Since the substations are in a fixed line (distinct positions), and each is assigned one mode, the total number is exactly the number of such binary strings: 13. - Deep Underground Poetry

Understanding the Context

Fixed Positions and Binary Assignment: The Foundation

Imagine a power transmission line stretching across a region, with substations placed at perfectly distinguishable physical points. Each substation is isolated and non-overlapping—no two occupy the same location. This unambiguous spatial arrangement allows engineers to define unique identifiers based solely on substation position.

When each substation is assigned one mode from a binary choice (often “active” / “non-active” or “on/off”), and no two substations share the same mode assignment, the core combinatorial problem reduces to how many distinct ways we can assign binary values across these fixed positions—ensuring each mode assignment is unique per substation.

Key Insights

This setup is equivalent to generating binary strings of length n, where n is the number of fixed substations. Each bit reflects the mode (0 or 1) at that specific position.

Why exactly 13 configurations?

While many systems allow more complex assignments or overlapping constraints, the uniqueness of 13 arises from a constrained combinatorial scenario: when the physical layout enforces exactly 13 distinct binary strings consistent with fixed mode assignments.

In practice, this count may originate from:

• A predefined substation network of 6 nodes (common in regional grid models),
  
• Each node assigned one of two modes (e.g., operational vs. standby),
  
• But mutual exclusivity rules—such as limiting “active” states to specific substations or enforcing parity conditions—reduce the total number of valid binary strings to precisely 13.

Final Thoughts

This reduction reflects realistic engineering constraints where not all binary combinations are feasible due to topology, load balancing, fault tolerance, or maintenance protocols.

Practical Implications

Understanding the exact count—13—enables better modeling of:

• System reconfigurations: During maintenance or emergencies, knowing how many valid operational modes exist helps optimize switch transitions.
  
• State monitoring: Binary string identification supports automated diagnostics and real-time grid state tracking.
  
• Capacity planning: The bounded number of configurations informs scalability limits and redundancy requirements in substation network design.