Introduction

Apple is one of the most widely cultivated and economically valuable fruits worldwide, highly favored by growers due to its strong adaptability to various environmental conditions1. However, cold spells in late spring often occur around the flowering period of apple trees. This abnormal climatic phenomenon leads to a sudden drop in temperature, which can cause irreversible damage to flower buds and young shoots. Consequently, both the yield and quality of apples are significantly affected, resulting in substantial economic losses for growers. These impacts pose serious challenges to the sustainable development of agricultural production2,3. As a result, cold spells in late spring are globally recognized as one of the major agro-meteorological disasters4. For instance, in 2018, a cold spell in Luochuan County, Shaanxi Province, China, led to an approximate 40% reduction in apple production across the region5. Similarly, on April 19, 2017, a widespread cold spell in Europe caused an estimated 24% decline in apple yields6. In recent years, the increasing frequency of cold spells has resulted in substantial yield losses—and even total crop failures—in many orchards, inflicting severe economic damage on fruit growers. Against this backdrop, growing attention has been directed toward the development of effective solutions for mitigating the impact of cold spells. Among various approaches, the design of a frost prevention strategy that is efficient, cost-effective, and environmentally sustainable is of particular importance. Currently, frost protection methods are generally classified into two categories: passive measures (e.g., plastic film coverage, delayed flowering techniques) and active measures (e.g., artificial smoke, frost fans). Due to the high cost, limited effectiveness, and environmental concerns associated with passive methods, they have not been widely adopted7. As a result, most orchards rely primarily on active frost prevention strategies.

Currently, active frost prevention measures mainly include artificial smoke, frost fans, hot air blowers, sprinkler systems, and heating pipelines. Jiajun Qin8 designed and implemented a LoRa-based frost prevention system for mountainous orchards, which raises the orchard temperature and reduces heat loss through the release of artificial smoke. Yi Dai9 conducted a quantitative three-dimensional evaluation of the frost mitigation effect of frost fans and concluded that such fans significantly increase the temperature in the canopy airspace, thereby alleviating frost damage. Hecheng Wu10 developed a self-heating frost fan tailored for hilly and mountainous agricultural regions and demonstrated in-orchard experiments that it can provide effective frost protection within a radius of 10 m. Ricardo Yauri11 proposed an automated system combining machine learning and predictive models, which was applied to sprinkler irrigation systems for frost protection. This approach significantly reduced water consumption while safeguarding crops. Leah Sklar12 introduced a solar-powered hydronic radiant heating system, where water heated by solar energy circulates through pipelines within the orchard, creating convective heat that protects crops from frost injury. Although these frost prevention measures have made significant contributions to mitigating frost damage, each method also presents certain limitations. The use of artificial smoke generated by burning dead branches and leaves is cost-effective, but it poses serious environmental pollution concerns. Frost fans offer broad protective coverage; however, their high installation and maintenance costs make them unaffordable for most small-scale growers, and their effectiveness largely depends on the intensity of the temperature inversion. Hot air blowers, while effective, consume large amounts of energy and entail high operational costs. Sprinkler systems and heating pipelines require extensive infrastructure deployment, leading to increased labor costs. Moreover, the installation of pipelines within the orchard may interfere with other agricultural operations.

In mountainous orchards, artificial smoke remains the dominant method for frost prevention. This is primarily due to the complex and undulating terrain, which reduces the effectiveness of frost fans and other equipment compared to flatland areas, while also significantly increasing installation and maintenance costs. In contrast, artificial smoke is considered a more practical and cost-effective solution for such environments13. However, conventional smoke generation methods raise serious environmental concerns, as they typically rely on diesel fuel and burning of plant debris, which emit large amounts of particulate matter as well as harmful gases such as CO, SO₂, and NOₓ during combustion14.

Reducing agricultural production costs can significantly increase growers’ economic returns and encourage the adoption of new technologies, thereby promoting agricultural modernization15. In mountainous orchards, the complex terrain substantially raises the installation and wiring costs of smoke machines, making the optimization of their layout essential. Optimizing smoke machine deployment involves addressing four key objectives: smoke coverage area, number of deployed machines, wiring length, and uniformity of smoke distribution. To tackle these challenges, a multi-objective robust optimization framework is developed, which includes orchard discretization, smoke dispersion modeling, an outer-layer search based on the Hippopotamus Optimization Algorithm (HOA), and an inner-layer optimization using a k-node Minimum Spanning Tree (k-MST) model formulated via Mixed-Integer Linear Programming (MILP).

Materials and methods

New anti-frost smoke machine combined with anti-frost system

Yunnan Province in China is endowed with abundant solar energy resources, with an annual total solar irradiation ranging from 1200 kWh/m2 to 1800 kWh/m2. Notably, the northwestern region of Yunnan ranks among the highest in the province in terms of solar resource availability16. In areas with such rich solar radiation, solar thermal systems offer significant advantages over other regions. Therefore, the frost prevention system proposed in this study prioritizes solar energy as its primary power source. Figure 1 presents a simplified schematic of the novel smoke-machine-based frost prevention system. The system operates as follows: it is powered by a solar photovoltaic system that harnesses natural sunlight to achieve energy self-sufficiency. Excess electricity is stored in batteries to ensure uninterrupted power supply during periods of low sunlight or adverse weather conditions. Upon the onset of frost events, the system automatically ignites the smoke agents. The smoke machines then rapidly emit large volumes of smoke, forming a protective layer above the orchard. This smoke layer increases the orchard temperature and reduces heat loss, thereby achieving frost protection and thermal insulation.

Fig. 1
figure 1

Principle and configuration diagram of the orchard anti-frost smoke machine system.

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To address the drawbacks of traditional artificial smoke fuels and ensure orchard ecological health and fruit quality, the smoke agents used in the smoke machines must possess environmentally friendly characteristics. Compared to conventional fuels, the novel eco-friendly fuel is based on a biochar and vegetable oil formulation. This smoke agent replaces diesel with vegetable oil, significantly reducing harmful gas emissions, and substitutes plant debris with vegetable carbon powder, which greatly enhances combustion efficiency. The combustion products mainly consist of CO₂, CO and minimal particulate matter, while the resulting residue can be returned to the field to improve soil fertility17. Compared with traditional fossil fuel-based smoke agents, this formulation effectively achieves a green and sustainable goal characterized by being non-toxic, non-corrosive, and free of acidic gases.

The ejection velocity of smoke is a key factor influencing its diffusion effectiveness. Since the natural diffusion rate of smoke from the smoke agent into the environment is relatively low, relying solely on passive diffusion is insufficient to achieve the desired frost protection. Therefore, this study introduces an auxiliary smoke diffusion device to the smoke machines. The device operates as follows: an air delivery pipe with uniformly distributed small holes is installed beneath the smoke agent, and an axial fan is connected to the pipe. The axial fan continuously draws in external air, which is then forced upward through the small holes, mixing thoroughly with the smoke. This process significantly enhances the diffusion velocity of the smoke.

As illustrated in Fig. 1, the smoke machines are powered by two distinct supply methods. The first method utilizes a regional high-voltage power supply, where the regional high-voltage grid is connected to the smoke machine network via transformers. The transformed electrical energy is distributed through the orchard’s internal power grid to each smoke machine, ensuring a stable power supply for their operation. The second method is a more environmentally friendly photovoltaic (PV) power system, in which each smoke machine is equipped with its own PV generation unit to supply electrical energy.

In the smoke-machine-based frost prevention system proposed in this study, multiple smoke machines operate collaboratively to achieve effective frost protection across large-scale orchards. In designing such a system, it is essential to balance economic cost with frost prevention performance. To this end, an advanced multi-objective robust optimization framework is developed to optimize the system configuration. A schematic diagram illustrating the components of the proposed model is presented in Fig. 2.

Fig. 2
figure 2

Multi-objective optimization flow diagram.

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Case study description

This study selected a typical large-scale mountainous apple orchard located in Ninglang County, Lijiang City, Yunnan Province, China, as the research site (see Fig. 3). The orchard is situated in the northwestern mountainous region of Yunnan, which is one of the main apple-producing areas of the province and also a region frequently affected by frost disasters. Although frost protection measures such as wind machines and sprinkler irrigation have been widely applied in plain orchards, their effectiveness is significantly limited in mountainous orchards due to complex terrain as well as high construction and labor costs. Consequently, traditional manual smoke-based frost protection remains the primary method in this region. In contrast, the integrated frost protection system based on smoke machines proposed in this study offers better economic and environmental performance, making it particularly suitable for application in mountainous orchards.

Fig. 3
figure 3

Panorama and close-up of the orchard.

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Situated at an altitude of approximately 2700 m, the region experiences high levels of solar radiation and large diurnal temperature variations, natural conditions that make it particularly prone to cold spells in late spring. The orchard parameters during the cold spell period are listed in Table 1.

Table 1 Orchard related parameters during frost period.
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Smoke diffusion model based on slope correction

In this study, the Gaussian plume model is adopted as the fundamental model for smoke dispersion. The Gaussian model is a mathematical representation based on Gaussian distribution and has been widely used in recent years for modeling the dispersion of atmospheric pollutants. For example, Deyi Chen18 applied the Gaussian dispersion model to predict tritium concentration near the ground surface and achieved notable results. Wang Yanjiao19 improved the accuracy of pollutant prediction by modifying the dispersion parameters of the Gaussian plume model using a hybrid genetic simulated annealing algorithm. Similarly, Shi Baojun20 employed the Gaussian model to analyze the dispersion characteristics of liquefied petroleum gas (LPG) leakage. The anti-frost smoke machine meets the following assumptions during the smoke release process: (1) The smoke source is a continuous and stable point source. When the smoke particles diffuse in the atmosphere with the wind, the horizontal and vertical concentration distributions obey the Gaussian distribution; (2) The wind speed remains uniform throughout the entire area and constant along the dominant direction; (3) The emission process is under steady-state conditions and the atmospheric environment is relatively stable. The formula for the three dimensional steady state Gaussian plume model is given as follows21:

$$\begin{array}{*{20}c} \begin{aligned} & C\left( {x,y,z} \right) = \frac{Q}{{2\pi \sigma_{y} \sigma_{z} u}}\exp \left( { - \frac{{y^{2} }}{{2\sigma_{y}^{2} }}} \right) \\ & \quad \times \left\{ {exp\left[ {\frac{{ - \left( {z - h} \right)^{2} }}{{2\sigma_{z}^{2} }}} \right] + exp\left[ {\frac{{ - \left( {z + h} \right)^{2} }}{{2\sigma_{z}^{2} }}} \right]} \right\} \\ \end{aligned} \\ \end{array}$$
(1)

where \(\text{C}\) represents the concentration at a given location; \(\text{Q}\) denotes the emission source strength; \(\text{x},\text{ y},\text{ z}\) correspond to the downwind, crosswind, and vertical directions, respectively; \(\text{u}\) is the wind speed at the emission height h; where \({\sigma }_{y}\) and \({\sigma }_{z}\) are the diffusion coefficients in the y and z directions, and the diffusion coefficient is determined by the Pasquill–Gifford equation.

The traditional Gaussian plume model is widely used to describe the dispersion of particulate matter and other pollutants in wind fields. However, this model is based on the assumption of flat terrain, neglecting the influence of topographic slope on dispersion pathways and concentration distribution. In mountainous orchards, where slopes are often significant, the dispersion path of smoke particles is altered during upslope movement due to the combined effects of terrain lifting and gravitational settling. This results in reduced dispersion velocity and a faster decay rate of particle concentration compared to flat terrain22. Therefore, it is necessary to introduce a slope factor \(\eta (\alpha )\), reflecting the influence of slope, into the original dispersion model to correct the concentration. The construction of the slope factor should satisfy the following basic assumptions: (1) When the slope angle \(\alpha = 0\), the correction factor should satisfy \(\eta (\alpha ) = 1\), that is, no additional correction effect is produced under flat terrain; (2) As the slope angle increases, the vertical drift effect of smoke gradually increases, resulting in a decaying trend in the effective coverage concentration. Therefore, the correction factor should be a monotonically decreasing function of the slope angle; (3) Smoke has a relatively gentle attenuation characteristic at a small slope angle, while it exhibits a significant attenuation effect at a large slope angle. This is consistent with the physical law that smoke gradually accumulates and decays at an accelerated rate under the influence of terrain in practice, and the mathematical characteristics of the exponential function match this physical law. Based on the above constraints, the exponential decay form is introduced, and the specific equation is as follows:

$$\begin{array}{c}\eta \left(\alpha \right)=\text{exp}\left(-\gamma \times tan\alpha \right)\end{array}$$
(2)

where α represents the orchard slope, γ represents the empirical weakening coefficient, and \(\gamma =1\) is obtained through CFD simulation and exponential model fitting.

The slope factor \(\eta (\alpha )\) is embedded into the original concentration model, yielding a slope corrected Gaussian dispersion model:

$$\begin{array}{c}{C}_{sl}\left(x,y,z\right)=\eta \left(\alpha \right)\times C\left(x,y,z\right)\end{array}$$
(3)

where \({C}_{sl}\left(x,y,z\right)\) represents the smoke concentration at location \(\left(x,y,z\right)\) under varying slope conditions.

In Eq. (1), the values of \({\sigma }_{y}\) and \({\sigma }_{z}\) depend on meteorological conditions. Two primary methods exist for calculating these dispersion parameters: the Pasquill atmospheric stability classification scheme and empirical formula methods. This study employs the Pasquill stability classification scheme to determine the dispersion parameters. Proposed by British scientist Pasquill in 1961, this method assesses atmospheric stability based on site-specific meteorological conditions during pollutant dispersion, including solar radiation intensity, cloud cover, and cloud height. As documented in23, and considering the typical weather conditions during cold spells in late spring (characterized by calm winds and clear skies at night), stability class E was selected, the values ​​of diffusion parameters \({\sigma }_{y}\) and \({\sigma }_{z}\) are:

$${\sigma }_{y}=0.11x\sqrt{1+0.0004x} {\sigma }_{z}=0.18x\sqrt{1+0.0015x}$$

Substituting the values ​​of diffusion parameters \({\sigma }_{y}\) and \({\sigma }_{z}\) into Eq. (3), we can obtain the following equation by simplifying:

$$\begin{aligned} & C_{sl} \left( {x,y,z} \right) = \exp \left( { - \gamma \times tan\alpha } \right) \times \frac{Q}{{0.124u\sqrt {1 + 0.0019x} }} \\ & \;\; \times \exp \left( { - \frac{{y^{2} }}{{0.0242x^{2} }}} \right) \times \left[ {exp\left( { - \frac{{\left( {z - h} \right)^{2} }}{{0.0648x^{2} }}} \right) + exp\left( { - \frac{{\left( {z + h} \right)^{2} }}{{0.0648x^{2} }}} \right)} \right] \\ \end{aligned}$$
(4)

where \(x\) denotes the downwind diffusion distance (m) from the smoke source.

According to the Gaussian plume diffusion model based on slope correction, the relevant parameter values are substituted into formula (4) and Matlab is used for calculation and simulation. When the smoke diffuses along the x and y directions, the simulation results of the smoke concentration distribution and the smoke diffusion radius on the flat ground are shown in Fig. 4.

Fig. 4
figure 4

Calculation of smoke diffusion surface using the Gaussian plume model based on slope correction.

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Multi-objective optimization algorithm

This section involves multiple variables and parameters. To facilitate clear and unified representation, the key variables and parameters utilized in this study are summarized in Table 2.

Table 2 Explanation of main symbols.
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Orchard discretization

Discretization is a fundamental technique in computational modeling, transforming continuous spatial or temporal domains into finite sets of discrete points or time steps, thereby enabling numerical computation and solution. Building upon established applications—such as Zhichao Han’s24 use of discretization for spatiotemporal optimal trajectory planning in autonomous vehicles, and Yongze Song’s25 discretization of continuous geographical variables (e.g., temperature, precipitation, soil moisture) into discrete levels to partition continuous space into spatial units. This study proposes a novel discretization-based approach to solve the multi-objective nonlinear programming problem for anti-frost smoke machine path planning. The schematic representation of this methodology is illustrated in Fig. 5. First, a set of uniform discrete points that meet the constraints are created in the internal area of ​​the orchard as candidate locations for the anti-frost smoke machine to arrange the heater. The candidate point set is defined as \(V\), where the number of candidate points is not less than the number of layouts. Then, in order to accurately evaluate the coverage effect of the smoke, the tree points in the orchard are used as checkpoints to evaluate the smoke coverage effect. Finally, this paper uses a regular grid division method to spatially discretize the orchard area, and on this basis constructs an undirected weighted graph, defined as \(G=(V,E)\), where \(V\) represents the set of candidate points, \(E\) represents the set of undirected edges with weights, representing the possible routing paths between any two candidate points, and the weight of the edge in the graph is defined by the Euclidean distance between the two points, which is used to model the anti-frost smoke machine routing problem. The advantage of using this discretization method is that by dividing the entire orchard area into a regular grid, the orchard is converted from a continuous space to a finite, regular set of points, so that the difficult nonlinear problem is converted into a linear problem on a finite set, which is convenient for subsequent use of the MILP based k-MST model for solution.

Fig. 5
figure 5

Orchard discretization: (a) Discretized locations of candidate points and checkpoints; (b) Undirected weighted graph.

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Optimization problem

Constructing sustainable agriculture not only has a significant impact on ecological stability and food security, but also serves as an important measure for future societal development26. This paper proposes a multi-objective robust optimization strategy to address the placement and wiring optimization of multiple anti-frost smoke machines in large scale orchards, aiming to achieve effective frost prevention while minimizing agricultural production costs. The optimization is subject to the following constraints: (i) To avoid overlapping the anti-frost smoke machine candidate points with the locations of fruit trees, each candidate point must maintain a distance of at least 1 m from the trees. (ii) Based on the smoke concentration distribution characteristics of an individual anti-frost smoke machine, the spacing between adjacent machines must be 5 m to ensure optimal diffusion. (iii) To ensure the formation of a significant smoke layer within the orchard, the heating intensity at each checkpoint must not fall below a predefined concentration threshold27. The subsequent optimization will proceed under these three constraints.

The frost prevention system involves the following optimization objectives: (i) Minimize the number of anti-frost smoke machines to reduce installation costs; (ii) Maximize the coverage area of the entire anti-frost smoke machine system to ensure the greatest possible protection for the orchard; (iii) Ensure uniform smoke concentration within the orchard to avoid significant concentration imbalances; (iv) Optimize the wiring layout of the anti-frost smoke machines, which require electrical wiring connections, to minimize the cost of cable installation.

$$\left\{ {\begin{array}{*{20}l} {f_{1} = \mathop \sum \limits_{i = 1}^{N} \phi_{i} } \hfill \\ {f_{2} = - \mathop \sum \limits_{s}^{M} \left\| {\left( {C_{s} \ge C_{\min } } \right)} \right.} \hfill \\ {f_{3} = std\left( {C_{1} ,C_{2} , \ldots C_{M} } \right)} \hfill \\ {f_{4} = \mathop \sum \limits_{i,j \in R,i < j} z_{ij} \cdot d_{ij} } \hfill \\ \end{array} } \right.$$
(5)

where The notations used in the objective functions \({f}_{1}\)–\({f}_{4}\), corresponding to optimization goals (i) to (iv) ; \({\upphi }_{i}\in [\text{0,1}]\): Binary decision variable indicating whether a anti-frost smoke machine is deployed at candidate point i; 1 means deployed, 0 means not deployed; \(N\): Total number of candidate deployment points; \({C}_{s}\): Smoke concentration at the s-th checkpoint. \({C}_{min}\): Minimum concentration threshold required at each checkpoint; \(M\): Total number of checkpoints; \(std\): Standard deviation of concentration values across all checkpoints, used to measure uniformity; \(R\): Set of locations where anti-frost smoke machines are actually deployed; \({d}_{ij}\): Euclidean distance between points i and j; \({z}_{ij}\in [\text{0,1}]\): Binary variable indicating whether the undirected edge (i,j) is selected in the wiring network; 1 means selected, 0 means not selected. The negative sign in objective function \({f}_{2}\) is introduced to transform the original goal of maximizing the number of checkpoints meeting the concentration threshold into a minimizable form. This ensures alignment in the optimization direction with the other objective functions in the model that require minimization, thereby conforming to the standard framework of multi-objective optimization algorithms.

The function that minimizes the cost of the entire system is as follows:

$$\begin{aligned} & \begin{array}{*{20}c} {Cost = w_{1} \cdot \mathop \sum \limits_{s = 1}^{M} \lambda_{s} + w_{2} \cdot std\left( {C_{1} ,C_{2} , \ldots C_{M} } \right)} \\ \end{array} \\ & + w_{3} \cdot \mathop \sum \limits_{i = 1}^{N} \phi_{i} + w_{4} \cdot \mathop \sum \limits_{i,j \in R,i < j} z_{ij} \cdot d_{ij} \\ \end{aligned}$$
(6)

where \({w}_{1}\), \({w}_{2}\), \({w}_{3}\), \({w}_{4}\) represent different weight coefficients, \({\lambda }_{s}\in \left\{\text{0,1}\right\}\), indicating whether the concentration reaches the threshold at the checkpoint s, 0 for yes, 1 for no; The weight coefficients \({w}_{1}\), \({w}_{2}\), \({w}_{3}\), \({w}_{4}\) control the importance of each target item in the comprehensive optimization and can be adjusted according to the application scenario. For example, when focusing on cost, \({w}_{3}\) and \({w}_{4}\) can be increased, and when emphasizing coverage and uniformity, the proportion of \({w}_{1}\) and \({w}_{2}\) can be increased.

k-MST modeling based on MILP

To optimize the wiring layout of anti-frost smoke machines in the orchard frost prevention system, this paper employs a Mixed-Integer Linear Programming (MILP) approach to model and solve the k-Node Minimum Spanning Tree (k-MST) problem28. The model aims to construct a minimum spanning tree that connects k selected candidate anti-frost smoke machine deployment points, thereby minimizing the total wiring length and reducing installation costs.

The following introduces the essential constraints required for the k-MST model. An undirected spanning tree containing k nodes must include exactly k − 1 edges. Therefore, the following constraint is added to the k-MST model to ensure that the number of selected edges conforms to the structure of a spanning tree, as shown in Eq. (7). Subsequently, the constraints specifying that exactly k nodes are selected, along with the domain constraints, are defined in Eqs. (8)-(9).

$$\begin{array}{c}\sum_{i,j\in E}{z}_{ij}=k-1\end{array}$$
(7)
$$\begin{array}{c}\sum_{i\in V}^{N}{\zeta }_{i}=k\end{array}$$
(8)
$$\begin{array}{*{20}c} {\zeta i \in \left\{ {0,1} \right\},z_{ij} \in \left\{ {0,1} \right\}} & {\forall i,j \in V} \\ \end{array}$$
(9)

To ensure that the connection structure among the selected anti-frost smoke machine deployment points satisfies the requirements of a minimum spanning tree, this paper incorporates the MTZ (Miller-Tucker-Zemlin) subtour elimination constraints28 into the MILP formulation. MTZ is a classical MILP based method originally developed by Miller et al. for solving the Traveling Salesman Problem (TSP), and is widely used to address cycle elimination in combinatorial optimization problems29. A continuous variable \({u}_{i}\) is introduced to represent the topological order of each candidate point \(i\in V\cup \tau\). By constraining the relative values of these topological indices, the model ensures that no closed cycles are formed in the resulting spanning tree. Specifically, for any directed edge (i, j) in the tree, the condition \({u}_{i}> {u}_{j}\) is imposed to enforce a logical order and eliminate potential loops. The corresponding constraints are formulated as follows:

$$\begin{array}{c}{u}_{i}\ge {u}_{j}+{w}_{ij}-k\left(1-{w}_{ij}\right) \forall i\in V-\left\{j\right\}, \forall j\in V\cup \left\{\tau \right\}, {u}_{\tau }=0\end{array}$$
(10)

where \({\text{w}}_{\text{ij}}\in \{\text{0,1}\}\): Indicates whether the directed edge (i,j) is selected in the MTZ model; \(\tau\): A virtual root node introduced in the MTZ formulation to serve as the origin of the tree; \(k\): The number of nodes to be included in the target tree (this value is provided by the outer layer optimizer); \({u}_{i}\) ​: A continuous variable representing the topological order of node ii, used to enforce acyclic structure and eliminate subtours.

Hippo optimization algorithm

The Hippo Optimization Algorithm (HOA) is a novel population based metaheuristic algorithm proposed by Mohammad Hussein Amiri in February 202430. Inspired by the behavior of hippopotamuses in their natural habitat, HOA simulates the processes of position updating, defensive strategies, and predator avoidance to achieve global search and optimization. HOA is characterized by fast convergence speed and high solution accuracy, and has demonstrated notable superiority in performance when compared to other mainstream population based optimization algorithms31. Similar to traditional optimization algorithms, HOA begins with an initialization phase, where a set of random candidate solutions is generated. The decision variable vectors for the initial population are created using the following formula:

$$\begin{array}{c}{\chi }_{i}:{\chi }_{i,j}=l{b}_{j}+r\cdot \left({ub}_{j}-l{b}_{j}\right) i=\text{1,2},3,,,n;j=\text{1,2},3,,,m\end{array}$$
(11)

where \({\chi }_{i}\): Represents the position of the i-th candidate solution ; \(r\): A random number uniformly distributed in the range [0,1]; \({ub}_{j},l{b}_{j}\)​: Represent the upper and lower bounds of the j-th decision variable, respectively; \(m\): Represents the number of decision variables; \(n\): Represents the number of individuals in the population.

The natural state of the hippo population can be represented by the initial matrix \(\chi\) of the random medium. The specific expression is:

$$\begin{array}{c}\chi =\left[\begin{array}{c}{\chi }_{1}\\ \vdots \\ {\chi }_{n}\end{array}\right]=\left[\begin{array}{ccc}{\chi }_{11}& \dots & {\chi }_{1m}\\ \vdots & \ddots & \vdots \\ {\chi }_{n1}& \dots & {\chi }_{nm}\end{array}\right]\end{array}$$
(12)

(1) Hippopotamus position update.

Since hippopotamuses are social animals, individuals with higher status within the group are responsible for protecting the herd and repelling intruders. When male hippos reach maturity, they are often expelled by the dominant leader of the group. These expelled males have two choices: either attempt to attract females or compete with the dominant leader in order to replace it and gain control over the group. their location information is expressed in Eqs. (13) and (14):

$$\left\{ {\begin{array}{*{20}l} {X_{i,j}^{Mhippo} = X_{i,j} + y_{1} (X^{Dhippo} - I_{1} X_{i,j} ) } \hfill \\ {i = 1,2,3,,,\frac{n}{2};j = 1,2,3,,,m} \hfill \\ \end{array} } \right.$$
(13)
$$\begin{array}{l}h=\left\{\begin{array}{l}{I}_{2}\times {r}_{1}+\left(\sim {p}_{1}\right)\\ 2\times {r}_{2}-1\\ {r}_{3}\\ {I}_{1}\times {r}_{4}+\left(\sim {p}_{2}\right)\\ {r}_{5}\end{array}\right.\end{array}$$
(14)

where: \({X}_{i,j}^{Mhippo}\): Represents the position of the i-th male hippopotamus in the j-th dimension; \({y}_{1}\): A random number uniformly distributed in the range [0,1]; \({X}^{Dhippo}\): Denotes the position of the dominant hippopotamus, i.e., the best solution found in the current iteration; \(h\): Represents a randomly selected scenario or context (e.g., competitive or attractive behavior); \(\sim {p}_{1}\), \(\sim {p}_{2}\)​: Random integers used to control the influence of competing or attractive forces; \({r}_{1}\), \({r}_{2}\), \({r}_{3}\), \({r}_{4}\): Random vectors with each element uniformly distributed in [0,1], used to add randomness in position updates; \({r}_{5}\): A random scalar uniformly distributed in [0,1]; \({I}_{1}\)​,\({I}_{2}\): Random integers within the range [1,2], used for switching between behavioral modes or selecting interaction strategies.

The position of female or immature hippos in the population is expressed by Eqs. (15) to (17). Most immature hippos stay near their mothers, but occasionally some curious ones may leave the group. If \(T\) > 0.6, it indicates that the immature hippo has already moved away from its mother. At this point, if the random number \({r}_{6}\)> 0.5, it means that although the immature hippo has left the group, it is still nearby; otherwise, it has completely left the population.

$$\begin{array}{c}T=\text{exp}\left(-\frac{t}{\tau }\right)\end{array}$$
(15)
$$X_{i,j}^{FBhippo} = \left\{ {\begin{array}{*{20}l} {X_{i,j} + h_{1} \left( {X^{Dhippo} - I_{2} G_{i} } \right)} \hfill & {T > 0.6} \hfill \\ {\left[\kern-0.15em\left[ I \right]\kern-0.15em\right]} \hfill & {else} \hfill \\ \end{array} } \right.$$
(16)
$$\left[\kern-0.15em\left[ I \right]\kern-0.15em\right] = \left\{ {\begin{array}{*{20}l} {X_{i,j} + h_{2} \left( {G_{i} - X^{Dhippo} } \right)} \hfill & {r_{6} > 0.5} \hfill \\ {lb_{j} + r_{7} \left( {ub_{j} - lb_{j} } \right)} \hfill & {else} \hfill \\ \end{array} } \right.$$
(17)

where: \(T\) represents the distance between the current iteration and the maximum number of iterations; \(t\) is the current iteration number; τ is the maximum number of iterations; \({X}_{i,j}^{FBhippo}\)​ denotes the position of the immature hippo; \({G}_{i}\) is the average position of some randomly selected hippos; \({h}_{1}\)​ and \({h}_{2}\)​ are numbers or vectors randomly selected from the scenarios in stage h; \(\left[\kern-0.15em\left[ I \right]\kern-0.15em\right]\) indicates the state of the immature hippo’s separation from the group, which includes staying near the population or completely leaving it; \({r}_{7}\)​ is a random number between 0 and 1.

Based on the above equations, the position update formulas for male or immature hippos in the hippo population are as follows:

$$X_{i} = \left\{ {\begin{array}{*{20}l} {X_{i}^{Mhippo} } \hfill & {F_{i}^{Mhippo} < F_{i} } \hfill \\ {X_{i} } \hfill & {else} \hfill \\ \end{array} } \right.$$
(18)
$$X_{i} = \left\{ {\begin{array}{*{20}l} {X_{i}^{FBhippo} } \hfill & {F_{i}^{FBhippo} < F_{i} } \hfill \\ {X_{i} } \hfill & {else} \hfill \\ \end{array} } \right.$$
(19)

where \({F}_{i}\)​ represents the objective function value of the current individual;\({F}_{i}^{Mhippo}\)​ denotes the objective function value of the male hippo; \({F}_{i}^{FBhippo}\)​ denotes the objective function value of the female hippo.

(2) Hippopotamus’s defensive strategy.

Hippos form large groups that can effectively defend against predator invasions. However, some curious immature hippos may stray from the group, becoming potential targets for predators. Compared to adult hippos, they are relatively weaker, and together with sick individuals within the population, are more likely to be preyed upon. To fend off predators, hippos adopt a defense strategy by swiftly turning toward the predator and emitting loud vocalizations to deter its approach. The position of the predator in the search space is given by:

$$\begin{array}{l}{X}_{j}^{Pred}=l{b}_{j}+{r}_{8}\cdot \left({ub}_{j}-l{b}_{j}\right) j=\text{1,2},3,,,m\end{array}$$
(20)
$$\begin{array}{c}D=\left|{X}_{j}^{Pred}-{X}_{i,j}\right|\end{array}$$
(21)

where \(D\) represents the distance between the hippo and the predator. During the defense process, the hippopotamus uses a defense mechanism based on the \({{F}_{j}}^{pred}\). \({{F}_{j}}^{pred}\) is the hippopotamus’s target value during the defense exploration phase. When this target value is less than \({F}_{i}\), it indicates that the predator is close to the hippopotamus, and the hippopotamus will drive away the predator. If \({{F}_{j}}^{pred}\) is greater than or equal to \({F}_{i}\), it indicates that the predator is far away from the hippopotamus. In this case, the hippopotamus turns to the predator, but due to the distance, its range of movement is limited. Equation (22) is the hippopotamus’s final position in this phase, and \({F}_{\text{i}}^{\text{Hippo R}}\) is the target value corresponding to the hippopotamus’s final position. If \({F}_{\text{i}}^{\text{Hippo R}}\) ≥ \({F}_{i}\), it indicates that the hippopotamus has been killed and replaced by another hippopotamus. Otherwise, the predator escapes and the hippopotamus returns to the population.

$$X_{i} = \left\{ {\begin{array}{*{20}l} {X_{i}^{HippoR} } \hfill & {F_{i}^{HippoR} < F_{i} } \hfill \\ {X_{i} } \hfill & {F_{i}^{HippoR} \ge F_{i} } \hfill \\ \end{array} } \right.$$
(22)

(3) Hippopotamus’s avoidance strategy.

When hippos are confronted by a group of hunters or are unable to repel a predator through defensive responses, they adopt an evasion strategy. In such cases, the hippo attempts to leave the area, typically fleeing to the nearest lake or pond. Consequently, the hippo seeks a safe location in the vicinity of its current position. The formula is as follows:

$${X}_{i,j}^{Hippo \varepsilon }={X}_{i,j}+{r}_{10}[{lb}_{j}^{local}+{S}_{1}{(ub}_{j}^{local}-{lb}_{j}^{local})]$$
(23)
$$\left\{ {\begin{array}{*{20}l} {lb_{j}^{local} = \frac{{lb_{j} }}{t}} \hfill \\ {ub_{j}^{local} = \frac{{lb_{j} }}{t}} \hfill \\ \end{array} } \right.\quad t = 1,2,3,,,\tau$$
(24)
$$\begin{array}{*{20}c} {S_{1} = \left\{ {\begin{array}{*{20}l} {2 \times r_{11} - 1} \hfill \\ {r_{12} } \hfill \\ {r_{13} } \hfill \\ \end{array} } \right.} \\ \end{array}$$
(25)

where: \({ub}_{j}^{local}\),​ \({lb}_{j}^{local}\) represent the upper and lower bounds of the current safe location in the j-th dimension, respectively; \({X}_{i,j}^{Hippo \varepsilon }\)​ denotes the position of the hippo searching for the nearest safe place; \({r}_{10}\)​ is a random number in the range [0,1]; \({S}_{1}\) is randomly selected from the three scenarios defined in Eq. (25); \({r}_{11}\)​ is a random vector with values in [0,1]; \({r}_{12}\)​ is a random number following a normal distribution; \({r}_{13}\) is a random number in the range [0,1].

The position update for the optimal population is given by the following expression:

$$\begin{array}{*{20}c} {X_{i} = \left\{ {\begin{array}{*{20}l} {X_{i}^{Hippo \varepsilon } F_{i}^{Hippo \varepsilon } < F_{i} } \hfill \\ {X_{i} F_{i}^{Hippo \varepsilon } \ge F_{i} } \hfill \\ \end{array} } \right.} \\ \end{array}$$
(26)

where: \({F}_{i}^{Hippo \varepsilon }\)​ represents the objective function value corresponding to the position of the optimal population.

The Hippo Optimization Algorithm (HOA) updates all population members after each iteration according to three stages: position update, defense strategy, and predator evasion. These updates are performed using the builtin formulas within the algorithm. Following the updates, the “dominant hippos,” representing the current iteration’s optimal solutions, are selected. This population update process continues based on the population rules until all iterations are completed. The flowchart of the HOA search and optimization process is illustrated in Fig. 6.

Fig. 6
figure 6

Flowchart of the HOA search and optimization process.

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HOA and k-MST inner–outer joint optimization framework

Since the k-MST requires a preset number of nodes k, it inherently lacks the capability to optimize the number of deployment points. To address this limitation, this paper proposes an inner outer joint optimization framework combining the Hippo Optimization Algorithm (HOA) with the knode Minimum Spanning Tree (k-MST) model. In this framework, HOA serves as the outer optimizer, responsible for searching the optimal number of anti-frost smoke machine deployment points within the candidate set. The MILP-based k-MST model acts as the inner solver, generating connection paths and minimizing wiring costs for each set of deployment points. This framework leverages the global search advantages of the swarm intelligence algorithm and the precision of k-MST in network connectivity control, achieving a comprehensive optimization of the number of anti-frost smoke machines deployed, coverage performance, and wiring efficiency.

The outer optimization employs the Hippo Optimization Algorithm (HOA), whose core task is to select an optimal subset from the predefined set of anti-frost smoke machine candidate points \(V\) based on the objective function in Eq. (5). Each candidate solution is represented by a binary vector \(X={\left[{x}_{1},{x}_{2},,,{x}_{n} \right]}^{T}\), where n denotes the total number of anti-frost smoke machine candidate points. This vector has a one-to-one mapping with the candidate set \(V\), where \({x}_{i}\) indicates that a anti-frost smoke machine is deployed at the i-th position, and \({x}_{i}\)=0 indicates no deployment. Each deployment scheme generated by the outer optimizer corresponds to a deployment point set \(S=\left\{{v}_{i}\in V\mid {x}_{i}=1\right\}\). The cardinality \(\mid S\mid\) of this set is then used as the number of nodes k input into the inner optimizer for further solving.

The inner layer utilizes a MILP-based k-MST model to optimize the connectivity structure among the deployment points selected by the outer HOA. The core task of this model is to construct a minimum spanning tree within the deployment point set S, ensuring all points remain connected while minimizing the total wiring length. In this framework, the number of deployment points is entirely determined by the outer optimizer, and the inner model performs wiring optimization based solely on the outer layer’s decisions. This approach not only overcomes the limitation of k-MST in adjusting the node scale but also ensures that wiring planning is conducted on the basis of minimal resource allocation, thereby improving the overall optimization efficiency and practical applicability of the system.

Results

Optimization results

For the case study in Section "Case study description", this paper compares the results of the traditional experience-based design with those of the multi-objective optimization-based design. The traditional experience-based design refers to schemes primarily relying on growers’ judgment or intuitive analysis of the terrain, lacking scientific and systematic theoretical support. Typically, such traditional designs adopt rule-based grids or fixed spacing layouts32. Therefore, this paper uses a traditional design scheme based on fixed spacing as a control experiment. In the candidate anti-frost smoke machine location set, anti-frost smoke machines are evenly deployed inside the orchard at fixed intervals, as illustrated in Fig. 7.

Fig. 7
figure 7

Anti-frost smoke machine layout plan based on traditional experience.

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To solve this multi-objective optimization problem, this paper utilizes MATLAB R2024b with the YALMIP interface to formulate the multi-objective problem and employs Gurobi 10.0.1 as the solver. In the HOA, the hippo population size is set to 200, with 500 iterations. Since smoke coverage is the primary factor affecting frost prevention effectiveness, other weight values \(w\) are kept constant while adjusting \({w}_{1}\) to identify the optimal \({w}_{1}\) that allows the optimization scheme to both satisfy frost protection requirements and minimize costs. The optimization results are presented in Table 3.

Table 3 Multi-objective optimization results based on different weights.
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In order to understand the impact of different weight settings on the anti-frost scheme, the Pareto front diagram of the multi-objective optimization problem is drawn according to the optimization results under different weights shown in Table 3. The Pareto front diagram of the multi-objective robust optimization problem is shown in Fig. 8.

Fig. 8
figure 8

Pareto chart for multi-objective optimization problems.

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From Table 3 and Fig. 8, it can be observed that when \({w}_{1}\) = 1000, the frost prevention system achieves the maximum smoke coverage. When \({w}_{1}\) = 1, the smoke coverage only decreases by 0.02% compared to \({w}_{1}\) = 1000, while the number of anti-frost smoke machines and the wiring length are reduced by 23.07% and 28.49%, respectively. Therefore, considering both frost protection effectiveness and deployment cost, \({w}_{1}\) = 1 achieves the optimal balance between frost prevention and deployment cost, demonstrating high engineering practicality. Subsequent calculations adopt \({w}_{1}\) = 1. Slope is one of the important terrain features and significantly affects smoke diffusion, as well as equipment deployment and wiring strategies. To enhance the adaptability and practicality of the optimization scheme, this paper introduces different slope conditions for comparative analysis during the optimization design process. The anti-frost smoke machine deployment and wiring diagrams under different slopes are shown in Fig. 9.

Fig. 9
figure 9figure 9

Layout and wiring diagram of anti-frost smoke machines at different slopes: (a) Layout plan with a slope of 5°; (b) Layout plan with a slope of 10°; (c) Layout plan with a slope of 15°; (d) Layout plan with a slope of 20°; (e) Layout plan with a slope of 25°; (f) Layout plan with a slope of 30°

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Table 4 presents the optimized results of anti-frost smoke machine deployment under different slope conditions. It can be observed that the optimized smoke coverage remains above 85% in all cases, which generally meets the requirements for frost prevention. As the slope increases, the coverage gradually decreases while the number of anti-frost smoke machines increases. This is because steeper slopes enhance gravitational effects, causing smoke particles to settle more easily along the slope surface, thereby weakening their horizontal diffusion capacity and reducing effective coverage of the target area. Consequently, more anti-frost smoke machines are required to compensate for the reduced diffusion. The wiring length does not show a strictly proportional relationship with the number of anti-frost smoke machines, as it is mainly influenced by their specific deployment locations. The concentration standard deviation under each slope condition is less than or close to 1, indicating low dispersion between the concentrations at different checkpoints and the average concentration, which is beneficial for forming a uniform and stable smoke layer throughout the orchard.

Table 4 Optimization results of different slopes.
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To more intuitively demonstrate the superiority of the proposed design scheme, a comparative analysis is conducted against the traditional fixed spacing design approach. The comparison results under different slope conditions are shown in Table 5. As illustrated, although the smoke coverage of the optimized scheme shows a slight decrease compared to the traditional method, both the number of anti-frost smoke machines and the total wiring length are significantly reduced, making the overall outcome more desirable. As the slope increases, the smoke diffusion capacity weakens, leading to a relative reduction in the advantage of the optimized scheme. However, it still exhibits clear benefits in terms of economic efficiency.

Table 5 Comparison of multi-objective optimization and traditional solutions based on different slopes.
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Verification of optimization results based on CFD

To verify the effectiveness of the optimized smoke machine deployment scheme, smoke diffusion was simulated using the ANSYS software, focusing on analyzing the smoke concentration distribution at the canopy level (2 m in height). To assess the correlation between the numerical simulation results and the mesh size of the model, a mesh sensitivity validation experiment was conducted. Under environmental wind conditions of 4 m/s and an initial smoke velocity of 3 m/s, the mesh cell sizes were set to 0.1 m, 0.3 m, 0.5 m, 0.7 m, and 1 m. The variation in smoke concentration at vertical distances of 1 to 3 m above the smoke machine is shown in Fig. 10. As observed, changes in mesh size lead to only minor differences in smoke concentration and distribution. When the mesh cell size is set to 0.3 m, it achieves a good balance between computational accuracy and resource consumption. In ANSYS 2022 R2, the fluid domain was discretized using a hexahedral meshing strategy. Boundary conditions such as wind inlet, particle inlet, pressure outlet, and wall surfaces were defined. Mesh refinement was applied near the smoke machine and at the boundary regions to improve simulation accuracy and ensure the mesh quality meets the requirements of fluid dynamics computation. The fluid property parameters used in the simulation model are listed in Table 6.

Fig. 10
figure 10

Grid independence verification.

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Table 6 Attribute parameters of the computational domain.
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As shown in Fig. 11, the overall smoke concentration within the orchard under different slope conditions remains around 20 μg/cm3, indicating the formation of a distinct smoke layer and achieving a desirable frost prevention effect. The smoke concentration distribution characteristics obtained through ANSYS simulation are highly consistent with those derived earlier using MATLAB, validating the applicability and effectiveness of the proposed optimization scheme under complex terrain conditions. However, it is observed in Fig. 11 that the smoke diffusion direction differs from that in Fig. 9. Specifically, the smoke in the central area diffuses linearly along the wind direction, while the smoke released from the machines on both sides spreads not only downwind but also diagonally outward. This is due to the higher smoke concentration in the central region compared to the sides, resulting in a concentration gradient that induces diagonal diffusion and forms a conical diffusion pattern. It is important to note that the MATLAB model involves certain idealized assumptions in the simulation of the smoke diffusion mechanism, and does not fully account for all influencing factors. This leads to some local discrepancies between the smoke distribution shown in Fig. 9 and the ANSYS simulation results. Nevertheless, the overall distribution characteristics in both simulations remain highly consistent, exhibiting similar coverage areas and diffusion patterns. This further confirms that the frost prevention scheme optimized using the MATLAB-based approach is effective and reliable under practical conditions.

Fig. 11
figure 11

Optimization results based on CFD verification: (a) Smoke density at a slope of 5°; (b) Smoke density at a slope of 10°; (c) Smoke density at a slope of 15°; (d) Smoke density at a slope of 20°; (e) Smoke density at a slope of 25°; (f) Smoke density at a slope of 30°

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Discussion

To evaluate the effectiveness of the proposed Hippo Optimization Algorithm (HOA) in the multi-objective optimization of anti-frost smoke machines deployment and wiring, the commonly used Genetic Algorithm (GA) is adopted as a comparison baseline models. In this section, a series of comparative experiments are conducted, in which HOA and GA are respectively employed as the outer-layer optimizers within the inner–outer joint optimization framework, in order to compare and analyze the performance differences of the multi-objective optimization algorithms under different outer optimizers. This setup allows for a direct comparison and analysis of the performance of different optimizers under the same multi-objective optimization context. In 1998, Jang Sung Chun applied GA to optimization problems using various search strategies, and it has since been widely adopted in many types of optimization tasks33. For example, Liang Zhao optimized the parameters of a spiral fertilizer applicator using GA, effectively addressing issues such as low precision and poor fertilizer utilization34. Kunlin Zou proposed an apple image segmentation algorithm based on color index and thresholding, and applied GA to optimize the parameters related to the color index and threshold segmentation35. HOA and GA are experimentally compared under the same optimization objectives, constraints, and related parameters, in order to analyze the performance of the proposed algorithm in terms of convergence speed, solution quality, and stability.

As a comparative method, the Genetic Algorithm (GA) follows a core process that includes selection, crossover, and mutation operations. The parameters are set as follows: population size of 200, crossover probability of 0.8, mutation probability of 0.15, and a maximum of 500 iterations. All experiments are conducted on the same computer to ensure fairness. Table 7 presents a comparison of the results between GA and the proposed HOA under flat terrain conditions in the context of multi-objective optimization. As shown in Table 7, the advantages of HOA become more evident, particularly in the design of frost prevention schemes under multi-objective settings, demonstrating stronger robustness and adaptability. In terms of overall performance, HOA significantly outperforms GA. Specifically, the HOA achieves a 29.17% and 35.96% improvement over GA in reducing the number of anti-frost smoke machines and the total wiring length, respectively, thereby substantially lowering hardware costs. Furthermore, while using fewer devices and shorter wiring, HOA still achieves a 5.56% increase in smoke coverage compared to GA. Therefore, HOA proves to be more suitable for addressing the optimization objectives of this design scheme.

Table 7 Results of GA and HOA algorithms under multi-objective optimization.
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As shown in Fig. 12, the objective function value in the HOA decreases rapidly within the first 10 iterations, indicating a fast initial convergence rate. In contrast, the convergence process of the Genetic Algorithm (GA) exhibits a typical stepwise decline, suggesting that GA relies heavily on mutation and crossover operations to approach the optimal solution. However, due to roulette wheel selection and a reduction in population diversity, GA tends to fall into local optima during the later stages, ultimately yielding only a suboptimal solution. With continued iterations, HOA maintains a steady downward trend and gradually converges after the 80th iteration, ultimately reducing the objective function value to approximately 330, which is significantly better than the final convergence value of GA, around 360. These results demonstrate that HOA possesses strong global exploration and local exploitation capabilities. Therefore, in multi-objective optimization problems, HOA, with its superior global search ability, proves to be more suitable for the optimization design of the smoke machine-based frost prevention system proposed in this study.

Fig. 12
figure 12

Convergence curve comparison of the two algorithms: (a) Iteration graph of HOA; (b) Iteration graph of GA.

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Conclusions

This paper provides a feasible solution to address the high cost and low efficiency of frost prevention in large mountainous orchards. The relevant conclusions are as follows:

  1. 1.

    This paper constructs an inner and outer joint optimization framework based on HOA and k-MST to optimize the number and location of anti-frost smoke machines, evenly heat the orchard, and achieve the maximum anti-frost effect. This framework fully utilizes the advantages of HOA in global search and jump exploration to find the optimal location and number of anti-frost smoke machines; the inner layer wiring model with k-MST as the core can achieve the lowest cost connectivity path planning based on the determined layout points, thus effectively taking into account the economy of system deployment. This optimization framework greatly improves the optimization speed through hierarchical collaborative calculation of the inner and outer layers.

  2. 2.

    The optimized anti-frost smoke machine layout scheme can achieve good frost prevention effect in orchards with different slopes and greatly reduce the layout cost. The results show that as the slope gradually increases from 5° to 30°, the multi-objective optimization method can save up to 55.55% in the number of anti-frost smoke machines compared with the traditional method, and the saving in wiring length is more significant, up to 44.16%. Especially under the condition of smaller slope, the optimization effect is more prominent. This paper compares the results of HOA with those of traditional GA and concludes that HOA has advantages in multi-objective optimization applications.

Although this study proposes a joint optimization method based on HOA and k-MST that demonstrates significant advantages in the layout of anti-frost smoke machines, certain limitations remain. On the one hand, the Gaussian plume model, even with the inclusion of a slope correction factor, cannot fully capture the complex interactions between microclimate and terrain in mountainous orchards. On the other hand, the proposed multi-objective optimization method has only been validated in a single apple orchard and still needs to be extended to other agricultural scenarios, such as tea plantations and rapeseed fields, to enhance its general applicability. Moreover, the optimization design scheme requires systematic field experiments to verify its effectiveness and reliability. Future work will focus on conducting economic feasibility analyses (evaluating installation costs and investment payback period), building orchard sensor networks for real-time acquisition of temperature, humidity, and wind field data to support intelligent decision-making, and developing a dual-layer control system (one for managing the operation of smoke machine fans and the other for energy system power supply strategies), there by promoting the construction of an intelligent and efficient frost protection system.