2.1. Description of the WT-DFIG System

The suggested DPC technique for a wind power conversion system based on DFIG is demonstrated in Figure 1. The system consists of a WT and a gearbox, which work together to transform the kinetic energy of wind into mechanical energy. The DFIG subsequently converts this mechanical energy into electrical energy. Whereas the rotor is indirectly linked to the AC grid via an AC frequency converter that includes dual power converters, AC/DC and DC/AC, the DFIG’s stator is directly linked to the grid. The suggested DPC controls are applied to the converter that is linked to the DFIG rotor side to deal with the production of active power and compensation of local reactive power. The converter that is linked to the AC grid side is controlled to ensure a unit power factor and a sinusoidal current with a steady frequency of 50 Hz.

2.2. WT Modeling

With $$

where Ω_t represents the speed of the turbine rotor.

2.3. WT Systems’ Operating Zones

As illustrated in Figure 2, WTs with variable speeds are operating in a variety of areas, including three different zones. Very low wind speeds, present in the zone before the first, prevent the turbine from turning. The “cut-in” speed is when the wind speed triggers the turbine to start producing power. The wind speed rises in the first zone when the nominal power is greater than the produced power. Zone two is where zone I’s generated power transitions to zone III’s maximum power. WT extracts the maximum power in both zones at a 0° blade pitch angle β using the maximum power point tracking (MPPT) algorithm.

In Zone 3, the speed of the wind surpasses the nominal speed of the wind, and by modifying the blade pitch angle β, the nominal output power limit is established by the pitch angle control mechanism. As a precaution, the DFIG rotor is disconnected from the turbine when the wind speed in the third operating zone reaches its maximum. This power interruption is required to avoid damage and safeguard the turbine [4, 30].

2.4. DFIG Modeling

The DFIG’s mathematical model in the dq frame is given as follows:

3.1. DPC Overview

Based on these objectives, the control system generates optimal rotor voltage vectors for the rotor-side converter (RSC).

3.2. Mathematical Foundation

3.2.2. Rotor Flux Dynamics

The rotor flux is defined by the following:

$$ (12)

After neglecting rotor resistance, the following describes the rotor flux fluctuation:

$$ (13)

According to Equation (13), the rotor flux variance is influenced by the choice of rotor voltage vector and the length of time it is applied. Consequently, by selecting the proper voltage vector, the rotor flux’s amplitude and angle can be adjusted. Moreover, a voltage vector is generated by the two-level inverter between two switching processes out of a group of eight voltage vectors, six of which have the same amplitude and two of which have zero amplitude. In contrast, the difference in the Q_AC and the P_s powers is impacted by the outcome of the rotor flux variation. Figure 3 displays how every rotor voltage vector affects the local reactive power compensation and active power.

3.3. C-DPC

3.3.1. System Architecture

C-DPC is a control that uses a switching table and hysteresis comparators. The WT-DFIG C-DPC basic scheme system is depicted in Figure 4. The main advantages of this approach are its fast dynamic response and straightforward control structure.

3.3.2. Hysteresis Power

In order to provide compensated local reactive power switching states and produced active power, three and two-level hysteresis bands were chosen for eP_s and eQ_AC, respectively, as shown in Figure 5.

In order to obtain the powers switching states. First, a comparison is made between the estimated values of active and reactive powers and their references. Then, digitizing the errors via hysteresis comparators. Therefore, the output is a value representing the effect that occurs to both the active power HP_s (−1,0 or 1) and the local reactive power compensation HQ_AC (−1 or 1). Furthermore, errors are reduced inside of the bands of the hysteresis comparators.

HP_s represents the active power regulator output, so when it is an output that represents (HP_s = 1), it means that it is a power that needs to be increased, (HP_s = −1) to decrease, or (HP_s = 1) does not affect it. Therefore, (HQ_AC) represents the local reactive power compensation regulator output, so when it is an output that represents (HQ_AC = 1), it means its power needs to be increased and (HQ_AC = −1) to decrease.

4.1. FH-DPC

4.1.1. System Overview

The objective of the proposed controls is to enhance the performance of C-DPC and eliminate issues associated with the response delay of hysteresis comparators. The system diagram for the FH-DPC control is illustrated in Figure 6.

4.1.2. Fuzzy Hysteresis Power

The hysteresis comparators principle served as the basis for the design of the FH-DPC controller. This one has two inputs with a single output. It replaces the compensated local reactive and generated active powers hysteresis comparators with a single fuzzy logic controller, as shown in Figure 7. Indeed, the two inputs are powers errors (eP_s and eQ_AC) and the output is a single digital value, that represents both powers errors (E_powers). The proposed FH-DPC design was implemented using these steps.

4.1.2.1. Fuzzification

Input variables are transformed into linguistic variables using membership functions (MFs). Active power error has three MFs (P, Z, and N), while reactive power error has two MFs (P and N) (as presented Figures 8 and 9).

4.1.2.2. Fuzzy Rules

Six rules are designed based on both power hysteresis comparators, using the Mamdani inference method, as shown in Table 2.

Table 2. Fuzzy rule of FH-DPC.

eQAC/ePS	P	Z	N
P	E1	E2	E3
N	E4	E5	E6

• Abbreviation: FH-DPC, fuzzy hysteresis-direct power control.

4.1.2.3. Defuzzification

To convert the fuzzy inference results into measurable output is the aim of the defuzzification process. The “Bisector” approach is used in this work to defuzzify.

The output (E_powers) is the digital error powers, which has six singletons MFs (E₁ … E₆), as presented in Figure 10.

4.2. Artificial Neural Network (ANN) Control Based on DPC (NH-DPC)

Electrical optimization and control are two of the many sectors that employ ANNs, which are approaches for machine learning. Their capacity to gain knowledge from data, process information swiftly, and provide precise predictions is what makes them so popular. ANNs consist of multiple layers (input, hidden, and output) with interconnected neurons linked by weighted connections, as illustrated in Figure 11.

4.2.1. Architecture Design

This section proposes a new direct power neural network control that utilizes a single neural network as a control instead of two hysteresis comparators. Instead of hysteresis or fuzzy logic, the proposed DPC employs a neural network to generate digital power errors. The structure of the WT-DFIG with the suggested control (NH-DPC) applied to the WT-DFIG system is shown in Figure 12.

4.2.2. The Neural Network Operating Principle

4.2.2.1. Feed-Forward Algorithm

The activation function is applied to the sum of the input vector $$ multiplied by the associated weight vectorw¹ = [w_k1¹, w_k2¹, ⋯, w_kM¹] in addition to the bias value. For a single neuron in the first hidden layer, as described in Figure 13. It is mathematically described as follows [27, 32, 33]:

$$ (16)

The initial hidden layer is indicated by the superscript (w, b,  and h).

Information flows sequentially through the network, with each layer’s outputs serving as inputs for the subsequent layer. The general form for layers beyond the first hidden layer is as follows:

$$ (17)

where $$ is the previous layer’s vector output.

The neural network’s final output is provided by the following:

$$ (18)

Among the activation functions that are employed are as follows:

• •

  The praline function

$$ (19)

• •

  The sigmoid function

$$ (20)

4.2.2.2. Back Propagation Algorithm

Used for training, this algorithm adjusts weights and biases to minimize the cost function, typically the mean squared error (MSE):

$$ (21)

where N is the number of outputs, y_k is the network output, and d_kis the target output.

The Levenberg–Marquardt method, a form of backpropagation, is employed for supervised learning. By propagating through hidden layers to the input layer, this approach calculates the gradient of the output layer’s cost function.

4.2.3. Neural Network Hysteresis Power

The appropriate data must be prepared before establishing and learning a network. For that, the input features and output must be defined. Thus, there are two input features, namely, the compensated local reactive and produced active powers errors (eQ_AC, eP_s), and the output is a digital powers error (E_powers). Some samples used for this neural network are shown in Table 4.

Table 4. The training samples for neural networks.

No. of samples	eQAC	ePs	Epowers
1	−0.4	−0.82	6
2	−0.18	+0.05	4
3	+0.2	+0.92	1
4	+0.98	−0.24	3

The proposed topology for NH-DPC is shown in Figure 14. After conducting multiple experiments, the most optimal structure was determined. The network architecture includes an input feature layer, three hidden layers consisting of three, seven, and five neurons, respectively, and a layer of output that employs the praline function. The activation function used across all hidden layers is the sigmoid function.

Proper training is essential for achieving high performance in the NH-DPC neural network. To accomplish this, 70% of the total 14,400 data samples were used for training, while 15% were assigned for validation and another 15% for testing. The Levenberg–Marquardt approach, which is based on the MSE cost function, was used during the learning phase.

The development of the neural network was carried out using MATLAB’s Neural Network Toolbox. Multiple experiments were conducted, and all of them resulted in similar outcomes in terms of efficiency and time. Figure 15 demonstrates the validation performance of the proposed neural network (NH-DPC), which achieved a value of 4.17e−13 after 430 iterations, taking approximately 1.5 min.

5.1. A Profile of Step Wind Speed

This section’s goal is to assess how well the five controllers used for reference tracking performed, namely C-DPC, NH-DPC, FH-DPC, and two methods suggested in [23], NF-DPC and NN-DPC. Furthermore, C-DPC, NH-DPC, and FH-DPC’s performance in local reactive power compensation modes, which were not assessed in [23], will also be compared.

In order to evaluate the performance and effectiveness of the proposed DPC approach in a system WT-DFIG, tests were conducted using a step-like wind profile with wind speeds ranging between 6 and 9.1 m/s. This profile of wind was designed to have fast changes in wind speed at specific time points and constant wind speed in between changes to satisfy the standards for mechanical speed of the three DFIG system operating modes. Following the system’s 1000 rpm synchronous speed, these modes were selected. For the first 3 s, the sub-synchronous mode is in on (the mechanical speed that is less than 1000 rpm), then synchronous mode for 3–5 s, and between 8 and 9 s (mechanical speed of exactly 1000 rpm), and finally super-synchronous mode between 5 and 8 s (mechanical speed above 1000 rpm). Figure 17 displays the mechanical velocity profiles and wind velocity profiles for each operating mode.

The active power reference P_s−ref is affected by changes in wind speed. Indeed, the suggested control techniques (NH-DPC, FH-DPC) and C-DPC have an effect on the active power produced by WT-DFIG. It appropriately conforms to its reference value, as illustrated in Figure 18. Then, P_s the power is always delivered into the AC grid and has a negative indication throughout every DFIG operating mode.

In Figure 18, we have magnified the transitional periods between different operating modes to assess the effectiveness of every approach by assessing the dynamic response. These zooms demonstrate that the system responds quickly and tracks the references perfectly across all periods of transitional between the different modes of operating. The finding is valid for two strategies proposed in this study (NH-DPC and FH-DPC) and C-DPC.

To analyze the steady-state performances of the proposed strategies. The active power for each operating mode is displayed individually in Figures 19–21 together with its reference. It is evident from these figures that the ripples in the active power of the C-DPC approach are relatively wide and significantly reduced by adopting the proposed methods NH-DPC and FH-DPC throughout every operating mode.

To validate the efficacy of every strategy, Table 6 shows the results of estimating the mean ripple range using the standard deviation. Compared to C-DPC, both NH-DPC and FH-DPC exhibit a substantial enhancement, with more than 70% enhancement achieved in every mode. Moreover, FH-DPC and NH-DPC perform similarly, with FH-DPC slightly outperforming NH-DPC in subsynchronous and synchronous modes. With an average rise in performance of 10%, the two suggested controls (NH-DPC and FH-DPC) likewise perform better than the alternatives listed in [23].

The reference values determine how NH-DPC, FH-DPC, and C-DPC affect the local reactive power compensation. This one has three operating modes. Indeed, the WT-DFIG system absorbs or generates it, or it operates with a unit power factor. Q_AC−ref power is determined by the demand on the AC grid, as shown in Figure 22.

To execute local reactive power compensation, reactive power from the grid is absorbed by the system when a positive reference signal is employed. It acts like an inductor. This happens between 0 and 1 s, between 4 and 5 s, and between 6 and 7 s. Between 2 and 4 s as well as 7 and 8 s, the system collects reactive power from the AC grid when the local reactive power compensation reference value (Q_AC−ref) is negative. It functions as a capacitor. By ensuring that there is no communication between the WT-DFIG and the AC grid, the control mechanism ensures that the AC grid runs at the unit power factor. This conducts like a resistor. From [1s, 2s] to [5s, 6s] to [8s, 9s], this is evident.

Figure 22 has been zoomed in to observe how NH-DPC, FH-DPC, as well as C-DPC methods, impact the compensated local reactive power system’s response time during brief intervals between various compensation modes. The dynamic response is adequate for all three control approaches, as seen in the zoomed figures.

Figures 23–25 are used to analyze the evolution of compensated local reactive power during steady-state operation, separately for each mode of operation. The suggested NH-DPC and FH-DPC approaches have importantly reduced the ripples in local reactive power compensation contrasted to the C-DPC control technique. The ripples in the local reactive compensation power of the C-DPC control are relatively large. This finding holds true for each operating mode of DFIG, including synchronous, sub-synchronous, and super-synchronous modes, in addition to all compensated local reactive power modes, including absorption, unit power factor operation, or generation.

To illustrate each control method’s efficacy, the results of the average ripple range are summarized in Table 7. Importantly, NH-DPC and FH-DPC both perform better than C-DPC, with a 77% average enhancement throughout all different compensation and operating modes. Moreover, FH-DPC performs slightly better than NH-DPC in all modes except for synchronous mode operation, where the roles are reversed. Both proposed controls also outperform the options listed in [23], with a 23% increase on average.

Under C-DPC, NH-DPC, and FH-DPC control approaches, the currents produced by the DFIG system are shown in Figure 26a–c. It is evident from Figure 26b,c that even while switching between operating modes, the system keeps the operating frequency at 50 Hz. In terms of signal quality, the results are highly acceptable: the suggested NH-DPC and FH-DPC approaches greatly enhance the quality of the generated current signals in comparison to C-DPC. Given that the DFIG stator is straight linked to the AC grid, it is imperative to provide high-quality produced current signals. Consequently, this study analyzed the harmonic content of the current produced by phase (A) across every DFIG operating mode via FFT analysis.

The spectra obtained are presented in Figure 27a–c, which indicates that the proposed control methods substantially reduce total harmonic distortions (THDs) in every investigated mode. The various methods suggested in this study are contrasted in Table 8, as well as those examined in [22]. Particularly, NH-DPC and FH-DPC exhibited an average THD enhancement of 77% contrasted to C-DPC. Furthermore, both proposed methods exceeded the options listed in [23] by an average of 11%.

Table 9 provides a comparative analysis of the two proposed control methods (NH-DPC and FH-DPC) based on a thorough review of the methodologies discussed in this paper. Upon closely examining the outcomes, it is evident that the FH-DPC method outperforms the NH-DPC method, primarily in terms of design efficiency and reducing the ripples of produced active and compensated local reactive powers. Specifically, FH-DPC improves efficiency by 77.22% compared to the 77.32% improvement of NH-DPC. Although the performance improvement of NH-DPC is slightly lower, its effect on improving THDs is noticeable.

Even with excellent data, training an AI-based DPC approach was difficult. The best network architectures for neural networks (NH-DPC) were found by overcoming computational constraints and using trial-and-error optimization in conjunction with powerful processors (10th gen Intel Core i3 CPUs, 12 GB RAM). The training process for NH-DPC neural networks is more complex due to the need to determine optimal weights and biases, resulting in longer training times (80–140 s). Nevertheless, after training, this network produces extremely effective mathematical models for simulations of the WT-DFIG system, operating with minimal computational cost. In contrast, FH-DPC while less computationally intensive during training than neural networks. The fuzzification and defuzzification processes in FH-DPC introduce additional complexity that may potentially increase computational load compared to traditional DPC methods.

5.2. A Profile of Random Variable Wind Speed

This section goals to evaluate the ability of the proposed control strategies to withstand random and sudden changes in the DFIG system’s manners and to track references. To achieve this, all three operating modes of the DFIG system will be randomly and continuously explored in addition to the overspeed mode. The choice of randomly and suddenly varying the profile behavior of wind velocity, continuously and successively, is intended to replicate a realistic wind system operation. This aspect was not examined in [23].

The mechanical velocity and wind velocity profile for every DFIG operating mode are shown in Figure 28. Wind speed ranging from 4.96–12 m/s is used. It is critical to remember that changes in wind speed have a direct influence on the mechanical velocity required for DFIG’s four operating modes. As represented in Figure 27, the mechanical velocity is set to its maximum by triggering the pitch angle control when the power reaches its maximum value. In this last part, the power extraction maximization is transformed into a limitation of generated power at its nominal value. In the remainder of Figure 28, the DFIG velocity evolution tracks exactly that of the wind velocity profile, where it is denominated as an MPPT zone. In the MPPT zone, the extracted power is maximized, in Zones 1 and 2, where the remaining value of λ on optimum and value of C_p on highest, and β = 0. As shown in Figure 29, the value β rises in the over-speed mode, where the blade pitch angle is regulated, while the values of λ and C_p decrease.

Figure 30 illustrates the generated active power by the WT based on the DFIG system under the C-DPC, NH-DPC, and FH-DPC techniques. The zoomed-in sections in Figure 30 show the fast response of the system and excellent active power reference tracking despite its sudden and random variations. This is accurate for all control approaches proposed in this study. Moreover, the NH-DPC and FH-DPC techniques substantially decrease the generated active power ripples in every operating mode, especially in the over-speed mode. However, in the case of C-DPC, the generated active power ripples are relatively large.

The results depicted in Figure 31 clearly indicate that local reactive power that is compensated by WT-DFIG accurately follows its reference. Therefore, the C-DPC, NH-DPC, and FH-DPC have effectively controlled the Q_AC power. Moreover, the NH-DPC and FH-DPC strategies have substantially reduced the ripples in the Q_AC across all operation modes, as compared to the C-DPC technique.

In Figure 32a–d, the generated currents of the WT based on the DFIG employed C-DPC, NH-DPC, and FH-DPC methods are illustrated. Figure 32b–d illustrates that despite the sudden fluctuations and random behavior in wind velocity, the operating frequency remains constant at 50 Hz. The outcomes are remarkable, with the NH-DPC and FH-DPC strategies greatly surpassing the C-DPC and producing substantially improved current signal quality.

To evaluate the performance of the NH-DPC and FH-DPC strategies, a comparison is made with several existing methods in terms of their ability to reduce the THD of the current in a WT-DFIG system. The results are presented in Table 10.