Prediction of Dielectric Loss Factor of Wood in Radio Frequency Heating and Drying Based on IPOA-BP Modeling

Abstract

In this paper, an Improved Pelican Optimization Algorithm (IPOA) was proposed to optimize a BP neural network model to predict the dielectric loss factor of wood in the RF heating and drying process. The neural network model was trained and optimized using MATLAB 2022b software, and the prediction results of the BP neural network with POA-BP and IPOA-BP models were compared. The results show that the IPOA-optimized BP neural network model is more accurate than the traditional BP neural network model. After the BP neural network model with IPOA optimization was used to predict the dielectric loss factor of wood, the value increased by 4.3%, the MAE decreased by 68%, and the RMSE decreased by 67%. The results provided by the study using the IPOA-BP model show that the prediction of the dielectric loss factor of wood under different macroscopic conditions in radio frequency heating and drying of wood can be realized without the need for highly costly and prolonged experimental studies.

1. Introduction

Radio frequency heating is an essential nontraditional technology for the high-quality dehydration and phytosanitation of wood products, and for the past decade or so, the radio frequency vacuum (RFV) drying of wood has become a valuable tool for drying high-quality wood, panels, and logs [1]. The dielectric loss factor of wood during RFV drying is a parameter that describes the ability of a material to transform and dissipate energy in an electric field. The efficiency of RF drying depends largely on the efficiency of RF heating. Because the drying process is mainly achieved by heating the water in the material to evaporate [2]. Therefore, when the dielectric loss factor is high, the RF heating efficiency is high, and the drying process will be faster and more efficient. Woods with high dielectric loss factors, such as poplar and pine, can effectively absorb and convert RF energy into thermal energy under the action of RF energy, thus realizing rapid and uniform drying effects. Wood with a high dielectric loss factor is able to absorb and convert RF energy into heat when exposed to RF energy, resulting in fast and uniform drying. On the contrary, wood with a low dielectric loss factor, such as birch and maple, has a relatively weak ability to absorb RF energy, which may lead to less than optimal drying results. Koumoutsakos et al. [3,4,5] showed that in the radio frequencies heating and drying process, the dielectric loss factor (ε″) is proportional to the heat energy delivered to the wood, which represents the dissipation of electromagnetic field energy, and a higher value of ${\mathsf{\epsilon }}^{″}$ indicates that more electrical energy is converted into heat energy. The formula for converting electrical energy into heat is as follows:

$$\mathrm{P}\mathrm{D}=2\mathsf{\pi }\mathrm{f}{\mathsf{\epsilon }}^{″}{\mathsf{\epsilon }}_0{\mathrm{E}}^2(\mathrm{W}/{\mathrm{m}}^3)$$

(1)

In Formula (1), PD is the power density (W/m), f is the frequency (MHz), ${\mathsf{\epsilon }}_0$ is the dielectric constant, and E is the dielectric field strength (V/M). PD directly determines the rate of heat production in the material and the amount of power entering the material. Macrophysical factors such as moisture content, temperature, texture direction, and density of the wood, as well as a variety of variables such as the electrochemical composition of the wood, work together to cause natural changes in the dielectric loss factor (ε″) while keeping other controllable conditions unchanged. This change will directly impact the behavior of the wood drying heating unit, thus limiting the maximum power that the wood can absorb during heating. Therefore, the dielectric loss factor (ε″) is an important parameter in designing and developing hardware for RF heating and drying logs and wood. In addition, the dielectric properties of wood and the spatial and temporal variations they produce result in a great deal of time and money in estimating the dielectric properties of wood. Consequently, establishing a robust test program to validly predict the dielectric loss factor of wood can optimize the RF heating drying process and improve drying efficiency and quality.

Past studies have shown a close relationship between dielectric properties and factors such as the properties of the wood under study and the applied field frequency [6]. Norimoto et al. [7] studied the dielectric loss factor of the wood chemical composition of different tree species as a function of frequency and temperature. Peyskens et al. [8] performed dielectric measurements of three softwood species (European pine, spruce, and hemlock) with different texture directions and moisture content at a frequency of 3 GHz. The results show that there was a positive correlation between water content and loss factor, and the longitudinal dielectric loss factor was significantly higher than the transverse dielectric loss factor. Meanwhile, the specific effects of wood water content and grain direction on the dielectric loss factor also depended on the species of wood. Many scholars have also concluded in the study of wood RF heating that the dielectric loss factor is closely related to the change in wood average moisture content and fiber saturation point.

Currently, the value of the dielectric loss factor (ε″) has been modeled using the effect of wood chemistry with the macroscopic physical properties of the wood. Avramidis et al. [9] established an artificial neural network capable of predicting the dielectric loss factor of wood through the function of ambient electrothermal conditions and introductory wood chemistry. Iliadis et al. [10] then applied hybrid soft computing techniques to predict wood loss factors and developed an innovative fuzzy logic method to evaluate experimental data from two different wood species to determine the optimal model. However, the current research on the prediction of the dielectric loss factor value of wood rarely involves the dielectric loss factor (ε″) model under different macroscopic physical characteristics, and the choice of tree species is comparatively straightforward.

Given this, in order to better investigate the influence of different relative humidity, water content, temperature, frequency, tree species, texture direction, and density on the dielectric loss factor (ε″), we selected four representative tree species of Douglas fir, Sitka spruce, western hemlock, and western red cedar. These four species are important conifer species on the West Coast, and conifers have great potential economic value as an abundant natural resource. The study of dielectric loss factors can help evaluate the economic value of conifers as dielectric materials and provide strong support for the development of related industries. The traditional measurement of dielectric loss factor usually requires complicated experimental equipment and complicated experimental steps, which are not only time-consuming and laborious but also costly. By using a machine learning model to predict the dielectric loss factor of wood, the experimental measurement can be eliminated, and the prediction results can be obtained quickly, thus saving a lot of time and cost. Improved Pelican Optimization algorithm (IPOA) is used to optimize the weight and threshold of the BP network to achieve a more accurate prediction of the wood dielectric loss factor. The dielectric loss factor of wood is one of the important parameters of the electrical properties of wood, and it has an important effect on the behavior of wood in electromagnetic fields. The prediction model is helpful to further understand the dielectric properties of wood under various environmental conditions and different physical characteristics, aiming at optimizing the design and development of the wood RF heating process, as well as providing references for wood processing and utilization.

2. The IPO-BP Neural Network Prediction Model

2.1. BP Neural Network Model

The learning process of the BP (Back Propagation) neural network consists of two processes: forward propagation of information and backpropagation of errors [11]. During the forward pass, information travels from the input layer to the hidden layer and ultimately reaches the output layer. If, upon comparison with experimental data, the actual output does not align with the expected output, the error signal triggers the backpropagation stage. Errors pass through the output layer, the weights of each layer are corrected according to the error gradient descent method, and the hidden layer and input layer are propagated backward layer by layer [12].

Neural network predictive modeling is a powerful tool that learns and automatically extracts features from large amounts of data by simulating the way neurons in the human brain connect and interact with each other to classify, predict, or generate new data. Such models have a wide range of applications in several fields, such as in biomaterials [13] and in evaluating the mechanical properties of wood [14], demonstrating their unique advantages. A significant disadvantage of the BP neural network is its tendency to get trapped in local minima, preventing it from achieving a globally optimal solution. Additionally, excessive training iterations can significantly decrease learning efficiency and hinder the convergence speed, making the optimization process less effective.

This paper used the machine learning toolbox of MATLAB (2022b) to build a BP model to predict the dielectric loss factors of 4 different tree species. A wood dielectric loss factor prediction model was established with temperature, relative humidity, moisture content, texture direction, frequency, density, species of heated wood, and tree species as input nodes. The model performance is shown in Figure 1:

2.2. Pelican Optimization Algorithm

In wood treatment processes, such as RF heating processes, there may be multiple critical parameters that affect wood properties. Pelican Optimization Algorithms (POA) can be used for data analysis to improve efficiency and accuracy. The POA is selected to optimize the BP neural network model because the POA’s global search capability, adaptive adjustment mechanism, and effectiveness in dealing with complex nonlinear problems enable it to intelligently guide the weight update of the neural network and effectively avoid local optimization, thus showing significant advantages in improving training efficiency and final accuracy. The POA is a heuristic algorithm proposed by Pavel Trojovsky and Mohammad Dehghani [15] in 2022 that simulates the natural behaviors and strategies displayed by pelicans during attack and hunting. This unique hunting method consists of two stages: approaching the prey and surface flight.

2.2.1. Prey Approach Phase

In the first stage, the pelicans identify their prey’s location and then move towards the target area. By modeling this pelican search strategy, we efficiently scanned the search space and leveraged the POA’s search capabilities to explore different regions in the search space. The above concepts and the mathematical model of pelicans moving towards prey position are detailed in Formula (2).

$${\mathrm{x}}_{\mathrm{i},\mathrm{j}}^{{\mathrm{P}}_1}=\left\{\begin{array}{ll}{\mathrm{x}}_{\mathrm{i},\mathrm{j}}+\mathrm{rand}\cdot \left({\mathrm{p}}_{\mathrm{j}}-\mathrm{I}·{\mathrm{x}}_{\mathrm{i},\mathrm{j}}\right),& {\mathrm{F}}_{\mathrm{p}}<{\mathrm{F}}_{\mathrm{i}};\\ {\mathrm{x}}_{\mathrm{i},\mathrm{j}}+\mathrm{rand}·\left({\mathrm{x}}_{\mathrm{i},\mathrm{j}}-{\mathrm{p}}_{\mathrm{j}}\right),& \mathrm{else},\end{array}\right.$$

(2)

In phase 1, ${\mathrm{X}}_{\mathrm{i},\mathrm{j}}^{\mathrm{P}1}$ represents the new state of the i th pelican in dimension j, which is determined by the random parameter I, where the value of I can be 1 or 2. ${\mathrm{p}}_{\mathrm{j}}$ represents the position of the prey in dimension j, and ${\mathrm{F}}_{\mathrm{p}}$ is the objective function value of the prey. The parameter I is chosen at random for each iteration and for each Pelican member. When the value of I is equal to 2, it prompts the pelican members to make a larger shift, leading them to explore newer areas of the search space that have not yet been visited.

In the POA, if the objective function of a location is worth improving, then the new location of the pelican is determined. This update is an efficient update because it ensures the algorithm does not fall into a nonoptimal region. The whole updating process is modeled by Formula (3). In this formula, ${\mathrm{X}}_{\mathrm{i}}^{\mathrm{P}1}$ is the new position of the i th pelican, and ${\mathrm{F}}_{\mathrm{i}}^{\mathrm{P}1}$ is based on the objective function value of the first stage.

$${\mathrm{X}}_{\mathrm{i}}=\left\{\begin{array}{c}{\mathrm{X}}_{\mathrm{i}}^{{\mathrm{P}}_1},{\mathrm{F}}_{\mathrm{i}}^{{\mathrm{P}}_1}<{\mathrm{F}}_{\mathrm{i}};\\ {\mathrm{X}}_{\mathrm{i}},\mathrm{else},\end{array}\right.$$

(3)

2.2.2. Surface Flight Stage

In the second stage, once the pelicans reach the surface, they spread their wings to lift the fish from the water into the air and eventually collect their prey in throat bags. This unique hunting strategy allows the pelicans to catch more fish in their attack area. By mathematically modeling this behavior of pelicans, the proposed POA can be ensured to converge to a better solution for the hunting area. This behavior of pelicans during hunting is mathematically simulated by Formula (4).

$${\mathrm{x}}_{\mathrm{i},\mathrm{j}}^{{\mathrm{P}}_2}={\mathrm{x}}_{\mathrm{i},\mathrm{j}}+\mathrm{R}·\left(1-\frac{\mathrm{t}}{\mathrm{T}}\right)·\left(2·\mathrm{rand}-1\right)·{\mathrm{x}}_{\mathrm{i},\mathrm{j}},$$

(4)

In phase 2, ${\mathrm{X}}_{\mathrm{i},\mathrm{j}}^{\mathrm{P}1}$ represents the new state of the i th pelican in dimension j. R is a constant whose value is set to 0.2. R·(1 − t/T) defines the neighborhood radius, where t is the current iteration counter, and T is the maximum number of iterations. This expression represents the range of neighborhoods where a local search is performed near each member in the expectation of convergence to a better solution.

At the same time, in the current iteration stage, the algorithm can be effectively updated and accept or reject the new population location according to specific criteria, as shown in Formula (5). In this formula, ${\mathrm{X}}_{\mathrm{i}}^{\mathrm{P}2}$ is the new position of the i th pelican, and ${\mathrm{F}}_{\mathrm{i}}^{\mathrm{P}2}$ is based on the objective function value of the first stage.

$${\mathrm{X}}_{\mathrm{i}}=\left\{\begin{array}{c}{\mathrm{X}}_{\mathrm{i}}^{{\mathrm{P}}_2},{\mathrm{F}}_{\mathrm{i}}^{{\mathrm{P}}_2}<{\mathrm{F}}_{\mathrm{i}};\\ {\mathrm{X}}_{\mathrm{i}},\mathrm{else},\end{array}\right.$$

(5)

However, the POA has some significant drawbacks. First of all, it uses an entirely random method to generate the starting population, which often results in uneven distribution of the initial population, lack of necessary population diversity, and limited spatial location for optimization, thus affecting the algorithm’s convergence rate [16]. Secondly, the POA tends to choose the location with the best fitness value in each iteration, which makes the algorithm easily fall into the local optimal region and reduces the accuracy of optimization. In addition, the global exploration ability of the POA is relatively general in the early stage of the algorithm, and the local development ability is weak in the later stage, which also increases the risk of the algorithm falling into the local optimal.

2.3. Improved Pelican Optimization Algorithm

Although there are differences between wood varieties, their structures usually have certain commonalities at the macrolevel, such as fiber arrangement, cell structure, and pore distribution. These commonalities make it possible for certain structure-based properties, such as dielectric loss factors, to exhibit similar patterns across wood species. The Pelican Optimization Algorithm itself does not directly consider the structural characteristics of wood. However, by selecting the appropriate input characteristics and constructing the appropriate prediction model, the structural characteristics of wood can be reflected, such as wood density, moisture content, texture direction, etc., so as to indirectly consider the influence of wood structural characteristics on dielectric loss factor. In the prediction model, the weights of different input features reflect the degree of their influence on the dielectric loss factor. Therefore, the Pelican Optimization Algorithm needs to be improved in order to reflect the effect of the structural characteristics of wood on the dielectric loss factor more accurately.

2.3.1. Logistic Chaos Mapping

In the swarm intelligence optimization algorithm, population initialization is a significant part, which affects the convergence speed of the algorithm, the coverage of the search space, and the quality of the final optimization results [17]. The traditional random initialization method can easily lead to the individual distribution of the population needing to be more concentrated or dispersed, thus affecting the algorithm’s overall performance. To overcome these shortcomings, we introduced the logistic mapping method for population initialization [18]. The calculation formula is as follows:

$$\mathrm{X}\left(\mathrm{t}\right)=\mathsf{\alpha }·\mathrm{X}\left(\mathrm{t}\right)·(1-\mathrm{X}\left(\mathrm{t}\right))$$

(6)

$\mathrm{X}\left(\mathrm{t}\right)$ is used to represent the location of individuals in the current population, while $\mathsf{\alpha }$ is a random number. After many experiments, we determined that the optimal value of $\mathsf{\alpha }$ is 4, so we set $\mathsf{\alpha }$ = 4 here. By introducing logistic mapping into the population initialization process, we can effectively prevent the population from being too concentrated or too dispersed while ensuring that each individual has enough space to explore. In addition, the randomness and unpredictability of Logistic mapping can fully guarantee the diversity and independence of individuals so the algorithm can avoid getting stuck in a locally optimal solution.

2.3.2. Reverse Differential Evolution Mechanism

A strategy of differential evolution learning based on reverse learning is proposed to balance the accuracy of global search and convergence. Reverse learning refers to dividing the solution space of a problem into two opposite subspaces, namely “forward solution” and “reverse solution”, and expanding the search space through the reverse population to increase the diversity and exploration of the algorithm [19].

The formula for calculating the reverse solution is

$${\mathrm{O}\text{\_}\mathrm{X}}_{\mathrm{i}}={\mathrm{l}\mathrm{b}}_{\mathrm{i}}+{\mathrm{u}\mathrm{b}}_{\mathrm{i}}-{\mathrm{X}}_{\mathrm{i}}$$

(7)

${\mathrm{O}\text{\_}\mathrm{X}}_{\mathrm{i}}$ is the inverse solution on the th dimension, ${\mathrm{l}\mathrm{b}}_{\mathrm{i}}$ is the lower bound on the th dimension i, and ${\mathrm{u}\mathrm{b}}_{\mathrm{i}}$ is the upper bound on the th dimension i. ${\mathrm{X}}_{\mathrm{i}}$ is the initial solution of the th dimension i, i = 1, 2, 3 … D. Merge the random population and the reverse population to obtain a new population $\mathrm{X}\text{\_}\mathrm{new}$:

$$\mathrm{X}\text{\_}\mathrm{new}=\left[\begin{array}{c}\mathrm{O}\text{\_}\mathrm{X}\\ \mathrm{X}\end{array}\right]$$

(8)

We first obtain high-quality populations by applying the reverse learning strategy [20]. Then, in order to further enhance the diversity of the population, we adopted the differential evolution algorithm (DE). The core of the differential evolution algorithm is to explore the search space and finally find the optimal solution to the problem by iterating through the cross, variation, and selection operations in the solution space [21]. The formula of the mutation operator is as follows:

$${\mathrm{v}}_{\mathrm{i}}^{\mathrm{t}}={\mathrm{X}\text{\_}\mathrm{new}}_{{\mathrm{r}}_1}^{\mathrm{t}}+\mathrm{m}({\mathrm{X}\text{\_}\mathrm{new}}_{{\mathrm{r}}_2}^{\mathrm{t}}-{\mathrm{X}\text{\_}\mathrm{new}}_{{\mathrm{r}}_3}^{\mathrm{t}})$$

(9)

v is the variation solution, m is the ratio column of variation because it is a random number between [0,1], t is the number of iterations, and r₁,r₂,r₃ are different numbers representing different pelicans. The crossover operator is an important operation in the differential evolution algorithm, which can increase the diversity of the population after mutation operation for individuals in the population. The formula is as follows:

$${\mathrm{u}}_{\mathrm{i},\mathrm{j}}^{\mathrm{t}}=\left\{\begin{array}{c}{\mathrm{v}}_{\mathrm{i},\mathrm{j}}^{\mathrm{t}},\mathrm{if}\mathrm{rand}\le \mathrm{p}\\ {\mathrm{X}\text{\_}\mathrm{new}}_{\mathrm{i},\mathrm{j}}^{\mathrm{t}},\mathrm{otherwise}\end{array}\right.$$

(10)

In Formula (10), u is the crossover solution, p is the crossover probability, p = 0.5, and j is the population individual (j = 1, 2, 3 … 2N). Finally, Formula (11) is used to carry out the greedy selection mechanism and select the optimal N individuals.

$${\mathrm{X}\text{\_}\mathrm{new}}_{\mathrm{i}}^{\mathrm{t}+1}=\left\{\begin{array}{c}{\mathrm{u}}_{\mathrm{i}}^{\mathrm{t}},\mathrm{if}\mathrm{f}\left({\mathrm{u}}_{\mathrm{i}}^{\mathrm{t}}\right)\le \mathrm{f}\left({\mathrm{X}\text{\_}\mathrm{new}}_{\mathrm{i}}^{\mathrm{t}}\right)\\ {\mathrm{X}\text{\_}\mathrm{new}}_{\mathrm{i}}^{\mathrm{t}},\mathrm{otherwise}\end{array}\right.$$

(11)

2.3.3. Firefly Algorithm Disturbance

In order to improve the algorithm’s exploration ability in local search and prevent solutions from falling into local optimality, we introduced the firefly disturbance strategy. This strategy helps to find better solutions by increasing the diversity of algorithms. In the firefly algorithm, each firefly position represents a potential solution, and the firefly disturbance simulates the behavior of the firefly position change to try to jump out of the current local optimal solution and search in the direction that may be better [22]. This strategy enables the algorithm to explore the search space more comprehensively and then increases the probability of finding the global optimal solution. The specific perturbation formula is as follows:

$$\mathrm{F}\text{\_}\mathrm{x}=\mathrm{X}+\mathsf{\beta }\ast \left(\mathrm{X}-\mathrm{Best}\text{\_}\mathrm{X}\right)+0.2\mathsf{\theta }$$

(12)

$$\mathsf{\beta }=2\ast \mathrm{e}(-2\ast \frac{‖\mathrm{X}-\mathrm{Best}\text{\_}\mathrm{X}‖}{\left(\mathrm{u}\mathrm{b}-\mathrm{l}\mathrm{b}\right)\ast \sqrt{\mathrm{D}}})$$

(13)

In the perturbation formula, $\mathsf{\theta }$ is a random number between $⌈\frac{-1}{2},\frac{1}{2}⌉$. $\mathrm{F}\text{\_}\mathrm{x}$ is the position after the perturbation to the pelican population, $\mathrm{X}$ is the current position, and $\mathrm{Best}\text{\_}\mathrm{X}$ is the current optimal solution.

2.4. IPOA-BP Model

Pelican Optimization Algorithm (POA) simulates pelican predation behavior to find the best solution. In the algorithm, the multidimensional information of pelican predation is used as the search node. The fitness of predation is analogous to the solution of the objective function. The predation process corresponds to the optimization process’s iteration and optimal solution search process. However, the original POA quickly falls into local optimal after updating the optimal position of the population, and it faces the challenges of slow convergence and low accuracy. To overcome these limitations, we implemented several improvements.

First, Logistic chaotic mapping is introduced, and random sequences are incorporated to enhance the accuracy and convergence speed of the POA. The ergodicity of chaotic mapping brings diversity to the initial pelican population, improving the algorithm’s broad area search ability. Secondly, in order to overcome the problem of premature convergence and slow convergence of the differential evolution algorithm (DE), we adopt the reverse differential evolution algorithm (DE), introduce the concept of reverse learning (OBL) into DE, and optimize the convergence rate of the algorithm by the concept of reverse number. Finally, we implemented the firefly disturbance strategy, that is, after the individual position of each generation of pelicans is updated, the firefly algorithm is used to randomly disturb the position of the current optimal solution to update its location so as to give the algorithm the ability to jump out of the local optimal and further improve the search efficiency.

The core idea of optimizing the BP neural network by Improved Pelican Optimization Algorithm (IPOA) is to treat the weights and thresholds of the BP neural network as the location information of the pelican and dynamically optimize these parameters through the IPOA until the global optimal location is found, thus significantly improving the prediction ability and efficiency of BP neural network. In this way, we can obtain more stable and accurate prediction results.

Figure 2 shows the flow chart of the BP model improved by IPOA. First, in order to build the initial neural network, the data need to be normalized to achieve the division of the data layers. Then, the improved BP model of IPOA is used to initialize the operation parameters. On this basis, the chaotic sequence generated by Logistic chaotic mapping is used to set the initial location of the pelican population. Next, the exploration and discovery phases of pelican populations are performed according to specific formulas. When the worst-placed individuals in the population are identified through the reverse differential evolution strategy, they are updated.

In order to further improve the searching ability, we introduce a firefly disturbance mechanism to carry out random disturbance to the current optimal solution. Finally, (w) and (b) are determined by assessing each individual’s fitness. If the individual fitness of a generation cannot be further optimized, the system checks whether the maximum number of iterations has been achieved. When the number of iterations and fitness are optimal, the neural network’s hidden layer weight is calculated.

3. Experimental Analysis

3.1. Data Collection

The data presented in this study are available in Drying Technology at 10.1080/07373937.2015.1072719. These data were derived from the following resources, which are available in the public domain: http://dx.doi.org/10.1080/07373937.2015.1072719.

Based on the experimental research data of Avramidis [1], the dielectric properties of four kinds of wood (Douglas fir, Sika spruce, western hemlock, and western red cedar) at low radio frequency are intensely discussed in this paper. In this study, samples were carefully cut into 37 mm in diameter and 5 mm thick circular specimens covering three tree species (Douglas, spruce, and hemlock), two wood regions (sapwood and heartwood), two grain directions (radial and longitudinal), and two different grain directions for western red cedar heartwood.

In order to ensure the accuracy of the experiment, the surface of the sample is evenly coated with conductive nickel paint to eliminate surface electrochemical differences. The samples were then subjected to up to 72 h of conditioning and reached equilibrium at constant temperatures of 45 °C, 25 °C, and 70 °C, as well as in conditioning chambers with 57%, 77%, and 87% relative humidity. At relative humidities of 57 percent and 77 percent, the researchers performed repeated dielectric property measurements at 0.1, 1, and 10 MHz, respectively, for each sample. As the moisture content of the sample increased, the measurement frequency was adjusted to 0.5 MHz, 1 MHz, and 10 MHz at a relative humidity of up to 87%.

Table 1. Experimental measurements of dielectric loss factor values for wood, ${\mathsf{\rho }}_0$ is the oven-dry density, $M_{xx}$ as moisture content corresponding to xx relative humidity). Specimen marking protocol: X: D = Douglas fir, S = spruce, H = hemlock, W = cedar; Y: H = heartwood, S = sapwood; Z: R = radial, L = longitudinal.

					${\mathsf{\epsilon }}_{57}^{\mathbf{″}}$				${\mathsf{\epsilon }}_{77}^{\mathbf{″}}$				${\mathsf{\epsilon }}_{87}^{\mathbf{″}}$
Specimen Species (XYZ)	Temperature (°C)	$$\begin{gathered} {\mathsf{\rho }}_{\mathbf{0}} \\ \mathsf{(}\mathbf{kg}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}\mathsf{)} \end{gathered}$$	${\mathbf{M}}_{57}$ (%)		(MHz)		${\mathbf{M}}_{77}$ (%)		(MHz)		${\mathbf{M}}_{87}$ (%)		(MHz)
DHR	25	521 ± 19	7.6	0.092	0.110	0.085	10.8	0.440	0.184	0.224	15.5	3.061	2.053	0.623
	45		7.2	0.120	0.091	0.055	10.2	0.779	0.239	0.197	14.2	5.077	2.305	0.508
	70		6.2	0.167	0.081	0.078	9.9	1.652	0.385	0.209	14.3	7.048	3.482	0.755
DHL	25		8.0	0.133	0.199	0.369	11.0	0.442	0.258	0.396	15.7	1.853	1.263	0.516
	45		7.5	0.131	0.142	0.285	10.5	0.785	0.263	0.302	14.5	3.004	1.823	0.454
	70		6.6	0.141	0.101	0.184	9.9	1.136	0.310	0.268	13.6	3.432	1.544	0.498
DSR	25		8.5	0.298	0.175	0.194	12.0	1.934	0.568	0.366	17.2	10.04	6.585	1.363
	45		8.0	0.514	0.211	0.188	11.4	4.222	0.932	0.445	15.8	16.92	10.12	1.661
	70		7.0	0.932	0.269	0.172	11.0	6.626	1.577	0.538	14.2	31.99	14.18	3.448
DSL	25		8.7	0.455	0.326	0.451	12.1	2.120	0.690	0.5s5	16.8	7.880	5.336	1.399
	45		8.2	0.824	0.337	0.381	11.6	5.451	1.081	0.571	15.9	16.65	10.10	1.451
	70		7.8	1.400	0.433	0.312	11.2	10.68	1.996	0.643	15.2	23.18	11.00	2.526
SHR	25	416 ± 11	8.2	0.065	0.092	0.167	11.2	0.200	0.124	0.193	16.2	0.913	0.817	0.306
	45		7.7	0.072	0.070	0.128	10.5	0.350	0.130	0.168	15.0	1.375	0.875	0.289
	70		6.7	0.092	0.055	0.098	9.5	0.763	0.136	0.139	14.6	2.948	1.205	0.336
SHL	25		8.2	0.204	0.270	0.484	11.2	0.599	0.369	0.533	16.2	2.458	2.269	0.832
	45		7.7	0.233	0.212	0.382	10.5	1.337	0.395	0.408	14.9	3.695	2.921	0.798
	70		6.6	0.292	0.173	0.263	9.4	2.009	0.494	0.369	14.6	6.381	2.987	0.966
SSR	25		8.4	0.444	0.220	0.209	11.5	2.619	0.674	0.374	16.9	9.573	7.113	1.487
	45		8.0	0.784	0.279	0.228	10.9	5.699	1.052	0.471	15.5	18.23	8.970	1.657
	70		6.8	1.406	0.379	0.205	9.8	12.30	1.784	0.511	14.8	45.08	13.17	3.167
SSL	25		8.6	1.676	0.702	0.555	11.6	7.969	1.944	0.976	17.2	26.18	30.99	4.435
	45		8.1	3.698	0.959	0.632	11.0	12.66	3.831	1.291	16.1	55.00	29.80	5.793
	70		6.9	5.431	1.531	0.629	9.7	42.95	7.339	1.449	15.6	93.78	39.80	7.688
HHR	25	481 ± 12	8.0	0.079	0.106	0.187	11.4	0.343	0.191	0.235	16.1	1.429	1.290	0.445
	45		7.5	0.095	0.086	0.153	10.9	0.651	0.226	0.215	14.9	2.665	1.558	0.476
	70		6.6	0.136	0.075	0.110	10.3	1.167	0.355	0.219	14.7	15.22	2.268	0.735
HHL	25		8.0	0.248	0.285	0.483	11.2	0.984	0.487	0.569	16.1	4.915	3.571	1.185
	45		7.5	0.294	0.244	0.394	10.6	2.105	0.654	0.501	14.8	7.307	4.404	1.264
	70		6.5	0.414	0.249	0.298	10.0	3.090	0.847	0.478	14.3	12.67	5.968	1.710
HSR	25		8.3	0.089	0.098	0.169	11.5	0.379	0.183	0.218	166	2.092	1.381	0.439
	45		7.8	0.113	0.085	0.138	11.0	0.710	0.235	0.198	15.3	2.962	1.985	0.467
	70		6.8	0.154	0.084	0.106	10.9	1.001	0.324	0.192	14.9	5.416	2.543	0.764
HSL	25		8.8	0.305	0.309	0.507	12.1	1.122	0.533	0.590	16.9	5.459	3.779	1.159
	45		8.3	0.382	0.274	0.409	11.5	2.674	0.708	0.529	15.5	8.356	4.385	1.326
	70		7.3	0.633	0.269	0.326	10.9	4.699	0.579	0.550	14.2	15.62	6.927	1.844
WHR	25	341 ± 9	6.6	0.042	0.055	0.091	8.8	0.144	0.073	0.108	12.4	0.722	0.562	0.200
	45		6.2	0.042	0.045	0.076	8.4	0.194	0.077	0.103	11.8	1.188	0.617	0.208
	70		5.5	0.049	0.036	0.059	7.5	0.320	0.086	0.079	10.6	2.089	0.138	0.265
WHL	25		6.4	0.108	0.153	0.217	8.6	0.273	0.179	0.256	12.5	1.791	1.204	0.440
	45		6.0	0.104	0.111	0.180	8.2	0.484	0.178	0.217	11.5	3.502	1.324	0.471
	70		5.3	0.135	0.089	0.150	6.3	1.092	0.288	0.180	10.6	4.588	1.823	0.538

details the average values of all dielectric loss factors (ε″) after 50 repeated measurements of the sample at three different combinations of relative humidity and temperature (nine environmental conditions in total).

In this study, the BP neural network model is optimized by using the Improved Pelican Optimization Algorithm to achieve an accurate prediction of the dielectric loss factor. We use the MATLAB (2022b) software platform to train the model and optimize the parameters. In the whole process of the experiment, we scientifically divided the collected 378 sets of experimental data, among which 265 sets of data were used as training sets for the learning and adaptation of the model, and the remaining 113 sets of data were used as test sets to evaluate the generalization ability and prediction accuracy of the model on unknown data. We expect to construct an efficient and accurate dielectric loss factor prediction model through this rigorous data processing and model training method.

3.2. Data Preprocessing

In order to optimize the network architecture and parameter configuration, we adopt the trial-and-error method as the solution strategy [23]. During this process, thousands of experimental tests were performed on various BP (backpropagation) neural network structures, parameter Settings, and data sets using MATLAB software. We continue these tests until the error between the measured value and the model prediction is within an acceptable threshold range. After a series of repeated attempts and adjustments, it was finally determined that the number of nodes in the hidden layer of the BP network model should be 5, and the mean square error of the model under this configuration was the best and reached the minimum value.

The sample data and the timeline must correspond precisely to ensure the accuracy and validity of the data. In view of the problem of uneven dimensionality of sample data, we need to normalize the sample data, so as to eliminate the dimensional differences between different indicators so that indicators of different units of magnitude can be compared and weighted under the same scale [24]. This is performed to ensure that the IPOA-BP model has the best generalization potential and performance. The normalization processing formula is shown in Formula (14).

$$X_{norm}=\frac{X-X_{min}}{X_{max}-X_{min}}$$

(14)

In the formula, X_norm is the normalized data, X is the original data, and X_max and X_min are the maximum and minimum values of the original data set, respectively.

3.3. Model Parameter Setting

In this paper, transit is used as the transfer function for all nodes of the hidden layer, and Purelin is used as the linear transfer function for all output layer nodes. The mathematical expression of the tansig function is shown in Formula (15), which is designed to meet the model’s specific requirements for data processing and output.

$$\tan \mathrm{sig}(\mathrm{x})=\frac{2}{1+e^{-2x}}-1$$

(15)

In this paper, we choose the trainlm() training function with the fastest convergence speed to construct the BP neural network simulation prediction model. We set the learning rate (lr) to 0.01, the target accuracy to 0.00001, and the maximum number of training iterations to 500 to ensure that the model can efficiently and accurately complete the training process.

4. Result and Discussion

4.1. Comparative Analysis of Model Performance

The error rate can judge the accuracy of the model. Common regression evaluation indexes include mean absolute error (MAE), mean square error (MSE), root mean square error (RMSE), and goodness of fit (R²), where RMSE is the square root of the MSE and is of the same order of magnitude as the true value. The MSE and the RMSE are essentially the same, but the RMSE is used to provide a better description of the data. Therefore, this paper only adopts RMSE as the evaluation index. RMSE is the square root of the ratio of the squared deviation of the predicted value to the true value of the number of observations. It can be used to measure the degree of dispersion of a set of numbers and better reflect the actual situation of the prediction value error. RMSE ∈[0,+∞); RMSE equals 0 when the predicted value is exactly the same as the true value; that is, the model is perfect, and the error is larger. R² represents the degree of precision with which the model fits the data. Generally, R² varies from 0 to 1. The nearer the value is to 1, the better the fit is and the more effective the fit is, and conversely, the worse the fit is. In order to minimize the controversy, the average value of the IPOA-BP model run for five times is taken as the final result. As shown in

Table 2. Model evaluation results.

		IPOA-BP	POA-BP	BP
MAE	Training	0.56362	1.2951	1.7489
	Testing	0.4104	1.4235	1.3496
RMSE	Training	1.4693	3.398	4.5163
	Testing	0.71495	2.7346	3.0069
${\mathrm{R}}^2$	Training	0.95728	0.89332	0.67513
	Testing	0.918	0.80706	0.73834

below, the RMSE value of the IPOA-BP model used for dielectric loss factor prediction is much smaller than that of the BP model, indicating that the optimized model has a good effect.

From Table 2, the RMES values for the training and test sets of the IPOA-BP model are 1.4693 and 0.71495, and the MAE values are 0.56362 and 0.4104, respectively. In addition, the R² of training and test sets are 0.95728 and 0.918, respectively. The results show that the measured results are in good agreement with the model predictions. Compared with the POA-BP and BP models, the IPOA-BP model training set data RMSE decreased by 67% and 93%, and the MAE decreased by 68% and 70%. This indicates that the optimization effect of the IPOA-BP model is noticeable, and the prediction effect of the IPOA-BP model is better than the other two models in all cases.

Figure 3 shows the convergence curves of the adaptations of the IPOA-BP model and the POA-BP model. As can be seen from Figure 3, setting the maximum number of model iterations to 1000, the IPOA-BP model reaches the optimum in the 22nd generation, while the POA-BP model reaches the optimum in the 27th generation. In addition, the adaptation degree value of the IPOA-BP model is always smaller than that of the POA-BP model, proving that the IPOA-BP model is better than the POA-BP model in terms of convergence speed and accuracy.

The improvement of evaluation indexes such as MAE and MSE can be translated into many practical benefits in RF heating technology, including improving prediction accuracy, improving energy utilization efficiency, and improving product quality. The realization of these benefits depends on accurate predictive models and optimized heating strategies, which together work on the RF heating process to drive continuous improvements in production efficiency and product quality. Based on more accurate prediction results, parameters such as the power and time of RF heating can be adjusted to make the heating process more accurate and efficient. For example, in wood processing, the heating power can be precisely controlled according to the real-time moisture content and temperature of the wood, avoiding overheating or underheating.

4.2. Performance Analysis in CEC Tests

The test results of CEC2005 have been widely recognized and can be used as a standard for evaluating the performance of high-authority optimization algorithms [25]. In order to objectively assess the effectiveness of the metaheuristic algorithms and to verify the practicality of the IPOA improvement strategy, this paper uses the CEC2005 test set to evaluate the performance of the IPOA, which is selected to be compared with four widely used metaheuristic algorithms on 14 benchmark test functions: the PSO (Eberhart et al. [26]), the GWO (Mirjalili et al. [27]), WSO (Braik et al. [28]), and the POA (Trojovský P et al. [15]). The CEC2005 test set contains 23 benchmark functions, including the single-peak test function, F1-F5, which has a relatively simple structure and is often used to test the convergence performance of the algorithm. The basic multimodal test function, F6-F12, with multiple local optimal solutions is used to test the algorithm’s ability to balance global, exploration, and local development of the decision space.

Parts of the analysis results are shown in Figure 4. The convergence speed of the improved IPOA on the unimodal function is significantly better than that of other algorithms, while its standard deviation and variance are maintained at a low level. Because the multimodal function has the characteristics of multiple locally optimal solutions, it can evaluate the searchability of the algorithm and the global optimal solution more accurately. In many independent experiments, the function verifies the excellent performance of the algorithm in search efficiency and achieves an excellent convergence effect.

The experimental results show that the IPOA can quickly approach the optimal solution after only a limited number of iterations, and its speed is significantly better than other algorithms. Through the trend analysis of curve fitting, we find that the IPOA performs better in local optimization ability than other algorithms. We conclude that the IPOA shows excellent search ability and has more rapid convergence when dealing with high-dimensional issues. In the solution of 14 benchmark test functions, the IPOA has reached the highest level in both solution accuracy and solution stability, and its fast convergence speed and accurate solution results show significant competitive advantages. This makes the IPOA competitive, which cannot be ignored in competition with other meta-heuristic algorithms.

4.3. Model Prediction Results

By carefully analyzing the line chart, we can observe different prediction models’ performances. As shown in Figure 5, among the prediction results of the three prediction models, the IPOA-BP model is significantly better than the other three models in terms of RMSE and MAE, and its R² value is closer to 1, which clearly indicates that the IPOA-BP model has the highest fitness value. In contrast, the BP model shows large fluctuations and high errors from the original data. After some training, the IPOA-BP model can effectively reduce its error and fit the original data more accurately.

This paper studies the prediction of the dielectric loss factor of wood during dielectric heating. The results show that a series of macroscopic physical factors significantly affect the dielectric loss factor, including wood moisture content, temperature, grain direction, frequency, and the species of wood being heated. In order to overcome the limitations of the traditional heuristic algorithm, which may stop iteration prematurely and easily fall into local optimal, this paper not only adopts the Pelican Optimization Algorithm (POA) proposed in 2022 but also combines Logistic chaotic mapping, reverse differential evolution, and firefly perturbation optimization strategies to optimize the BP model. Combining the above methods, the built IPOA-BP model shows a good fitting effect with the actual value when predicting the dielectric loss factor of wood in the RF heating and drying process, thus achieving a more accurate prediction. This research not only provides strong theoretical support and technical reference for RF heating technology in the wood processing industry but also provides new ideas for solving the balance between the scalability problem and the calculation cost problem. By reasonably adjusting and optimizing the heating parameters, the heating effect can be ensured while reducing energy consumption and operating costs, promoting the green and sustainable development of the wood processing industry.

5. Conclusions

In order to optimize the RF heating and drying process of wood, this paper focuses on the study of wood varieties, namely Douglas fir, Sika spruce, western hemlock, and western red cedar, and predicts their dielectric loss factors under different physical characteristics and macroscopic conditions. Wood’s dielectric loss factor prediction model was constructed with temperature, relative humidity, equilibrium moisture content, texture direction, frequency, density, and the species of wood being heated as input variables.

The experimental results show that the root mean square error (RMSE) of the training set is 1.4693, while the RMSE of the test set is 0.71495, which indicates that the IPOA-BP prediction model can accurately predict the dielectric loss factor of wood after RF heating. In order to comprehensively evaluate the performance of the model, the IPOA-BP model is compared with the BP model and POA-BP model, and a variety of regression evaluation indexes and CEC tests are selected for comprehensive analysis. The analysis results show that compared with the improved model, the proposed IPOA-BP model not only has higher prediction accuracy but also has faster convergence speed, without extra costs or time investments. Therefore, the model can be used to estimate the dielectric loss factor of wood, but its generalization ability (i.e., its prediction ability under different wood species and conditions) still needs further validation. In addition, how to effectively adjust various other parameters to optimize the model performance is an issue that requires continuous research.

By using this method to predict the dielectric loss factor of wood, the RF heating and drying process can be optimized to improve the efficiency and quality of wood processing. At the same time, the model for predicting the dielectric loss factor can be combined with other models for predicting wood properties (e.g., density, strength, etc.) to form a multiproperty prediction system. This will provide a more comprehensive understanding of wood properties and uses, better utilization of wood resources, and improved sustainability of the wood processing industry.