Optimization of Hybrid Energy Systems Based on MPC-LSTM-KAN: A Case Study of a High-Altitude Wind Energy Work Umbrella Control System

Abstract

This paper presents an optimization method for hybrid energy systems based on Model Predictive Control (MPC), Long Short-Term Memory (LSTM) networks, and Kolmogorov–Arnold Networks (KANs). The proposed method is applied to a high-altitude wind energy work umbrella control system, where it aims to enhance the stability and efficiency of energy utilization. The work umbrella system integrates wind and solar energy sources, with energy stored in a battery and used to control the umbrella’s operations. The MPC framework is employed to optimize control actions by solving a finite-horizon optimization problem, ensuring the battery State of Charge (SOC) remains within an optimal range. The LSTM network provides accurate predictions of environmental conditions, including wind speed and solar irradiance, which are essential for MPC’s decision-making process. To address complex nonlinearities in the system, the KAN is utilized to model and approximate these dynamics, refining the LSTM predictions. The integration of these advanced control strategies enables the system to handle varying operational conditions and maintain optimal performance. The case study demonstrates the effectiveness of the MPC-LSTM-KAN approach, revealing improvements in the SOC stability, energy efficiency, and operational endurance of the high-altitude wind energy work umbrella system. The results indicate that this hybrid optimization method offers a robust solution for managing hybrid energy systems in dynamic environments.

1. Introduction

1.1. Motivation

The advancement of high-altitude wind energy generation has emerged as a promising avenue for renewable energy production due to the consistent and powerful wind currents available at higher altitudes. Unlike conventional ground-based wind turbines, which are often limited by variable wind conditions and geographic constraints, high-altitude wind energy systems have the potential to capture energy more efficiently and consistently. However, the deployment and operation of these systems introduce complex challenges, particularly in the reliable control of the energy generation process, which is intrinsically linked to the management of power supply and storage.

One of the critical components of these systems is the work umbrella controller, responsible for the precise and timely opening and closing of the high-altitude energy generation umbrella. This controller plays a pivotal role in optimizing the capture of wind energy, yet its operation is heavily dependent on a stable power supply. The power for the controller is typically derived from an off-grid battery system, supplemented by wind and solar energy harvested locally by the controller. Ensuring that the battery’s State of Charge (SOC) remains within an optimal range, particularly around 80%, is vital for maintaining the controller’s functionality over extended periods. A stable SOC not only prevents premature battery degradation but also ensures that the controller can perform the necessary operations to maximize energy capture without interruptions.

Maintaining the SOC within this desired range poses significant technical challenges, particularly due to the dynamic nature of energy generation and consumption in such systems. Fluctuations in wind and solar power, coupled with the variable demands of the controller, can lead to rapid changes in the SOC, risking both undercharging and overcharging scenarios. These risks, if not properly managed, can lead to reduced system efficiency, decreased operational endurance, and, potentially, the failure of the energy generation system during critical periods.

To address these challenges, the integration of advanced control strategies is necessary. In this context, Model Predictive Control (MPC) combined with Long Short-Term Memory (LSTM) networks offers a robust solution. The LSTM network’s ability to model and predict temporal sequences allows it to anticipate future variations in power generation and consumption, providing valuable foresight to the MPC framework. This predictive capability enables MPC to make more informed and accurate control decisions, adjusting the power allocation to maintain the SOC around the target 80% level. Such an approach not only stabilizes the SOC but also enhances the overall efficiency and reliability of the high-altitude wind energy system.

The Kolmogorov–Arnold Network (KAN) provides a powerful mathematical tool for approximating multidimensional continuous functions. Although MPC and LSTM are capable of handling optimization control and time-series prediction, respectively, they may encounter limitations when dealing with complex nonlinear dynamic relationships, especially in high-dimensional, multivariable, dynamic systems. The introduction of the KAN addresses this shortcoming by efficiently approximating complex nonlinear functions, thereby enhancing the modeling capability of the overall framework.

By employing MPC-LSTM-KAN in the control strategy, the system is better equipped to handle the inherent uncertainties and dynamic conditions of renewable energy generation. This, in turn, leads to an increase in the number of operational cycles the work umbrella can perform, extending the system’s operational life and ensuring consistent energy production. The successful implementation of this approach could significantly improve the feasibility and scalability of high-altitude wind energy generation, contributing to the broader goal of transitioning to sustainable energy sources.

1.2. Literature Review

Yi He’s review [1] highlights the importance of optimizing off-grid hybrid renewable energy systems (HRESs) for rural electrification and decarbonization. It provides a comprehensive analysis of system configurations, energy management strategies, and sizing methodologies, with a focus on the dominance of solar photovoltaic and battery systems and the widespread use of rule-based management strategies. The study identifies future research directions, including hybrid energy storage systems, deep reinforcement learning, and distributionally robust optimization techniques. M. Thirunavukkarasu’s review [2] discusses the challenges in optimizing hybrid renewable energy systems (HRESs) due to their complexity and unpredictability. The paper compares classical and AI-based optimization techniques, highlighting the promise of AI methods, though it suggests hybrid algorithms as a more reliable solution. The review emphasizes the growing use of optimization tools like HOMER and calls for increased focus on sources like hydro, geothermal, and biomass. Jinpeng Ma’s study [3] focuses on optimizing hybrid solar energy systems with energy storage for green buildings, comparing PV/battery and off-grid PV/hydrogen systems. Using particle swarm optimization (PSO) and real meteorological data, the research finds that the PV/battery system is more economical than the PV/hydrogen system in remote areas, especially as reliability decreases and interest rates increase. The study demonstrates the effectiveness of PSO in minimizing total system costs and highlights the benefits of PV/battery systems in reducing overall capacity requirements. Yigeng Huangfu’s study [4] proposes a subsection bi-objective optimization dynamic programming strategy for managing a photovoltaic/battery/hydrogen hybrid energy system (HES). The strategy combines rule-based judgments with dynamic programming to optimize power distribution, addressing the intermittency of renewables. A multi-objective genetic algorithm is used for comparison, with results showing improved photovoltaic utilization and significant fuel economy improvements, demonstrating the effectiveness of the proposed approach in optimizing HES performance. Mohammad Reza Maghami’s paper [5] reviews the challenges faced by hybrid energy systems (HESs), particularly in control, efficiency, and reliability. The study highlights how artificial intelligence (AI) techniques have been applied to improve HES performance, addressing issues like classification, forecasting, optimization, and control. It also explores energy management, system sizing, and storage, offering insights into current challenges and suggesting areas for future research. A. Mohammed’s paper [6] focuses on optimizing hybrid renewable energy systems for powering a large-scale desalination plant in Jubail, Saudi Arabia. Using the HOMER tool, it evaluates different combinations of photovoltaic, wind, and grid systems, with wind–grid identified as the most cost-effective and environmentally friendly solution. The study highlights the system’s potential to reduce emissions and provide energy security for desalination processes.

Hanyou Liu’s study [7] proposes a hierarchical distributed MPC method for hybrid energy management, combining forward dynamic programming (FDP) and MPC to optimize power allocation. The method improves power tracking, voltage stability, and energy savings, demonstrating its effectiveness in real-time hybrid power system control. Javier Arroyo’s study [8] introduces a reinforced Model Predictive Control (RL-MPC) algorithm for building energy management, merging the strengths of MPC and reinforcement learning (RL). The RL-MPC approach combines state estimation, dynamic optimization, and learning, outperforming pure RL by meeting constraints while maintaining MPC-level performance in uncertain environments. Keke Huang’s study [9] proposes an LSTM-based MPC method for multimode process control, using LSTM networks to predict system behavior without requiring mode-switching strategies. By integrating this with an MPC framework and adaptive gradient descent for optimization, the method improves control accuracy and reduces overshoot by 10% compared to traditional learning-based MPC methods, demonstrating enhanced stability and practical applicability. Jingyang Yan’s work [10] introduces an LSTM-based MPC for piezoelectric motion stages in autofocus systems, addressing hysteresis effects through a learning-based model that integrates focus measurements. The proposed method optimizes focus positioning and incorporates an enhanced backpropagation algorithm to speed up the MPC process, reducing autofocus time by at least 30% compared to traditional rule-based and other learning-based methods. Hong Wang’s work [11] proposes a low-risk, high-efficiency path planning method for autonomous vehicles, leveraging LSTM-based trajectory prediction combined with vehicle-to-vehicle (V2V) information. The approach includes a risk assessment and mitigation algorithm, validated in active lane-change and collision-avoidance scenarios. Results show improved risk mitigation and driving efficiency compared to constant-velocity and nonlinear input-output prediction methods, particularly for sudden changes in velocity and trajectory. Rafn Gunnarsson’s work [12] introduces a novel LSTM architecture, the Complete Remaining Trace Prediction (CRTP) LSTM, designed to predict both the complete remaining trace and runtime of ongoing cases. Unlike previous models that predict single events, the CRTP-LSTM utilizes all available event attributes, making it data-aware and more robust. Experimental results show that CRTP-LSTM outperforms other approaches in trace and runtime prediction, highlighting the benefits of using comprehensive event information for more accurate predictions. András Sasfi’s work [13] introduces a robust adaptive MPC framework for nonlinear continuous-time systems with parametric uncertainty and disturbances. By utilizing control contraction metrics (CCMs), a homothetic tube is formed around nominal predictions, ensuring all uncertain trajectories are contained. The approach also incorporates model adaptation via set-membership estimation, reducing conservatism and guaranteeing robust constraint satisfaction. The framework allows for online optimization over parameters and is demonstrated through a numerical example involving a planar quadrotor, highlighting its effectiveness in handling uncertainty.

1.3. Contributions of the Paper

1. Novel Control Framework: This paper introduces a unique control framework that combines Model Predictive Control (MPC), Long Short-Term Memory (LSTM) networks, and Kolmogorov–Arnold Networks (KANs) for optimizing hybrid energy systems. This integration enhances the control system’s ability to handle complex dynamics and environmental uncertainties.

2. Application to High-Altitude Wind Energy: The proposed framework is applied to a high-altitude wind energy work umbrella, demonstrating its effectiveness in managing wind and solar energy sources. It successfully maintains battery State of Charge (SOC) stability and improves the system’s operational efficiency and endurance.

3. Enhanced Predictive Accuracy and Control: By leveraging LSTM for environmental forecasting and a KAN for modeling nonlinearities, the framework significantly boosts predictive accuracy and control performance. The paper provides a thorough evaluation, showing the framework’s advantages over traditional control methods and contributing to the advancement of renewable energy management.

1.4. Structure of the Paper

The rest of the paper is organized as follows. Section 2 presents the design of the MPC-LSTM-KAN controller, detailing its integration of Model Predictive Control, Long Short-Term Memory networks, and Kolmogorov–Arnold Networks. Section 3 applies this controller to a high-altitude wind energy work umbrella system, demonstrating its effectiveness in optimizing a hybrid energy system. Section 4 analyzes experimental results, focusing on performance metrics like SOC stability and energy efficiency. Section 5 concludes the paper by summarizing key findings and suggesting future research directions to enhance the controller’s application in renewable energy systems.

2. Methodology

2.1. MPC

Model Predictive Control (MPC) is a powerful and flexible control strategy widely used in various engineering applications [14], particularly where system dynamics are complex and subject to constraints. In this study, MPC is employed as the core component of the control strategy for optimizing the operation of the high-altitude wind energy work umbrella system. The design and implementation of the MPC framework are tailored to meet the specific requirements of this hybrid energy system, with a focus on maintaining battery State of Charge (SOC) stability and ensuring efficient energy utilization.

MPC operates by solving an optimization problem at each control step over a finite prediction horizon. The control inputs are calculated by minimizing a cost function that typically reflects the deviation of the system’s predicted behavior from the desired reference trajectory, (Figure 1) as well as the energy consumption or other performance metrics. In this context, the MPC model is designed to minimize the error between the predicted and desired SOC while also considering constraints such as the physical limits of the battery, the power output of the energy generation hardware, and the operational requirements of the work umbrella.

The prediction model used within the MPC framework is based on a state–space representation of the system, which captures the dynamic behavior of the hybrid energy model, including wind and solar inputs, battery dynamics, and energy consumption. This model allows the MPC to anticipate future states of the system and adjust the control inputs accordingly to optimize performance over the entire prediction horizon.

$$\dot{x}\left(k\right)=Ax\left(k\right)+Bu\left(k\right)$$

(1)

$$x\left(k\right)=\left[\begin{array}{c}SOC\left(k\right)\end{array}\right]$$

(2)

$$u\left(k\right)=\left[\begin{array}{c}{P}_{Wind}\left(k\right)\\ P_{Solar}\left(k\right)\\ I_1\left(k\right)\\ I_2\left(k\right)\\ I_3\left(k\right)\end{array}\right]$$

(3)

$$A=\left[\begin{array}{c}1\end{array}\right]$$

(4)

$$\begin{array}{cc}\hfill B& ={\displaystyle \frac{{\eta }_{in}\Delta t}{C_n}}{\left[\begin{array}{ccccc}1& 1& 0& 0& 0\end{array}\right]}^T\hfill \\ \hfill & -{\displaystyle \frac{{\eta }_{out}{V}_{bat}\Delta t}{C_n}}{\left[\begin{array}{ccccc}0& 0& 1& 1& 1\end{array}\right]}^T\hfill \end{array}$$

(5)

In the MPC model, the state variable x represents the current State of Charge (SOC) of the battery [15]. The control input vector u consists of five parameters: wind power generation $P_{\mathrm{Wind}}$; photovoltaic power generation $P_{\mathrm{Solar}}$; and the electrical currents $I_1$, $I_2$, and $I_3$ for the three DC motors. These parameters serve as the key inputs to the MPC controller, where $P_{\mathrm{Wind}}$ and $P_{\mathrm{Solar}}$ represent the uncontrollable renewable energy inputs, while the motor currents $I_1$, $I_2$, and $I_3$ are controllable inputs. The objective of MPC is to optimize these inputs to maintain the SOC within a desired range while also considering the energy demand of the motors. This formulation of B captures the relationship between the system’s inputs and the change in the battery’s State of Charge. The first term accounts for the positive energy input from wind power and solar power, scaled by the efficiency of power input ${\eta }_{\mathrm{in}}$, time step $\Delta t$, and battery capacity $C_n$. The second term reflects the energy consumption by the three DC motors, scaled by the output efficiency ${\eta }_{\mathrm{out}}$, battery voltage $V_{\mathrm{bat}}$, and time step $\Delta t$. The structure of B distinguishes between the renewable energy sources (wind and solar) and the energy demands from the motors, ensuring that the MPC model accurately represents the energy dynamics of the system.

A key advantage of MPC is its ability to handle multivariable control problems with constraints, making it particularly well-suited for managing the dynamic and uncertain nature of renewable energy systems. The MPC controller in this study is designed to operate within predefined constraints, such as maintaining the SOC within an optimal range to prevent overcharging or deep discharging of the battery. These constraints are crucial for ensuring the longevity and reliability of the system.

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& 0\leqq x\leqq 100\hfill \\ & 0\leqq I_{1,2,3}\leqq 10\hfill \\ & P_{Wind}\ge 0\hfill \\ & P_{Solar}\ge 0\hfill \\ & I_{1,2,3}=\left\{\begin{array}{cc}{I}_{1,2,3}\hfill & ,x\ge 10\hfill \\ 0\hfill & ,x<10\hfill \end{array}\right.\hfill \end{array}\right.\end{array}$$

(6)

In (6), We constrain the battery SOC between 0 and 100, representing the actual remaining battery capacity from 0% to 100%. The motor driver can withstand a maximum current of 15 A; however, for safety, we limit the motor current to the range of 0 to 10 A. Wind and photovoltaic power, as positive inputs, are always non-negative ($\ge 0$). Additionally, when the battery charge falls below 10%, we suspend device operation, setting the motor current to 0A to reserve power for the system’s main control and communication modules.

To enhance the predictive accuracy and robustness of MPC, it is integrated with Long Short-Term Memory (LSTM) networks, which are capable of capturing and predicting temporal dependencies in the environmental data. This hybrid MPC-LSTM approach leverages the strengths of both methods, enabling the controller to make more informed decisions by accurately predicting future environmental conditions and system states.

2.2. LSTM

Long Short-Term Memory (LSTM) networks (Figure 2) are a specialized type of recurrent neural network (RNN) that excel in modeling and predicting sequences over time, particularly in scenarios where long-term dependencies are critical. In the context of this study, LSTM networks are integrated with Model Predictive Control (MPC) to enhance the predictive capabilities of the control strategy for the high-altitude wind energy work umbrella system.

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill i_t& =\sigma \left(W_{xi}{x}_t+W_{hi}{h}_{t-1}+b_i\right)\hfill \\ \hfill o_t& =\sigma \left(W_{xo}{x}_t+W_{ho}{h}_{t-1}+b_o\right)\hfill \\ \hfill f_t& =\sigma \left(W_{xf}{x}_t+W_{hf}{h}_{t-1}+b_f\right)\hfill \\ \hfill c_t& =f_t{c}_{t-1}+i_{t}tanh\left(W_{xf}{x}_t+W_{hf}{h}_{t-1}+b_f\right)\hfill \\ \hfill h_t& =o_{t}tanh\left(c_t\right)\hfill \end{array}\right.\end{array}$$

(7)

In (7), $i_t$ represents the input gate, $o_t$ denotes the output gate, and $f_t$ is the forget gate. These gates are parameterized by weight matrices $W_{xi}$, $W_{xo}$, and $W_{xf}$ and biases $b_i$, $b_o$, and $b_f$. The cell state $c_t$ is updated by combining the previous cell state $c_{t-1}$ weighted by the forget gate $f_t$ and the new information modulated by the input gate $i_t$. The hidden state $h_t$ is computed using the output gate $o_t$ and the current cell state $c_t$. This formulation allows LSTM networks to effectively capture long-term dependencies and temporal dynamics by maintaining and updating both the cell state and hidden state through gated mechanisms.

LSTM networks are designed to overcome the limitations of traditional RNNs, particularly the vanishing gradient problem, which hinders the ability of standard RNNs to learn from long-term dependencies in the data. This is achieved through a unique architecture that includes memory cells and three key gates: the input gate, forget gate, and output gate. These gates regulate the flow of information into and out of the memory cells, enabling the LSTM network to retain relevant information over extended periods and discard irrelevant data.

In this study, the LSTM network was employed to model and predict the environmental conditions that directly influence the operation of the hybrid energy system, such as wind speed, solar irradiance, and other weather-related variables. The ability of LSTM networks to capture temporal patterns in these data are crucial for accurately forecasting future environmental conditions, which in turn informs the decisions made by the MPC framework.

The LSTM network is trained on historical environmental data, allowing it to learn the complex temporal relationships and dependencies inherent in the data. Once trained, the LSTM network generates predictions over the same time horizon used by the MPC. These predictions are then used by the MPC to anticipate future states of the system and optimize control inputs accordingly. By incorporating LSTM-based predictions, MPC is better equipped to handle the inherent variability and uncertainty in renewable energy generation, leading to more reliable and efficient system operation.

The integration of LSTM with MPC leverages the strengths of both approaches: the predictive accuracy of LSTM in modeling time-dependent environmental factors and the optimization capabilities of MPC in managing system dynamics and constraints. This hybrid MPC-LSTM approach is particularly beneficial in the context of high-altitude wind energy generation, where environmental conditions can change rapidly and unpredictably, making accurate forecasting essential for optimal control.

2.3. KAN

During the learning process, the weights of the network are optimized while the activation functions for each node remain fixed. In contrast to Multi-Layer Perceptrons (MLPs), the Kolmogorov–Arnold Network (KAN) learns the activation functions for each node individually [16]. Unlike MLPs, which utilize linear weights, the KAN employs univariate, spliced functions. To achieve high-precision function approximation, the KAN model must learn to approximate both the univariate functions and their compositional structure. Figure 3 illustrates the KAN structure and the learnable activation functions as proposed in [17]. This paper adopts a similar KAN architecture specifically tailored for estimating the wind and solar power generation.

$${\displaystyle f\left(x\right)=f\left(x_1,\cdots ,x_n\right)=\sum _{q=1}^{2n+1}{\Phi }_q\left({\displaystyle \sum _{p=1}^n{\varphi }_{q,p}\left(x_p\right)}\right)}$$

(8)

Let ${\phi }_{q,p}:[0,1]\to \mathbb{R}$ and ${\Phi }_q:\mathbb{R}\to \mathbb{R}$. The activation value of a single neuron in the $(l+1)$-th layer, denoted as neuron j, is computed as the sum of all post-activation values from the preceding layer l, where i refers to the neurons in the l-th layer [18].

$${\displaystyle x_{l+1,j}=\sum _{i=1}^{n_l}{\varphi }_{l,j,i}\left(x_{l,i}\right),j=1,\cdots ,n_{l+1}}$$

(9)

Previous studies [19,20] have identified the Kolmogorov–Arnold Network (KAN) as highly suitable for time-series data characterized by significant interdependencies. These studies highlight the KAN’s superior interoperability and accuracy in predicting time-series data. Additionally, compared to similar architectures applied in estimation and prediction tasks involving time-series data, such as the MLP and LSTM, the KAN utilizes fewer parameters while maintaining comparable or even higher accuracy. In this work, a customized KAN model is employed to forecast wind and solar power generation within a single network.

2.4. MPC-LSTM-KAN

In the proposed methodology, the LSTM-KAN architecture enhances traditional LSTM by replacing its linear output layer with a Kolmogorov–Arnold Network (KAN) (Figure 4). This modification leverages KAN’s capability to approximate complex functions more efficiently, allowing the model to capture nonlinear relationships within the control system more effectively. The integration of a KAN into the LSTM framework is designed to improve the accuracy and adaptability of the system’s predictions, ensuring more precise control in dynamic environments ([21]).

The formula is as follows:

$${\displaystyle h_t=\sum _{q=1}^{2n+1}{\Phi }_q\left({\displaystyle \sum _{p=1}^n{\varphi }_{q,p}\left(o_t{tanh\left(c_t\right)}_p\right)}\right)}$$

(10)

The LSTM-KAN framework utilizes historical environmental data, including wind speed, solar irradiance, and precipitation, as input variables (

Table 1. Partial environmental data.

Date	Time	T (°C)	RH (%)	PRCP (mm)	WS(m/s)	SR (${\mathbf{W}/\mathbf{m}}^2$)
2023-06-01	02:00	21.7	95.4	0	3.8	0
2023-06-01	03:00	21.7	95.3	0.1	3.9	0
2023-06-01	04:00	21.7	95.4	0.1	3.9	0
2023-06-01	05:00	22.1	93.9	0.1	4.4	50.5
2023-06-01	06:00	23.2	90.6	0.1	4.5	200.9
2023-06-01	07:00	25	86	0.1	4.3	352.1
2023-06-01	08:00	26.5	81	0.1	4.1	496.3
2023-06-01	09:00	27.6	77	0.1	3.9	590
2023-06-01	10:00	28.4	73.5	0.1	3.8	673
2023-06-01	11:00	29.3	69.2	0.1	3.5	811.7
2023-06-01	12:00	30.1	65.4	0.1	3.3	869
2023-06-01	13:00	30.6	63	0.1	3.3	807.5
2023-06-01	14:00	30.6	62.7	0.1	3.5	702.2
2023-06-01	15:00	29.9	64.8	0.1	3.9	465.9
2023-06-01	16:00	28.8	69.3	0.1	4.6	300.3
2023-06-01	17:00	27.3	76	0.1	5.5	144.7
2023-06-01	18:00	25.6	83.3	0.1	5.9	25.1
2023-06-01	19:00	24.3	88.2	0	5.8	0
…	…	…	…	…	…	…

) to predict future trends in wind power and solar power generation. By employing LSTM for temporal sequence learning, combined with the Kolmogorov–Arnold Network (KAN) for enhanced nonlinear mapping, the model improves the accuracy of these predictions. This refined prediction of future wind speeds and solar irradiance leads to more reliable forecasts of potential power generation, enabling a more precise estimation of future increases in the system’s State of Charge (SOC). The integration of these two methods creates a robust approach to handling the variability and uncertainty inherent in renewable energy sources, thereby facilitating more efficient energy management.

In this study, following the prediction of future power generation using the LSTM-KAN model, the projected wind and photovoltaic (PV) generation outputs were incorporated as uncontrollable inputs into the Model Predictive Control (MPC) framework. Concurrently, the real-time states of the three motors were treated as controllable inputs, which were fed into the MPC for optimization. The MPC algorithm then calculated an optimal control strategy based on these inputs, taking into account both the anticipated fluctuations in renewable power generation and the current operational conditions of the motors. The optimized control signals were subsequently applied to the system, guiding motor operation and power distribution in a manner that maximized efficiency. The optimization process was performed in a receding horizon fashion, ensuring continuous recalibration of control actions as new environmental data became available. This rolling optimization approach enhances the system’s ability to adapt to changing external conditions, improving the stability and responsiveness of energy management while maintaining a balance between generation and consumption to ensure optimal system performance. Figure 5 briefly illustrates this process:

By incorporating the predicted wind and photovoltaic (PV) power outputs as uncontrollable inputs, the methodology effectively addresses two critical aspects: it mitigates the unpredictability associated with treating these predictions as disturbance inputs, and it circumvents the need for controllability that would be required if they were considered controllable inputs. This approach allows the Model Predictive Control (MPC) framework to handle these inputs with greater accuracy and reliability, as the predictions of renewable energy generation are directly integrated into the optimization process without introducing the complexities associated with their inherent variability and control requirements. Consequently, this strategy enhances MPC’s ability to adapt to forecast changes in power generation while simplifying the control process and improving overall system performance.

3. Case Study

The MPC-LSTM-KAN framework was validated using a high-altitude wind energy generation umbrella controller as a case study.

3.1. Work Umbrella Model

This section introduces the detailed model of the work umbrella (Figure 6), which encompasses various components essential for hybrid energy generation.

The hybrid energy model integrates wind and solar energy sources to power the umbrella’s operations. The battery model is designed to simulate the charge–discharge cycles, capturing the dynamics of energy storage and usage within the system. Additionally, the generation hardware, including the wind turbines and solar panels, is described alongside the key parameters governing their operation, such as efficiency rates, power output, and environmental dependencies. The power consumption model further details the energy requirements of the work umbrella during various operational phases, such as opening, closing, and maintaining stability, providing a comprehensive view of the energy dynamics involved.

3.2. Dataset

The environmental data utilized in this study is sourced from NASA, covering the period from 1 June 2023 to 31 July 2024. This dataset includes temperature, humidity, precipitation, surface wind speed, and surface radiation for Fengxian District, Shanghai, China. For training the LSTM-KAN model, data from June 2023 to June 2024 were used, while the data from July 2024 served as the testing set. Table 1 provides a subset of this dataset, and Figure 7 presents a more comprehensive visualization of the data.

Shanghai is located on the eastern coast of China and has a typical subtropical monsoon climate, characterized by distinct seasonal patterns. The annual temperature range is between −10 °C and 40 °C, with hot and humid summers and relatively cold winters. The region experiences high humidity, with an average annual relative humidity of around 70%, which can exceed 80% during the summer months. Precipitation is concentrated during the plum rain season and the summer typhoon period, with an annual rainfall of approximately 1100–1200 mm. Wind speeds are generally moderate, averaging 3–5 m/s throughout the year, with extreme winds rarely exceeding 16 m/s. Solar irradiance varies seasonally, reaching up to 1031 W/m² on clear summer days.

The model developed in this study is specifically tailored to the climatic characteristics of Shanghai. For application in regions with different climatic features, it is recommended to retrain the model using local data to ensure optimal performance. The configuration of photovoltaic panels and wind turbines significantly affects the overall energy generation of the system. Therefore, when utilizing this model, it is advisable to adjust the proportion of wind and solar energy generation according to local geographical conditions. For instance, in areas with relatively strong wind resources, increasing the number of wind turbines may enhance the system’s overall efficiency.

3.3. Controller Model

The MPC framework operates based on a state–space model, as detailed in Equations (1) through (5). In this context, the state–space matrix captures the dynamics of the system, with the state variable x representing the battery’s State of Charge (SOC). The control inputs u include wind power generation $P_{\mathrm{wind}}$, solar power generation $P_{\mathrm{solar}}$, and the currents from the three motors $I_1,I_2,I_3$. These inputs are used to optimize the system’s performance while maintaining operational constraints on SOC and motor currents. In the equation for the matrix B (5), the efficiency values ${\eta }_{\mathrm{in}}$ and ${\eta }_{\mathrm{out}}$ are both set to 0.9, representing 90% efficiency for both the energy input and output processes. The battery voltage, $V_{\mathrm{bat}}$, is specified as 24 V, chosen to meet the operational requirements of the DC motors in the system. The battery pack consists of 18,650 cells arranged in a 4-parallel, 7-series configuration. The total battery capacity, $C_n$, is set to 14 Ah, reflecting the combined capacity of the parallel-connected cells. This configuration ensures efficient energy supply and utilization for the system while maintaining compatibility with the motor specifications.

The uncontrollable inputs, $P_{\mathrm{wind}}$ and $P_{\mathrm{solar}}$, are derived using Equations (11) and (12):

$$P_{Wind}={\displaystyle \frac{1}{2}}\rho A_{wind}{v}^3{C}_p$$

(11)

$$P_{Solar}=E{A}_{solar}{\eta }_{solar}{P}_R$$

(12)

The wind power generation Formula (11) calculates the electrical power produced by a wind turbine. In this equation, the air density $\rho$ is set to $1.225{\mathrm{kg}/\mathrm{m}}^3$, representing typical conditions at sea level. The swept area $A_{\mathrm{wind}}$, through which the wind interacts with the turbine blades, is taken as $2{\mathrm{m}}^2$. Wind speed $vs.$ significantly influences power output, as the power generated is proportional to the cube of the wind speed $v^3$. The power coefficient $C_p$, set to $0.3$, reflects the turbine’s efficiency in converting wind energy into mechanical energy, with 30% of the wind’s kinetic energy being harnessed for power generation.

The solar power generation Formula (12) is used to estimate the electrical power output from a photovoltaic (PV) system. In this equation, E represents the solar irradiance, or the power per unit area received from the sun, measured in ${\mathrm{W}/\mathrm{m}}^2$. The total area of the solar panels, $A_{\mathrm{solar}}$, is set to $0.64{\mathrm{m}}^2$, corresponding to the physical size of the PV panels. The photovoltaic conversion efficiency, ${\eta }_{\mathrm{solar}}$, is set to 0.2, indicating that 20% of the solar energy received is converted into electrical energy. The performance ratio $P_R$, taken as 0.75, accounts for losses in the PV system, such as temperature effects and other inefficiencies, meaning 75% of the theoretically available power is effectively utilized.

The LSTM-KAN configuration follows the parameters outlined in

Table 2. LSTM-KAN configuration.

Parameter	Value
Epochs	100
LSTM-layers	512-512
LSTM-Input-size	5
LSTM-sequence	24
KAN-layers	4-16-4

:

4. Results and Discussion

4.1. Simulation Environment

The simulation environment is outlined as follows:

• Simulation Software: Python 3.12;
• Development Environment: PyCharm 2023.3.4 (Community Edition);
• Operating System: Windows 10 (Version 22H2);
• Processor: Intel i5-12400F;
• RAM: 64 GB;
• Storage: 1 TB SSD;
• Graphics Card: NVIDIA GeForce GTX 3060 (Driver Version: 537.13).

4.2. LSTM-KAN

To validate the effectiveness of the LSTM-KAN model, we compared its performance against a traditional LSTM model configured with 16 layers and 512 dimensions per layer. The evaluation focuses on both predictive accuracy and computational efficiency, as shown in Figure 8:

We observed that by integrating a simple KAN network with three layers and a total of 24 neurons, the LSTM model required only two layers with 512 dimensions each to achieve a lower loss value compared to the standalone 16-layer LSTM model. This demonstrates that the KAN network significantly enhances the LSTM’s capability, enabling more efficient and accurate predictions with a much smaller architecture.

During the initial 30 epochs, the loss value of the LSTM-KAN model was higher than that of the standalone LSTM model. This slower convergence can be attributed to the KAN’s need to accurately learn and approximate the complex nonlinear relationships present in the data. In the early stages of training, the KAN focuses on refining its modeling of these nonlinear features, leading to a more gradual reduction in loss. However, after the first 30 epochs, the LSTM-KAN model’s loss stabilized and consistently outperformed the LSTM model. This precise nonlinear modeling enhances the model’s ability to provide higher accuracy and stability in long-term predictions. Although the training process may take slightly longer, the overall improvement in system performance is well worth the extended training time. Furthermore, it is noteworthy that despite its smaller size, the LSTM-KAN model achieves superior performance, demonstrating that the integration of the KAN network enables more precise and efficient learning, resulting in a more compact and effective model.

Table 3. Algorithm performance: MAE and RMSE.

Algorithm	MAE	RMSE
LSTM	0.1967	0.2651
LSTM-KAN	0.1894	0.2527
LSTM-MLP	0.1952	0.2615

presents a comparative analysis between LSTM-KAN, LSTM, and LSTM-MLP. The results demonstrate that the LSTM enhanced with the KAN outperforms both the standard LSTM and the LSTM-MLP configurations.

4.3. LSTM-KAN Prediction

Using the environmental data from June 2023 to June 2024 as the training set, the LSTM-KAN model was trained to predict future wind and solar power generation based on historical data such as wind speed, solar irradiance, precipitation, temperature, and humidity. The trained model was then tested on the data from July 2024, which were not used during training, to evaluate its performance. After training the model on the historical data, the prediction results were as follows. Figure 9 illustrates the predicted and actual wind speeds for July 2024.

Figure 10 presents the predicted and actual solar irradiance values for the same period.

These figures demonstrate the performance of the LSTM-KAN model in accurately forecasting environmental variables critical for renewable energy generation, such as wind and solar power. The close alignment between the predicted and actual values in both cases highlights the model’s effectiveness in capturing complex temporal patterns in the data.

The LSTM-KAN model, trained on environmental data from June 2023 to June 2024, successfully forecasts wind and solar power generation for July 2024. The predicted values closely match the actual values, capturing both the trend and variations in renewable energy inputs. Key performance metrics, such as Mean Absolute Error (MAE) and Root Mean Square Error (RMSE), were computed to assess the accuracy of the model (

Table 4. Prediction outcome evaluation.

Prediction Parameter	MAE	RMSE
Solar irradiation	0.298080	0.292719
Wind speed	0.217400	0.288138

).

These accurate predictions lay a solid foundation for the subsequent MPC optimization process, where reliable future renewable energy inputs are essential for effective control and energy management in the system.

4.4. MPC-LSTM-KAN Simulation

In this subsection, we demonstrate the integration of LSTM-KAN predictions with the MPC control framework to optimize the operation of the system. The wind and solar power generation predictions from the LSTM-KAN model are treated as uncontrollable inputs in MPC, while the current states of the three DC motors serve as controllable inputs. MPC optimization is executed in a receding horizon fashion, with each step utilizing the predicted future renewable energy inputs to ensure system efficiency and stability.

The simulation was conducted using the proposed MPC-LSTM-KAN controller, and the system’s performance was evaluated in terms of battery SOC stabilization, motor speed control, and overall energy balance. Figure 11 presents the SOC levels, motor currents, and renewable energy inputs during the simulation, highlighting the effectiveness of the proposed control strategy.

After 500 h, the battery’s State of Charge (SOC) stabilizes, primarily due to a significant increase in wind power. Throughout the optimization process, the battery SOC was optimized to stay above 60% to ensure extended device endurance. Our goal was to maintain the SOC above this threshold while maximizing energy consumption to reduce wind and solar curtailment rates. Although there were instances when the SOC dipped below 60%, these dips were a result of insufficient power generation. Despite these occasional fluctuations, the overall SOC was maintained at around 60%.

The three motor currents (I1, I2, I3) shown in the second plot were consistently above 3 A, ensuring that the system remained fully operational. There were no instances of shutdown due to low battery levels, highlighting the effectiveness of MPC-LSTM-KAN optimization in managing energy inputs and motor control.

In contrast, a system operating without the optimization framework would encounter difficulties under identical conditions. As illustrated in Figure 12, the non-optimized system ceased functioning after 22 h due to insufficient battery charge.

5. Conclusions

In this study, we introduce the LSTM-KAN method for short-term photovoltaic power generation forecasting. This study utilizes the LSTM-KAN method for short-term forecasting of photovoltaic power generation. Utilizing environmental data from a designated period in Shanghai, China, this approach enhances the learning and prediction of photovoltaic power output. Compared to the standard LSTM model, LSTM-KAN demonstrates a faster convergence rate during training and achieves fewer prediction errors. The study findings provide refined technical support for short-term power regulation and introduce advanced methodologies for forecasting short-term time-series data.

The successful implementation of the MPC-LSTM-KAN framework underscores its potential for improving energy management in high-altitude wind energy systems. The ability to predict future power outputs with high accuracy and incorporate these predictions into the MPC optimization process is crucial for maintaining system stability and efficiency.

In practical terms, the framework’s ability to effectively manage energy resources and maintain SOC within desired ranges could lead to increased system endurance and reliability. This has significant implications for the operation of high-altitude wind energy systems, as it enhances their capability to respond to fluctuating environmental conditions and ensures more efficient use of available energy.

The results also suggest that integrating advanced prediction models like LSTM-KAN into MPC frameworks can provide substantial benefits in terms of control accuracy and system performance. Future work could explore further refinements of the LSTM-KAN model and its integration with other optimization techniques to enhance the robustness and efficiency of energy management systems.

When applying this model to other hybrid energy systems, adjustments are required. The system inputs consist of uncontrollable yet predictable environmental data and controllable system power. To ensure the model’s applicability, the input environmental data should be adapted to predict energy generation specific to the new context, and the system model within the MPC controller must be replaced with one that accurately reflects the characteristics of the target system, including the appropriate state–space matrices, cost function, and constraints.

In future research, the model’s applicability will be enhanced by integrating additional renewable energy sources, thereby increasing the diversity of hybrid energy systems. Furthermore, plans include scaling the model to accommodate larger energy networks, such as regional and national grids, which will address the complexities inherent in energy management at this broader scale. Additionally, dynamic elements will be incorporated, including frequent updates of environmental data and the implementation of real-time control mechanisms, to improve system responsiveness and adaptability to rapidly fluctuating energy demands.