Home Deep neural network application in real-time economic dispatch and frequency control of microgrids
Article Open Access

Deep neural network application in real-time economic dispatch and frequency control of microgrids

  • Jun Liu EMAIL logo
Published/Copyright: March 20, 2025
Download Article (PDF)
Become an author with De Gruyter Brill
Submit Manuscript Author Information
Nonlinear Engineering
From the journal Nonlinear Engineering Volume 14 Issue 1

Abstract

In recent years, the development of microgrids has driven the reform of the electricity market, breaking the monopoly of traditional power grids and promoting the healthy development of the electricity market. However, the stability of microgrids is significantly impacted by the integration of various energy sources and numerous users. This study explores the application of an intelligent dynamic programming algorithm based on the deep neural network algorithm, combined with adaptive dynamic programming. Subsequently, an intelligent real-time power generation control algorithm (IRPGC) is obtained by introducing rejection operation improvement. Finally, a real-time integrated scheduling and control framework for microgrids is constructed. The research results showed that the IRPGC algorithm had an average error of less than 10−5 after 5,000 iterations. Compared with mainstream algorithms, this algorithm achieved favorable results in frequency deviation evaluation indicators, with a frequency deviation fluctuation range of −0.073 to 0.013 Hz, an average error integral of 51.45, an absolute error integral of 0.54, and a time-weighted absolute error integral of 1.58 × 105. In the practical application of real-time microgrid power generation scheduling and control framework, the optimal rejection threshold range was found to be [0.94, 0.97]. The aforementioned results indicate that the proposed method exhibits good control performance and application effectiveness, providing a reference for real-time power generation scheduling and control in microgrids.

Keywords: deep neural network; microgrid; economic dispatch; frequency control; rejection operation

1 Introduction

In recent years, with the rapid growth of renewable energy generation, the introduction of renewable energy, integration of emerging loads and power electronic devices, the source load fluctuations, and control complexity of microgrids have increased, posing severe challenges to the stability and safe operation of power systems [1,2,3]. In the future, developing comprehensive smart energy is one of the important directions to promote energy transformation, upgrading, and innovative development models. The development of comprehensive smart energy requires continuous innovation breakthroughs and policy support. From an economic perspective, the development of microgrids is currently facing enormous challenges. The parity of new energy on the grid does not necessarily mean parity utilization, which imposes a huge burden on the regulation costs of the power grid system. Meanwhile, only about 30% of low-carbon, zero-carbon, and negative-carbon technologies have entered commercial operation, making it very difficult or costly to regulate the power grid solely using advanced technologies. Yao and Sheng and Santra et al. found that as fossil fuels continue to pose serious threats to the global environment, new energy has attracted widespread attention from countries around the world due to its green and renewable advantages [4,5]. In the application of artificial intelligence technology in microgrids, deep neural network (DNN) algorithms have significant advantages. They can release the massive multisource data accumulated in the power system to greater value and quickly respond and adjust energy optimization management. Kumar et al. and Shakti et al. found that the DNN algorithms have strong data processing and prediction capabilities, as well as good flexibility, which are very suitable for different operating modes of microgrids. However, their computational complexity is high, and they are prone to problems such as overfitting and poor generalization ability [6,7]. To address these issues, this study combines the DNN algorithm with adaptive dynamic programming (ADP) to develop an intelligent dynamic programming algorithm (DNN-ADP). Subsequently, the reject operation is performed to optimize the DNN-ADP algorithm, resulting in the intelligent real-time power generation control algorithm (IRPGC). Finally, a real-time integrated scheduling and control framework for microgrids is established. The research aims to achieve superior control performance while addressing the divergence issues in microgrid systems and the shortcomings in economic dispatch and frequency control in microgrid applications.

The two main innovations of the study are as follows. First, the IRPGC algorithm is designed to address the real-time generation scheduling and control challenges posed by the large-scale integration of new energy and users into microgrids. Second, a real-time integrated scheduling and control framework is established for microgrids to guide for the intelligent construction of integrated energy systems. The structure of the research is divided into four main sections. Section 2 presents a review of relevant research findings. Section 3 presents the design of the DNN-ADP algorithm and the IRPGC algorithm, along with the construction of the real-time integrated scheduling and control framework for microgrids. Section 4 focuses on validating the effectiveness and feasibility of the proposed method. Section 5 summarizes the research.

2 Related works

With the continuous growth in energy demand, traditional fossil fuels are increasingly facing a crisis of depletion, leading to serious ecological and environmental pollution issues. Therefore, the development of renewable energy and sustainable practices has become a research focus. Real-time power generation control has become a key means to achieve these goals. Numerous scholars conducted in-depth analyses and discussions on this subject. Zeinal-Kheiri et al. proposed a real-time adaptive microgrid energy scheduling method based on Lyapunov optimization, combined with a stochastic day-ahead French real-time energy management system. The research results demonstrated a reduction in real-time operational costs for the microgrid, confirming the effectiveness of the method in improving performance [8]. Wang et al. designed a model predictive control algorithm to address optimal energy management issues, utilizing two different power sources to achieve fuel economy and emission reduction goals. Experimental results indicated that the algorithm could ensure optimal results, with an approximately 5.2% improvement in fuel economy compared with rule-based control strategies [9]. Due to a lack of precise assessment and configuration studies on the control capabilities of variable-speed constant-frequency pumped storage hydropower, Zhang et al. focused on a cascade hydropower-photovoltaic-variable-speed constant-frequency pumped storage hydropower system. A rule-based method for determining adjustment capacity was proposed. The results indicate that this method is effective in suppressing rapid fluctuations in photovoltaic systems in real time [10]. Fang et al. identified that frequent adjustments in real-time power output from concentrated solar power could impact the durability and profitability of factories. Therefore, an optimal bidding strategy for energy and frequency regulation was developed. It was applied to the coordinated operation of concentrated solar and wind farms in the market recently [11].

DNN has broad application prospects in the field of the power grid, which can be used to monitor the operation status of the power grid in real time, predict future electricity demand, and forecast the generation of renewable energy. By employing real-time power generation regulation, DNNs enhance the operational efficiency and reliability of the power grid. Zhao and Hanglin explored compression techniques without compromising DNN performance. A DNN optimization compression algorithm based on a hybrid mechanism was proposed. Experimental results on the mini-ImageNet dataset indicated that even with a 6.3% reduction in compression accuracy, the algorithm achieved a remarkable 98.5% reduction in the capacity of the compressed AlexNet [12]. Chen et al. introduced an effective method for detecting internal voids in wood to ensure structural safety. The DNN was used for the analysis. The experimental results demonstrated high accuracy and generality in identifying the severity of voids [13]. In response to the demand for high-precision indoor positioning in location-based services and emerging Internet-of-things applications, Chen et al. designed a scene analysis positioning solution based on a multidetector DNN architecture. The results showed that this method effectively addressed the complex linear relationship between fingerprints and spatial locations [14]. To tackle challenges in resource allocation for energy-saving in multicell carrier nonorthogonal multiple access under interference and other factors, Adam et al. proposed a real-time power allocation using a dual-channel enhanced DNN. Simulation results indicated the suitability of this method for real-time applications [15].

To sum up, it is obvious that the current research mainly focuses on the control of power systems and the application of the DNN algorithms. Traditional intelligent control strategies face challenges in addressing economic dispatch and frequency control in microgrids. To provide effective control methods for large-scale microgrid systems that handle numerous energy sources, the study designs the DNN-ADP algorithm and the IRPGC algorithm, establishing a real-time integrated scheduling and control framework for microgrids.

3 Real-time integration framework for microgrid based on DNN

In the context of dual carbon, the installed capacity and power generation of new energy in China have grown rapidly. At the same time, the high proportion of renewable energy and emerging loads connected to the power grid will cause complex control, source load fluctuations, and mechanism ambiguity, which poses a huge challenge to the stable and safe operation of the power grid. Therefore, the research first introduces the ADP algorithm to construct the DNN-ADP algorithm based on the DNN algorithm. Then, it is optimized by introducing rejection operations to construct the IRPGC algorithm. Finally, a real-time integrated scheduling and control framework for microgrids is established.

3.1 Design of DNN-ADP

Microgrid is a complex distributed system with various types of power sources and loads. The corresponding power supply has characteristics such as indirectness and uncertainty, with significant frequency fluctuations [16,17]. Traditional control strategies struggle to adapt to the rapid changes in microgrids and cannot achieve the economic and real-time operation of microgrids, resulting in poor performance in economic dispatch and frequency control in microgrids. Due to the limitation of the DNN algorithm on the real-time scheduling accuracy of microgrids, the DNN-ADP algorithm is designed based on the DNN algorithm combined with the ADP algorithm. The algorithm includes four multioutput DNNs: execution network, model prediction network, evaluation network 1, and evaluation network 2. The ADP algorithm can effectively handle optimization problems for discrete and continuous systems and stop when obtaining the optimal control law or reaching the set maximum number of iterations. The specific process of the DNN-ADP algorithm is as follows. First, the ADP algorithm iteratively approximates the true solution of dynamic programming, thereby approximating the optimal control solution of nonlinear systems in microgrids. The ADP algorithm has four basic structures: heuristic dynamic programming (HDP), performing dependency-heuristic dynamic programming (PDHDP), bi-heuristic dynamic programming (Bi-HDP), and performing dependent dual heuristic dynamic programming (PDDHDP) [18]. The schematic diagram of the four basic structures of the ADP algorithm is shown in Figure 1.

Figure 1 Schematic diagram of four basic structures of the ADP algorithm. (a) HDP structure. (b) PDHDP structure. (c) Bi-HDP structure. (d) PDDHDP structure.
Figure 1

Schematic diagram of four basic structures of the ADP algorithm. (a) HDP structure. (b) PDHDP structure. (c) Bi-HDP structure. (d) PDDHDP structure.

The basic structural diagrams of HDP, PDHDP, Bi-HDP, and PDDHDP are shown in Figure 1(a)–(d). For the k + 1 segment, x ( k ) is the starting state, which is jointly generated by the u ( k ) controlled by the previous k segment. U ( x ( k ) ) is the feedback control variable. The input variables are mapped from the state variables to the control variables through the execution network. Then, the model network constructs a model for the nonlinear system. Finally, the performance indicator function of the neural network can be output by evaluating the network. Assuming the weight of the evaluation network is ω , differential processing can be performed. The corresponding continuous partial derivatives can be obtained through ω and the output of different network structures. The next state is calculated through x ( k + 1 ) = d [ x ( k ) , u ( k ) ] , and d is the control process calculation function. In HDP, there are three components: the execution network, model network, and evaluation network, corresponding to the input and output of state variables and performance indicator functions. In PDHDP, a part of the execution network serves as the input to the evaluation network. The remaining functions are similar to the ADP algorithm. Bi-HDP includes the model network, evaluation network, and control network. Unlike HDP, it approximates the objective function, which makes the training of the evaluation network more complex. PDDHDP is similar to Bi HDP in maintaining consistency, but has higher control accuracy because it takes control variables and system states as inputs. In the ADP algorithm, it starts with assuming an initial point as the estimated cost. After each optimization iteration, adjustments are made to the control law and cost function correction values. After multiple iterations, the policy improvement procedure and value determination operation generate corresponding suboptimal control law c l and cost function J l . If they converge to optimal control and optimization functions, the process can be stopped. In the policy improvement procedure, a cost function J ( ∗ , c l ) corresponding to a certain state c l is given. The optimized control law c l + 1 can be obtained, as shown in Eq. (1).

(1) c l + 1 ( x k ) = arg min u k { U ( x k , u k ) + J ( x k + 1 , c l ) } .

In Eq. (1), x k represents state variables, u k belongs to the controlled variables, k denotes a change in time during dynamic variations, U is the control set, and l represents the l th iteration in the optimization process. In the value determination operation, the expression for transforming the cost function for any control law is given by Eq. (2).

(2) J l + 1 ( x k , c ) = U ( x k , u k ) + J ( x k + 1 , c l ) .

When dealing with the control problem of microgrid power generation, the control area is simplified and treated as having only one generator. This results in the state space equation for region i , as shown in Eq. (3).

(3) x ̇ = A i x i + B i u i + ∑ j = 1 , j ≠ i N A i j x j + G i Δ p L i , i = 1 , 2 , … , N x i = [ Δ g i , Δ p m i , Δ p v i , Δ p t e i j ] T u i = Δ p c i A i j = 0 0 0 0 0 0 0 0 0 0 0 0 − T i j 0 0 0 .

In Eq. (3), A i , B i , A i j , and G i are state matrices; Δ p L i , Δ g i , Δ p m i , Δ p v i , and Δ p c i represent the load, frequency deviation, generator mechanical power deviation, turbine governor position deviation, and power at the load reference point for region i , respectively; Δ p c i represents the exchange power between regions i and j ; and T is the time constant. The dynamic model expressions for Δ g i , Δ p m i , Δ p v i , and Δ p c i are given in Eq. (4).

(4) Δ g ̇ i = − D i M i Δ g i + 1 M i Δ p m i − 1 M i Δ p t e i j − 1 M i Δ p L 1 Δ p ̇ m i = − 1 T e h i Δ p m i + 1 T c h i Δ p v i Δ p ̇ v i = − 1 R i T f i Δ g i − 1 T f i Δ p v i + 1 T f i Δ p e i Δ p ̇ t e i j = ∑ j = 1 , j ≠ i N T i j ( Δ g i − Δ g j ) .

In Eq. (4), D i , M i , and T f i are the damping coefficient, rotational inertia, and time constant of the generator, respectively; R i and T c h i are the speed damping coefficient and turbine time constant for region i , respectively; and T i j represents the synchronization time factor. The convergence analysis of the aforementioned dynamic model is conducted through previous research methods [19,20]. In the reinforcement learning part of the DNN-ADP algorithm, the maximum cumulative reward is achieved through the interaction between intelligent agents and the environment. Initially, the policy π ( a ∣ s ) is defined, representing the relationship between state S t and action A t . The probability P of the outputting action a for any input s is given in Eq. (5):

(5) π ( a ∣ s ) = P { A t = a ∣ S t = s } .

The input s in Eq. (5) belongs to the state set S . The quality judgment of the state–action pair formed by s t and a t at time t is determined through the accumulated rewards. Therefore, the discount factor γ is used to calculate the cumulative reward value R t , as shown in Eq. (6):

(6) R t = ∑ k = 0 ∞ γ k r t + k .

In Eq. (6), r t and r t + k represent immediate rewards and subsequent rewards at step k , respectively. The state–action value function Q π ( s , a ) is shown in Eq. (7) [21].

(7) Q π ( s , a ) = E π [ R t ∣ S t = s , A t = a ] .

Based on Eq. (7), the optimal policy, represented by the maximized Q ⁎ ( s , a ) , can be learned. For ease of subsequent solution, the Bellman equation is introduced for iterative computation, as shown in Eq. (8) [22,23].

(8) Q ⁎ ( s , a ) = R s a + γ ∑ s ′ ∈ S P s s ′ a max a ′ Q ⁎ ( s ′ , a ′ ) .

In Eq. (8), s and a are the current state and actions, respectively; s ′ and a ′ are the next-step state and action; and P s s ′ a represents the probability of transitioning from state s to state s ′ using action a . The most representative algorithm in reinforcement learning is the Q-learning algorithm, as shown in Eq. (9) [24,25].

(9) Q t + 1 ( s t , a t ) = Q t ( s t , a t ) + α [ r t + 1 + γ max a ′ Q ( s t + 1 , a ′ ) − Q ( s t , a t ) ] .

In Eq. (9), α represents the learning rate. The flowchart of the Q-learning algorithm is illustrated in Figure 2.

Figure 2 Q-learning algorithm flowchart.
Figure 2

Q-learning algorithm flowchart.

In Figure 2, to ensure the convergence of the current algorithm by calculating the tolerance between the estimated value and the old value, the study introduces the ε-graddy algorithm, which can effectively solve the contradiction between exploration and utilization in reinforcement learning and improve the convergence efficiency of the algorithm. Finally, in the DNN section, the perceptual and decision-making capabilities of deep learning are combined with reinforcement learning to achieve better results. Combining all the aforementioned factors, the DNN-ADP algorithm is obtained. Its corresponding structural diagram is shown in Figure 3.

Figure 3 Schematic diagram of the DNN-ADP algorithm structure.
Figure 3

Schematic diagram of the DNN-ADP algorithm structure.

In Figure 3, the DNN-ADP algorithm is derived from DNN by combining the ADP algorithm. This helps improve the optimization and control performance of microgrids and accelerates the convergence and real-time updating of learning results.

3.2 Construction of the IRPGC algorithm and real-time integrated scheduling and control framework for microgrids

To effectively address the stability, flexibility, and reliability in microgrid generation scheduling control, as well as uncertainties and disturbances from the external environment, the study introduces the rejection operation. Based on the DNN-ADP algorithm, the IRPGC algorithm is proposed, which can output multiple power generation commands at once for microgrid control. This addresses the shortcomings in economic scheduling and frequency control. Traditional reinforcement learning methods require training or reinforcing each action, which consumes a long time. Although the accuracy of the trained IRPGC algorithm may not be entirely correct, even if some actions are trained by the algorithm, it can provide more accurate output, which requires more computer memory. The IRPGC algorithm based on DNN utilizes two or more stacked-constrained Boltzmann machines. In the training phase, unsupervised layerwise greedy training is performed, followed by supervised learning to train the network after offline training completion. The IRPGC algorithm does not require the collaborative coordination of other optimization algorithms and can optimize computational efficiency and memory space. The schematic diagram illustrating the specific convergence acceleration principle of the IRPGC algorithm is shown in Figure 4.

Figure 4 Schematic diagram of the IRPGC algorithm accelerating convergence principle.
Figure 4

Schematic diagram of the IRPGC algorithm accelerating convergence principle.

In Figure 4, the predictive network can perform advance predictions on the microgrid system to assess the effects after a specific action is executed. This ultimately accelerates the convergence process. In the non-recognition (NR) operation part of the IRPGC algorithm, to address the low reliability of action commands when the probability values at the output layer corresponding to evaluation network 1 are low, the IRPGC algorithm employs the DNN-ADP algorithm to obtain new power generation commands. For ease of subsequent control, the NR operation relies solely on a simple rejection threshold, as calculated in Eq. (10).

(10) Δ P ⁎ = Δ P , T DP > T y PID , T DP ≤ T .

In Eq. (10), Δ P ⁎ represents the power generation command corrected through the NR operation, Δ P represents the power generation command output by the IRPGC algorithm, and T and T DP are the rejection threshold and the output layer probability value of the DNN-ADP algorithm, respectively. These are proportional-integral-derivative (PID) control algorithm outputs. The research utilizes the DNN to output the results of evaluation network 1 when it knows it is more important. In cases where it is unaware of its importance, the rejection operation is chosen, and y PID is output. The power generation command for intelligent power generation control in the microgrid is output through the controller, where the system state of the microgrid environment is represented by Δ g , the action values are set as an n ⁎ m matrix, and n and m correspond to the quantity of units and the quantity of instructions generated by each unit, respectively. The action matrix A is given by Eq. (11).

(11) A = A 1 1 A 2 1 ⋯ A n 1 A 1 2 A 2 2 ⋯ A n 2 ⋮ ⋮ ⋱ ⋮ A 1 m A 2 m ⋯ A n m .

According to the principle of a single hidden layer feedforward neural network used in the algorithm, it can be known that for any number of different samples, there is no limit to the interval and the function is infinitely differentiable. In the case of uncertain assignment, the corresponding action matrix is reversible. In this way, the execution process of the IRPGC algorithm is obtained, as shown in Figure 5.

Figure 5 IRPGC algorithm execution process.
Figure 5

IRPGC algorithm execution process.

In Figure 5, after the four DNNs undergo computation in the algorithm, a comparison is made between the output layer probability values of evaluation network 1. Finally, a judgment is made, and action values are output. Based on the aforementioned content, to obtain better control performance indicators and the stability of microgrid systems, this study is similar to the stability of numerical analysis, that is, the sensitivity of algorithms to rounding errors, and investigates the regulation from economic dispatch and frequency control. In addition, to address the shortcomings of traditional control strategies in realizing regulation in these two aspects of microgrid control, the study constructs a combined scheduling and control framework for the microgrid. This framework coordinates economic scheduling with automatic power generation control and droop control. In the droop control section, the transformer at the interface of distributed power sources is controlled to exhibit generator active power and frequency characteristics, as calculated in Eq. (12) [26,27].

(12) f = f 0 + m p q ( p ref q − p q ) .

In Eq. (12), f and f 0 represent the frequency and rated frequency of the microgrid, respectively; q denotes the q th distributed power source; and m p q , P ref q , and P q correspond to the droop coefficient, active power reference value, and output value associated with the q th distributed power source, respectively. Since the system frequency in this part is determined by the system load, the study investigates the automatic generation control within the framework to reduce frequency deviation. In the intelligent generation control section, it is necessary to satisfy the balance constraint ∑ n = 1 n P q = P d . When the total load changes to Δ P , the corresponding power allocation expression can be obtained, as shown in Eq. (13).

(13) Δ P q = m P MG m P Q Δ P .

In Eq. (13), m P MG represents the equivalent value of the droop coefficient in the microgrid system. Finally, the value of P ref q is assigned to P ref q ⁎ , enabling coordination between economic dispatch and intelligent generation control at the same time scale. In the economic dispatch section, the calculated output of the generation controller combining droop control, intelligent generation control, and economic dispatch is given by Eq. (14).

(14) P ED + Δ P q + P ref q .

In Eq. (14), P ED represents the dispatchable active power corresponding to q . To minimize the economic cost of generation control, the expression for generation cost is derived, as shown in Eq. (15) [28]:

(15) min C c t = ∑ q = 1 n ( a q P ED q 2 + b q P ED q + c q ) s . t . ∑ q = 1 n P ED q = P d P q min ≤ P ED q ≤ P q max .

In Eq. (15), a q , b q , and c q are the cost coefficients corresponding to q ; P d represents the predicted energy demand of the microgrid; and P q min and P q max are the minimum and maximum active power, respectively, corresponding to q . Combining the aforementioned content, a real-time integrated scheduling and control framework for the microgrid can be constructed, as illustrated in Figure 6.

Figure 6 Real-time integrated scheduling and control framework for microgrids.
Figure 6

Real-time integrated scheduling and control framework for microgrids.

In Figure 6, the IRPGC algorithm takes into account both long-time scale economic dispatch information and real-time control frequency deviation information, replacing traditional dispatch control frameworks.

4 Results analysis of real-time economic dispatch and frequency control in microgrid based on IRPGC algorithm

To assess the effectiveness and feasibility of real-time economic dispatch and frequency control in microgrids based on the IRPGC algorithm, comparative experiments are conducted on the performance of the algorithm. Subsequently, simulations are carried out in practical applications.

4.1 Performance analysis of real-time economic dispatch and frequency control in microgrid based on IRPGC algorithm

To validate the performance of the proposed IRPGC algorithm, the simulation environment is set up using the Windows 10 operating system on a computer with 16GB of RAM. The experiments are conducted using MATLAB software. In addition, to demonstrate the feasibility and superiority of the proposed real-time integrated scheduling and control framework for microgrids, the study selects microgrid data from Hainan power grid for experimentation. This microgrid serves 200 million households in an area of 3.4 × 1011 m2. The microgrid includes three energy sources and power loads: wind power generation, photovoltaic power generation, electric vehicles, household loads, and eight automatic generators. The frequency reference coefficient is set to 70. The economic dispatch and intelligent power generation control periods are set at 300 and 5 s, respectively. The time constants for the governor, generator, and turbine are set at 0.08, 0.03, and 10, respectively. Finally, the economic dispatch parameters for the microgrid data are presented in Table 1.

Table 1

Economic dispatch parameter settings corresponding to microgrid data

Parameter Alternator
1 2 3 4 5 6 7 8
a q ($/MW2) 0.677 0.452 0.565 0.565 0.4521 0.565 0.565 0.339
b q ($/MW) 370 250 310 310 250 310 310 191
c q ($) 11,260 7,610 9,490 9,490 7,610 9,490 9,490 5,630
P q min (MW) 20 110 160 30 20 60 20 20
P q max (MW) 80 220 745 150 70 242 62 90

To scientifically evaluate the performance of the proposed algorithm and the real-time integrated framework for microgrids, the proposed method is with mainstream frameworks. The intelligent power generation control algorithms include PID, fractional order PID (FO-PID), active disturbance rejection controller (ADRC), sliding mode control (SMC), and fuzzy logic control (FLC). Correspondingly, the economic dispatch optimization algorithms include simulated annealing, particle swarm optimization, genetic algorithm, multivariate optimization algorithm (MOA), and grey wolf algorithm (GWA). MOA is a swarm intelligence optimization algorithm with clear individual division of labor and collaborative cooperation, which has the diversified characteristics of different divisions of labor. GWA is a traditional GWA optimized by improving the convergence factor strategy and dynamic weight strategy. To assess the robustness of the IRPGC algorithm under complex conditions, the study introduces 10% random disturbance power and load, adds to wind and photovoltaic power, considers five different charging behaviors for electric vehicles, and overlays them on the electric vehicle power curve. Finally, typical household loads from real-life scenarios are introduced.

Figure 7(a)–(d) corresponds to the power variation curves of photovoltaic power generation, wind power generation, electric vehicle power, and household load in the Hainan microgrid data, respectively. From Figure 7, the four power variation curves exhibited overall periodic changes. Photovoltaic power generation, wind power generation, and household load power reached their peaks around 4104 s, while the power of electric vehicles reached its peak around 7104 s. When training the DNN of the IRPCC algorithm, the visible layer input and output units are set to 3 and 8, respectively, and the number of hidden layer units is set to [24, 60, 64, 8]. The maximum number of iterations is set to 1,000, and the learning rate is set to 0.1.

Figure 7 Different types of power variation curves in microgrid data. (a) Photovoltaic power. (b) Wind power. (c) Electric vehicle power. (d) Household load power.
Figure 7

Different types of power variation curves in microgrid data. (a) Photovoltaic power. (b) Wind power. (c) Electric vehicle power. (d) Household load power.

First, it is necessary to verify the improvement effect of the number of hidden layers in different neural network structures on system computation time, as shown in Figure 8. From Figure 8, as the number of hidden layers increased, both DNN and IRPGC algorithms fluctuated within a certain calculation time range, and the variation curve of the IRPGC algorithm was smoother and the fluctuation range was smaller. The average computation time for different neural network structures is calculated for the analysis. The average computation time for DNN was 81.963 s, while the average computation time for the IRPGC algorithm was 67.245 s, with a reduction of 14.718 s. In terms of computation time, the IRPGC algorithm increased by 17.96%, indicating that the research algorithm greatly improves computation time and efficiency.

Figure 8 The variation curve of the number of hidden layers and computation time for different neural network structures.
Figure 8

The variation curve of the number of hidden layers and computation time for different neural network structures.

The average error results during the training phase of the IRPGC algorithm based on Hainan microgrid data are shown in Figure 9. From Figure 9, after 5,000 iterations, the average error of the IRPGC algorithm remained within 10−5. In addition, by conducting the stability analysis based on mathematical foundations, Eq. (16) is obtained.

(16) lim i → ∞ u i ( x k ) = u ∗ ( x k ) lim i → ∞ J i ( x k ) = J ∗ ( x k ) .

Figure 9 The average error results during the training phase of the IRPGC algorithm based on Hainan microgrid data.
Figure 9

The average error results during the training phase of the IRPGC algorithm based on Hainan microgrid data.

Eq. (16) indicates that the system based on the IRPGC algorithm framework proposed in the study is stable. To better evaluate the control performance of different algorithms, common metrics such as mean error integral (MEI), absolute error integral (AEI), and time-weighted absolute error integral (TWAEI) are chosen for evaluation. TWAEI is mainly an integral expression of a function that represents the deviation between the expected output and the actual output of the system, which is used to measure the performance of the control system. The AEI index is used to evaluate the role of the system during the transition process.

The comparison of frequency deviation results for different algorithms based on Hainan microgrid data is shown in Figure 10. According to Figure 10, the frequency deviation ranges corresponding to PID, FO-PID, ADRC, SMC, FLC, and IRPGC were −0.04 to 0.037 Hz, −0.046 to 0.019 Hz, −0.047 to 0.0187 Hz, −0.049 to −0.02 Hz, −0.043 to −0.0191 Hz, and −0.073 to 0.013 Hz, respectively. The above results may be due to the fact that the research algorithm can effectively replace traditional microgrid power generation scheduling control frameworks and be applied in real-time microgrid power generation scheduling control frameworks. Moreover, from the long-term overall change curve, the frequency deviation values were mostly negative, indicating that the frequency deviation is opposite to the power generation instructions obtained through economic dispatch and intelligent power generation controllers.

Figure 10 Comparison of frequency deviation results of different algorithms based on Hainan microgrid data.
Figure 10

Comparison of frequency deviation results of different algorithms based on Hainan microgrid data.

Figure 11(a) and (b) presents the results of different frequency deviation evaluation indicators and rejection operation outcomes based on the IRPGC algorithm, respectively. In Figure 11(a), the MEI indicators corresponding to PID, FO-PID, ADRC, SMC, FLC, and IRPGC were 62.21, 254.24, 243.96, 309.61, 244.89, and 51.45, while the AEI indicators were 1.73, 6.11, 6.09, 9.72, 6.24, and 0.54, respectively. The TWAEI indicators corresponded to 2.15 × 105, 11.96 × 105, 11.95 × 105, 14.85 × 105, 11.99 × 105, and 1.58 × 105. In the aforementioned error integration results, all indicators of the research algorithm have the lowest values, with the AMI indicator reaching 0.54. This means that the proposed system has minimal errors in mathematical analysis, while the MEI indicator indicates that the research method has a good transient response and appropriate damping. This is because the intelligent power generation controller and economic dispatch in the system framework will optimize their respective optimization objectives separately, providing a precise adjustment command with a unified time scale. Therefore, the performance of all aspects can be significantly optimized. As shown in Figure 11(b), the rejection operation using the IRPGC algorithm is simulated 69 s earlier. Afterward, the probability value of the output layer corresponding to network 1 in the algorithm always exceeds the set threshold. Overall, the IRPGC algorithm effectively fits random disturbances in the power flow, resulting in a significant improvement in control performance. In conclusion, the proposed IRPGC algorithm exhibits excellent control performance and stability, making it suitable for integrated scheduling and control in microgrids.

Figure 11 Evaluation index results and rejection operation results of different frequency deviations based on the IRPGC algorithm. (a) Results of evalutation indicators for different frequency deviations based on the IRPGC algorithm. (b) Reject operation result.
Figure 11

Evaluation index results and rejection operation results of different frequency deviations based on the IRPGC algorithm. (a) Results of evalutation indicators for different frequency deviations based on the IRPGC algorithm. (b) Reject operation result.

4.2 Application analysis of real-time economic dispatch and frequency control in microgrid based on IRPGC algorithm

The aforementioned results confirmed the superior control performance of the IRPGC algorithm. To further explore the practical application effects of the algorithm in real-time generation scheduling and control frameworks in microgrids, experiments were conducted using data from the Sanya microgrid. The DNN of the IRPGC algorithm and the corresponding hidden layer ranges were [3, 10] and [8, 400], respectively, taking root-mean-square error (RMSE) as an indicator.

Figure 12(a)–(d) shows the RMSE results for stages 1–4 of the hidden layers based on the IRPGC algorithm. From Figure 12, in the application of real-time generation scheduling and control, the optimal rejection threshold range based on the IRPGC algorithm was [0.94, 0.97]. These results indicate that the IRPGC algorithm has good feasibility and applicability in practical applications.

Figure 12 RMSE results of hidden layer for different applications based on the IRPGC algorithm. (a) Stage 1. (b) Stage 2. (c) Stage 3. (d) Stage 4.
Figure 12

RMSE results of hidden layer for different applications based on the IRPGC algorithm. (a) Stage 1. (b) Stage 2. (c) Stage 3. (d) Stage 4.

5 Conclusion

In recent years, microgrids have been widely applied. However, ensuring the frequency stability of microgrids and maintaining a balance between economic and environmental benefits has been a research focus. To address these issues, the study initially integrated the ADP algorithm into the DNN, resulting in the DNN-ADP algorithm. Subsequently, the rejection operation was introduced to obtain the IRPGC algorithm. Finally, a real-time integrated scheduling and control framework for microgrids was established. From experimental results, after 5,000 iterations during the training phase, the IRPGC algorithm exhibited an average error within the range of 10−5. The frequency deviation ranges corresponding to PID, FO-PID, ADRC, SMC, FLC, and IRPGC were −0.04 to 0.037 Hz, −0.046 to 0.019 Hz, −0.047 to 0.0187 Hz, −0.049 to −0.02 Hz, −0.043 to −0.0191 Hz, and −0.073 to 0.013 Hz, respectively. In terms of the three frequency deviation evaluation indicators, MEI indicators for PID, FO-PID, ADRC, SMC, FLC, and IRPGC were 62.21, 254.24, 243.96, 309.61, 244.89, and 51.45, while AEI indicators were 1.73, 6.11, 6.09, 9.72, 6.24, and 0.54, respectively. The TWAEI indicators corresponded to 2.15105, 11.96105, 11.95105, 14.85105, 11.99105, and 1.58105. In the practical application of real-time generation scheduling and control in the microgrid, the optimal rejection threshold range based on the IRPGC algorithm was found to be [0.94, 0.97]. In conclusion, the proposed IRPGC algorithm demonstrated good economic control performance and frequency control performance, effectively addressing complex control issues in microgrids. However, there are still limitations in the research. There is a significant computational load in practical applications. Therefore, in future research, this method can be further lightweight to improve computational efficiency.

  1. Funding information: The research was supported by Natural Science Research Project of Shangluo University (No. 24KYZX03).

  2. Author contributions: All the contributions related to the paper are attributed to Jun Liu. The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.

  3. Conflict of interest: Author states no conflict of interest.

  4. Data availability statement: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

[1] Kumar S, Das P, Kumar K. Adaptive mesh based efficient approximations for Darcy scale precipitation–dissolution models in porous media. Int J Numer Methods Fluids. 2024;96(8):1415–44.10.1002/fld.5294Search in Google Scholar

[2] Saini S, Das P, Kumar S. Computational cost reduction for coupled system of multiple scale reaction diffusion problems with mixed type boundary conditions having boundary layers. RACSAM. 2023;117(2):66.10.1007/s13398-023-01397-8Search in Google Scholar

[3] Shiromani R, Shanthi V, Das P. A higher order hybrid-numerical approximation for a class of singularly perturbed two-dimensional convection-diffusion elliptic problem with non-smooth convection and source terms. Comput Math Appl. 2023;142:9–30.10.1016/j.camwa.2023.04.004Search in Google Scholar

[4] Yao A, Sheng W. Interaction of elementary waves on a boundary for a hyperbolic system of conservation laws. Math Methods Appl Sci. 2008;31(11):1369–81.10.1002/mma.977Search in Google Scholar

[5] Santra S, Mohapatra J, Das P, Choudhuri D. Higher order approximations for fractional order integro-parabolic partial differential equations on an adaptive mesh with error analysis. Comput Math Appl. 2023;150:87–101.10.1016/j.camwa.2023.09.008Search in Google Scholar

[6] Kumar K, Podila PC, Das P, Ramos H. A graded mesh refinement approach for boundary layer originated singularly perturbed time-delayed parabolic convection diffusion problems. Math Methods Appl Sci. 2021;44(16):12332–50.10.1002/mma.7358Search in Google Scholar

[7] Shakti D, Mohapatra J, Das P, Vigo-Aguiar J. A moving mesh refinement based optimal accurate uniformly convergent computational method for a parabolic system of boundary layer originated reaction–diffusion problems with arbitrary small diffusion terms. J Comput Appl Math. 2022;404:113167.10.1016/j.cam.2020.113167Search in Google Scholar

[8] Zeinal-Kheiri S, Shotorbani AM, Khardenavis A, Mohammadi-Ivatloo B, Sadiq R, Hewage K. An adaptive real - time energy management system for a renewable energy - based microgrid. IET Renew Power Gener. 2021;15(13):2918–30.10.1049/rpg2.12223Search in Google Scholar

[9] Wang B, Peng X, Zhang L, Su P. Real time power management strategy for an all-electric ship using a predictive control model. IET Gener Transm Distrib. 2022;16(9):1808–21.10.1049/gtd2.12419Search in Google Scholar

[10] Zhang S, Xiang Y, Liu J, Liu J, Yang J, Zhao X, et al. A regulating capacity determination method for pumped storage hydropower to restrain PV generation fluctuations. CSEE J Power Energy Syst. 2022;8(1):304–16.Search in Google Scholar

[11] Fang Y, Zhao S, Du E, Li S, Li Z. Coordinated operation of concentrating solar power plant and wind farm for frequency regulation. J Mod Power Syst Clean Energy. 2021;9(4):751–9.10.35833/MPCE.2021.000060Search in Google Scholar

[12] Zhao X, Hanglin LI. Deep neural network compression algorithm based on hybrid mechanism. J Comput Appl. 2023;43(9):2686–91.Search in Google Scholar

[13] Chen L, Xiong H, Sang X, Yuan C, Li X, Kong Q. An innovative deep neural network–based approach for internal cavity detection of timber columns using percussion sound. Struct Health Monit. 2022;21(3):1251–65.10.1177/14759217211028524Search in Google Scholar

[14] Chen CY, Lai IC, Wu PY, Wu RB. Optimization and evaluation of multidetector deep neural network for high-accuracy Wi-Fi fingerprint positioning. IEEE Internet Things J. 2022;9(16):15204–14.10.1109/JIOT.2022.3147644Search in Google Scholar

[15] Adam ABM, Wang Z, Wan X, Xu Y, Duo B. Energy-efficient power allocation in downlink multi-cell multi-carrier NOMA: Special deep neural network framework. IEEE Trans Cogn Commun Netw. 2022;8(4):1770–83.10.1109/TCCN.2022.3198652Search in Google Scholar

[16] Das P, Rana S, Vigo-Aguiar J. Higher order accurate approximations on equidistributed meshes for boundary layer originated mixed type reaction diffusion systems with multiple scale nature. Appl Numer Math. 2020;148:79–97.10.1016/j.apnum.2019.08.028Search in Google Scholar

[17] Das P. An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh. Numer Algorithms. 2019;81(2):465–87.10.1007/s11075-018-0557-4Search in Google Scholar

[18] Das P. Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems. J Comput Appl Math. 2015;290:16–25.10.1016/j.cam.2015.04.034Search in Google Scholar

[19] Das P, Vigo-Aguiar J. Parameter uniform optimal order numerical approximation of a class of singularly perturbed system of reaction diffusion problems involving a small perturbation parameter. J Comput Appl Math. 2019;354:533–44.10.1016/j.cam.2017.11.026Search in Google Scholar

[20] Saini S, Das P, Kumar S. Parameter uniform higher order numerical treatment for singularly perturbed Robin type parabolic reaction diffusion multiple scale problems with large delay in time. Appl Numer Math. 2024;196:1–21.10.1016/j.apnum.2023.10.003Search in Google Scholar

[21] Kumar S, Kumar S, Das P. Second-order a priori and a posteriori error estimations for integral boundary value problems of nonlinear singularly perturbed parameterized form. Numer Algorithms. 2024;1–28.10.1007/s11075-024-01918-5Search in Google Scholar

[22] Das P, Rana S. Theoretical prospects of fractional order weakly singular Volterra Integro differential equations and their approximations with convergence analysis. Math Methods Appl Sci. 2021;44(11):9419–40.10.1002/mma.7369Search in Google Scholar

[23] Srivastava HM, Nain AK, Vats RK, Das P. A theoretical study of the fractional-order p-Laplacian nonlinear Hadamard type turbulent flow models having the Ulam–Hyers stability. RACSAM. 2023;117(4):160.10.1007/s13398-023-01488-6Search in Google Scholar

[24] Das P, Rana S, Ramos H. On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis. J Comput Appl Math. 2022;404:113116.10.1016/j.cam.2020.113116Search in Google Scholar

[25] Das P, Rana S, Ramos H. A perturbation-based approach for solving fractional-order Volterra–Fredholm integro differential equations and its convergence analysis. Int J Comput Math. 2020;97(10):1994–2014.10.1080/00207160.2019.1673892Search in Google Scholar

[26] Das P, Natesan S. Higher-order parameter uniform convergent schemes for Robin type reaction-diffusion problems using adaptively generated grid. Int J Comput Methods. 2012;9(4):1250052.10.1142/S0219876212500521Search in Google Scholar

[27] Das P. Richardson extrapolation method for singularly perturbed convection-diffusion problems on adaptively generated mesh. CMES. 2013;90(6):463–85.Search in Google Scholar

[28] Das P, Natesan S. A uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion Robin type boundary-value problems. J Appl Math Comput. 2013;41:447–71.10.1007/s12190-012-0611-7Search in Google Scholar
Received: 2024-08-02
Revised: 2024-10-10
Accepted: 2024-10-24
Published Online: 2025-03-20

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Research Articles
  2. Generalized (ψ,φ)-contraction to investigate Volterra integral inclusions and fractal fractional PDEs in super-metric space with numerical experiments
  3. Solitons in ultrasound imaging: Exploring applications and enhancements via the Westervelt equation
  4. Stochastic improved Simpson for solving nonlinear fractional-order systems using product integration rules
  5. Exploring dynamical features like bifurcation assessment, sensitivity visualization, and solitary wave solutions of the integrable Akbota equation
  6. Research on surface defect detection method and optimization of paper-plastic composite bag based on improved combined segmentation algorithm
  7. Impact the sulphur content in Iraqi crude oil on the mechanical properties and corrosion behaviour of carbon steel in various types of API 5L pipelines and ASTM 106 grade B
  8. Unravelling quiescent optical solitons: An exploration of the complex Ginzburg–Landau equation with nonlinear chromatic dispersion and self-phase modulation
  9. Perturbation-iteration approach for fractional-order logistic differential equations
  10. Variational formulations for the Euler and Navier–Stokes systems in fluid mechanics and related models
  11. Rotor response to unbalanced load and system performance considering variable bearing profile
  12. DeepFowl: Disease prediction from chicken excreta images using deep learning
  13. Channel flow of Ellis fluid due to cilia motion
  14. A case study of fractional-order varicella virus model to nonlinear dynamics strategy for control and prevalence
  15. Multi-point estimation weldment recognition and estimation of pose with data-driven robotics design
  16. Analysis of Hall current and nonuniform heating effects on magneto-convection between vertically aligned plates under the influence of electric and magnetic fields
  17. A comparative study on residual power series method and differential transform method through the time-fractional telegraph equation
  18. Insights from the nonlinear Schrödinger–Hirota equation with chromatic dispersion: Dynamics in fiber–optic communication
  19. Mathematical analysis of Jeffrey ferrofluid on stretching surface with the Darcy–Forchheimer model
  20. Exploring the interaction between lump, stripe and double-stripe, and periodic wave solutions of the Konopelchenko–Dubrovsky–Kaup–Kupershmidt system
  21. Computational investigation of tuberculosis and HIV/AIDS co-infection in fuzzy environment
  22. Signature verification by geometry and image processing
  23. Theoretical and numerical approach for quantifying sensitivity to system parameters of nonlinear systems
  24. Chaotic behaviors, stability, and solitary wave propagations of M-fractional LWE equation in magneto-electro-elastic circular rod
  25. Dynamic analysis and optimization of syphilis spread: Simulations, integrating treatment and public health interventions
  26. Visco-thermoelastic rectangular plate under uniform loading: A study of deflection
  27. Threshold dynamics and optimal control of an epidemiological smoking model
  28. Numerical computational model for an unsteady hybrid nanofluid flow in a porous medium past an MHD rotating sheet
  29. Regression prediction model of fabric brightness based on light and shadow reconstruction of layered images
  30. 10.1515/nleng-2025-0171
  31. Review Article
  32. Haar wavelet collocation method for existence and numerical solutions of fourth-order integro-differential equations with bounded coefficients
  33. Special Issue: Nonlinear Analysis and Design of Communication Networks for IoT Applications - Part II
  34. Silicon-based all-optical wavelength converter for on-chip optical interconnection
  35. Research on a path-tracking control system of unmanned rollers based on an optimization algorithm and real-time feedback
  36. Analysis of the sports action recognition model based on the LSTM recurrent neural network
  37. Industrial robot trajectory error compensation based on enhanced transfer convolutional neural networks
  38. Research on IoT network performance prediction model of power grid warehouse based on nonlinear GA-BP neural network
  39. Interactive recommendation of social network communication between cities based on GNN and user preferences
  40. Application of improved P-BEM in time varying channel prediction in 5G high-speed mobile communication system
  41. Construction of a BIM smart building collaborative design model combining the Internet of Things
  42. Optimizing malicious website prediction: An advanced XGBoost-based machine learning model
  43. Economic operation analysis of the power grid combining communication network and distributed optimization algorithm
  44. Sports video temporal action detection technology based on an improved MSST algorithm
  45. Internet of things data security and privacy protection based on improved federated learning
  46. Enterprise power emission reduction technology based on the LSTM–SVM model
  47. Construction of multi-style face models based on artistic image generation algorithms
  48. Special Issue: Decision and Control in Nonlinear Systems - Part II
  49. Animation video frame prediction based on ConvGRU fine-grained synthesis flow
  50. Application of GGNN inference propagation model for martial art intensity evaluation
  51. Benefit evaluation of building energy-saving renovation projects based on BWM weighting method
  52. Deep neural network application in real-time economic dispatch and frequency control of microgrids
  53. Real-time force/position control of soft growing robots: A data-driven model predictive approach
  54. Mechanical product design and manufacturing system based on CNN and server optimization algorithm
  55. Application of finite element analysis in the formal analysis of ancient architectural plaque section
  56. Research on territorial spatial planning based on data mining and geographic information visualization
  57. Fault diagnosis of agricultural sprinkler irrigation machinery equipment based on machine vision
  58. Closure technology of large span steel truss arch bridge with temporarily fixed edge supports
  59. Intelligent accounting question-answering robot based on a large language model and knowledge graph
  60. Analysis of manufacturing and retailer blockchain decision based on resource recyclability
  61. Flexible manufacturing workshop mechanical processing and product scheduling algorithm based on MES
  62. Exploration of indoor environment perception and design model based on virtual reality technology
  63. Tennis automatic ball-picking robot based on image object detection and positioning technology
  64. A new CNN deep learning model for computer-intelligent color matching
  65. Design of AR-based general computer technology experiment demonstration platform
  66. Indoor environment monitoring method based on the fusion of audio recognition and video patrol features
  67. Health condition prediction method of the computer numerical control machine tool parts by ensembling digital twins and improved LSTM networks
  68. Establishment of a green degree evaluation model for wall materials based on lifecycle
  69. Quantitative evaluation of college music teaching pronunciation based on nonlinear feature extraction
  70. Multi-index nonlinear robust virtual synchronous generator control method for microgrid inverters
  71. Manufacturing engineering production line scheduling management technology integrating availability constraints and heuristic rules
  72. Analysis of digital intelligent financial audit system based on improved BiLSTM neural network
  73. Attention community discovery model applied to complex network information analysis
  74. A neural collaborative filtering recommendation algorithm based on attention mechanism and contrastive learning
  75. Rehabilitation training method for motor dysfunction based on video stream matching
  76. Research on façade design for cold-region buildings based on artificial neural networks and parametric modeling techniques
  77. Intelligent implementation of muscle strain identification algorithm in Mi health exercise induced waist muscle strain
  78. Optimization design of urban rainwater and flood drainage system based on SWMM
  79. Improved GA for construction progress and cost management in construction projects
  80. Evaluation and prediction of SVM parameters in engineering cost based on random forest hybrid optimization
  81. Museum intelligent warning system based on wireless data module
  82. Special Issue: Nonlinear Engineering’s significance in Materials Science
  83. Experimental research on the degradation of chemical industrial wastewater by combined hydrodynamic cavitation based on nonlinear dynamic model
  84. Study on low-cycle fatigue life of nickel-based superalloy GH4586 at various temperatures
  85. Some results of solutions to neutral stochastic functional operator-differential equations
  86. Ultrasonic cavitation did not occur in high-pressure CO2 liquid
  87. Research on the performance of a novel type of cemented filler material for coal mine opening and filling
  88. Testing of recycled fine aggregate concrete’s mechanical properties using recycled fine aggregate concrete and research on technology for highway construction
  89. A modified fuzzy TOPSIS approach for the condition assessment of existing bridges
  90. Nonlinear structural and vibration analysis of straddle monorail pantograph under random excitations
  91. Achieving high efficiency and stability in blue OLEDs: Role of wide-gap hosts and emitter interactions
  92. Construction of teaching quality evaluation model of online dance teaching course based on improved PSO-BPNN
  93. Enhanced electrical conductivity and electromagnetic shielding properties of multi-component polymer/graphite nanocomposites prepared by solid-state shear milling
  94. Optimization of thermal characteristics of buried composite phase-change energy storage walls based on nonlinear engineering methods
  95. A higher-performance big data-based movie recommendation system
  96. Nonlinear impact of minimum wage on labor employment in China
  97. Nonlinear comprehensive evaluation method based on information entropy and discrimination optimization
  98. Application of numerical calculation methods in stability analysis of pile foundation under complex foundation conditions
  99. Research on the contribution of shale gas development and utilization in Sichuan Province to carbon peak based on the PSA process
  100. Characteristics of tight oil reservoirs and their impact on seepage flow from a nonlinear engineering perspective
  101. Nonlinear deformation decomposition and mode identification of plane structures via orthogonal theory
  102. Numerical simulation of damage mechanism in rock with cracks impacted by self-excited pulsed jet based on SPH-FEM coupling method: The perspective of nonlinear engineering and materials science
  103. Cross-scale modeling and collaborative optimization of ethanol-catalyzed coupling to produce C4 olefins: Nonlinear modeling and collaborative optimization strategies
  104. Special Issue: Advances in Nonlinear Dynamics and Control
  105. Development of a cognitive blood glucose–insulin control strategy design for a nonlinear diabetic patient model
  106. Big data-based optimized model of building design in the context of rural revitalization
  107. Multi-UAV assisted air-to-ground data collection for ground sensors with unknown positions
  108. Design of urban and rural elderly care public areas integrating person-environment fit theory
  109. Application of lossless signal transmission technology in piano timbre recognition
  110. Application of improved GA in optimizing rural tourism routes
  111. Architectural animation generation system based on AL-GAN algorithm
  112. Advanced sentiment analysis in online shopping: Implementing LSTM models analyzing E-commerce user sentiments
  113. Intelligent recommendation algorithm for piano tracks based on the CNN model
  114. Visualization of large-scale user association feature data based on a nonlinear dimensionality reduction method
  115. Low-carbon economic optimization of microgrid clusters based on an energy interaction operation strategy
  116. Optimization effect of video data extraction and search based on Faster-RCNN hybrid model on intelligent information systems
  117. Construction of image segmentation system combining TC and swarm intelligence algorithm
  118. Particle swarm optimization and fuzzy C-means clustering algorithm for the adhesive layer defect detection
  119. Optimization of student learning status by instructional intervention decision-making techniques incorporating reinforcement learning
  120. Fuzzy model-based stabilization control and state estimation of nonlinear systems
  121. Optimization of distribution network scheduling based on BA and photovoltaic uncertainty
https://doi.org/10.1515/nleng-2024-0074
Keywords for this article
deep neural network; microgrid; economic dispatch; frequency control; rejection operation
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. Generalized (ψ,φ)-contraction to investigate Volterra integral inclusions and fractal fractional PDEs in super-metric space with numerical experiments
  3. Solitons in ultrasound imaging: Exploring applications and enhancements via the Westervelt equation
  4. Stochastic improved Simpson for solving nonlinear fractional-order systems using product integration rules
  5. Exploring dynamical features like bifurcation assessment, sensitivity visualization, and solitary wave solutions of the integrable Akbota equation
  6. Research on surface defect detection method and optimization of paper-plastic composite bag based on improved combined segmentation algorithm
  7. Impact the sulphur content in Iraqi crude oil on the mechanical properties and corrosion behaviour of carbon steel in various types of API 5L pipelines and ASTM 106 grade B
  8. Unravelling quiescent optical solitons: An exploration of the complex Ginzburg–Landau equation with nonlinear chromatic dispersion and self-phase modulation
  9. Perturbation-iteration approach for fractional-order logistic differential equations
  10. Variational formulations for the Euler and Navier–Stokes systems in fluid mechanics and related models
  11. Rotor response to unbalanced load and system performance considering variable bearing profile
  12. DeepFowl: Disease prediction from chicken excreta images using deep learning
  13. Channel flow of Ellis fluid due to cilia motion
  14. A case study of fractional-order varicella virus model to nonlinear dynamics strategy for control and prevalence
  15. Multi-point estimation weldment recognition and estimation of pose with data-driven robotics design
  16. Analysis of Hall current and nonuniform heating effects on magneto-convection between vertically aligned plates under the influence of electric and magnetic fields
  17. A comparative study on residual power series method and differential transform method through the time-fractional telegraph equation
  18. Insights from the nonlinear Schrödinger–Hirota equation with chromatic dispersion: Dynamics in fiber–optic communication
  19. Mathematical analysis of Jeffrey ferrofluid on stretching surface with the Darcy–Forchheimer model
  20. Exploring the interaction between lump, stripe and double-stripe, and periodic wave solutions of the Konopelchenko–Dubrovsky–Kaup–Kupershmidt system
  21. Computational investigation of tuberculosis and HIV/AIDS co-infection in fuzzy environment
  22. Signature verification by geometry and image processing
  23. Theoretical and numerical approach for quantifying sensitivity to system parameters of nonlinear systems
  24. Chaotic behaviors, stability, and solitary wave propagations of M-fractional LWE equation in magneto-electro-elastic circular rod
  25. Dynamic analysis and optimization of syphilis spread: Simulations, integrating treatment and public health interventions
  26. Visco-thermoelastic rectangular plate under uniform loading: A study of deflection
  27. Threshold dynamics and optimal control of an epidemiological smoking model
  28. Numerical computational model for an unsteady hybrid nanofluid flow in a porous medium past an MHD rotating sheet
  29. Regression prediction model of fabric brightness based on light and shadow reconstruction of layered images
  30. 10.1515/nleng-2025-0171
  31. Review Article
  32. Haar wavelet collocation method for existence and numerical solutions of fourth-order integro-differential equations with bounded coefficients
  33. Special Issue: Nonlinear Analysis and Design of Communication Networks for IoT Applications - Part II
  34. Silicon-based all-optical wavelength converter for on-chip optical interconnection
  35. Research on a path-tracking control system of unmanned rollers based on an optimization algorithm and real-time feedback
  36. Analysis of the sports action recognition model based on the LSTM recurrent neural network
  37. Industrial robot trajectory error compensation based on enhanced transfer convolutional neural networks
  38. Research on IoT network performance prediction model of power grid warehouse based on nonlinear GA-BP neural network
  39. Interactive recommendation of social network communication between cities based on GNN and user preferences
  40. Application of improved P-BEM in time varying channel prediction in 5G high-speed mobile communication system
  41. Construction of a BIM smart building collaborative design model combining the Internet of Things
  42. Optimizing malicious website prediction: An advanced XGBoost-based machine learning model
  43. Economic operation analysis of the power grid combining communication network and distributed optimization algorithm
  44. Sports video temporal action detection technology based on an improved MSST algorithm
  45. Internet of things data security and privacy protection based on improved federated learning
  46. Enterprise power emission reduction technology based on the LSTM–SVM model
  47. Construction of multi-style face models based on artistic image generation algorithms
  48. Special Issue: Decision and Control in Nonlinear Systems - Part II
  49. Animation video frame prediction based on ConvGRU fine-grained synthesis flow
  50. Application of GGNN inference propagation model for martial art intensity evaluation
  51. Benefit evaluation of building energy-saving renovation projects based on BWM weighting method
  52. Deep neural network application in real-time economic dispatch and frequency control of microgrids
  53. Real-time force/position control of soft growing robots: A data-driven model predictive approach
  54. Mechanical product design and manufacturing system based on CNN and server optimization algorithm
  55. Application of finite element analysis in the formal analysis of ancient architectural plaque section
  56. Research on territorial spatial planning based on data mining and geographic information visualization
  57. Fault diagnosis of agricultural sprinkler irrigation machinery equipment based on machine vision
  58. Closure technology of large span steel truss arch bridge with temporarily fixed edge supports
  59. Intelligent accounting question-answering robot based on a large language model and knowledge graph
  60. Analysis of manufacturing and retailer blockchain decision based on resource recyclability
  61. Flexible manufacturing workshop mechanical processing and product scheduling algorithm based on MES
  62. Exploration of indoor environment perception and design model based on virtual reality technology
  63. Tennis automatic ball-picking robot based on image object detection and positioning technology
  64. A new CNN deep learning model for computer-intelligent color matching
  65. Design of AR-based general computer technology experiment demonstration platform
  66. Indoor environment monitoring method based on the fusion of audio recognition and video patrol features
  67. Health condition prediction method of the computer numerical control machine tool parts by ensembling digital twins and improved LSTM networks
  68. Establishment of a green degree evaluation model for wall materials based on lifecycle
  69. Quantitative evaluation of college music teaching pronunciation based on nonlinear feature extraction
  70. Multi-index nonlinear robust virtual synchronous generator control method for microgrid inverters
  71. Manufacturing engineering production line scheduling management technology integrating availability constraints and heuristic rules
  72. Analysis of digital intelligent financial audit system based on improved BiLSTM neural network
  73. Attention community discovery model applied to complex network information analysis
  74. A neural collaborative filtering recommendation algorithm based on attention mechanism and contrastive learning
  75. Rehabilitation training method for motor dysfunction based on video stream matching
  76. Research on façade design for cold-region buildings based on artificial neural networks and parametric modeling techniques
  77. Intelligent implementation of muscle strain identification algorithm in Mi health exercise induced waist muscle strain
  78. Optimization design of urban rainwater and flood drainage system based on SWMM
  79. Improved GA for construction progress and cost management in construction projects
  80. Evaluation and prediction of SVM parameters in engineering cost based on random forest hybrid optimization
  81. Museum intelligent warning system based on wireless data module
  82. Special Issue: Nonlinear Engineering’s significance in Materials Science
  83. Experimental research on the degradation of chemical industrial wastewater by combined hydrodynamic cavitation based on nonlinear dynamic model
  84. Study on low-cycle fatigue life of nickel-based superalloy GH4586 at various temperatures
  85. Some results of solutions to neutral stochastic functional operator-differential equations
  86. Ultrasonic cavitation did not occur in high-pressure CO2 liquid
  87. Research on the performance of a novel type of cemented filler material for coal mine opening and filling
  88. Testing of recycled fine aggregate concrete’s mechanical properties using recycled fine aggregate concrete and research on technology for highway construction
  89. A modified fuzzy TOPSIS approach for the condition assessment of existing bridges
  90. Nonlinear structural and vibration analysis of straddle monorail pantograph under random excitations
  91. Achieving high efficiency and stability in blue OLEDs: Role of wide-gap hosts and emitter interactions
  92. Construction of teaching quality evaluation model of online dance teaching course based on improved PSO-BPNN
  93. Enhanced electrical conductivity and electromagnetic shielding properties of multi-component polymer/graphite nanocomposites prepared by solid-state shear milling
  94. Optimization of thermal characteristics of buried composite phase-change energy storage walls based on nonlinear engineering methods
  95. A higher-performance big data-based movie recommendation system
  96. Nonlinear impact of minimum wage on labor employment in China
  97. Nonlinear comprehensive evaluation method based on information entropy and discrimination optimization
  98. Application of numerical calculation methods in stability analysis of pile foundation under complex foundation conditions
  99. Research on the contribution of shale gas development and utilization in Sichuan Province to carbon peak based on the PSA process
  100. Characteristics of tight oil reservoirs and their impact on seepage flow from a nonlinear engineering perspective
  101. Nonlinear deformation decomposition and mode identification of plane structures via orthogonal theory
  102. Numerical simulation of damage mechanism in rock with cracks impacted by self-excited pulsed jet based on SPH-FEM coupling method: The perspective of nonlinear engineering and materials science
  103. Cross-scale modeling and collaborative optimization of ethanol-catalyzed coupling to produce C4 olefins: Nonlinear modeling and collaborative optimization strategies
  104. Special Issue: Advances in Nonlinear Dynamics and Control
  105. Development of a cognitive blood glucose–insulin control strategy design for a nonlinear diabetic patient model
  106. Big data-based optimized model of building design in the context of rural revitalization
  107. Multi-UAV assisted air-to-ground data collection for ground sensors with unknown positions
  108. Design of urban and rural elderly care public areas integrating person-environment fit theory
  109. Application of lossless signal transmission technology in piano timbre recognition
  110. Application of improved GA in optimizing rural tourism routes
  111. Architectural animation generation system based on AL-GAN algorithm
  112. Advanced sentiment analysis in online shopping: Implementing LSTM models analyzing E-commerce user sentiments
  113. Intelligent recommendation algorithm for piano tracks based on the CNN model
  114. Visualization of large-scale user association feature data based on a nonlinear dimensionality reduction method
  115. Low-carbon economic optimization of microgrid clusters based on an energy interaction operation strategy
  116. Optimization effect of video data extraction and search based on Faster-RCNN hybrid model on intelligent information systems
  117. Construction of image segmentation system combining TC and swarm intelligence algorithm
  118. Particle swarm optimization and fuzzy C-means clustering algorithm for the adhesive layer defect detection
  119. Optimization of student learning status by instructional intervention decision-making techniques incorporating reinforcement learning
  120. Fuzzy model-based stabilization control and state estimation of nonlinear systems
  121. Optimization of distribution network scheduling based on BA and photovoltaic uncertainty
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 24.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/nleng-2024-0074/html
Scroll to top button