Adaptive control strategy for microgrid inverters based on Narendra model | Scientific Reports

Scientific Reports volume 14, Article number: 21389 (2024) Cite this article

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In recent years, microgrid technology has been widely studied and applied. However, with times developing, the installed capacity of distributed power generation devices has been improved, and work is being carried out in increasingly complex situations, resulting in a decline in the control performance of microgrids. In view of this, to effectively improve inverter’s control performance, research is conducted on the fusion of Narendra model and adaptive control strategies for real-time voltage correction and compensation in complex situations. Compared to traditional inverters, inverters under research methods have faster voltage recovery speed when encountering load switching, and can recover in about one cycle, with good control performance. In the comparison between the improved inverter adaptive control system and the inverter adaptive system, the improved inverter voltage recovery speed is faster, can be restored within one cycle, and the control effect of the inverter is better. The harmonic rate of the port voltage has decreased from 10.43 to 1.92%. The applicability of the research method was verified. It indicats that the research method can improve inverter’s control effect and solve problems such as voltage deviation, three-phase asymmetry, harmonic pollution, etc. that are easily generated by the output terminal voltage. Simultaneously, research has provided theoretical basis and data support for the research of microgrids.

Microgrid refers to a small power grid composed of small distributed power sources that can operate independently. It can be operated separately or connected to an external power grid. Microgrids can achieve local power supply, reduce dependence on external power grids, and improve power supply reliability and flexibility1. As economy takes off at a high speed and continues to develop, China’s demand for fossil fuels such as coal and oil is increasing. “2030 carbon peak” and “2060 carbon neutrality” have a long way to go, and there is an urgent need to find and develop new clean and renewable energy sources. Proton exchange membrane fuel cell (PEMFC) cogeneration system for residential use uses fuel cell power generation technology, using pure hydrogen/natural gas to produce direct current, which is converted into household alternating current through an inverter. And the waste heat generated during power generation process is further recycled, stored and used in various heat consumption scenarios such as home bathing and floor heating, with an overall energy efficiency of over 80% and significant energy-saving and emission reduction effects2. Therefore, ensuring the power quality of fuel cell microgrids is of great significance, and power electronic converters are usually used to achieve power control of microgrids. The inverter is a key link in the power electronic converter, which affects the power quality of entire microgrid3. However, conventional inverter control methods can easily lead to poor control performance in complex engineering conditions, which can have adverse effects on the power quality of microgrids. The inverter commonly used at present, due to factors such as three-phase voltage imbalance and grid side frequency distortion, is very easy to cause frequency doubling interference in the system. This in turn affects the phase-locked accuracy and direct current side control quality of the inverter, thereby affecting the overall control performance (CP) of overall inverter. Adaptive control strategy (ACS) is a control method that dynamically adjusts control parameters to adapt to changes in the system by monitoring variables in real time, thereby keeping the system in optimal state4. ACS has applications in many fields, such as industrial production, healthcare, aerospace, etc. It has advantages such as strong adaptability, high accuracy, and fast running speed5. In view of this, it is necessary to further improve CP of inverters in complex environments, thereby improving the power quality of microgrids. In response, this project proposes a new adaptive control method suitable for microgrid inverters under specific conditions. This method can fully utilize the flexibility of power electronic converters and effectively reduce the design cost of microgrid systems. Therefore, this study has significant academic value and engineering application prospects.

The main contributions are:

Presenting the adaptive control method suitable for microgrid inverters under specific conditions based on improving the power quality of microgrids,

Adaptive control method uses the flexibility of power converters and effectively reduce the design cost of microgrid.

This research content mainly includes four parts. Firstly, a brief introduction is given to the research on microgrid inverters and ACS. The second part first introduces the adaptive control system (ACSY) for microgrid inverters that integrate Narendra model, and then makes improvements based on this. Next, the performance of Narendra based microgrid inverter ACS was verified, and performance testing and comparative analysis experiments were conducted. Finally, a summary and outlook of the research content were provided.

When studying microgrid inverters, Mongrain R S and Ayyanar R used real-time simulation to model microgrid and grid connected inverters in their research on continuous penetration of distributed energy. Research has confirmed the stability of microgrids, with a 100% penetration rate for photovoltaic power generation when operating on isolated islands6. Salim O M et al. solved the problem of improving power quality by using two cascaded voltage regulation schemes. The research results were compared with other simulation experiments. It has better performance in terms of output voltage, frequency, response time, and THD7. Johansen K uses a secondary network approach to protect the system in his research on power services. Unlike the protection of microelectronic networks, the protection of microelectronic networks is usually distributed throughout the entire system, with low fault currents and non unidirectional current flow. Therefore, if a microgrid is deployed on a secondary network, significant system protection challenges need to be overcome8. Vemula NK and Parida SK have studied the control of microgrids in China. In order to improve the stability of the system and the small signal stability and transient response of inverters under different operating conditions, a droop control scheme based on optimal internal model control is proposed. Compared with existing methods, the research method has the best effect on improving system stability9. In the study of self-regulation in microgrids, Dheer D K and Gupta Y have enhanced the self-regulation droop control method in order to improve the voltage distribution in microgrids. The proposed topology structure methods for microgrids are feasible10. Zhang F and Kang J analyzed the characteristics of parallel hybrid inverters using droop control in microgrids. An adaptive power allocation method is proposed to identify the imbalance between each inverter on the DC bus. This research method adjusts the droop parameters and the output power of each inverter. After numerical simulation, experimental tests were conducted. The proposed method improves system stability while ensuring its power capability11.

And the development of ACS is also relatively mature. Yang N et al. proposed nonlinear ACS for synchronous generators in single machine infinite bus systems. Compared to traditional methods, the research method has a higher CP and stronger robustness12. Amini F and Shahri N N N considered the variable characteristics of the structure and the soil structure interaction in the study of liquid column dampers. The research method calculates the gas pressure inside the column by regularly adjusting and updating the frequency and damping. Intelligent dampers can more effectively reduce structural displacement13. Agha Kashkooli M R and Jovanovic M G proposed a new model reference adaptive system for speed estimation and sensorless operation of brushless doubly fed reluctance generators. By using the voltage and current measurements under the adaptive model, the corresponding static frame current components are estimated to provide higher accuracy. The controller has good active and reactive power performance14. In the study of unmanned aerial vehicles, Derrouaoui S H et al. proposed a simple mechanism with a lightweight mechanical structure to simplify and reduce the prototype. Considering the aerodynamic effect, a new general model was developed in the experiment. The proposed method has improved the performance and efficiency of the controller15.

In summary, the research on microgrid inverters has become relatively mature, and the stability of their control systems is a hot topic of research. ACS has also achieved good results in various fields. In view of this, research will introduce ACS based on the integration of Narendra, hoping to improve microgrid inverters’ control stability.

Microgrid16,17,18,19,20 inverter ACSY is an intelligent control system that can automatically adjust control strategies based on changes in network parameters. The system can automatically adjust control strategies based on network conditions and load changes by monitoring and analyzing network parameters in real-time, thereby improving the stability and efficiency of the system. The study first introduced ACSY, and then based on adaptive strategies, derived self adaptive control laws and designed adaptive controllers. Finally, the idea of auto disturbance rejection controller (ADRC) is introduced to improve CP of inverter. Inverter in this paper is based on insulated-gate bipolar transistor (IGBT) full bridge.

To change the isolation of inverter system under complex working conditions and improve CP of the inverter, Narendra was integrated into the study and a new adaptive strategy was proposed. Based on the adaptive strategy, a self adaptive control law was derived, and an adaptive controller was designed to achieve real-time compensation control in the event of three-phase imbalance and voltage deviation. Figure 1 is a schematic diagram of the model ACSY.

Schematic diagram of adaptive control system.

In Fig. 1, if \(r\) is input, then \(y_{m}\) is the reference output, \(y_{p}\) is the actual output, and \(e\) represents the generalized error. According to the schematic diagram, model ACSY consists of three parts: an adjustable system, a reference model, and an adaptive mechanism. Among them, feedback object controller and controlled object form an adjustable system with their feedforward21. The purpose of the adaptive controller is to use a classical stability theory to design the adaptive control rate, and adjust the actual system according to the adaptive control law, so that the final actual output is equal to the reference model output. Equation (1) represents two measurement indicators.

In Eq. (1), \(e_{x}\) represents the generalized error vector. \(e_{y}\) represents the output error vector. \(x_{m}\) represents the state of the reference model. \(x_{p}\) represents the state of the adjustable system22. When designing Narendra controllers, Eq. (2) represents the state description of the controlled system.

In Eq. (2), \(u_{adj}\) represents the control23,24,25,26,27 quantity and also represents the adjusted voltage. \(u_{oq}\) represents the output rate of the system, which is the output voltage on inverter’s \(q\) axis. \(u_{oq}\) represents a state vector. Equation (3) is its transfer function.

In Eq. (3), \(G_{p} (s) = c^{T} (sI - A_{p} )^{ - 1} b_{p} = k_{p} \frac{{n_{p} (s)}}{{d_{p} (s)}}\) represents the first polynomial of order \(m\), \(d_{p} (s)\) represents the first polynomial of order \(n\), and the state description expression in Eq. (4) is obtained based on the reference model.

In Eq. (4), \(u_{{_{refq} }}^{*}\) represents the control quantity as the input voltage. \(u_{om}\) indicates that the output has reached the output voltage of the reference model. \(v_{m}\) represents a state vector. At this point, Eq. (5) is the transfer function.

In Eq. (5), \(G_{m} (s)\) represents strictly positive-real function. Equation (6) represents the error between the actual system and the controlled system.

In Eq. (6), \(e_{u} (t)\) satisfies \(\mathop {\lim }\limits_{t \to \infty } e_{u} (t) = 0\). After this design is completed, it is necessary to understand the adaptive control law. Its goal is to adjust adjustable parameters based on actual working conditions, so that the performance of the controlled object is consistent with that of the reference model. Because it is difficult to obtain state variables in actual control28,29,30,31,32 systems, it is necessary to integrate Narendra. Narendra principle is to add auxiliary signals to the input and output of the actual system to assist in obtaining the state variables of the actual system33. Figure 2 is a schematic diagram of adaptive control for Narendra fusion.

Schematic diagram of adaptive control integrating Narendra model.

In Fig. 2, Auxiliary equation, abbreviated as AE, has two auxiliary equations. Adjustable parameters are four adjustable parameters, abbreviated as ap. The adjustable variables are named \(k_{0}\), \(d_{0}\), \(d_{1}\), \(d_{2}\), \(u_{a1}\) and \(u_{a2}\), respectively, representing the input and output of the inverter after auxiliary equation processing34. The control quantity composed of all adjustable parameters is represented by \(u_{adj}\). The processed input signal and output signal are input into the adaptive mechanism, and the adjustable parameters of the adaptive mechanism are automatically adjusted. Among them, Eq. (7) is the expression for constructing auxiliary equations.

In Eq. (7), \(A_{a}\) represents the stable matrix of \(n - 1\) order, and Eq. (8) is the transfer function of two auxiliary equations.

In Eq. (8), the parameters of Huwitz polynomial with \(d_{a}\) first can be expressed as Eq. (9).

In Eq. (9), the denominator of the transfer function of the auxiliary signal generator is the same polynomial. The order of the numerator is one order lower than the denominator, and the order is \((n - 1)\). \(d_{0}^{T}\) and \(d_{1}^{T}\) represent their adjustable gains, respectively. The auxiliary function makes the real-time state of the controlled object easier to observe and can provide feedback compensation for the controlled object. Figure 3 shows the structure of two controlled systems and two auxiliary equations.

The structure of controlled systems and auxiliary equations.

In Fig. 3, in adaptive control, if this method is adopted, the input and output signals of the controlled object are treated as auxiliary equations. In this way, real-time adjustment can be made based on actual working conditions, making the parameters of the controlled object perfectly match those of the reference model, achieving zero output error. Based on the above analysis, constructing auxiliary equations to process the input and output signals of the controlled object can also achieve perfect performance matching between the controlled object and the reference model, thereby achieving adaptive control of the actual system.

After integrating Narendra, CP of the inverter has been improved in both simple and complex situations. For grid connected inverter power supply systems with a single inverter structure, current control mode needs to be adopted for inverter control during grid connected operation to meet the requirements of the power grid35,36,37 and grid load for power quality. The voltage of the local load is approximately the grid voltage. When the power system switches to independent load operation due to grid abnormalities or other reasons, the inverter control must adopt a voltage control mode to meet the requirements of the load on the power supply voltage waveform, amplitude, and frequency. Therefore, this type of system not only needs to have two control and operation modes, but also must have a switching function between two operation modes and two control modes. Moreover, for sensitive and important loads, it is necessary to maintain continuous power supply and stable voltage during the operation mode switching process to meet the load’s requirements for power supply quality. However, there are still some shortcomings that need to be improved. The dynamic response of the control system will be affected by the initial values of adjustable parameters, which will affect the construction of the adaptive controller. During the algorithm validation process, its harmonic distortion rate is high, its stability is poor, and it is not enough to face complex working situations. ADRC is a kind of controller, which can control the system without precise mathematical model or prior knowledge. Compared with the traditional control methods, ADRC can adjust adaptively according to the changes of the system, rather than relying on the accurate mathematical model, which makes ADRC more robust. ADRC also has the advantages of high adaptive degree, high control precision and easy realization. In view of this, ADRC idea is introduced to improve the inverter’s CP. Figure 4 shows the main framework of ADRC technology.

Main framework of ADRC technology.

ADRC is a nonlinear control system used to control industrial processes. It combines nonlinear control technology with modern control theory to achieve better control effects by observing, compensating, and adjusting the controlled object. In Fig. 3, ADRC consists of three main parts: tracking differentiator, extended State observer and nonlinear state error feedback control law. A tracking differentiator is a function that decomposes errors and their rate of change into a series of signals. These signals are called differentials and are used as sources of estimation error. By processing these signals, real-time compensation and adjustment of errors can be achieved38. The extended state observer is a nonlinear control system composed of multiple filters and observers. Through these filters, the state of the controlled object can be observed in real-time and possible future changes can be estimated to eliminate or mitigate the impact of external interference and internal parameter changes on control effectiveness. The nonlinear state error feedback control law can decompose errors and error rate of change into smaller and simpler signals, and implement them in the controller. These signals can be used as compensation signals or as a way to eliminate interference, thereby improving control effectiveness39.

ADRC can treat the internal instability and external unknown interference as total interference, observe in this form, and compensate based on the degree of interference. To some extent, it improves CP of the inverter, but in practical applications, it is necessary to distinguish between internal and external disturbances. Therefore, linear ADRC is adopted in this research to better combine Active disturbance rejection control with adaptive control. Figure 5 shows its control framework.

Linear ADRCler framework.

In Fig. 5, linear ADRC is a linear control technology used to control nonlinear systems. It consists of two controllers: a linear error feedback control law and an extended State observer. Both controllers are linear and can be directly associated with the state of the controlled object. Linear Active disturbance rejection control is a simple and effective method to control nonlinear systems. It can effectively eliminate the impact of external disturbances and internal uncertainties on control effectiveness, thereby improving control accuracy and stability. Among them, Eq. (10) represents the controlled object.

In Eq. (10), \(y\) represents the output of the system, \(\omega\) represents the external unknown interference detected by the system, and \(t\) represents the time-varying state of the system. Equation (10) can be further represented as Eq. (11).

In Eq. (11), \(b_{0}\) represents the unstable part within the system. At this point, \(f\) represents the total interference situation inside and outside the system. It can be converted to a state equation in Eq. (12).

In Eq. (12), \(A = \left[ {\begin{array}{*{20}c} \begin{gathered} 0 \hfill \\ 0 \hfill \\ \end{gathered} & \begin{gathered} 1 \hfill \\ 0 \hfill \\ \end{gathered} \\ \end{array} } \right]\), \(B = \left[ \begin{gathered} B_{0} \hfill \\ 0 \hfill \\ \end{gathered} \right]\), \(E = \left[ \begin{gathered} 0 \hfill \\ 1 \hfill \\ \end{gathered} \right]\), \(C = \left[ {1,0} \right]\) and \(D = \left[ 0 \right]\). Equation (13) is a state observer designed for the controlled state object.

In Eq. (13), \(L = \left[ {\begin{array}{*{20}c} {\beta_{01} } & {\beta_{02} } \\ \end{array} } \right]^{T}\) represents the gain coefficient of error feedback, \(z_{1}\) represents observer’s state variable, and tracking output is \(y\). Equation (14) represents the control quantity.

In Eq. (14), substituting the controlled object can obtain Eq. (15).

In Eq. (15), the error feedback control law can be regarded as an integrator, and its expression is Eq. (16).

In Eq. (16), the bandwidth of the system is represented by \(\omega_{c}\). Figure 6 shows the improvement effect of the self anti-interference device in the fusion of Narendra’s ACSY.

Improved adaptive control system.

In Fig. 6, the basic idea of the extended state observer is to simultaneously estimate the internal disturbance caused by the internal uncertainty of the system and the external unknown external disturbance of the system, and track and compensate them. These two parts are realized by two controllers, namely adaptive controller and extended State observer. In theory, this improved adaptive controller can combine both controllers’ advantages.

The proposed scheme includes the mathematical model40,41,42,43,44. Mathematical formulation is as equality or inequality format45,46,47,48. Mathematical model shows the system behavior49,50,51,52. In the different applications, to apply the mathematical model on the system, it is needed to smart coordination, algorithms and devices53,54,55,56.

To reduce experimental errors, the study used the same equipment for simulation analysis, and MATLAB/Simulink was used to build a simulation model for comparative experiments. Under the condition of constant inverter parameters, three-phase asymmetric load and three-phase symmetric resistive inductive load were used to simulate different operating conditions in reality. The three-phase asymmetric load value is set to \(6\Omega /12\Omega /18\Omega + 11.5\;{\text{mH}}\), and the three-phase symmetric resistive load is set to \(20\Omega + 11.5\;{\text{mH}}\). Other data such as data of control method, filter, switching frequency is reported in22,33,34,38,39. Noted that the switching frequency used for the proposed control technique is fixed.

To verify the superiority of ACS, a traditional control strategy will be introduced, and comparative experiments will be conducted under the same operating conditions to compare the voltage CP of the two inverters. Firstly, a three-phase symmetrical load57,58,59,60,61,62 is connected to the output end of the inverter, and after running for 0.25 s, another set of loads is connected. Figure 7 shows its comparative experiment.

Comparative experiment on switching three-phase symmetrical loads.

Figure 7 shows the changes in voltage under different control strategies. In Fig. 7a, the voltage63,64,65,66,67,68,69 variation under traditional control is shown. Before switching, the voltage is a standard sine wave, and its CP is stable. After switching, the voltage fluctuates to a certain extent, showing a downward trend, indicating that its CP has been affected. In Fig. 7b, the voltage variation under adaptive control is shown. Before switching, the voltage is also a standard sine wave, and CP is stable. After switching, there is no voltage fluctuation, indicating that the port voltage CP under the adaptive strategy is good. A comparative experiment of switching three-phase asymmetric loads was continued. Firstly, a three-phase symmetric load was connected to the output end of the inverter, and after running for 0.25 s, another set of loads was connected. Figure 8 shows its comparative experiment.

Comparative experiment on switching of three phase asymmetric load.

In Fig. 8, the voltage changes under different control strategies are shown. In Fig. a, the voltage variation under traditional control is shown. Before switching, the voltage is a standard sine wave, and its CP is stable. After switching, the voltage began to fluctuate and an imbalance fault occurred, which affected CP, and at this time, the inverter CP was poor. In Fig. 8b, the voltage changes under adaptive control are shown. Before switching, the voltage is also a standard sine wave, and CP is stable. After switching, there is no voltage fluctuation, indicating that the port voltage CP under the adaptive strategy is good. In order to simulate complex working conditions, a set of three-phase symmetrical loads was switched on and off at 0.25 s.

It is noteworthy that load changes have an effect on the voltage of system, so that load increase leads to voltage drop. Therefore, in checking the capability of the control system, it is necessary to check the voltage status of the system. So that if the voltage is improved, i.e. the voltage drop is reduced, the capability of the control system is favorable. But the current curve does not show the characteristics of the control system well. Therefore, only the voltage curve was drawn in Figs. 7 and 8.

At 0.5 s, a set of three-phase asymmetric loads were switched on and off in Fig. 9.

Comparative experiment under complex working conditions.

Figure 9 shows the voltage variation under the ACS proposed in this study under complex working conditions. The red line represents the reference output value, the blue line represents the actual output value, and the orange line represents the error line. The reference output has a high degree of fit with the actual output, with only a small fluctuation occurring after 0.5 s, and quickly returning to normal after the fluctuation. By adopting research methods, inverters can also have high stability under complex operating conditions.

To verify the improved ACS superiority, the pre improved ACS will be introduced and compared with two inverters’ voltage CP under the same working conditions through comparative experiments. Firstly, a set of symmetrical resistive loads is connected to the output end of the inverter. After running for 0.25 s, a set of nonlinear loads is connected in Fig. 10.

Comparison of voltage harmonic control effects before and after improvement.

Figure 10a shows the voltage variation under adaptive control. At 0.25 s, the voltage undergoes distortion and slowly returns to normal after a cycle. Figure 10b shows the voltage variation under adaptive control. At 0.25 s, the voltage undergoes distortion and quickly returns to normal after less than one cycle. Figure 10c shows the changes in harmonics under adaptive control. At 0.25 s, the harmonics showed a significant decrease, dropping to 10.43%. Figure 10b shows the voltage variation under adaptive control, with harmonics decreasing to 1.92% at 0.25 s. The improved inverter has better control effect because it effectively solves the problem of high harmonics. Figure 11 shows the control effect of voltage.

Comparison of voltage control effects.

In Fig. 11a, the voltage fluctuated at 0.25 s before improvement and returned to normal after one cycle. In Fig. 11b, the improved voltage fluctuates at 0.25 s and returns to normal within one cycle. The improved inverter CP further enhances its anti-interference ability and stability. To simulate complex working conditions, a set of three-phase symmetrical loads was switched on and off at 0.25 s. At 0.5 s, a set of three-phase asymmetric loads was switched on and off. Figure 12 compares the control effect of the inverter before and after the improvement.

Comparison of voltage control effects before and after improvement under complex working conditions.

In Fig. 12, the voltage changes under ACS before and after this improvement are shown. The red line represents the reference output value, the blue and yellow lines represent the actual output value of the inverter before and after the improvement, and the orange and purple lines represent the error lines before and after the improvement. The reference output has a high degree of fitting with the actual output. The voltage under ACS fluctuates slightly after 0.5 s and quickly returns to normal after the fluctuation. The voltage under the improved ACS showed no significant fluctuation after 0.5 s. The improved inverter has a better CP, further verifying the effectiveness of the improved method.

To improve CP of inverters in microgrid, enhance system stability, and fully utilize the flexibility of power electronic converters, a new adaptive control method suitable for microgrid inverters is proposed. In the comparison between inverter ACSY and traditional inverter, the curve of traditional inverter fluctuates and CP significantly decreases when encountering load switching. Compared to traditional inverters, inverters under research methods have faster voltage recovery speed when encountering load switching, and can recover in about one cycle with good CP. In the comparison between the improved inverter ACSY and the inverter adaptive system, the improved inverter has a faster voltage recovery speed and can be restored within one cycle, resulting in better control performance of the inverter. The harmonic rate of the port voltage has decreased from 10.43 to 1.92%. The mentioned items are among the advantages of the proposed plan. The study validated the applicability and superiority of the research method. However, there are shortcomings in this study. In actual microgrid systems, multiple inverters are usually operated in parallel, and the method and number of parallel connections can affect system stability. Due to limitations in experimental time and conditions, no research has been conducted in this direction. Therefore, in future research, theoretical analysis and experimental verification will be conducted on the impact of inverters number and parallel connection methods on this system.

In this paper, the capability of the scheme has not addressed the failure due to occurrence of sudden faults. But, the scheme based on mentioned issue is considered as future work. Also, the economic indicators for the proposed plan have not been done, which is considered as future work. In the proposed plan, the advantages of the proposed control method over new method such as model reference adaptive control (MRAC) have not been investigated, but this issue was considered as a future work.

All data generated or analysed during this study are included in this published article.

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This paper is supported by the national key R & D program “Key technologies for integrated energy supply systems for residential Proton exchange membrane fuel cell”, project number 2021YFB4001700, submitted to your publication.

Marketing Service Center (Metrology Center), State Grid Shandong Electric Power Company, Jinan, 250000, Shandong, China

Qing Wang, Zhiru Chen, Zhen Jing & Zhi Zhang

Shandong Electric Power Company Marketing Service Center, Jinan, 250000, Shandong, China

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Wang, Q., Li, G., Chen, Z. et al. Adaptive control strategy for microgrid inverters based on Narendra model. Sci Rep 14, 21389 (2024). https://doi.org/10.1038/s41598-024-71584-z

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Received: 17 June 2024

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Published: 13 September 2024

DOI: https://doi.org/10.1038/s41598-024-71584-z

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