###### Abstract  
As a range-free localization algorithm, DV-Hop has gained widespread attention due to its advantages of simplicity and ease of implementation. However, this algorithm also has some defects, such as poor localization accuracy and vulnerability to network topology. This paper presents a comprehensive analysis of the factors contributing to the inaccuracy of the DV-Hop algorithm. An improved proportional integral derivative (PID) search algorithm (PSA) DV-Hop hybrid localization algorithm based on weighted hyperbola (IPSA-DV-Hop) is proposed. Firstly, the first hop distance refinement is employed to rectify the received signal strength indicator (RSSI). In order to replace the original least squares solution, a weighted hyperbolic algorithm based on the degree of covariance is adopted. Secondly, the localization error is further reduced by employing the improved PSA. In addition, the selection process of the node set is optimized using progressive sample consensus (PROSAC) followed by a 3D hyperbolic algorithm based on coplanarity. This approach effectively reduces the computational error associated with the hopping distance of the beacon nodes in the 3D scenarios. Finally, the simulation experiments demonstrate that the proposed algorithm can markedly enhance the localization precision in both isotropic and anisotropic networks and reduce the localization error by a minimum of 30% in comparison to the classical DV-Hop. Additionally, it also exhibits stability under the influence of a radio irregular model (RIM). wireless sensor network; DV-Hop; anisotropic networks; RIM; PSA 2024123908  
nodes [5]. Therefore, this approach offers better performance in terms of hardware cost and energy consumption. The DV-Hop algorithm is a classic range-free localization method. Its core mechanism is based on distance vector routing. With advantages such as low communication overhead and ease of deployment, it demonstrates significant potential for application in various scenarios. Currently, the majority of research on DV-Hop is concentrated on integrating novel algorithms to enhance performance during specific phases of the original DV-Hop process. [PERSON] et al. proposed an improved DV-Hop algorithm based on node negotiation and multiple communication radii. This method utilized a variable communication radius for refinement and considered only the beacon node with the least number of hops for computation, thereby improving localization accuracy [6]. In order to further enhance localization accuracy, [PERSON], [PERSON] et al. introduced a power function interpolation-based approach to rectify the RSSI values and substitute the least squares estimation with a disk-based multiplication algorithm [7]. Since WSNs are often deployed in complex terrain, many scholars have proposed DV-Hop algorithms for 3D scenarios. [PERSON], [PERSON] et al. proposed a swarm intelligent localization algorithm based on hybrid gray wolves and fireflies to assist localization with the assistance of virtual beacon nodes, which improved localization accuracy in 3D scenarios [8]. [PERSON], [PERSON] et al. achieved significant improvement in localization errors in 3D scenarios by correcting hop distances with correction factors and CTO-based orthogonal learning [9]. Although numerous improvements have been made in WSN localization, most algorithms continue to face challenges in adapting to diverse scenarios. They often suffer from complex localization processes, poor robustness, and low accuracy in various network topologies. To address these issues, a hybrid PSA-DV-Hop localization algorithm based on weighted hyperbolic improvement is proposed in this paper. In the 2D scenario, the weighted hyperbolic PSA based on collinearity can effectively improve localization accuracy while maintaining low calculation complexity. In order to extend the algorithm to 3D scenarios, the PROSAC algorithm is employed to analyze the correlation degree among nodes. In addition, the coplanarity strategy is used to correct the positions of unknown nodes.