Power systems with utility-scale solar photovoltaic (PV) can significantly influence the operating point (OP) of synchronous generators, particularly during periods of high solar PV generation. A sudden drop in solar PV output, due to cloud cover or other transient conditions, will alter the generation of synchronous generators, shifting their OPs. These shifted OPs can become a challenge for stability, as the system may operate closer to its stability limits. If a disturbance occurs at this new OP, the system may experience reduced stability margins, making it more vulnerable to instability, increased oscillations, or even loss of synchronism. This study introduces a scalable Delta-automatic generation control (Delta-AGC) logic method designed to address the stability challenges arising from shifts in the OPs of synchronous generators during abrupt drops in PV generation. By temporarily adjusting the OPs of synchronous generators through modification of their participation factors (PFs) in the AGC logic dispatch, the proposed method enhances power system stability. The proposed Delta-AGC logic method focuses on the optimal determination of Delta PFs in power systems with large number of generators, using the concept of coherency and employing a hierarchical optimization strategy that includes both inter-coherent and intra-coherent group optimization. Additionally, a new electromechanical oscillation index (EMOI), integrating both time response analysis (TRA) and frequency response analysis (FRA), is utilized as an online situational awareness tool (SAT) for optimizing the system's stability under various conditions. This online SAT has been implemented in a decentralized manner at the area level, limiting wide-area communication overheads and any cybersecurity concerns. The Delta-AGC logic method is illustrated on a modified IEEE 68 bus system, incorporating large utility-scale solar PV plants, and is validated through real-time simulation. Various cases, including high-loading conditions with and without power system stabilizers, conventional AGC logic, and Delta-AGC logic, are carried out to evaluate the effectiveness of the proposed Delta-AGC logic method. The results illustrate the performance and benefits of the Delta-AGC logic method, highlighting its potential to significantly enhance power system stability.