[BUG] Observability in radial grid with phasor voltage measurement is not correct

[TonyXiang8787]

Background

In #914 and #1024 we have implemented the observability check in radial grid with phasor voltage measurement. However, the mathematical workout is not correct. We assumed that the phasor voltage sensor can be arbitrarily assigned to a branch, just as an injection measurement. This assumption does not hold.

Rationale

We should not assign phasor voltage sensor to any branches. Instead, we should solely rely on branch and injection sensor to connect the graph. After trying to make maximum connected graph, we check if the graph is fully connected. If yes, there should be at least one voltage sensor (phasor or not) for the grid to be observable. If no, each island should have at least one voltage phasor sensor for the grid to be observable.

Mathematical work-out

Graph

We still use the DFS graph built by the topology and YBus. The vertex order is in the reverse order of the DFS search.

During the vertex traversal, we try to propagate the availability of voltage phasor sensor in the current sub-graph. When we want to steal the upstream injection sensor to connect the graph, we choose not to steal it if the current leaf sub-graph already has voltage phasor sensor. In this way, the upstream injection sensor can be reserved for another leaf sub-graph.

Check observability

Initialization

Voltage phasor sensor

As proposed in #914, we have a separate vertex array for phasor voltage sensor.

1. Set the vertices with phasor voltage sensor to phasor_voltage_sensor_available and other vertices to phasor_voltage_sensor_unavailable.
2. If there is no phasor voltage sensor at all, set the first vertex which has voltage magnitude sensor to phasor_voltage_sensor_available.

Flow sensor

For flow sensor we re-use the YBusStructure as data array to set injection sensor and branch sensor. Note we need to do bi-directional traversal, but we still set only the upper triangle of the matrix. When we do downstream edges search, we use the transpose lookup to find the relevant information about measured or not_measured.

Graph search

We traverse the vertices of the above grid in the reverse order of the DFS results. For each vertex, we do the following:

1. If the predecessor edge is not_measured .
   1. If the current vertex has injection_available
      1. change the predecessor edge to measured.
      2. change the current vertex to injection_unavailable.
   2. Else if the predecessor vertex has injection_available AND current vertex has phasor_voltage_sensor_unavailable. The second condition prevents the upstream vertex's injection gets claimed by an already phasor voltage reachable leaf sub-graph.
      1. change the predecessor edge to measured.
      2. change the predecessor vertex to injection_unavailable.
2. Change the current vertex to injection_unavailable, regardless of the original state.
3. If the predecessor edge is measured and current vertex has phasor_voltage_sensor_available.
   1. change predecessor vertex to phasor_voltage_sensor_available, this propagates the phasor voltage reachability.

Now we do a forward DFS traverse to spread the phasor voltage reachability to downstream leaves. This means a reverse loop from YBus order. For each vertex, we do the following:

1. If the current vertex has phasor_voltage_sensor_available. We loop all its downstream edges and vertices. For each downstream edge and vertex, we do:
   1. If the downstream edge has measured. Note we are now in the lower triangle of the data array which is always ignored. You need to look at its transpose position.
      1. change downstream vertex to phasor_voltage_sensor_available.

After forward DFS traverse, we check if all the vertices have phasor_voltage_sensor_available. If yes, the system is observable. If no, the system is not observable.

NOTE: the system without phasor voltage sensor is covered by artificially change one vertex with voltage magnitude sensor to voltage phasor sensor.