ {a, b, c} and the current through the filter capacitor i x cf .Then, further apply the dq-transformation from (1) to get i dq g and i dq cf , where the grid phase angle θg is generated from inner ADRC.
Step 4: To control the current dynamics, a reduced-order ADRC is employed in the inner loop, accounting for disturbances associated with system parameter variations and grid voltage uncertainties.The outer loop provides a current reference (i d * g ) to the inner loop for precise grid current control of i dq g , with i q * g considered zero since no reactive power is supplied through this controller.The proposed controller's performance is supported by the following assumptions: 1) Nonlinearities in saturation and deformity in line filters are considered disturbances.2) All disturbances and their higher derivatives must be uniformly bounded [22].3) The current i d g is regarded as 'zero' due to the faster inner current loop dynamics compared to the outer voltage loop; the PV-GFI system will malfunction if i q g exceeds a safety threshold of 1.2 pu.4) Large variations in load demand can cause current constraint violations, as the current is directly affected by power demand changes; thus, it is assumed that the required load power p L varies within a range related to i q g [23].
Step 5: Generate vdq cf by multiplying a constant gain k cf by i dq cf .This implements an active damping concept to compensate for the loss component in the filter capacitor.
Step 6: Implement an SVPWM strategy to generate switching pulses (S a • • • S c ) for the GFI system, where the input to SVPWM is the compensated switching function s dq * c (s dq * i + vdq cf ).The switching function s dq * i is generated from the inner ADRC as explained later in this section.

A. Inner ADRC Strategy for GFI Control
In this section, an inner ADRC is designed for the GFI system model presented in (12).Initially, injected currents i dq g are measured at PCC using three different current sensors and converted to i dq g through abc → dq transformation by using an estimated grid phase angle θg .The estimation of θg is explained in the latter part of this section.
1) Inner ADRC Design: Based on i dq g , the internal state variables {x dq 1 , x dq 2 , x dq 3 } of this fourth-order perturbed system are further evaluated using RESO.The extended state variable x dq 4 corresponds to the lumped unknown disturbance ψ dq .The purpose here is to control i dq g through the control input s dq i , where b 0 is the calculated input gain based on the LCL parameters.Consider ε dq g (= x dq 1 − xdq 1 ) as a small steady-state error while tracking the estimated state variable xdq 1 with the actual state variable Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Fig. 6.Grid phase angle estimation method.
x dq 1 of the GFI system.Again, are the estimated state variables and xdq 4 → x dq 4 is the estimated unknown external disturbance in dq-plane.As a consequence, the inner ADRC in the dq-plane can now be derived using the RESO and PLLO based external disturbance observer, as given in (19) and (20).

Known disturbance estimation
PLLO design:

Known disturbance estimation
where j = 0 1 −1 0 is the equivalent matrix form of the complex operator j defined previously and the superscript [ ] dq corresponds to a vector [[ ] d [ ] q ] T .The error margins and system stability criteria of the GFI system are used to define the observer gains {β dq 1 , β dq 2 , β dq 3 , β dq 4 }.Internal and external disturbances can be employed as feed-forward compensation terms for the GFI system control input, according to the ADRC theory mentioned in [8].Consequently, it is possible to redefine the final control law of the GFI system, as given in (21).
where s dq i0 is the initial value of the control input of the GFI system.Substituting (21), the perturbed system in ( 9) is reduced to (22).
2) Stability Analysis and Performance Tracking: With a simple proportional current controller given by ( 23), the desired inner closed-loop performance can be ascertained.
where the proportional gain k dq p is selected as a trade-off between steady state and transient performance.The transfer function corresponding to the estimated disturbance ψdq and the actual disturbance ψ dq is defined in (24).ψdq (s) For the steady state response of the inner ADRC, the error transfer function of the disturbance may be used as in (26).
Φ dq e (s) = ψdq e (s)  3) Grid Frequency and Phase Angle Estimation: This section presents an estimation method for the phase and frequency of the grid voltage.Fig. 6 exhibits such estimation process in dq-plane.To accurately estimate the q-axis unknown external disturbance state ψq e , the grid voltage phase error information ψdq (= ψd + j ψq ) must be normalized as given in (27).Equation (27) also indicates that ψq e depends on change in fundamental grid angular frequency (Δω g ), the normalized angular frequency amplitude (Ω n±3 ) and corresponding phase (θ ±3 ) of the (±3 )th grid voltage harmonic.
Harmonic content in ψq e must be suppressed to obtain ωg by using a proportional integral resonant (PIR) tracking controller with a low-bandwidth loop filter to bring the normalized ψq to zero.As demonstrated in Fig. 6, the transfer function of these tracking controller assemblies is obtained as shown in (28).
where  of the GFI system; here, the grid's nominal frequency f gn is considered as 50 Hz, and a damping term is added to the traditional resonant controller to avoid instability due to grid frequency variation and discretization errors.
The linear dynamic response of the PIR tracking controller to predicted disturbances can accommodate any fundamental operating frequency.The proposed estimation approach, which employs a dq reference frame instead of a stationary αβ frame, is less complex than PLL-based grid voltage phase estimation, as it avoids the need for a low pass filter and phase delay associated with intricate cut-off frequency selection [24].A stable closed-loop transfer function of the PIR tracking controller can be achieved with the necessary and sufficient criteria of ω c + k p > 0, allowing k i to be set to 20 Hz and enabling filter bandwidth adjustment via k p .The low-pass filter cutoff frequency (ω c ) is set to five times the value of ω gn , and the grid phase angle can be estimated as given in (29).LESO design: PLLO design: 2) Stability and Tracking performance Analysis: When LESO/PLLO given in (30)-( 31) is stable, the estimation error ε dc v converges to zero.Let ε dc e be the DC-link voltage tracking error and can be defined as ε dc e = v * dc − vdc .The DC-link voltage tracking error dynamics can be represented as (32) by utilizing (31).
In order to track the reference DC-link voltage (v * dc ), an appropriate linear feedback control law is implemented in (33).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where k dc p denotes the proportional gain of the outer ADRC DC-link voltage regulator, as shown in Fig. 7. Equation (34) represents the final control law of the outer ADRC by solving (32) and (33). where ) is considered as the initial value of the control input of the PV system.The derivative term v * dc serves as a feedforward term that tracks the variation in reference speed to reduce oscillation and overshoot during speed adjustment.v * dc can be generated by using a tracking differentiator (TD).By considering vdc as the DC-link voltage feedback, the closed-loop transfer function of the DC-Link controller can be expressed as (35).
The outer ADRC must be able to accurately track external disturbance during grid power injection.Eq. (36) represents the transfer function corresponding to the estimated value ψdc v (s) and the actual value ψ dc v (s) of the external disturbance that affects the voltage of the DC-link.
where β dc 1 and β dc 2 can be chosen from the characteristic polynomial of LESO/PLLO i.e. s 2 + β dc 1 s + β dc 2 = (s + ω dc v ) 2 .Consider ω dc v as the undamped natural frequency of the outer voltage control loop.Now, through equating the above relation, 2 can be obtained.The stability of the outer ADRC with LESO and PLLO can be achieved with ω dc v varies from 40 rad/s to 400 rad/s with an incremental value of 40 rad/s, which can be seen from Figs. 8 and 9.The outer ADRC performance with LESO (PLLO) is evaluated from Fig. 8 (Fig. 9), which assists in selecting bandwidth based on settling time and resonant peak.This evaluation is conducted similarly to the approach in Section III-A2.Fig. 8(c) and (d) (Fig. 9(c  Stability analysis of the proposed techniques is studied through pole-zero plots with assumptions that the inner loop control is ideal with a short settling time, no transients, and nonlinearities are ignored.Under these assumptions, the inner loop is considered to have unity gain for outer loop controller analysis [25].The pole-zero plot in Fig. 10 shows stability analysis for outer loop controller using LESO and PLLO approaches.For both approaches, all poles lie on the left half of the s-plane, indicating system stability.System zeros may exist due to the error transfer function study, but they don't influence stability, only affecting settling time and overshoot.When studying inner-loop ADRC with RESO and PLLO methods individually, all poles also lie on the left-hand side of the s-plane, as shown in Fig. 11, confirming system stability.

A. Prototype Description
The effectiveness of the proposed ADRC is validated on a single-state PV-GFI laboratory experimental prototype, as shown in Fig. 12.The system parameter descriptions are provided in Table I.The PV-GFI prototype includes a voltage source inverter, a PV emulator, current sensors, voltage sensors, a rectifier with RL load, and a dSPACE MicroLabBox DS1202.The DSP controller DS1202 is utilized for digital pulse generation to control the GFI.The signals from current and voltage sensors are given as feedback to DS1202 through the internal ADCs.The sampling time of the DS1202 is set to 50 μs.

B. Experimental Verification Through Test Scenarios
The performance of the proposed ADRC controller is examined in five different scenarios are listed below.
r Scenario-1: Sudden change in solar irradiance.r Scenario-2: Sudden change in solar temperature.r Scenario-3: Filter parameter variation.r Scenario-4: Grid voltage is subjected to voltage sag.r Scenario-5: Grid voltage is subjected to phase jump. 1) Scenario-1: Fig. 13 shows the experimental results for GFI based on the proposed ADRC controller in Scenario-1.In Scenario 1, the dynamic change in solar irradiance can be observed from 1000 W/m 2 to 500 W/m 2 500 W/m 2 to 1000 W/m 2 .The solar temperature is kept constant at 25 • C during this irradiance change.The performance of phase voltage v a g and three-phase grid current i a g , i b g , i c g are shown in Fig. 13(a).It can be seen that v a g and i a g are in phase, which indicates that the system can maintain the unity power factor in scenario 1. Fig. 13(a) (bottom) represents a zoomed portion of v a g , i a g , i b g , and i c g waveforms during the sudden change of solar irradiance.It can be seen that the grid current is sinusoidal in nature.Moreover, the THD of i a g is kept within the limit of the IEEE-519 standard.The PV voltage v pv , the PV current i pv , the PV power p pv and grid active power p g can be shown in Fig. 13(b).It can be observed that a small change in v pv occurs during this scenario-1, which tracks the reference MPPT voltage generated by P &amp; O algorithm.Due to a sudden change in irradiance from 1000 W/m 2 to 500 W/m 2 , i pv falls from 12A to 6A.Moreover, the injected grid active power is 7.86 kW and 3.86 kW for 1000 W/m 2 and 500 W/m 2 irradiance, respectively.
2) Scenario-2: Fig. 14   3) Scenario-3: The robustness of the proposed ADRC controller-based three-phase grid inverter is observed under +10% change in filter parameters, as shown in Fig. 15.Simultaneously, solar irradiation changes from 500 W/m 2 to 1000 W/m 2 .It can be seen in Fig. 15(a) that solar irradiation increases as the grid current increases and maintains in sinusoidal in nature.The THD of i a g is kept within the limit of the IEEE-519 standard.In Fig. 15(b), v pv change from 649 V to 650 V and i pv change from 6 A to 12 A at 500 W/m 2 to 1000 W/m 2 .The PV voltage, v pv maintains stable and tracks the maximum voltage during this Scenario-3.Moreover, the active grid power injects maximum power into the grid.
4) Scenario-4: Fig. 16 illustrates the experimental results during the balance grid voltage sag condition.The grid voltage with 20% voltage sag is illustrated in Fig. 16(a).The system is operated in a constant power region, which increases the grid current i a g , i b g , i c g under voltage sag condition.During this voltage sag, the % THD of the grid currents are observed to be 1.68%.It can be observed from Fig. 16(b), there is no effect on v pv , i pv and p pv during the voltage sag condition.It indicates that v pv remains stable during this Scenario-4.In addition, the active power of the injected grid tracks the maximum power of the PV array.

5) Scenario-5:
The performance analysis under Scenario-5 is shown in Fig. 17, which is depicted from the experimental evaluation.This scenario-5 considers the grid voltage with a 60 • phase jump, as indicated in Fig. 17(a).During this phase jump scenario, the grid currents i a g , i b g ,and i c g return to normal condition after a few cycles.As seen in Fig. 17, there is no influence on v pv , i pv and p pv throughout this scenario, which indicates that v pv is stable.Furthermore, the injected grid active power is obtained as the maximum power of the PV array.

C. Comparative Analysis
In scenarios 4 and 5, the proposed inner ADRC approach is compared to ADRC- [9].In ADRC- [9], the fixed DC-link voltage is employed as an input source to analyze the performance of the grid scenario.To offer a fair comparison with the proposed controller, the input voltage of ADRC- [9] is linked to the PV array.Furthermore, both current-controller approaches are combined with the same outer-loop controller.Figs.16 and    and (d) in Scenario-4 that the performance on the input side is identical, which has no influence on v pv , i pv and p pv .According to Fig. 16(a) and (c), the %THD of i a g using the proposed inner ADRC controller and ADRC- [9] are 1.68% and 2.48%, respectively.Similarly, %THD of i a g with the proposed inner ADRC controller and the ADRC-RESO technique mentioned [9] in Fig. 17(a) and (c) in Scenario-5 are 2% and 2.78%, respectively.Figs.18 and 19 shows a performance comparison of the proposed ADRC and traditional PI controller [3] for the DC link under input side scenario and grid side conditions, respectively.Table II   ADRC with standard PI controllers in the presence of various disturbances of the grid.The proposed ADRC techniques have a faster dynamic response than the traditional PI controller under 0.2 pu voltage sag, changes in solar irradiation, and temperature.Furthermore, for various disturbances in the grid, the suggested approach exhibits a lower peak overshoot error than the typical PI controller.As demonstrated in the Fig. 20 , performances of the estimated frequency and phase angle of the proposed technique are compared with MSTOGI-PLL- [26].In this analysis, the grid voltage is subjected to a 0.2 pu voltage sag and a 60 • phase angle jump.Table III displays the results of the proposed techniques and MSTOGI-PLL- [26].Under the 0.2 pu voltage sag, 60 • phase angle jump, the proposed techniques provide a quicker transient response with less settling time than MSTOGI-PLL- [26].V. CONCLUSION This paper presents an initial model of the PV-GFI system with LCL filter, taking into account the system parameter uncertainties and unknown grid disturbances.The current inner loop is decoupled in the dq-domain via detecting disturbances and then applying RESO/PLLO for the implementation of inner loop ADRCs.With the use of discrete-time frequency analysis and pole-zero maps, the suggested inner ADRCs have been shown to be robust and stable under internal uncertainties.Through the use of this internally generated disturbance, the grid voltage and frequency phase angles are determined without the use of Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
an external PLL.To control the DC-link voltage, a combination of the LESO and PLLO with the suggested outer ADRC is examined, taking into account the uncertainties relating to photovoltaics and grid voltages.Under Various disturbances and uncertainties, the proposed ADRC systems demonstrate their feasibility and effectiveness.It can be observed that the proposed decoupling control strategy requires a lesser number of sensors, insensitive to control settings, and highly resilient.Furthermore, it is likely to significantly improve the quality of grid-side current.
for LESO (PLLO) in Figs.4(a), (b), 5(a), and 5(b).These observations aid in selecting the ADRC bandwidth considering the steady-state error.In comparison to RESO, PLLO achieves the desired performance with approximately 1/6 th of RESO's bandwidth.However, PLLO exhibits a greater overshoot.Consequently, a trade-off must be made between bandwidth and overshoot.Ultimately, bandwidths of 100 rad/s (PLLO) and 300 rad/s (RESO) are chosen.The parameter k dq p , selected between 20% and 33% of ω dq o , enhances ADRC's current performance, provided settling time specifications for transient response are considered.Frequency domain analysis (Bode plot) in Figs.4(c) and 5(c) reveals that increasing bandwidth does not significantly improve ADRC performance for either RESO or PLLO.PLLO has a phase margin of −180 • , while RESO exhibits a phase margin of −129.69 • .PLLO's resonant peak is higher than RESO's.The Nyquist plots, shown in Figs.4(d) and 5(d), indicate the open-loop performance of RESO and PLLO, with bandwidths not greatly impacting ADRC's open-loop stability.As bandwidth increases, disturbance estimation errors decrease, and tracking speed increases.
θg = ωg dt.(29) B. Outer ADRC Strategy for DC-Link Voltage Control 1) Outer ADRC Design: The perturbed first-order system in (18) includes feedback from the estimated vdc and measurement noise δ n .The DC-link voltage controller can be designed as an external disturbance observer utilizing a LESO or PLLO-based observer, as described in (30) and (31).Here, [v dc , ψdc v ] → [v dc , ψ dc v ], are the estimated value of the DC-link voltage and unknown total disturbances, ε dc v = v dc − vdc is the DC-link voltage error.
) and (d)) present the Bode Diagram and Nyquist plot for LESO

Fig. 13 .
Fig. 13.Experimental results under Scenario-1.(a) v a g , i a g , i b g , and i c g .(b) v pv , i pv , p g , and p pv .
shows the experimental results for proposed ADRC controller based three-phase grid inverter under Scenario-2.At constant 1000 W/m 2 irradiance, the dynamic change in solar temperature from 25 • C to 35 • C and 35 • C to 25 • C can be observed in this Scenario-2.Fig. 14(a) depicts the performance of v a g , i a g , i b g ,and i c g during a rapid temperature

Fig. 14 .
Fig. 14.Experimental results under Scenario-2.(a) v a g , i a g , i b g , and i c g .(b) v pv , i pv , p g , and p pv .

Fig. 15 .
Fig. 15.Experimental results under Scenario-3.(a) v a g , i a g , i b g , and i c g .(b) v pv , i pv , p g , and p pv .
17 depict the performance of both current controller techniques under Scenarios 4 and 5, respectively.It can be shown from Fig. 16(b)Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 16 .
Fig. 16.Experimental results with comparative study under Scenario-4.(a) v a g , i a g , i b g , and i c g results with proposed ADRC.(b) v pv , i pv , p g , and p pv results with proposed ADRC.(c) v a g , i a g , i b g , and i c g results with proposed ADRC-[9].(d) v pv , i pv , p g , and p pv results with proposed ADRC-[9].

Fig. 17 .
Fig. 17.Experimental results with comparative study under Scenario-5.(a) v a g , i a g , i b g , and i c g results with proposed ADRC.(b) v pv , i pv , p g , and p pv results with proposed ADRC.(c) v a g , i g , i b g , and i c g results with proposed ADRC-[9].(d) v pv , i pv , p g , and p pv results with proposed ADRC-[9].

Fig. 20 .
Fig. 20.Experimental comparative analysis of the proposed method with [26] for frequency and phase error estimation.(a) During voltage sag.(b) During phase jump.
dq 1 s 3 +β dq 2 s 2 +β dq 3 s+β dq k p , k i , k r and k d are the proportional, integral, resonant gain, and damped gain of the PIR controller, respectively.The quality factor of the damped term of the resonant controller is denoted by Q d .The PIR controller achieves a desirable steady-state value of ωg with minimized steady-state error, but it struggles to track the angular frequency of the grid with high accuracy.To counteract the effects of LCL filter resonance on tracking ωg , a resonant term is added to the existing PI controller in (28).The resonant controller's angular frequency (ω rc ) is set to twice the nominal grid angular frequency (ω gn , i.e., 2πf gn ), or ω rc = 2ω gn , without affecting the high-frequency stability Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I SYSTEM PARAMETERS (
PLLO)to analyze disturbance rejection and noise suppression in the outer ADRC.When LESO and PLLO are employed together, the outer ADRC performance of PLLO surpasses that of LESO if noise suppression characteristics are comparable.

TABLE II COMPARISON
BETWEEN THE PROPOSED ADRC METHOD AND CONVENTIONAL PI CONTROLLER FOR DC LINK ERROR SIGNAL compares the effectiveness of the proposed