Robust Nonlinear Controller Design for Three-Phase ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 29, NO. 3, JUNE 2014

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Robust Nonlinear Controller Design for Three-Phase Grid-Connected Photovoltaic Systems Under Structured Uncertainties M. A. Mahmud, Member, IEEE, M. J. Hossain, Senior Member, IEEE, H. R. Pota, and N. K. Roy

Abstract—This paper presents a robust nonlinear controller design for a three-phase grid-connected photovoltaic (PV) system to control the current injected into the grid and the dc-link voltage for extracting maximum power from PV units. The controller is designed based on the partial feedback linearization approach, and the robustness of the proposed control scheme is ensured by considering structured uncertainties within the PV system model. An approach for modeling the uncertainties through the satisfaction of matching conditions is provided. The superiority of the proposed robust controller is demonstrated on a test system through simulation results under different system contingencies along with changes in atmospheric conditions. From the simulation results, it is evident that the robust controller provides excellent performance under various operating conditions. Index Terms—Grid-connected PV system, matching conditions, partial feedback linearization, robust nonlinear controller, structured uncertainty.

I. INTRODUCTION

I

N RESPONSE to global concerns regarding the generation and delivery of electrical power, photovoltaic (PV) technologies are gaining popularity as a way of maintaining and improving living standards without harming the environment. To extract maximum power from the PV system [1], a robust controller is required to ensure maximum power-point tracking (MPPT) [1]–[3] and deliver it to the grid through the use of an inverter [4]–[6]. Robustness is essential since the power output of PV units varies with changes in atmospheric conditions. Thus, the controller must be robust enough to provide a tighter switching scheme for the inverter to transfer maximum power

Manuscript received April 16, 2013; revised October 07, 2013 and January 30, 2014; accepted February 25, 2014. Date of publication April 29, 2014; date of current version May 20, 2014. Paper no. TPWRD-00156-2013. M. A. Mahmud is with the School of Software and Electrical Engineering, Swinburne University of Technology, Hawthorn, VIC 3122, Australia (e-mail: [email protected]). M. J. Hossain is with Griffith School of Engineering, Griffith University, Gold Coast Campus, Gold Coast, QLD 4222, Australia (e-mail: j.hossain@griffith. edu.au). H. R. Pota is with the School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT 2600, Australia (e-mail: [email protected]; [email protected]). N. K. Roy is with the School of Engineering, Deakin University, Waurn Ponds, VIC 3220, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRD.2014.2309594

into the grid over a wide range of operating conditions with a short transient period. In a grid-connected PV system, control objectives are met by using a pulse-width modulation (PWM) scheme based on two cascaded control loops [7]. The two cascaded control loops consist of an outer voltage-control loop to track the maximum power point (MPP) and an inner current control loop to control the duty ratio for the generation of a sinusoidal output current which needs to be in phase with the grid voltage for unity power factor operation [7]. The current loop is also responsible for maintaining power quality (PQ) and for current protection that has harmonic compensation. Linear controllers are widely used to operate PV systems at MPP [8]–[13]; however, most of these controllers do not account for the uncertainties in the PV system. Over the past few decades, one of the most important contributions in the field of control theory and applications has been the development of robust linear controllers for linear systems in the presence of uncertainties through the control scheme which is often obtained from linear matrix inequality (LMI) methods [14], [15]. A feedforward approach is proposed in [16] to control the current and dc-link voltage, and the robustness is assessed through modal analysis. A robust fuzzy-controlled PV inverter is presented in [17] for the stabilization of a grid-connected PV system where the robustness is achieved by using the Taguchi tuning algorithm. A minimax linear quadratic Gaussian (LQG) technique is proposed in [18] to design a robust controller for the integration of PV generation into the grid where the higher-order terms during the linearization are considered as modeling uncertainties. The controller design methods as presented in [8]–[13] and [16]–[18] are based on linearized models of nonlinear PV systems. In practice, PV sources are time varying, and the system is not linearizable around a unique operating point or a trajectory to achieve the desired performance over a wide range of changes in atmospheric conditions. To overcome the limitations of linear controllers, a nonlinear proportional-integral-derivative (PID) controller based on the model prediction is presented in [19], where improved performance is reported. A Lyapunov-based control scheme for a gridconnected PV inverter is presented in [20] where an adaption law is included to improve the robustness. However, it is well known that the adaption technique is useful for systems with slow parameter variations which is not the case for PV systems as the changes occur rapidly. A sliding mode controller for a nonlinear grid-connected PV system is proposed in [21] and

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[22] along with a new MPPT technique for providing robust tracking against uncertainties and unknown disturbances within the system. The performance of the sliding mode controller is confined to the sliding surface which is constructed from a linear combination of output injected errors. A feedback linearizing technique is proposed in [23] for PV applications where a superfluous complex model of the inverter is considered to design the controller. To overcome the complexity, a simple and consistent inverter model is used in [24]. In [23] and [24], a feedback linearizing controller is designed by considering the dc-link voltage and quadrature-axis grid current as output variables. Power-balance relationships are considered to express the dynamics of the voltage across the dc-link capacitor. However, this relationship cannot capture nonlinear switching functions between the inverter input and output; to accurately represent a grid-connected PV system but it is essential to consider these switching actions. The current relationship between the input and output of the inverter can be written in terms of switching functions rather than the power balance equation. The voltage dynamics of the dc-link capacitor include nonlinearities due to the switching actions of the inverter. The inclusion of these nonlinearities will improve the accuracy of the PV system model; however, the grid-connected PV system will be partially rather than exactly linearized as shown in [25]. Although the approaches presented in [23]–[25] ensure the MPP operation of the PV system, they do not account for inherent uncertainties in the system as well as the dynamics of the output filter. This paper aims to design a robust partial feedback linearizing controller where the robustness of the controller is ensured through the satisfaction of matching conditions. The stability of the internal dynamics of the PV system is discussed before deriving the proposed control law. Upper bounds are set on the modeling errors, which include both parametric and state-dependent uncertainties. The upper bounds on the uncertainties are such that the proposed controller guarantees stability and improved performance in the presence of all possible perturbations within the given bounds. Some directions about the implementation of the proposed controller are also discussed. The effectiveness of the proposed controller is tested for changes in atmospheric conditions as well as different system conditions, such as three-phase short-circuit faults and single-line-to-ground fault, and the results are compared with that of a partial feedback linearizing controller where robustness requirements are not included in the design [25]. II. PV SYSTEM MODEL A three-phase grid-connected PV system is shown in Fig. 1 where the PV array consists of a number of PV cells in a series and parallel combination to achieve the desired output voltage. The detailed modeling of a PV array and cell is given in [25]. The output voltage of the PV array is a dc voltage and, thus, the output dc power is stored in the dc-link capacitor . The output current of the PV array is and that of the dc-link capacitor is . The dc output power of the PV array is converted into ac power through the inverter. The inverter, shown in Fig. 1, is an

Fig. 1. There-phase grid-connected PV system with an output

filter.

insulated-gate bipolar transistor (IGBT)-based six-pulse bridge ( , and ) in a three-phase voltagesource converter (VSC) configuration since the PV system is connected to a three-phase grid supply point. The main purpose of this paper is to design a robust switching scheme for the IGBT-based six pulse bridge so that the PV system is capable of providing maximum power into the grid. The extraction of maximum power from the PV unit can be confirmed by monitoring the dc-link voltage . The output ac power of the inverter is supplied to the grid through the filters and connecting lines where the resistance of the connecting line is and are the inductances of the filters and connecting lines, and the filter capacitance is . In Fig. 1, the grid voltages for phase , phase , and phase are , and , respectively. The extracted maximum power from the PV system will be supplied to the grid if the output current of the inverter remains in phase with the grid voltage and, thus, it is essential to control this output current. The state-space form of a three-grid-connected PV system without an filter is given in [25]. In this section, a detailed model of a three-phase grid-connected PV system is developed based on the schematic shown in Fig. 1. From Fig. 1, it can be seen that there are two loops at the output side of the inverter. At first, by applying Kirchhoff’s voltage law (KVL) at the inverter-side loop, the following equations can be obtained:

(1)

where , and are the output currents of the inverter flowing though phase , phase , and phase , respectively; , and are the voltages across the filter capacitor for phase , phase , and phase , respectively; and , and are the input switching signals for phase , phase , and phase , respectively. By applying Kirchhoff’s current law (KCL) at the dc-link capacitor node, the following equation can be obtained: (2) The input current to the inverter

can be written as [22] (3)

MAHMUD et al.: ROBUST NONLINEAR CONTROLLER DESIGN

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which yields

(4) If KCL is applied at the node where the filter capacitor is connected, the following voltage–current relationships can be obtained

Fig. 2. PLL model.

where (5)

and

are the switching signals in the

transformation matrix

frame, and , and the

can be written as

(8) where , and are the output currents of the filter flowing though phase , phase , and phase , respectively. Finally by applying KVL at the grid-side loop, the state-space model can be written as

(6)

The complete model of a three-phase grid-connected PV system is given by (1) and (4)–(6), which are nonlinear and time variant and this makes it difficult for implementing a simple control scheme [22]. The control problem is simplified by transforming the PV system model into a time-invariant model. This can be done by applying transformation using the angular frequency of the grid and considering the rotating reference frame synchronized with the grid where the component of the grid voltage is , which means that is zero. By using transformation, (1) and (4)– can be written in the form of the following group of equations:

The group of (7) represents the complete mathematical model of a three-phase grid-connected PV system with an output filter, and this model is a nonlinear model due to the nonlinear behavior of the switching signals and the output current of the PV array. But in a practical system, it is essential to use a phase-locked loop (PLL) for instantaneous tracking of grid frequency. The dynamical of the PLL used in this paper is mainly based on the approach as presented in [26] which is shown in Fig. 2. In this PLL model, represents the reference value of the direct-axis grid voltage, indicates the measured angular frequency, and is the feedforward angular frequency. As mentioned earlier in this section, the robust nonlinear control scheme needs to be designed in such a way that it is capable of controlling the dc-link voltage and the output current of the inverter. Since the component of the grid voltage is zero, the component current will not affect the maximum power delivery and it is essential to control the component current. From the mathematical model as represented by (7), it can be seen that the dynamics of the component output current of the filter do not have the relationship with the switching signals. But the dynamics of the inverter output current have a coupling with the switching signal and, therefore, is chosen as another control objective. An overview of designing the partial feedback linearizing controller is discussed in the following section by considering and as control objectives. III. OVERVIEW OF FEEDBACK LINEARIZING CONTROLLER DESIGN

(7)

THE

Since the three-phase grid-connected PV system, as represented by the group of (7), has two control inputs ( and ) and the controller needs to be designed with two control objectives ( and ), the mathematical model can be represented by the following form of a nonlinear multi-input multioutput (MIMO) system

(9)

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coordinate transformation and some are not. The new states of a partially feedback linearized system can be written as

where

(13) where represents the state vector obtained from nonlinear coordinate transformation of order and is the state vector of the nonlinear (remaining) part of order . The dynamic of is called the internal dynamics of the system which need to be stable in order to design and implement a partial feedback linearizing controller for the following partially linearized system: (14)

and

The design of a partial feedback linearizing controller depends on the feedback linearizability of the system, and this feedback linearizability is defined by the relative degree of the system [27]. The relative degree of the system, in turn, depends on the output functions of the system. The nonlinear model of a three-phase grid-connected PV system as shown by (9) can be linearized using feedback linearization when some conditions are satisfied. Consider the following nonlinear coordinate transformation for the aforementioned three-phase grid-connected PV system:

where is the system matrix, is the input matrix, and is the new linear control input for the partially linearized system, and any linear controller design technique can be employed to obtain the linear control law for the partially linearized system. But before obtaining a robust control law through the proposed approach, it is essential to model the uncertainties and to ensure the partial feedback linearizability of the PV system along the stability of internal dynamics which are not linearized through the transformation. The details of uncertainty modeling are discussed in the following section. IV. UNCERTAINTY MODELING

where and are the relative degrees corresponding to output functions and , respectively; are the Lie derivative of along [27]. The change of coordinate (10) transforms the nonlinear system (9) from to coordinates provided that the following conditions are satisfied for:

As mentioned earlier, the output power of the PV system depends on the intensity of the solar irradiation which is uncertain because of unpredictable changes in weather conditions. These changes may be modeled as uncertainties in current out of the solar panels which, in turn, causes uncertainties in the current (in the -frame, , and ) injected into the grid. Since the uncertainty in the output power of inverters is related to the frequency of the grid, the proposed scheme has the capability of accounting for the uncertainty in the grid frequency. In addition, the parameters used in the PV model are, in most cases, either time varying or not exactly known and, therefore, parametric uncertainties exist too. Thus, it is essential to represent these uncertainties in PV system models. In the presence of uncertainties, a three-phase grid-connected PV system can be represented by the following equation:

(11)

(15)

(10)

where is the Lie derivative of along The linearized system can be expressed as follows:

. (12)

where is the system matrix, is the input matrix, and is the new linear control input for the feedback linearized system. When , only partial feedback linearization is possible, that is, some states are transformed through nonlinear

where


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then the upper bound on uncertainties system under study can be obtained as

and

and

for the PV

(20) which are uncertainties in and , respectively. To formulate the robust controller design problem, we assume that the structure of state-dependent uncertainties and the nature of parametric uncertainties satisfy the following matching conditions [28]: and

span

and

(21)

(16)

If these matching conditions hold, then the following condition holds too: (17) where is the total relative degree of the nominal system (9) which for the system under study is 2 [25], is the relative degree of uncertainty , and is the relative degree of uncertainty . In order to match the uncertainties with the PV system model, the relative degree of the uncertainty should be equal to the relative degree of the nominal system: 2. This can be calculated from the following equations:

The partial feedback linearizing scheme, as presented in [25], cannot stabilize the PV system appropriately if the aforementioned uncertainties are considered within the PV system model since the controller is designed to stabilize only the nominal system. However, in the robust partial feedback linearizing scheme, the aforementioned uncertainties need to be included to achieve the robust stabilization of the grid-connected PV system. The robust controller design by considering the aforementioned uncertainties within the grid-connected PV system model is shown in the following section.

V. ROBUST CONTROLLER DESIGN (18)

If the relative degree of is 1 for each output and , then the total relative degree of will be 2. This will be the case if and are not equal to zero. To match the uncertainties in , the relative degree of should be equal to or greater than the total relative degree of the nominal system. It will be 2 if the following conditions hold:

The following steps are followed to design the robust controller for a three-phase grid-connected PV system as shown in Fig. 1. Step 1: Partial feedback linearization of grid-connected PV systems. In this case, the partial feedback linearization for the system with uncertainties (15) can be obtained as

(22) (19)

Since the proposed uncertainty modeling scheme considers the upper bound on the uncertainties, these bounds should be set to proceed with the robust controller design. If the maximum-allowable changes in the system parameters are set to 30% and the variations in solar irradiation and environmental temperature are set to 80% of their nominal values,

For the PV system, the state equations are derived as

(23)

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If and are considered as the linear control inputs, then the feedback-linearized PV system with uncertainties (23) can be represented by the following linear systems:

(24)

(28)

can be expressed as

Thus, the dynamics of

(29)

where For

, the dynamics can be simplified as [25] (30)

(25)

The linear control laws and for the feedback linearized system (24) can be obtained using any standard linear control theory. The physical control inputs and of the PV system can be calculated from (25). But before obtaining these physical control laws, it is essential to analyze the stability of internal dynamics which is shown in the following step. Step 2: Stability of internal dynamics. In the previous step, the seventh-order PV system (7) is transformed into a second-order linear part (24), representing the linear dynamics of the system. For the transformed system, the desired performance can be obtained through the implementation of a partial feedback linearizing controller. But before implementing such controllers, it is essential to analyze the dynamics of the nonlinear part represented by the state , (13) and the dynamics of which are called internal dynamics. To ensure stability of any feedback-linearized system, the control law needs to be chosen in such a way that

which implies that the state of the linear system decays to zero as time approaches infinity, that is . For the PV system considered in this paper, this means that at steady state (26) Let us now consider the nonlinear state expressed by the function , see (13). To ensure the stability of PV systems, this needs to be selected to satisfy the following conditions [29]: (27) For the PV system, (27) is satisfied if we choose

which represents a stable system, and the dynamics of the remaining states can be calculated by using (29) as with

and

(31)

which represents marginal stability of the internal dynamics [30]. Therefore, the robust partial feedback linearized controller can be designed and implemented for a three-phase grid-connected PV system with an output filter. The derivation of the proposed robust control law is shown in the next step. Step 3: Derivation of the robust control law. From (25), the control law can be obtained as follows: (32)

Equation (32) is the final robust control law for a three-phase grid-connected PV system and the control law contains the model uncertainties. The main difference between the designed robust control law (32) and the control law as presented in [25] is the inclusion of uncertainties within the PV system model. Here, the linear control inputs and can be obtained in a similar way as discussed in [25]. The performance of the designed robust stabilization scheme is evaluated and compared in the following section to that without any uncertainty as presented in [25] along with the direction of practical implementation. VI. CONTROLLER PERFORMANCE EVALUATION To evaluate the performance of the three-phase grid-connected PV system with the designed robust controller, a PV array with 20 strings each characterized by a rated current of 2.8735 A is used. Each string is subdivided into 20 modules characterized by a rated voltage of 43.5 A and connected in series. The total output voltage of the PV array is 870 V, the output current is 57.47 A, and the total output power is 50 kW. The value of the dc-link capacitor is 400 F, the line resistance is 0.1 , and inductance is 10 mH. The grid voltage is 660 V and the frequency is 50 Hz. The switching frequency of the inverter is considered as 10 kHz. The inclusion of LCL filter dynamics affects the stability of the system for which a peak amplitude response at the resonance frequency of the LCL filter exists, and if the parameters of


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Fig. 3. Implementation block diagram.

LCL filters are not chosen appropriately, the dynamic stability of the system will be degraded. To perform the simulation, the inductor and capacitor values of the LCL filter are selected based on the filter design process as presented in [31] which are 10 mH and 3.1 F. Since partial feedback linearizing controllers are sensitive to system parameters, it is essential to have an exact system model in order to achieve good performance. But for real life gridconnected PV systems, there often exist inevitable uncertainties in constructed models. In addition, uncertain parameters exist that are not exactly known or difficult to estimate. Therefore, to evaluate the performance of the designed robust control scheme, it is essential to consider these uncertainties. The implementation block diagram of the designed robust control scheme is shown in Fig. 3 where the nominal model of the PV system is shown along with the uncertainties. From Fig. 3, it can also be seen that the output currents of the threephase inverter, voltages across the filter capacitor, output currents of the filter, and grid voltages are transformed into direct—and quadrature-axis components using transformation. Then, the control law is obtained in the frame and, finally, reverse transformation (i.e., ) is done to implement the controller through the PWM. The designed scheme can also be implemented in a discrete time approach using digital signal processing and control engineering (dSPACE). To digi-

tally implement the proposed scheme, it is essential to collect the data (output voltage and current of the PV array, output currents of the inverter and filter, voltage across the filter capacitor, grid voltage) from the PV system using different channels and convert these data through a dSPACE analog-to-digital converter (ADC). The proposed algorithm needs to be developed in a Matlab Simulink platform, and the scheme needs to be built in dSPACE. To verify the feasibility of the proposed controller, different operating conditions have been considered. The performance of the controller is tested for the following scenarios: 1) standard atmospheric conditions; 2) changing atmospheric conditions; and 3) different types of short-circuit faults. A. Controller Performance Under Standard Atmospheric Conditions In this case study, the standard values of the solar irradiation (1 kW ) and environmental temperature (298 K) are considered. Since the main control objective is to inject maximum power (50 kW) into the grid, the designed robust control scheme must be able to deliver this power into the grid by considering some uncertainties into the parameters and states of the system. To achieve this, the grid current and voltage remain in phase with each other which is already discussed in our previous work as presented in [25]. But with uncertainties, the voltage and

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Fig. 4. Controller performance at standard atmospheric conditions. Fig. 6. Controller performance during the three-phase short-circuit fault.

Fig. 5. Controller performance at changing atmospheric conditions.

current will not be in phase as the uncertainties are not considered with the technique as proposed in [25]. In such a situation, the designed robust controller ensures the operation of the three-phase grid-connected PV system at unity power factor which is shown in Fig. 4. In Fig. 4, it can be seen the partial feedback linearizing controller (PFBLC) is unable to transfer maximum power into the grid (green line) when 25% variations in the system parameters and 70% uncertainties in solar irradiation and cell temperature are considered within the PV system model. But the robust partial feedback linearizing controller (RPFBLC) maintains the operation of the PV system at a unity power factor (red line). B. Controller Performance Under Changing Atmospheric Conditions At this stage, it is considered that the PV unit operates under standard atmospheric conditions until 1 s. At 1 s, the atmospheric condition changes in such a way that the solar irradiation of the PV unit reduces to 70% from the standard value. Under this situation though, the PFBLC is able to maintain the stability of the system but still there are some phase differences between the grid current and voltage but with the RPFBLC, there are no phase differences. Thus, the designed controller performs well in changing conditions which is shown in Fig. 5 from which it can be seen that the PV unit operates under the standard atmospheric condition up to 1.1 s and changing atmospheric conditions up to 1.2 s. After that, it operates under standard conditions, and the designed controller maintains the operation of the system at unity power factor. C. Controller Performance During Short-Circuit Faults in the System In this subsection, the robust performance of the controller is evaluated by applying a three-phase short circuit and single-

line-to-ground fault at the output of the inverter. For both cases, the following fault sequence is considered: • fault occurs at 1.5 s; • fault is cleared at 1.6 s where the prefault and postfault operations of the PV system are considered at standard atmospheric conditions. With this fault sequence, the following two cases are considered to evaluate the performance of the designed controller. Case 1) Controller performance with a three-phase shortcircuit fault In this case study, a three-phase short-circuit fault is applied at the output terminal of the inverter with the aforementioned fault sequence. Since the current controller in the frame enables the appropriate current control in a balanced condition, the phase fault current generated by the PV inverter is limited by the reference values of - and -axis current [32]. The grid-connected PV system is unable to generate active power due to three-phase voltage collapse. During the faulted condition, the voltage of each phase will be reduced to zero, and the current of each phase will be increased. But after clearance of the fault, the controller reacts quickly. With the designed robust controller, the postfault voltage and current are in phase as shown in Fig. 6 (red line). But there is still a phase difference between voltage and current, see Fig. 6 (green line) when the uncertainties are not considered in the controller design. Case 2) Controller performance with a single line-to-ground fault. When a single-line-to-ground fault is applied at the output terminal of the inverter, there is a voltage imbalance due to the negative-sequence voltage component. Due to the stiffness of the grid, this voltage imbalance will also appear at the grid supply point [33]. In this situation, the proposed controller will inject negative-sequence current according to the principle of negative-sequence current injection as discussed in [33]. In this case study, a single-line-to-ground fault is applied between Phase and ground. The voltage of only phase will be reduced to zero and similarly the current of the same phase will increase up to the reference values. The voltage and current with the proposed controller are shown in Fig. 7. From this,


Fig. 7. Controller performance during the single-line-to-ground fault.

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bounds of the PV systems’ parameters and states need to be known rather than network parameters, system operating points, or natures of the faults. The resulting robust controller enhances the overall stability of a three-phase grid-connected PV system considering admissible network uncertainties. Thus, this controller has good robustness against the changes in parameters and variations in atmospheric conditions irrespective of the network parameters and configuration. Future work will include the implementation of the proposed control scheme on a practical system. REFERENCES

Fig. 8. Positive-sequence active single-line-to-ground fault.

Fig. 9. Negative-sequence active single-line-to-ground fault.

and reactive

and reactive

current during the

power during the

it can be seen that the designed controller ensures that unity power factor operation as the current (red line) and voltage (blue line) are in phase during the postfault period. From Fig. 7, it can also be seen that for the PFBLC, the phase difference between grid voltage and current is less compared to that of Fig. 6, which is due to the less severe single-line-to-ground fault compared to the three-phase short-circuit fault. The corresponding positive-sequence active and reactive current is shown in Fig. 8, and the negative-sequence active and reactive power is shown in Fig. 9. VII. CONCLUSION A robust controller is designed by modeling the uncertainties of a three-phase grid-connected PV system in a structured way based on the satisfaction of matching conditions to ensure the operation of the system at unity power factor. The partial feedback linearizing scheme is employed to obtain the robust control law and with the designed control scheme, only the upper

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M. J. Hossain (S’09–M’11–SM’13), photograph and biography not available at the time of publication.

H. R. Pota, photograph and biography not available at the time of publication.

N. K. Roy, photograph and biography not available at the time of publication.