Power regulation of variable speed multi rotor wind systems using fuzzy cascaded control

Power quality is a crucial determinant for integrating wind energy into the electrical grid. This integration necessitates compliance with certain standards and levels. This study presents cascadedfuzzy power control (CFPC) for a variable-speed multi-rotor wind turbine (MRWT) system. Fuzzy logic is a type of smart control system already recognized for its robustness, making it highly suited and reliable for generating electrical energy from the wind. Therefore, the CFPC technique is proposed in this work to control the doubly-fed induction generator (DFIG)-based MRWT system. This proposed strategy is applied to the rotor side converter of a DFIG to improve the current/power quality. The proposed control has the advantage of being model-independent, as it relies on empirical knowledge rather than the specific characteristics of the DFIG or turbine. Moreover, the proposed control system is characterized by its simplicity, high performance, robustness, and ease of application. The implementation of CFPC management for 1.5 MW DFIG-MRWT was carried out in MATLAB environment considering a variable wind speed. The obtained results were compared with the direct power control (DPC) technique based on proportional-integral (PI) controllers (DPC-PI), highlighting that the CFPC technique reduced total harmonic distortion by high ratios in the three tests performed (25%, 30.18%, and 47.22%). The proposed CFPC technique reduced the response time of reactive power in all tests by ratios estimated at 83.76%, 65.02%, and 91.42% compared to the DPC-PI strategy. Also, the active power ripples were reduced by satisfactory proportions (37.50%, 32.20%, and 38.46%) compared to the DPC-PI strategy. The steady-state error value of reactive power in the tests was low when using the CFPC technique by 86.60%, 57.33%, and 72.26%, which indicates the effectiveness and efficiency of the proposed CFPC technique in improving the characteristics of the system. Thus this control can be relied upon in the future.


References
Type of study www.nature.com/scientificreports/applied to the RSC only to demonstrate the extent of its ability to reduce energy fluctuations and improve the characteristics of the proposed energy system.The MATLAB, Matlab/simulink 2022 (https:// www.mathw orks.com/ produ cts/ new_ produ cts/ relea se202 2a.html) was used for verifying the validity, effectiveness, and ability of this strategy compared to the DPC strategy.Also, a comparison with existing papers was achieved in terms of minimizing the ratios of response time, energy ripple, SSE, and overshoot.In this work, a variable WS was used to study the effectiveness and robustness of the CFPC-PWM technique using a 1500 kW DFIG-MRWT system.Several objectives have been achieved through this work, which can be explained in the following points: • Reduction of energy ripples compared to the DPC-PI technique and other strategies recently proposed in the specialized literature; • Reducing the THD value of the current compared to several scientific works; • Significantly increasing the system robustness; • Reducing the values of both overshoot and SSE of DFIG power compared to several works.
• Developing a new strategy different from several existing strategies.
The paper is based on the following sections: The second section deals with the energy system proposed for the study, where the MM for both MRWT and DFIG is given, mentioning the negatives and positives of the proposed system.In addition, to gives a simulation of the MRWT, which demonstrates its effectiveness and distinctive performance compared to traditional turbines.In the third section, numerical and graphical results for both DPC and CFPC-PWM techniques are given, along with a comparison with other scientific papers.Finally, the paper ends with a section in which all obtained results are summarized.

Designed energy system
The energy system designed for this study is represented in Fig. 1, as it is characterized by simplicity, lowcharge, uncomplicated, and easy control.This system contains MRWT, DFIG, RSC, and GSC.In addition to the control strategy, this proposed system contributes significantly to significantly reducing the production bill and energy consumption while preserving the environment.
RSC control: The PWM was used for command because of its simplicity and ease of application, which makes the designed system less complex and, therefore less expensive and easy to maintain.GSC control: To simplify the system and show the extent of the impact of the designed command in reducing EE ripples, a GSC was proposed using a diode, as no control is used for it.In this way, it is possible to know the performance and efficiency of the CFPC-PWM technique compared to the DPC technique and some other papers in minimizing the value of current THD and the robustness of the system.
Power control: The CFPC-PWM technique is used as a suitable solution for power control because of its robustness and lack of relation to the MM system.
Energy estimation: This process is necessary forthis designed system since it requires knowledge of the error in the powers to calculate the voltage reference values.Although the estimation of power is related to the resistance (Rs), this process has been relied upon because it is necessary to extract the control pulses in the RSC.
Turbine control: The maximum power point tracking (MPPT) strategy based on the PI controller is used to control the MRWT and extract maximum power from the WE.This strategy was relied upon to simplify control and not complicate it.
The MPPT-PI strategy used in this work is the same strategy detailed in the work 7 , where the same characteristics of the MRWT were used.Using the MPPT-PI strategy to determine the reference value for Ps makes the current, torque, and Ps related to the nature of the WS.

DFIG model
The power in this proposed system is produced using DFIG.Due to the many advantages compared to other generators, it is necessary to complete a MM to be used in simulation.The Park transform is used for this purpose to extract the equations needed for DFIG 45,46 .Equation (1) represents both the flux and voltage of the rotor of the DFIG 50 .
where, Ѱ dr and Ѱ qr are the rotor fluxes, I dr and I qr are the rotor currents, M is the mutual inductance, V dr and V qr are the rotor voltages, L r is the inductance of the rotor, and R r is the rotor resistance.
In Eq. ( 2), the relationship between flux, voltage and DFIG stator current is shown.The fixed part is the one that is connected directly to the network without an intermediary 19 .
where, Ѱ ds and Ѱ qs are the stator fluxes, ω s is the electrical pulsation of the stator, L s is the inductance of the stator, V ds and V qs are the stator voltages, and R s is the stator resistance.The relationship between the speed and torque of the DFIG is shown in Eq. ( 3), as this equation shows the development of speed as a function of torque.This development turns the machine into a generator or engine state.In addition, the equation explains the torque expression used in this work.
where, T e is the torque, J is the inertia, p is the number of pole pairs, Ω is the mechanical rotor speed, f is the viscous friction coefficient, and T r is the load torque.
The DFIG energy is represented in Eq. ( 4), where power is closely related to voltage and current, and the quality of the energy is related to the current quality.
By making the flux a constant value and directing it along the d-axis and neglecting the resistance (R s ), the flux can be written according to Eq. ( 5).This equation is used to simplify the MM of the DFIG and control itself.
The Eq. ( 4) can be written in the following form: The torque expression in Eq. (3) becomes the form represented by Eq. (9).
The rotor currents of the DFIG can be written according to Eq. (10).
Hence, Vqr and Vdr can be written as: where, g = ω r ω s .

MRWT model
Turbines are in continuous and permanent development because of their great importance in the Resfield, as they have received great interest from researchers and manufacturers.Recently, it has been noted that the number of WFs on land and sea has increased to generate EE, as their use contributes significantly to protecting the environment and reducing toxic gas emissions.Additionally, the use of WTs plays a substantial role in diminishing the expenses associated with generating RE, which is a favourable outcome.MRWT is a new WT that has emerged as a suitable solution to leverage the power gained from WE and overcome the problems of traditional WTs.This technology is detailed in 12,45,51 , where several WTs of different sizes can be used to form a single turbine.
To control this WT, the MPPT strategy is used for this purpose, as the MPPT of MRWT is considered complex and difficult to implement compared to the MPPT strategy of traditional WTs.This technology is in continuous development despite its novelty and has shown outstanding performance in increasing the power gained from the WE.The energy gained from WE can be expressed by Eq. (12), where this power is the sum of the energy of each WT 42 .
(3) www.nature.com/scientificreports/where, the T MRWT is the torque of the mother WT and T n is the torque for the WTmother with n = 1 and 2.
The torque for each WT is represented in Eq. ( 13), where this torque is related to the dimensions of each WT and the WS 43 .
where, λ 1 and λ 2 are the tip speed ratio of the both rotors, ρ is the air density, w 2 and w 1 are the mechanical speed of both rotors, and R 2 and R 1 are the blade radius of both rotors.
Equation ( 14) expresses the power gained from the WE as it relates to the WT power.This resulting energy is related to a parameter called the coefficient of power (Cp), which is expressed by Eq. (15).
where, the P MRWT is the energy of the mother WT and P n is the power for the WT mother with n = 1 and 2.
where, β is the pitch angle.
Equation ( 16) represents the tip speed ratios of the both WTs 47 .
where,V 1 and V 2 are the tip speed ratios of the rotor, where But the WS of the second WT differs from V, so depending on the work 29 it can be calculated according to Eq. ( 17).This speed is related to the distance between the two WTs (x), which is estimated at 15 m, and to the WS of the first WT (V 1 ).
The WS of the Moroccan city of Dakhla is used for a comparative study between the two WTs, where the shape of the change in this speed is represented in Fig. 2a.The rotation speed of the DFIG used with the two WTs is represented in Fig. 2b, where the rotation speed of the DFIG changes according to the change in WS.This speed increases and decreases with the increase and decrease in WS.Also, the rotational speed of the DFIG in the case of MRWT is greater than the rotational speed of the DFIG in the case of a conventional WT.From Fig. 2b, the smallest value of the rotational speed of a DFIG in the case of a conventional WT was 8.60 s at the moment.Its value reached 931.33 rot/min, and in the case of the MRWT, the lowest speed of the DFIG was 997.80 rot/min at the moment of 8.60 s.Therefore, the MRWT provided a greater value for the lowest DFIG speed than the traditional SRWT.The largest DFIG speed value at the moment was 4.71 s for the two types, where the speed value reached 1732 rot/min and 1855.13 rot/min for both SRWT and MRWT, respectively.Therefore, MRWT provided the largest value compared to SRWT, with an estimated ratio of 6.63%.This percentage proves the superiority of the MRWT and its ability to improve the characteristics of the WE system.In Table 2, the speed values at different moments are given when using both SRWT and MRWT, where improvement percentages are calculated from the speed value during each time point.From Table 2, it is noted that the MRWT provided a rotation speed greater than the rotation speed provided by the ordinary WT during the various given moments.Accordingly, from the table, it is noted that the MRWT provided rotation speed improvement ratios ranging from 6.58% to 6.65% compared to the conventional WT.So, an MRWT can improve the rotational speed of the WT by an average of 6.63% compared to a regular WT, providing the distinctive performance of this WT and its effectiveness.
The energy gained from the WE for the two WTs is represented in Fig. 2c, where the value of this gained energy changes according to the change in WS.Also, the energy gained from WE is greater in the case of using an MRWT compared to a regular WT.From Fig. 2c, the largest value of energy gained from the WE for a conventional WT is estimated at about 0.72 MW in 4.71 s, and for an MRWT, the largest value of energy was 1500 kW at the same moment in time.The lowest value of the energy gained from the WE was at the moment of 8.60 s for the two WTs, where it was estimated at 0.126 MW and 0.831 MW for both SRWT and MRWT, respectively.The MRWT provided the greatest energy gain from the WE during different periods, where these periods are 1.4 s to 1.48 s, 4.64 s to 4.79 s, and 6.24 s to 6.36 s, which indicates that this WT has the ability to generate ( 12) Vol.:(0123456789)

A. DPC-PI technique
Traditionally, the DPC-PI strategy is considered one of the most famous solutions proposed to overcome the problems of the DPC, as it relies on using of a PI controller to regulate the distinct amounts.In the DPC-PI strategy, the SVM or PWM technique is used to convert the voltage reference values generated by the PI controller into the pulses needed to operate the DFIG inverter.The DPC-PI strategy is characterized by simplicity, ease of implementation, inexpensiveness, and rapid dynamic response.Also, this strategy uses a PI controller, which makes it one of the most prominent strategies that contains a small number of gains and is, therefore, easy to adjust and change the dynamic response to energies.A Fig. 3 represents the principle of the DPC-PI strategy of DFIG, where the PWM strategy was used to generate the pulses necessary to operate the DFIG inverter.In this strategy, the power error is calculated to determine the reference values for the voltage, as these reference values are the outputs of the PI controllers.Therefore, to determine the power error, it is necessary first to estimate the power.
Equation (18) represents the errors in the energies used in the DPC-PI technique.To calculate these errors, the energies must first be estimated, and the voltage and current are measured for this purpose.
The reference value of Ps is calculated according to the MPPT, where the value of Ps becomes largely related to the changes in the WS profile.
In the DPC-PI technique, the equations for estimating energies are the same as those used in the DPC.Estimating energies is linked to first estimating each of the rotor and stator fluxes.Equation ( 19) can be used to calculate the stator flux in the stator of the DFIG 21,47 20) can be used to estimate the flux in the moving part of the DFIG, as measuring voltage and current is necessary for this purpose.
Using the previous equations, the energies are estimated according to Eq. ( 22) 20,50 .
The DPC-PI technique's reliance on a PI-type controller to control power makes this strategy less robust, which is a negative.According to the work done in 59 , the DPC-PI technique is affected by the change in DFIG parameters, as an increase in the value of the power ripples and the value of the current THD is observed, which is an undesirable matter that makes it necessary to search for the best continuous control.Therefore, to overcome the problems and drawbacks of the DPC-PI technique and the DPC strategy of DFIG, the solution lies in the strategy proposed in the next subsection.

B. CFPC-PWM technique
In this section, a new technique is designed based on using the FL technique, where both the four FL controllers and the PWM strategy are used for this purpose.So the CFPC-PWM technique is the new control that was relied upon in this paper to overcome the problems of both DPC and DPC-PI, as it is considered a new strategy that relies on the use of the FL strategy because of its robustness, not being affected by the internal and external factors of the system, and does not use the MM of the system.This strategy is an innovative development of the DPC-PI strategy, where energy estimation is relied upon and the same equations found in both DPC and DPC-PI are used.Accordingly, Fig. 4 gives a clear picture of the principle of the designed technique for controlling the DFIG inverter, as this strategy is applied to the DFIG inverter only without the network inverter to simplify the system and reduce its total costs.Also, to demonstrate the ability of the proposed CFPC-PWM strategy to improve the quality of power and current without resorting to controlling the network inverter.
In this proposed CFPC-PWM strategy, the reference values of the rotor current (I dr * and I qr * ) are first calculated based on the error powers (e Ps and e Qs ).To calculate the reference values of the currents, Fuzzy 1 and Fuzzy 2 are used for this purpose, where Eq. ( 23) is used to calculate these reference values.
where, e Ps and e Qs are the surface of the DFIG power.
The MPPT strategy is used to determine the reference value for Ps.Using this strategy makes the reference value for Ps change according to the change in WS, and the same goes for the measured value of Ps.Also, using the MPPT strategy makes the torque and current change according to the change in the shape of the WS, which is a positive thing that allows obtaining maximum values.www.nature.com/scientificreports/ In the CFPC-PWM technique, Fuzzy 3 and Fuzzy 4 are used to calculate the voltage reference values(V dr * and V qr * ) according to Eq. (24).It is noted that the calculation of these reference values is not linked to the system parameters, which makes the CFPC-PWM technique more robust.
where, e Idr and e Iqr are the surface or error of the DFIG rotor current.
In the proposed CFPC-PWM strategy, the reference values of the voltage are calculated based on the line values of the currents, which makes this strategy completely different from other strategies, especially the DPC-PI of the DFIG strategy.Therefore, this designed CFPC-PWM strategy is considered a new strategy and has not been discussed before, as it does not use the parameters of the system under study, which makes it give excellent results in the event of a fault in the system.From a first look at this proposed CFPC-PWM strategy, it can be said that the number of gains is negative for this strategy, as there is a significant number of gains as a result of using four FL-type controllers, and in each controller there are 3 gains, making their total 12 gains.
The internal structure of the FL controller is represented in Fig. 5, where there are three gains (K 1 , K 2 , and K 3 ) used to adjust and change the proposed controller response.In Fig. 5 the surface, mesh and quiver of the used FL controller are also shown.The FL controller used has two inputs and one output.FL controller was chosen as a suitable solution because of its robustness, ease of application, outstanding performance, and unaffected by parameter changes.The method of experimentation and simulation was used to determine the gain values of the FL controllers and the gains that provided the best results in terms of power quality and current were taken.
All the FL controllers used have the same structure represented in Fig. 5a, using the same number of rules.49rules were used to implement the FL controls used in the designed technique, as Table 4 shows these rules.This number of rules was chosen to obtain a fast dynamic response and obtain the greatest efficiency and performance to reduce power ripples and the value of current THD. Figure 6 represents the seven FL controller membership functions (MFs) used for input variables (error and change in error) 55,56 .
In Table 5, the characteristics of the FL strategy used to achieve the proposed control are summarized.These characteristics can be changed depending on the system used for the study.

Results and discussions
In this section, the CFPC-PWM technique will be implemented using MATLAB software (https:// www.mathw orks.com/ produ cts/ new_ produ cts/ relea se202 2a.html), with results compared to the DPC-PI technique in terms of THD of current, response time, energy undulations minimization ratio, overshoot, and SSE.Therefore, three tests are proposed to study the comparison between the CFPC-PWM technique and DPC-PI.Also, two different WS profiles are used to study the efficiency, effectiveness, and performance of the proposed CFPC-PWM strategy, where the simulation time for the first and second tests was 0.63 s and for the third test was 2.2 s.To accomplish this work, ode4 (Runge-kutta) solver was used.Also, the type solver option is Fixed-step.For both controls, Fixed-step size: 1e-5 was used.Also, unconstrained and auto were used for Periodic sample time constraints and tasking mode for periodic sample times, respectively.This section ends with a comparative study between the completed work and some existing work.The values of the DFIG parameters used in the simulation are given in Table 6 1, 26 .

First test case
The CFPC-PWM technique is tested in the case of a variable WS, where the WS is used according to Fig. 7, and the characteristics of both techniques are studied in terms of tracking references.The necessary graphical and numerical results are extracted for this purpose.The results obtained are represented in Fig. 8, where it is noted that the energies follow the references well (Fig. 8a,b).Also, the Ps changes according to the change in WS, but the Qs is not affected by the change in WS and remain constant throughout the simulation period.In addition, ripples are observed in the case of both techniques.
A Fig. 8c,d represent torque and current, respectively.Through these two forms, torque and current change according to the change in WS, as their value increases with increasing WS and decreases with decreasing WS.The current takes a sinusoidal shape for both techniques.The THD value of the current for both techniques is represented in two Fig. 8e,f, where the THD for the CFPC-PWM technique was 0.21% and for the DPC-PI was 0.28%.Accordingly, the CFPC-PWM technique significantly minimized the THD by 25%, resulting in improved current quality compared to the DPC-PI technique.Also, it is noted that the two controls have almost the same amplitude value of the fundamental (50 Hz) of current signal with an advantage for the proposed CFPC-PWM strategy, as the amplitude value was 389.5 A and 389.8A for both the DPC-PI technique and CFPC-PWM strategy, respectively.
Figure 9 represents a zoom of the results of the first test, where it is noted that the power, current, and torque ripples are larger in the case of the DPC-PI compared to the CFPC-PWM technique.The power ripples, response time, SSE, and DFIG overshoot are listed in Table 7.The CFPC-PWM strategy reduced the SSE value of DFIG power compared to the DPC-PI strategy by ratios estimated at 85.71% and 86.60% for both Ps and Qs, respectively.Also, the power ripples were improved compared to the DPC-PI strategy by 37.50% and 39.02% for both Ps and Qs, respectively.In the case of overshoot, the proposed CFPC-PWM strategy provided satisfactory results for Ps(65.56%) and unsatisfactory results for Qs (−72.96%)compared to the DPC-PI strategy.The CFPC-PWM strategy provided a better response time for Qs than the DPC-PI strategy, as this time was reduced compared to the DPC-PI strategy by an estimated ratio of 83.76%.However, in the case of Ps, the CFPC-PWM strategy provided an unsatisfactory time compared to the DPC-PI strategy.The latter reduced the Ps response time by an

Second test case
In this test, the same WS change form used in the first test is used.The robustness of the CFPC-PWM technique is studied in this test, where the DFIG parameters are changed according to Table 8.This test aims to determine the robustness of the proposed CFPC-PWM strategy compared to the DPC-PI strategy.The graphical results   Table 5. Parameters of the FL controller.

Or technique Max Defuzzification Centroid
Aggregation Max are shown in Fig. 10.The powers continue to follow the references despite the change in the DFIG parameters for the two techniques (Fig. 10a,b), with an increase in ripples and the current THD being observed.Torque and current have the form of Psand their value is related to the change in WS despite the change in the DFIG parameters (Fig. 10c,d).Also, the current remains sinusoidal for the two strategies.In Fig. 10e,f, it is noted that the THD value was 0.53% and 0.37% for the DPC-PI and CFPC-PWM techniques, respectively.So, the CFPC-PWM technique minimized THD by an estimated 30.18% compared to the DPC-PI technique.On the other hand, it is noted that the proposed CFPC-PWM strategy gave a larger amplitude for the fundamental signal (50 Hz) of current compared to the DPC-PI strategy, as the amplitude value was 404 A and 405 A for both the DPC-PI and proposed CFPC-PWM strategy, respectively.So, according to these values, the proposed CFPC-PWM strategy can improve the current quality despite changing the DFIG-MRWT parameters, which is a positive thing.The energy ripples, torque, and current of DFIG-MRWT are represented in Fig. 11.It is noted that these ripples are large in the case of the DPC-PI technique compared to the CFPC-PWM technique.This showed that the CFPC-PWM technique has better performance in enhancing the features of the energy system.The numerical values of the energy ripples are listed in Table 9, where the necessary reduction ratios were calculated to show how much the proposed CFPC-PWM strategy can reduce these ripples compared to the DPC-PI strategy.Also, the values and ratios of reduction for response time, overshoot, and SSE of DFIG power are given in this table.When examining the table, it was noted that the CFPC-PWM technique reduced the size of the power ripples by ratios estimated at 32.20% and 41.66% for both Ps and Qs, respectively, compared to the DPC-PI strategy.Also, the SSE of DFIG power was reduced by ratios estimated at 83.67% and 57.33% for both Ps and Qs, respectively, compared to the DPC-PI strategy.The latter gave better results than the CFPC-PWM technique in terms of Ps response time, as this time was reduced by an estimated 40.42% compared to the proposed CFPC-PWM strategy.However, the proposed CFPC-PWM strategy gave a better Qs time than the DPC-PI strategy, as this reduction   was estimated at a ratio of 65.02% compared to the DPC-PI strategy.In terms of overshoot of DFIG power, the proposed CFPC-PWM strategy gave a better value for overshoot of Ps compared to the DPC-PI technique, as the reduction ratio was estimated at 41.39%, which is a good ratio.But the proposed CFPC-PWM strategy gave an unsatisfactory value for the overshoot of Qs compared to the DPC-PI technique, as the DPC-PI strategy reduced this value by an estimated ratio of 73.39% compared to the proposed CFPC-PWM strategy, which is a negative matter that can be attributed to the gains.This negativity can be overcome in the future by using smart strategies in determining gain values.In Table 10, a comparative study is completed between the two controls in terms of the degree to which the values of THD of current and amplitude of fundamental (50 Hz) are affected.From Table 10, it is noted that the value of THD has changed in the second test compared to the first test, as its value is noted to have increased in the case of the two controls.In the CFPC-PWM strategy, it increased by 0.16, and in DPC-PI technique, it increased by 0.25.Therefore, the CFPC-PWM strategy is better than DPC-PI technique in terms of changing the THD value, as the ratio change in the THD value was 47.16% and 43.24% for both the DPC-PI strategy and CFPC-PWM technique, respectively.Therefore, CFPC-PWM technique has higher durability in terms of improving current quality.On the other hand, the amplitude of the fundamental (50 HZ) signal changed its value in the second test for the two controls (Table 10), where an increase in the amplitude value was observed.This increase was estimated at 3.75% and 3.58% for both the proposed CFPC-PWM and DPC-PI strategies, respectively.Therefore, the proposed CFPC-PWM strategy provided a greater ratio, which indicates that the amplitude changed more if the proposed CFPC-PWM strategy was used compared to the DPC-PI strategy.www.nature.com/scientificreports/ In Table 11, the ratios of change in the values of ripples, response time,overshoot, and SSE of DFIG energy in the two tests relative to the two controls together are given.These ratios are calculated according to Eqs. (25)  to (28).This table gives a clear picture of the change in the values of ripple, response time, SSE, and overshoot for the two techniques used, where the ratios of change in these values between the first and second tests are calculated.These ratios demonstrate the extent to which the CFPC-PWM strategy is affected compared to the DPC-PI strategy in terms of changing system parameters.The CFPC-PWM strategy provided greater change in SSE of DFIG power compared to DPC-PI.Also, the proposed CFPC-PWM strategy provided a lower ratio of change in the Qs ripples than the DPC-PI strategy, as this ratio was estimated at 56.26% and 54.28% for both the DPC-PI and proposed CFPC-PWM strategies, respectively.However, the proposed CFPC-PWM strategy provided a greater ratio of change in the Ps ripples than the DPC-PI strategy, as this ratio was estimated at 68% and 70.50% for both the DPC-PI and proposed CFPC-PWM strategies, respectively.In the case of overshoot of DFIG power, it is noted that in the two controls, the value of overshoot of Qs decreased in the second test compared to the first test, as this decrease was estimated at a rate of 40.71% and 38.35% for both the DPC-PI and the proposed CFPC-PWM strategy, respectively.Therefore, the proposed CFPC-PWM strategy provided the lowest ratio.But in terms of the overshoot value of Ps, it is noted that this value increased in the second test compared to the first test for the two controls, where this increase was estimated at ratios of 11.72% and 48.12% for both the DPC-PI and proposed CFPC-PWM strategy, respectively.Therefore, the proposed CFPC-PWM strategy provided the largest ratio compared to the DPC-PI strategy.The response time values for the power in the case of DPC-PI control were noted to have decreased in the second test compared to the first test due to the change in the DFIG parameters.This decrease was estimated at ratios of 52.78% and 48.62% for both Ps and Qs, respectively.In the case of the proposed CFPC-PWM strategy, it is noted that the response time of the Psdecreased in value in the second test compared to the first test, as this decrease was estimated at 47.19%.Therefore, the proposed CFPC-PWM strategy provided a lower declineratio than the DPC-PI strategy.However, the response time of the Qs increased slightly in the second test compared to the first test, where this increase was estimated at 1.68%.According to these ratios presented, it can be said that the proposed CFPC-PWM strategy provided better impact rates and unsatisfactory impact rates compared to the DPC-PI strategy, which is normal, as each control strategy has negatives and positives.
where, X is a quantity that can be Ps or Qs.

Third test case
This test differs from the two tests above in terms of the form of WS change, as a form of WS change different from the form of WS change used in the first test is used.In Fig. 12, the WS profile used in this test is listed, where the WS variation profile is in steps.In Figs. 13 and 14 the graphical results are listed and the numerical results are listed in Table 12.According to Fig. 13, the capacities continue to follow the references well and are the same as the results of the previous tests, with the presence of ripples.The Ps of the two controllers changes according to the change in WS, and the Qs does not change according to the change in WS, as its value remains constant and equal to 0 VAR.The value of torque and current changes according to the change in the shape of the WS, as it decreases and increases as the WS decreases and increases with the presence of ripples in both the two controls.Also, the current has a sinusoidal shape in the case of the two controls, which is the same as the results of the previous two tests.(25)  A(%) = X s ripple (test2) − X s ripple (test1) X s responsetime (test2) www.nature.com/scientificreports/ The current THD for the two controls is listed in Fig. 13e,f.This value was 0.57% and 1.08% for both the proposed CFPC-PWM and DPC-PI strategies, respectively.So, the proposed CFPC-PWM strategy reduced the THD value compared to the DPC-PI strategy by an estimated rate of 47.22%, which is a high ratio and indicates that the quality of the current is better in this test if the proposed CFPC-PWM strategy is used compared to the DPC-PI strategy.Also, it is noted that the amplitude value of the fundamental signal (50 Hz) was 2424 A and 2463 A for both the proposed CFPC-PWM and DPC-PI strategies, respectively.So, the proposed CFPC-PWM strategy presented a lower duration in this test than the DPC-PI strategy, which is undesirable.
In Fig. 14, the ripples of power, torque, and current for the two controls are shown.Therefore, the ripples are low when using the proposed CFPC-PWM strategy compared to the DPC-PI strategy, as the values of these ripples are listed in Table 12.This table gives the reductions for response time, ripple, SSE, and DFIG overshoot.From this table, it is noted that the CFPC-PWM technique provided good results in terms of ripples and SSE of powers and this is shown by the calculated ratios.The CFPC-PWM strategy reduced power ripples compared to the DPC-PI strategy by ratios estimated at 31.51% and 38.46% for both Qs and Ps, respectively.Also, the SSE value of the compared powers was reduced compared to the DPC-PI strategy by ratios estimated at 72.26% and 55.55% for both Qs and Ps, respectively.On the other hand, the CFPC-PWM strategy reduced the response time of Qs and overshoot of active power compared to the DPC-PI strategy by ratios estimated at 91.42% and 2.29%, respectively.These ratios show the superiority of the proposed CFPC-PWM strategy in improving the characteristics of the studied energy system.However, this strategy provided unsatisfactory results in terms of response time to Ps and overshoot of Ps, which is a negative matter that can be overcome in the future by using smart strategies such as NNs.
In Table 13, the change in the values of both current THD and amplitude of fundamental (50 Hz) is studied, as it is noted that these two values changed in the third test compared to the first test.So, these two values are affected by the change in the shape of the WS, as it is noted that these two values increased significantly in the third test.In the case of the THD value, this increase was estimated at 70.07% and 63.15% for both the DPC-PI and the proposed CFPC-PWM strategy, respectively.Accordingly, the proposed CFPC-PWM strategy presented a lower ratio, which indicates that it is less affected than the DPC-PI strategy, which is a positive thing that indicates its superiority.In terms of the value of the amplitude, both strategies presented a significant increase in the value of this amplitude in the third test compared to the first test.These increases were estimated at 84.18% and 83.91% for both the DPC-PI and the proposed CFPC-PWM strategy, respectively.So the CFPC-PWM provided a lower ratio than the DPC-PI strategy, which indicates that this strategy is more robust and efficient than the DPC-PI strategy and, therefore, can be relied upon in the future in the field of control.
Finally, this proposed work is concluded with a comparative study of existing works related to DFIG.This analysis is highly significant as it provides a different perspective on the CFPC-PWM technique and its effectiveness in enhancing the features of DFIG in comparison to other studies.A comparison is made between other works in terms of ripple reduction ratios, overshoot, and SSE of DFIG power.Also, by comparing the response time of the powers, SSE, and current THD.The comparison results are listed in Tables 14, 15, 16 and 17.Through these completed tables, it is noted that the CFPC-PWM technique provided better reduction rates than several scientific works, which proves its distinguished performance.These tables give a clear picture of the superiority  of the CFPC-PWM technique and its great ability to improve energy quality compared to other strategies.So, this strategy can be relied upon in the future in the field of command.

Conclusions
A new control based on CFPC strategy and PWM technique was introduced to control the DFIG power and improve the characteristics of the energy system based on a MRWT.The proposed CFPC-PWM technique was compared with the DPC-PI technique and other existing techniques, where MATLAB software (https:// www.mathw orks.com/ produ cts/ new_ produ cts/ relea se202 2a.html) was used to implement this suggested technique using different WS profiles.The behavior of the CFPC-PWM technique was studied in terms of reference tracking, current THD, robustness, ripple reduction rates, response time, overshoot, and SSE of DFIG power.The results obtained from this work can be summarized in the following points:  www.nature.com/scientificreports/ • Reducing the THD value of the current compared to the DPC-PI technique, as the reduction ratio in thesug- gested tests was estimated at 25%, 30.18%, and 47.22%; • The proposed CFPC-PWM strategy is considered a new, more efficient and reliable strategy in the field of power control than the DPC-PI strategy; • Significantly increasing the robustness of the DFIG-MRWT system.
• Reducing the value of Ps ripples compared to DPC-PI, as the ratio of reduction in the proposed tests was estimated at 37.50%, 32.20%, and 38.46%; • Minimizing the both SSE and overshoot valuesofDFIG energy.
In the future, other new strategies based on the combination of different controls will be implemented to control MRWT systems.Also, in addition to this work, the strategy of the cascaded NN technique will be experimentally implemented to compare the simulation results with those obtained experimentally, including in other works.

Figure 4 .
Figure 4. Block diagram of the internal structure of the CFPC strategy.
https://doi.org/10.1038/s41598-024-67194-4www.nature.com/scientificreports/estimated 38.76% compared to the proposed CFPC-PWM strategy.This negativity of the proposed CFPC-PWM strategy can be attributed to the values of control gains, where the DPC-PI method was used to determine them.

Figure 10 .
Figure 10.Results in the second test.

Table 1 .
Some proposed solutions to overcome the problems of the DPC strategy of DFIG.

Table 2 .
Speed value of both WTs.

Table 3 .
. Mechanical power of both WTs.

Table 7 .
Zoom in the results of the first test.Numerical results in the first test.

Table 8 .
New values for the DFIG parameters.

Table 9 .
Zoom in the results of the second test.Numerical results of the second test case.

Table 10 .
Study of the change in the THD and amplitude value of the fundamental (50 Hz) for the both techniques.

Table 12 .
Zoom in the results of the third test.Numerical results in the third test.

Table 13 .
Study of the change in the amplitude value of the fundamental (50 Hz) and THD for both techniques (first and third tests).

Table 14 .
Comparison in terms of response time for DFIG power.

Table 15 .
Comparison in terms of power ripple minimization rates.

Table 16 .
Comparison in terms of current THD.

Table 17 .
Comparison in terms of SSE for DFIG energy.