## 1 Introduction  
Wireless sensor networks (WSNs) are a fundamental component in a wide range of applications, including environmental monitoring, object tracking, logistics management, and industrial operations [1]. These networks consist of small, cost-effective sensor nodes that communicate wirelessly, and the precise localization of these nodes within the network is essential for their effective operation. These networks, comprising small, cost-effective sensor nodes, may require precise localization for effective operation. A review published in [2] exposed location-aware schemes, as well as parameters affecting the performance of localization during the past few years. Within the realm of WSN localization, techniques are broadly categorized into two main types: anchor-based and anchor-free methods. Anchor-based methods rely on nodes with known positions to assist in the localization process, offering high accuracy but often at the cost of reduced flexibility and increased operational expenses. In contrast, anchor-free methods, which do not depend on predefined anchor positions, offer greater flexibility and scalability, making them increasingly popular in dynamic and large-scale deployment scenarios [3; 4; 5]. Traditionally, localization has often relied on anchor nodes with known positions. Although these algorithms have high localization accuracy, anchors introduce limitations in deployment flexibility and incur higher operational costs. Consequently, there is a growing interest in developing and refining anchor-free algorithms, which promise a flexible structure and reduced cost. Our research directly addresses the challenges and opportunities in anchor-free localization within WSNs [6; 7]. In our endeavor, we introduce the \"contextual received signal strength approach\" (CRSSA), an innovative method leveraging the widely utilized received signal strength (RSS) metric, compatible with a broad range of commercially available sensors. This universality is a key strength, enabling diverse sensors to efficiently process RSS data extracted from standard received communication packets. Although RSS-based localization is celebrated for its simplicity and widespread applicability, it grapples with accuracy challenges stemming from the complexities and unpredictability of signal propagation, especially in industrial environments. Our previous work in developing CRSSA for one-dimensional (1D) environments showed promising results [8]; however, the problem was confined to a basic linear context or string topology. In this article, the primary research question that we address is whether extending CRSSA to two-dimensional (2D) WSNs maintains its precision and avoids divergence. Moving beyond the linear limitations of the 1D context, we aim to explore the capabilities of CRSSA in the more complex and spatially diverse 2D environments, ensuring that the transition to a higher dimensional space does not compromise the method's accuracy or lead to algorithmic instability. By thoroughly investigating signal behavior complexities, including multipath effects and environmental interference, we strive to affirm the robustness and reliability of the CRSSA for diverse and demanding conditions in industrial WSN applications [6; 7]. Our research encompasses a comprehensive simulation evaluation, designed to rigorously evaluate the CRSSA in a 2D setting. We compare the performance of the 2D-extended CRSSA against established anchor-free algorithms, with a focus on determining whether the shift to 2D preserves the accuracy and stability observed in 1D environments. Special emphasis is placed on evaluating the precision of inter-node distance estimation, a critical factor in assessing the effectiveness of any localization method. The forthcoming sections will thoroughly explore the theoretical underpinnings of the 2D CRSSA, delve into the intricate signal propagation challenges in 2D environments, and present detailed results from simulations.