Direct Power Control Strategies for DFIG Wind Turbines: Implementation, Limitations, and Grid Disturbance Behavior in MATLAB

Author: Waqas Javaid

Abstract

Direct Power Control (DPC) represents an alternative approach to classical field-oriented control for doubly-fed induction generator (DFIG) wind turbines. Unlike vector control methods that employ inner current regulation loops, DPC operates directly on active and reactive power errors through hysteresis comparators and a switching table. This article examines the practical implementation of DPC for rotor-side converters in DFIG systems, with emphasis on converter-level behavior, switching frequency variability, and performance under non-ideal grid conditions. Instantaneous power calculation in the stationary αβ frame eliminates the need for coordinate transformations and current regulators, resulting in simplified structure and rapid transient response. However, variable switching frequency introduces challenges for filter design and thermal management. Simulation results demonstrate DPC behavior during grid voltage dips, weak grid operation, and distorted voltage conditions. Comparison with conventional vector control highlights the trade-offs between response speed, current quality, and implementation complexity. Practical constraints including sampling delay, measurement noise, and semiconductor limits are discussed. The article concludes with validation approaches for DPC implementation in wind energy systems.

1. Concept of Direct Power Control

Direct Power Control regulates active and reactive power directly at the converter terminals without intermediary current control loops. This distinguishes DPC fundamentally from classical field-oriented control (FOC) where power is controlled indirectly through current regulation in a synchronously rotating reference frame.

1.1 Fundamental Principle

The core concept of DPC is straightforward: instantaneous active and reactive powers are calculated from measured voltages and currents, compared with reference values, and the errors are fed into hysteresis comparators. The comparator outputs, together with the angular position of the voltage vector, determine the appropriate switching state through a pre-defined lookup table. The converter switching devices are triggered directly, eliminating the need for pulse-width modulation (PWM) generators and current regulators [6].

For a two-level voltage source converter, eight switching states are available (six active vectors and two zero vectors). The selection logic aims to drive the power errors toward zero within a hysteresis band. When active power falls below the reference minus the hysteresis band, a switching state that increases power is selected; when it exceeds the reference plus the band, a state that decreases power is applied. The same principle applies to reactive power, creating a coupled decision process managed by the switching table.

1.2 Motivation for DPC Adoption

DPC emerged from the need for faster transient response and reduced parameter sensitivity compared to vector control. In FOC systems, the cascaded structure with inner current loops and outer power loops introduces phase lag and limits bandwidth. PI regulator tuning requires accurate machine parameters, and performance degrades when parameters deviate from nominal values.

DPC addresses these limitations through several mechanisms. First, eliminating current regulators removes the associated tuning effort and parameter dependence. Second, direct power regulation achieves response times limited only by switching device capabilities and sampling frequency [4]. Third, the hysteresis-based approach inherently accommodates nonlinearities and saturation effects without additional compensation.

The structural simplification is substantial: a complete DPC implementation requires approximately 30-40% fewer blocks than an equivalent FOC system. This reduction in complexity translates to lower computational load on control processors and simplified software validation.

2. Power Estimation and Reference Frames

2.1 Instantaneous Power Calculation

DPC operates on instantaneous power quantities calculated in the stationary αβ reference frame. For a three-phase system, the instantaneous active and reactive powers are given by:

where vα​, vβ​ and iα​, iβ​ are the voltage and current components obtained through Clarke transformation:

The factor 3/2 arises from the power-invariant transformation convention. These calculations are performed at each sampling instant using the most recent measurements.

2.2 αβ Frame vs. dq Frame

The choice of αβ frame over dq frame is fundamental to DPC operation. In vector control, the dq frame rotates synchronously with the stator flux or grid voltage, transforming AC quantities to DC values that can be regulated with PI controllers. This rotation requires accurate knowledge of the flux or voltage angle, typically obtained through phase-locked loops (PLLs) or flux estimators.

DPC remains in the stationary αβ frame for two reasons. First, the instantaneous power equations are simpler and require no angular information beyond the voltage vector position used for sector determination. Second, avoiding rotational transformations eliminates PLL dynamics and associated delays, contributing to faster transient response.

The voltage vector angle is still needed for sector identification, but this calculation uses only the atan2 function of the αβ voltage components and does not require phase tracking over multiple cycles.

2.3 Sampling Frequency Constraints

DPC performance depends critically on sampling frequency. The hysteresis comparators and switching table operate on sampled measurements, and the time between samples determines the minimum achievable response time. Typical sampling frequencies for DPC implementations range from 20 kHz to 100 kHz, substantially higher than the 5-10 kHz sampling used in vector control with PWM.

Higher sampling frequencies improve power tracking accuracy and reduce the average power error but increase computational burden and may require faster analog-to-digital converters. The relationship between sampling frequency and power ripple is approximately inverse: doubling the sampling frequency halves the time before error detection and correction, reducing the magnitude of power excursions beyond the hysteresis band.

2.4 Measurement Noise Influence

Noise on voltage and current measurements propagates directly into power calculations and can cause spurious switching events. The derivative nature of the hysteresis comparator makes it particularly sensitive to high-frequency noise, which may trigger unnecessary switching and increase losses.

Practical implementations employ analog anti-aliasing filters before sampling and digital filtering after conversion. However, filtering introduces phase delay that affects the apparent timing of power errors relative to the actual system state. The trade-off between noise rejection and phase delay must be carefully managed, typically with filter cutoff frequencies an order of magnitude above the expected switching frequency.

2.5 Digital Delay Impact

The interval between measurement sampling and switching state application creates a inherent delay in digital implementations. During this computation time, the system state evolves, and the switching decision based on delayed information may no longer be optimal.

For a sampling period Ts​, the total delay from measurement to output is typically 1.5Ts​ to 2Ts​, accounting for sampling, computation, and gate drive propagation. This delay effectively adds phase lag to the control loop and can lead to increased power ripple or instability at high switching frequencies. Compensation techniques include predictive algorithms that estimate the system state at the switching instant based on the system model and measured derivatives [1] [2].

3. Switching Logic and Control Structure

3.1 Hysteresis Controllers

The hysteresis comparator is the decision element in DPC. For active power, the comparator operates as:

where Hp​ is the hysteresis band for active power. The reactive power comparator Sq​ operates identically with band Hq. The hysteresis bands define the allowable power error before corrective action is taken.

The width of the hysteresis band directly affects system behavior. Narrow bands produce smaller steady-state power ripple but increase switching frequency, raising losses and thermal stress. Wide bands reduce switching frequency and losses but allow larger power excursions and may cause visible torque pulsations. Typical bands range from 1% to 5% of rated power.

3.2 Switching Tables

The switching table maps the combination of hysteresis outputs (Sp, Sq) and the voltage vector sector to a specific converter switching state. The sector is determined by the angle of the stator voltage vector:

The voltage plane is divided into 12 sectors of 30° each, providing sufficient resolution for vector selection. The switching table is designed to select vectors that increase or decrease active and reactive power according to the error signs.

Table 1 presents a typical switching table for DFIG rotor-side converter DPC. The table entries correspond to switching states [a b c] where 1 indicates the upper switch conducting and 0 the lower switch conducting.

Table 1: DPC Switching Table for 12-Sector Operation

The zero vectors (000, and optionally 111) are used when both power errors are within bands, reducing switching losses during steady-state operation.

3.3 Switching Frequency Variation

Unlike PWM-based methods with fixed carrier frequency, DPC produces variable switching frequency that depends on operating conditions. Factors influencing instantaneous switching frequency include:

• Hysteresis band width: Narrow bands increase frequency
• Power level: Higher power levels typically increase frequency
• Machine speed: Rotor voltage magnitude affects vector effectiveness
• Grid conditions: Disturbances cause frequency excursions

Typical DPC implementations exhibit switching frequencies ranging from 2 kHz to 15 kHz over the operating range. This variation complicates filter design, as output filters must attenuate harmonics across a wide frequency spectrum rather than at a fixed carrier frequency and its multiples.

3.4 Trade-off Between Ripple and Losses

The hysteresis bands represent the primary design trade-off in DPC. Narrow bands minimize power ripple and improve current quality but increase switching frequency and semiconductor losses. Wide bands reduce losses but degrade power quality and may cause unacceptable torque pulsations in the generator.

For a given application, the bands are selected to satisfy both power quality requirements (grid code compliance) and thermal constraints of the semiconductor devices. The relationship is approximately:

where k is a machine-dependent constant, H is the hysteresis band, Vdc​ is DC-link voltage, and Lσ​ is leakage inductance. This inverse relationship means that halving the band doubles the switching frequency.

4. Practical Switching Constraints

4.1 Semiconductor Limits

Power semiconductor devices (IGBTs) have maximum current, voltage, and junction temperature ratings that constrain DPC operation. The variable switching frequency characteristic of DPC complicates thermal design because losses cannot be predicted as simply as with fixed-frequency PWM.

Switching losses in IGBTs are approximately proportional to switching frequency and current. For a given device, the maximum allowable junction temperature determines the acceptable combination of conduction loss and switching loss. DPC implementations must include thermal monitoring or conservative band selection to prevent device overheating under worst-case conditions [1].

4.2 Thermal Stress

The non-uniform distribution of switching events in DPC creates thermal cycling that differs from PWM operation. Periods of high power demand or grid disturbance may produce bursts of high-frequency switching, causing rapid temperature rise. The thermal time constants of the silicon die and module package determine whether such bursts remain within safe operating limits.

Active thermal management techniques, such as temporarily widening hysteresis bands during overload conditions, can protect devices while maintaining control. However, this degrades power quality during the very conditions where accurate control is most needed.

4.3 DC-Link Voltage Dependency

The effectiveness of each voltage vector in changing power depends on the DC-link voltage magnitude. Higher DC voltage produces larger voltage steps at the converter output, accelerating power changes but potentially causing overshoot. The switching table assumes nominal DC voltage; significant variations alter the vector magnitudes and may degrade performance [2].

Rotor-side DPC in DFIG systems must coordinate with grid-side converter control to maintain DC-link voltage within acceptable bounds. During transients, DC voltage fluctuations affect the available rotor voltage and may temporarily reduce control authority.

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4.4 Overcurrent Protection Interaction

DPC lacks explicit current regulation, so overcurrent protection must be provided separately. Hysteresis comparators acting on power errors do not directly limit currents; a large power error could theoretically command switching states that produce excessive currents [3].

Practical implementations include additional current monitoring with fast hardware comparators that override switching commands if currents exceed safety thresholds. This protection layer must coordinate with the DPC logic to prevent nuisance trips while ensuring device safety. The coordination is typically implemented at the gate drive level with independent current sensing.

5. Behavior Under Non-Ideal Conditions

5.1 Grid Voltage Dips

Voltage dips represent one of the most challenging conditions for DPC operation. When grid voltage magnitude decreases, the relationship between switching states and power changes is altered. The same vector that increased power under nominal voltage may be less effective or even counterproductive during a dip.

DPC responds to voltage dips through its inherent fast action. As soon as the power error exceeds the hysteresis band due to the disturbance, the switching table selects vectors appropriate for the current conditions. The response time is limited only by sampling and switching delays, typically within 1-2 sampling periods [4].

However, the switching table optimized for nominal operation may not be optimal during deep voltage sags. Advanced DPC implementations incorporate voltage magnitude information into the vector selection process, either through adaptive tables or by scaling the hysteresis bands proportionally to voltage.

Compared to current-controlled systems, DPC typically achieves faster initial response to voltage dips but may exhibit larger current spikes because the power-based logic does not directly limit currents. The absence of current regulation means that during the first few switching periods after a dip, currents can rise rapidly until the power error integrates sufficiently to trigger corrective action.

5.2 Rapid Torque Variation

Mechanical torque variations from wind gusts or tower shadow effects produce corresponding power reference changes. DPC tracks these variations with minimal phase lag because the power loop operates at switching frequency rather than the limited bandwidth of cascaded regulators [5].

For a step change in power reference, DPC typically achieves 90% of the new value within 2-3 switching periods, corresponding to 200-300 μs for typical implementations. Vector control with PWM and PI regulators requires 5-10 ms for similar response, limited by current regulator bandwidth and PWM carrier period.

5.3 Weak Grid Conditions

Weak grids with significant impedance affect DPC performance by introducing coupling between power variations and voltage magnitude/phase. When the converter injects power, the voltage at the point of common coupling changes, which in turn affects the power calculation and subsequent switching decisions.

This coupling can lead to instability in extreme cases because the power measurement includes the effect of the converter’s own action on grid voltage. The hysteresis comparator sees a combined effect of intended power change and grid voltage response, potentially causing incorrect decisions [6].

Solutions include using estimated grid voltage (excluding converter-induced drop) for power calculation or implementing additional stabilization terms in the switching logic. Some DPC variants employ virtual flux estimation to reduce sensitivity to grid impedance.

5.4 Distorted Voltage Waveforms

Harmonic distortion in grid voltage introduces corresponding oscillations in instantaneous power calculations. These oscillations can trigger spurious switching events if their magnitude exceeds the hysteresis band, increasing switching frequency and losses without contributing to useful power transfer.

For a grid with 5% fifth harmonic, the resulting power oscillation at six times fundamental frequency may reach several percent of rated power. If this exceeds the hysteresis band, the comparator will attempt to correct these oscillations, causing unnecessary switching.

Solutions include:

• Widening hysteresis bands beyond expected distortion levels
• Implementing notch filters on power measurements at harmonic frequencies
• Using fundamental-positive-sequence extraction for power calculation

The latter approach sacrifices some response speed for improved steady-state behavior in distorted grids.

6. DPC in DFIG Wind Turbines

6.1 Stator Grid Connection Implications

In DFIG systems, the stator connects directly to the grid while the rotor connects through the back-to-back converter. This configuration imposes specific requirements on rotor-side DPC that differ from full-power converter applications.

The rotor-side converter controls the rotor currents to achieve desired stator active and reactive power. The relationship between rotor quantities and stator power depends on machine parameters and operating point. For DPC implementation, this means that the switching table must account for the induction machine characteristics rather than simply applying vectors based on grid voltage alone.

6.2 Rotor Flux Dynamics

Rotor flux responds to applied rotor voltage with dynamics governed by rotor time constant. When a voltage vector is applied, the resulting rotor current change produces torque and power variations through interaction with the rotating stator field. The effectiveness of each vector depends on the relative orientation of rotor flux and applied voltage.

The standard DPC switching table for DFIG is derived from analysis of rotor flux trajectory under different voltage vectors. Vectors that increase rotor flux magnitude tend to increase reactive power, while vectors that advance rotor flux angle relative to stator flux increase active power. This coupling means that vector selection must consider both effects simultaneously.

6.3 Transient Rotor Current Spikes

During grid disturbances, the sudden change in stator voltage induces transient currents in the rotor circuit through transformer action. These currents can exceed the converter rating within microseconds, before DPC has time to respond.

Rotor-side DPC alone cannot prevent these spikes because they result from magnetic coupling rather than converter action [7].

Protection requires either:

• Fast hardware current limiting in the converter
• Crowbar circuits that short the rotor windings during severe events
• Series dynamic braking resistors

The coordination between DPC and protection circuits must ensure that the converter remains in control during recoverable disturbances while protecting devices during severe events.

6.4 Coordination with Grid-Side Converter

The grid-side converter maintains DC-link voltage and can contribute to reactive power support. DPC on the rotor side must coordinate with grid-side control to maintain overall system stability.

During transient events, both converters respond to the disturbance. The rotor-side DPC attempts to regulate stator power, while the grid-side converter regulates DC voltage. If their actions conflict, instability can result. For example, rotor-side action that increases rotor power during a voltage dip may draw energy from the DC link, causing DC voltage drop that the grid-side converter must correct.

Coordinated strategies include:

• Prioritizing DC voltage regulation during severe events
• Limiting rotor-side power commands during transients
• Implementing communication or feedforward between converters

7. Simulation and Validation Approaches

7.1 Time-Domain Simulation

DPC development relies heavily on time-domain simulation because the variable switching frequency and hysteresis nonlinearities preclude small-signal analysis. Simulation must capture switching events with sufficient temporal resolution to accurately represent power ripple and switching losses [8].

Typical simulation approaches include:

• Switching-level models: Detailed IGBT models with gate drive circuits, accurate for loss estimation and EMI studies but computationally intensive
• Averaged models: Represent switching effects through controlled voltage sources, faster but lose switching ripple information
• Piece wise linear models: Approximate machine behavior between switching events, balancing speed and accuracy

For DFIG DPC studies, switching-level simulation of 1-2 seconds of operation typically requires 10-30 minutes of computation time on desktop hardware. This limits the number of test cases but provides essential validation of switching behavior.

7.2 Hardware-in-the-Loop Testing

Hardware-in-the-loop (HIL) validation bridges simulation and physical implementation. The DPC algorithm runs on actual control hardware while the power circuit and machine are simulated in real time on FPGA-based platforms.

HIL testing reveals implementation issues not visible in offline simulation:

• Computation time delays and their effects
• ADC sampling and conversion limitations
• Interface timing between controller and gate drives
• Protection logic coordination

Modern HIL systems achieve time steps below 1 μs, sufficient to resolve switching events for frequencies up to 100 kHz. This enables thorough testing of DPC behavior under fault conditions without risk to actual hardware [9].

7.3 Stress Testing Under Fault Scenarios

DPC robustness is validated through systematic testing of fault scenarios:

• Symmetrical voltage dips: 0% to 90% remaining voltage
• Asymmetrical faults: Single-phase and phase-to-phase faults
• Frequency excursions: ±5 Hz from nominal
• Phase jumps: Up to 60° instantaneous phase change
• Harmonic distortion: Up to 10% individual harmonics

For each scenario, key metrics include:

• Peak current during transient
• Recovery time to steady-state
• Number of switching events during disturbance
• DC voltage deviation

These tests identify weaknesses in the switching table design and guide refinement of hysteresis bands and vector selection.

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7.4 MATLAB Simulation results

The MATLAB simulation results presented in Figures 2 through 15 demonstrate the complete operational characteristics and dynamic performance of the Direct Power Control system applied to a DFIG wind turbine. Figure 2 illustrates the stator voltage and current waveforms during steady-state operation at rated conditions, showing the three-phase sinusoidal quantities with proper phase alignment and minimal harmonic distortion, confirming that the DPC maintains acceptable power quality despite variable switching frequency operation. Figure 3 presents the rotor voltage and current waveforms on the rotor-side converter output, revealing the characteristic switching ripple imposed by the hysteresis-based control, with voltage pulses of varying width corresponding to the instantaneous power error conditions. Figure 4 displays the instantaneous active and reactive power tracking performance, overlaying the measured quantities (P, Q) with their respective references (P, Q), demonstrating that the hysteresis comparators maintain the power errors within the prescribed bands of ±2% for active power and ±3% for reactive power, with the switching actions clearly visible as power excursions are corrected within 1-2 sampling periods. Figure 5 shows the three-phase rotor currents with enhanced resolution, highlighting the current ripple magnitude of approximately 8% peak-to-peak, which represents the trade-off between switching losses and power quality inherent in DPC systems. Figure 6 presents the DC-link voltage behavior during the simulation period, demonstrating that the grid-side converter maintains the DC voltage within ±5% of the nominal 1150V setpoint, with transient deviations rapidly corrected through coordinated control action. Figure 7 illustrates the electromagnetic torque response, showing rapid torque tracking without overshoot and minimal steady-state ripple of approximately 3% of rated torque, confirming that direct power regulation translates effectively to torque control in the DFIG system. Figure 8 displays the rotor speed variation as the wind turbine transitions between operating points, demonstrating smooth acceleration and deceleration profiles consistent with the mechanical dynamics of a 2MW turbine system. Figure 9 provides detailed insight into the hysteresis comparator operation, showing the active power error (Perr) trajectory within the hysteresis band boundaries (±0.02 pu), with the comparator output (Sp) transitioning between states whenever the error attempts to exceed the band limits, clearly illustrating the bang-bang control nature of DPC. Figure 10 examines the switching frequency characteristics through a histogram of switching intervals, revealing the variable nature of DPC operation with frequencies ranging from 2.5 kHz to 8.5 kHz, centered around a mean of approximately 5.2 kHz, which confirms the expected frequency spread and informs filter design requirements. Figure 11 presents the harmonic spectrum of stator currents analyzed through FFT, showing the dominant harmonics distributed across a wide frequency range rather than concentrated at a single carrier frequency, with total harmonic distortion (THD) of 4.8% meeting grid code requirements despite the absence of fixed-frequency PWM. Figure 12 demonstrates the system response to a 20% step decrease in active power reference, capturing the transient behavior over a 100ms window; the active power reaches its new setpoint within 2ms (approximately 10 switching periods) with minimal overshoot, while reactive power remains undisturbed, confirming the decoupled control capability of the switching table design. Figure 13 shows the corresponding rotor current response during this transient, revealing a temporary 30% increase in current magnitude as the machine adjusts to the new operating point, with the current returning to steady-state within 5ms and no oscillatory behavior. Figure 14 illustrates the system behavior during a simulated grid voltage dip of 30% magnitude lasting 200ms, demonstrating that the DPC maintains synchronism and continues to deliver reduced power throughout the disturbance, with active power temporarily reducing to 0.3 pu while reactive power is prioritized to support grid voltage recovery, followed by rapid restoration to pre-fault levels when voltage returns to nominal. Finally, Figure 15 presents the three-dimensional visualization of the switching table operation over multiple fundamental cycles, showing the distribution of applied voltage vectors across the 12 sectors and four hysteresis state combinations, confirming that all available vectors are utilized appropriately and that zero vectors are preferentially selected during steady-state operation to minimize switching losses. Collectively, these fifteen figures provide comprehensive validation of the DPC implementation, demonstrating its fast transient response, acceptable steady-state performance, variable switching frequency characteristics, and robust behavior during grid disturbances, while also revealing the practical trade-offs between power quality, switching losses, and implementation complexity that designers must consider when applying DPC to DFIG wind turbine systems.

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8. Conclusions

Direct Power Control offers a fundamentally different approach to DFIG wind turbine control compared to classical field-oriented methods. By eliminating inner current loops and operating directly on power errors through hysteresis comparators and switching tables, DPC achieves simplified structure, reduced parameter sensitivity, and fast transient response.

The trade-offs inherent in DPC design center on the hysteresis bands, which determine the balance between power quality and switching losses. Narrow bands improve steady-state performance but increase losses; wide bands reduce losses at the expense of increased ripple. The variable switching frequency characteristic complicates filter design and thermal management but emerges naturally from the hysteresis principle rather than requiring active frequency modulation.

Under non-ideal grid conditions, DPC demonstrates behavior distinct from current-controlled systems. Response to voltage dips is extremely fast, limited only by sampling and switching delays, but the absence of explicit current limiting can produce larger transient spikes. Weak grid conditions introduce coupling effects that may require modified switching tables or additional stabilization measures. Grid voltage distortion causes power measurement ripple that can trigger unnecessary switching unless addressed through filtering or band adaptation [10].

For DFIG applications specifically, rotor-side DPC must account for the induction machine’s flux dynamics and coordinate with grid-side converter control. The switching table design requires careful consideration of rotor flux orientation and the coupling between active and reactive power responses to each voltage vector.

Validation of DPC implementations requires switching-level simulation to capture the detailed behavior arising from hysteresis nonlinearity and variable switching frequency. Hardware-in-the-loop testing bridges simulation and physical implementation, revealing practical constraints related to sampling delay, measurement noise, and computational timing.

DPC represents a viable alternative to vector control for DFIG wind turbines, particularly in applications where fast transient response and structural simplicity outweigh the challenges of variable switching frequency and harmonic content. Continued development of adaptive hysteresis bands, improved switching tables for disturbed grid conditions, and coordination strategies with grid-side converters will further enhance the applicability of DPC in modern wind energy systems.

References

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[2] R. Datta and V. T. Ranganathan, “Direct power control of grid-connected wound rotor induction machine without rotor position sensors,” IEEE Transactions on Power Electronics, vol. 16, no. 3, pp. 390-399, 2001.

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[4] J. Hu, H. Nian, B. Hu, Y. He, and Z. Q. Zhu, “Direct active and reactive power regulation of DFIG using sliding-mode control approach,” IEEE Transactions on Energy Conversion, vol. 25, no. 4, pp. 1028-1039, 2010.

[5] G. Abad, M. A. Rodríguez, and J. Poza, “Two-level VSC based predictive direct power control of the doubly fed induction machine with reduced power ripple at low constant switching frequency,” IEEE Transactions on Energy Conversion, vol. 23, no. 2, pp. 570-580, 2008.

[6] S. A. Zabihi and F. Zare, “Direct power control of a grid-connected photovoltaic system associated with a flywheel energy storage system,” IEEE Transactions on Power Electronics, vol. 30, no. 7, pp. 3785-3796, 2015.

[7] J. Eloy-García, S. Arnaltes, and J. L. Rodríguez-Amenedo, “Direct power control of voltage source inverters with unfolding technique,” IEEE Transactions on Power Electronics, vol. 22, no. 4, pp. 1356-1363, 2007.

[8] D. Zhi and L. Xu, “Direct power control of DFIG with constant switching frequency and improved transient performance,” IEEE Transactions on Energy Conversion, vol. 22, no. 1, pp. 110-118, 2007.

[9] C. Wessels, F. Hoffmann, and J. Möckel, “Direct power control for three-phase grid converters with active filtering function,” IEEE Transactions on Power Electronics, vol. 28, no. 8, pp. 3890-3901, 2013.

[10] M. Malinowski, M. P. Kazmierkowski, and A. M. Trzynadlowski, “A comparative study of control techniques for PWM rectifiers in AC adjustable speed drives,” IEEE Transactions on Power Electronics, vol. 18, no. 6, pp. 1390-1396, 2003.

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