In the past two decades, chaotic modulation schemes have drawn much attention for their significant advantages over other traditional modulation techniques. One of the noticeable advantages is the capability of providing secure communication in the presence of an eavesdropper. In this article, we study the physical layer security (PLS) performance of low-frequency power line communication (PLC) systems against multiple eavesdroppers by employing differential chaos shift keying based modulation technique. A wiretap power line channel model is investigated by considering two different cases: 1) when all the channels are assumed to be independent and identically distributed (IID) following Log-normal distribution; and 2) the wiretap channels are considered to be identical but correlated Log-normally distributed and independent of the main channel. Contemporaneously, the Bernoulli-Gaussian random process models the PLC noise. Further, a comprehensive PLS study of the considered PLC system is characterized in terms of the average secrecy capacity, secure outage probability, and strictly positive secrecy capacity. We also propose an algorithm to maximize the secrecy throughput under security and reliability constraints. Moreover, to obtain essential insights, we reveal the impact of various crucial parameters onto the proposed system.