Observability Analysis
In this example, we analyze a 6-bus power system, shown in Figure 1. The initial objective is to conduct an observability analysis to identify observable islands and restore observability. Later, we examine optimal PMU placement to ensure system observability using only phasor measurements.
Figure 1: The 6-bus power system.
Users can download a Julia script containing the scenarios from this section using the following link.
We define the power system, specify buses and branches, and assign the generator to the slack bus:
system = powerSystem()
addBus!(system; label = "Bus 1", type = 3)
addBus!(system; label = "Bus 2")
addBus!(system; label = "Bus 3")
addBus!(system; label = "Bus 4")
addBus!(system; label = "Bus 5")
addBus!(system; label = "Bus 6")
@branch(reactance = 0.22)
addBranch!(system; label = "Branch 1", from = "Bus 1", to = "Bus 2")
addBranch!(system; label = "Branch 2", from = "Bus 2", to = "Bus 3")
addBranch!(system; label = "Branch 3", from = "Bus 2", to = "Bus 4")
addBranch!(system; label = "Branch 4", from = "Bus 3", to = "Bus 5")
addBranch!(system; label = "Branch 5", from = "Bus 3", to = "Bus 4")
addBranch!(system; label = "Branch 6", from = "Bus 4", to = "Bus 6")
addGenerator!(system; label = "Generator 1", bus = "Bus 1")Notably, observability analysis and optimal PMU placement are independent of branch parameters, as well as measurement values and variances.
Identification of Observable Islands
Next, we define the measurement model. JuliaGrid employs standard observability analysis based on the linear decoupled measurement model. Active power measurements from wattmeters are used to estimate bus voltage angles, while reactive power measurements from varmeters estimate bus voltage magnitudes. In this example, the 6-bus power system is monitored by four meters, as shown in Figure 2.
Figure 2: The 6-bus power system monitoring with four power meters.
Notably, the island detection step relies only on wattmeters, meaning that if the goal is simply to identify observable islands, defining wattmeters alone is sufficient. However, for observability restoration followed by state estimation, varmeters are also needed. Therefore, the four meters represent both wattmeters and varmeters:
device = measurement()
addWattmeter!(system, device; label = "Meter 1", from = "Branch 1", active = 1.1)
addVarmeter!(system, device; label = "Meter 1", from = "Branch 1", reactive = -0.5)
addWattmeter!(system, device; label = "Meter 2", bus = "Bus 2", active = -0.1)
addVarmeter!(system, device; label = "Meter 2", bus = "Bus 2", reactive = -0.1)
addWattmeter!(system, device; label = "Meter 3", bus = "Bus 4", active = -0.3)
addVarmeter!(system, device; label = "Meter 3", bus = "Bus 4", reactive = 0.6)
addWattmeter!(system, device; label = "Meter 4", to = "Branch 6", active = 0.2)
addVarmeter!(system, device; label = "Meter 4", to = "Branch 6", reactive = 0.3)Attempting to solve nonlinear state estimation with these measurements would not be possible, as the gain matrix would be singular. The same issue arises with DC state estimation. To prevent this, users can perform observability analysis, which adds non-redundant measurements to ensure a nonsingular gain matrix and a unique state estimator.
Observability analysis begins with identifying observable islands. We have the ability to identify both flow observable and maximal observable islands, each of which can be used as the foundation for the restoration step. To provide a comprehensive analysis, we will explore both types of islands. In the first step, we focus on determining the flow observable islands:
islands = islandTopologicalFlow(system, device)As the result, four flow observable islands are identified:
julia> islands.island4-element Vector{Vector{Int64}}: [1, 2] [3] [4, 6] [5]
The first observable island consists of Bus 1 and Bus 2, the second island is formed by Bus 3, the third island includes Bus 4 and Bus 6, while Bus 5 constitutes the fourth island, as shown in Figure 3.
Figure 3: Flow observable islands in the 6-bus power system.
In addition to flow islands, we can also identify maximal observable islands:
islands = islandTopological(system, device)The results reveal the identification of two maximal observable islands:
julia> islands.island2-element Vector{Vector{Int64}}: [1, 2, 3, 4, 6] [5]
As observed, the devices Meter 2 and Meter 3 together merge the first, second, and third flow observable islands into one, as shown in Figure 4.
Figure 4: Maximal observable islands in the 6-bus power system.
From the standpoint of island identification, detecting power flow islands requires less computational effort compared to identifying maximal observable islands. However, when it comes to restoring observability, the process involving power flow islands tends to be more computationally demanding than with maximal observable islands.
Observability Restoration
To perform the observability restoration step, a new set of measurements, called pseudo-measurements, is needed. These typically hold historical data about electrical quantities. Let us define this set:
pseudo = measurement()
addWattmeter!(system, pseudo; label = "Pseudo 1", from = "Branch 5", active = 0.3)
addVarmeter!(system, pseudo; label = "Pseudo 1", from = "Branch 5", reactive = 0.1)
addWattmeter!(system, pseudo; label = "Pseudo 2", bus = "Bus 5", active = 0.3)
addVarmeter!(system, pseudo; label = "Pseudo 2", bus = "Bus 5", reactive = -0.2)Next, we can invoke the observability restoration function:
restorationGram!(system, device, pseudo, islands)This function identifies the minimal set of pseudo-measurements needed to make the system observable, which in this case is Pseudo 2. This pseudo-measurement is then transferred to the measurement model.
As a result, the final set of wattmeters measuring active power is:
printWattmeterData(system, device)|--------------------------------------------|
| Wattmeter Data |
|--------------------------------------------|
| Label | Active Power |
| | |
| | Measurement | Variance | Status |
| | [pu] | [pu] | |
|----------|-------------|----------|--------|
| Meter 1 | 1.1000 | 1.00e-04 | 1 |
| Meter 2 | -0.1000 | 1.00e-04 | 1 |
| Meter 3 | -0.3000 | 1.00e-04 | 1 |
| Meter 4 | 0.2000 | 1.00e-04 | 1 |
| Pseudo 2 | 0.3000 | 1.00e-04 | 1 |
|--------------------------------------------|Likewise, the final set of varmeters measuring reactive power is:
printVarmeterData(system, device)|--------------------------------------------|
| Varmeter Data |
|--------------------------------------------|
| Label | Reactive Power |
| | |
| | Measurement | Variance | Status |
| | [pu] | [pu] | |
|----------|-------------|----------|--------|
| Meter 1 | -0.5000 | 1.00e-04 | 1 |
| Meter 2 | -0.1000 | 1.00e-04 | 1 |
| Meter 3 | 0.6000 | 1.00e-04 | 1 |
| Meter 4 | 0.3000 | 1.00e-04 | 1 |
| Pseudo 2 | -0.2000 | 1.00e-04 | 1 |
|--------------------------------------------|As we can see, adding the Pseudo 2 measurement makes the system observable, which we can confirm by identifying observable islands with only one island:
islands = islandTopological(system, device)julia> islands.island1-element Vector{Vector{Int64}}: [1, 2, 4, 6, 5, 3]
To proceed with the nonlinear state estimation algorithm, we need one additional step to make the state estimation solvable. Specifically, it is crucial to ensure that the system has at least one bus voltage magnitude measurement. This requirement stems from the fact that observable islands are identified using wattmeters, which estimate voltage angles. Since the voltage angle at the slack bus is already known, the same approach should be applied to bus voltage magnitudes. To fulfill this condition, we add the following measurement:
addVoltmeter!(system, device; label = "Pseudo 3", bus = "Bus 1", magnitude = 1.0)Figure 5 illustrates the measurement configuration that makes our 6-bus power system observable and ensures a unique state estimator.
Figure 5: Measurement configuration that makes the 6-bus power system observable.
Optimal PMU Placement
The goal of the PMU placement algorithm is to determine the minimal number of PMUs required to make the system observable. In this case, we analyze a 6-bus power system without power meters and identify the smallest set of PMUs needed for full observability, ensuring a unique state estimator:
placement = pmuPlacement(system, HiGHS.Optimizer)The optimal PMU locations can be retrieved with:
julia> keys(placement.bus)KeySet for a OrderedCollections.OrderedDict{String, Int64} with 3 entries. Keys: "Bus 2" "Bus 3" "Bus 4"
Figure 6 illustrates the PMU configuration that ensures observability and guarantees a unique state estimator. Each installed PMU measures the bus voltage phasor and the current phasors of all connected branches.
Figure 6: PMU configuration that makes the 6-bus power system observable.
The configuration of phasor measurements includes voltage phasor measurements at Bus 2, Bus 3, and Bus 4. Additionally, there is a set of current phasor measurements at the from-bus ends of the branches, as shown below:
julia> keys(placement.from)KeySet for a OrderedCollections.OrderedDict{String, Int64} with 5 entries. Keys: "Branch 2" "Branch 3" "Branch 4" "Branch 5" "Branch 6"
To complete the measurement setup, the set should also include current phasor measurements at the to-bus ends of the branches, as specified in the following information:
julia> keys(placement.to)KeySet for a OrderedCollections.OrderedDict{String, Int64} with 4 entries. Keys: "Branch 1" "Branch 2" "Branch 3" "Branch 5"
These variables provide users with a convenient way to define phasor measurement values, whether based on AC power flow or AC optimal power flow analyses, which have been thoroughly explored in state estimation examples. Additionally, users have the option to manually specify measurement values, an example of this approach is presented below:
pmu = measurement()
addPmu!(system, pmu; label = "PMU 1-1", bus = "Bus 2", magnitude = 1.1, angle = -0.2)
addPmu!(system, pmu; label = "PMU 1-2", to = "Branch 1", magnitude = 1.2, angle = -2.7)
addPmu!(system, pmu; label = "PMU 1-3", from = "Branch 2", magnitude = 0.6, angle = 0.3)
addPmu!(system, pmu; label = "PMU 1-4", from = "Branch 3", magnitude = 0.6, angle = 0.7)
addPmu!(system, pmu; label = "PMU 2-1", bus = "Bus 3", magnitude = 1.2, angle = -0.3)
addPmu!(system, pmu; label = "PMU 2-2", to = "Branch 2", magnitude = 0.6, angle = -2.8)
addPmu!(system, pmu; label = "PMU 2-3", from = "Branch 4", magnitude = 0.3, angle = -2.8)
addPmu!(system, pmu; label = "PMU 3-1", bus = "Bus 4", magnitude = 1.2, angle = -0.3)
addPmu!(system, pmu; label = "PMU 3-2", to = "Branch 3", magnitude = 0.6, angle = -2.3)
addPmu!(system, pmu; label = "PMU 3-3", to = "Branch 4", magnitude = 0.3, angle = 0.3)
addPmu!(system, pmu; label = "PMU 3-4", from = "Branch 6", magnitude = 0.2, angle = 1.9)This set of phasor measurements ensures system observability and guarantees a unique state estimator. The defined phasor measurements can be displayed using:
printPmuData(system, pmu)|-----------------------------------------------------------------------------|
| PMU Data |
|-----------------------------------------------------------------------------|
| Label | Voltage Magnitude | Voltage Angle |
| | | |
| | Measurement | Variance | Status | Measurement | Variance | Status |
| | [pu] | [pu] | | [rad] | [rad] | |
|---------|-------------|----------|--------|-------------|----------|--------|
| PMU 1-1 | 1.1000 | 1.00e-08 | 1 | -0.2000 | 1.00e-08 | 1 |
| PMU 2-1 | 1.2000 | 1.00e-08 | 1 | -0.3000 | 1.00e-08 | 1 |
| PMU 3-1 | 1.2000 | 1.00e-08 | 1 | -0.3000 | 1.00e-08 | 1 |
|-----------------------------------------------------------------------------|
|-----------------------------------------------------------------------------|
| PMU Data |
|-----------------------------------------------------------------------------|
| Label | Current Magnitude | Current Angle |
| | | |
| | Measurement | Variance | Status | Measurement | Variance | Status |
| | [pu] | [pu] | | [rad] | [rad] | |
|---------|-------------|----------|--------|-------------|----------|--------|
| PMU 1-2 | 1.2000 | 1.00e-08 | 1 | -2.7000 | 1.00e-08 | 1 |
| PMU 1-3 | 0.6000 | 1.00e-08 | 1 | 0.3000 | 1.00e-08 | 1 |
| PMU 1-4 | 0.6000 | 1.00e-08 | 1 | 0.7000 | 1.00e-08 | 1 |
| PMU 2-2 | 0.6000 | 1.00e-08 | 1 | -2.8000 | 1.00e-08 | 1 |
| PMU 2-3 | 0.3000 | 1.00e-08 | 1 | -2.8000 | 1.00e-08 | 1 |
| PMU 3-2 | 0.6000 | 1.00e-08 | 1 | -2.3000 | 1.00e-08 | 1 |
| PMU 3-3 | 0.3000 | 1.00e-08 | 1 | 0.3000 | 1.00e-08 | 1 |
| PMU 3-4 | 0.2000 | 1.00e-08 | 1 | 1.9000 | 1.00e-08 | 1 |
|-----------------------------------------------------------------------------|