AC Power Flow

To perform the AC power flow analysis, we will first need the PowerSystem type that has been created with the AC model. Following that, we can construct the power flow model encapsulated within the ACPowerFlow type by employing one of the following functions:

• newtonRaphson,
• fastNewtonRaphsonBX,
• fastNewtonRaphsonXB,
• gaussSeidel.

These functions will set up the AC power flow framework. To obtain bus voltages and solve the power flow problem, we can use the following functions:

• mismatch!,
• solve!.

Additionally, the package provides two functions for reactive power limit validation of generators and adjusting the voltage angles to match an arbitrary bus angle:

• reactiveLimit!,
• adjustAngle!.

After obtaining the AC power flow solution, JuliaGrid offers post-processing analysis functions for calculating powers and currents associated with buses, branches, or generators:

• power!,
• current!.

Additionally, specialized functions are available for calculating specific types of powers or currents for individual buses, branches, or generators.

Bus Type Modification

Depending on how the system is constructed, the types of buses that are initially set are checked and can be changed during the construction of the ACPowerFlow type.

Assuming the Newton-Raphson method has been chosen, to explain the details, we can observe a power system:

system = powerSystem()

addBus!(system; label = "Bus 1", type = 3)
addBus!(system; label = "Bus 2", type = 2)
addBus!(system; label = "Bus 3", type = 2)

addGenerator!(system; bus = "Bus 2")

analysis = newtonRaphson(system)

Initially, Bus 1 is set as the slack bus (type = 3), and Bus 2 and Bus 3 are generator buses (type = 2). However, Bus 3 does not have a generator, and JuliaGrid considers this a mistake and changes the corresponding bus to a demand bus (type = 1).

After this step, JuliaGrid verifies the slack bus. Initially, the slack bus (type = 3) corresponds to Bus 1, but since it does not have an in-service generator connected to it, JuliaGrid recognizes it as another mistake. Therefore, JuliaGrid assigns a new slack bus from the available generator buses (type = 2) that have connected in-service generators. In this specific example, Bus 2 becomes the new slack bus.

As a result, we can observe the updated array of bus types:

julia> print(system.bus.label, system.bus.layout.type)Bus 1: 1
Bus 2: 3
Bus 3: 1

Note that, if a bus is initially defined as the demand bus (type = 1) and later a generator is added to it, the bus type will not be changed to the generator bus (type = 2). Instead, it will remain as a demand bus.

Setup Starting Voltages

To begin analyzing the AC power flow in JuliaGrid, we must first establish the PowerSystem type. Once the power system is set up, we can select one of the available methods for solving the AC power flow problem, such as newtonRaphson, fastNewtonRaphsonBX, fastNewtonRaphsonXB, or gaussSeidel.

Assuming we have selected the Newton-Raphson method, we can use the following code snippet:

system = powerSystem()

addBus!(system; label = "Bus 1", type = 3, magnitude = 1.0, angle = 0.0)
addBus!(system; label = "Bus 2", type = 1, magnitude = 0.9, angle = -0.1)
addBus!(system; label = "Bus 3", type = 2, magnitude = 0.8, angle = -0.2)

addGenerator!(system; bus = "Bus 1", magnitude = 1.3)
addGenerator!(system; bus = "Bus 2", magnitude = 1.1)
addGenerator!(system; bus = "Bus 3", magnitude = 1.2)

acModel!(system)
analysis = newtonRaphson(system)

Here, the function newtonRaphson generates starting voltage vectors in polar coordinates.

The starting voltage magnitudes are set to:

julia> print(system.bus.label, analysis.voltage.magnitude)Bus 1: 1.3
Bus 2: 0.9
Bus 3: 1.2

This vector is created based on the bus types by selecting voltage magnitude values from the PowerSystem type, using the vectors:

julia> [system.bus.voltage.magnitude system.generator.voltage.magnitude]3×2 Matrix{Float64}:
 1.0  1.3
 0.9  1.1
 0.8  1.2

The starting voltage angles are set to:

julia> print(system.bus.label, analysis.voltage.angle)Bus 1: 0.0
Bus 2: -0.1
Bus 3: -0.2

This vector is derived from the voltage angle values in the PowerSystem type:

julia> system.bus.voltage.angle3-element Vector{Float64}:
  0.0
 -0.1
 -0.2

Custom Starting Voltages

This method of specifying starting values has a significant advantage in that it allows the user to easily change the starting voltage magnitudes and angles, which play a crucial role in iterative methods. For instance, suppose we define our power system as follows:

system = powerSystem()

addBus!(system; label = "Bus 1", type = 3, magnitude = 1.0, angle = 0.0)
addBus!(system; label = "Bus 2", type = 1, magnitude = 0.9, angle = -0.1)
addBus!(system; label = "Bus 3", type = 2, magnitude = 0.8, angle = -0.2)

addGenerator!(system; bus = "Bus 1", magnitude = 1.1)
addGenerator!(system; bus = "Bus 3", magnitude = 1.2)

acModel!(system)

Now, the user can initiate a "flat start", this can be easily done as follows:

for i = 1:system.bus.number
    system.bus.voltage.magnitude[i] = 1.0
    system.bus.voltage.angle[i] = 0.0
end

analysis = newtonRaphson(system)

The starting voltage values are:

julia> print(system.bus.label, analysis.voltage.magnitude, analysis.voltage.angle)Bus 1: 1.1, 0.0
Bus 2: 1.0, 0.0
Bus 3: 1.2, 0.0

Consequently, the iteration begins with a fixed set of voltage magnitude values that remain constant throughout the iteration process. The remaining values are initialized as part of the "flat start" approach.

Power Flow Solution

To start, we will create a power system and define the AC model by invoking the acModel! function:

system = powerSystem()

addBus!(system; label = "Bus 1", type = 3, active = 0.5, magnitude = 0.9, angle = 0.0)
addBus!(system; label = "Bus 2", type = 1, reactive = 0.05, magnitude = 1.1, angle = -0.1)
addBus!(system; label = "Bus 3", type = 1, active = 0.5, magnitude = 1.0, angle = -0.2)

@branch(resistance = 0.02, conductance = 1e-4, susceptance = 0.04)
addBranch!(system; label = "Branch 1", from = "Bus 1", to = "Bus 2", reactance = 0.05)
addBranch!(system; label = "Branch 2", from = "Bus 1", to = "Bus 2", reactance = 0.01)
addBranch!(system; label = "Branch 3", from = "Bus 2", to = "Bus 3", reactance = 0.04)

@generator(active = 3.2)
addGenerator!(system; label = "Generator 1", bus = "Bus 1", magnitude = 1.1)
addGenerator!(system; label = "Generator 2", bus = "Bus 2", magnitude = 1.2)

acModel!(system)

Once the AC model is defined, we can choose the method to solve the power flow problem. JuliaGrid provides four methods: newtonRaphson, fastNewtonRaphsonBX, fastNewtonRaphsonXB, and gaussSeidel.

For example, to use the Newton-Raphson method to solve the power flow problem, we can use:

analysis = newtonRaphson(system)

analysis = newtonRaphson(system, QR)

This function sets up the desired method for an iterative process based on two functions: mismatch! and solve!. The mismatch! function calculates the active and reactive power injection mismatches using the given voltage magnitudes and angles, while solve! computes the voltage magnitudes and angles.

To perform an iterative process with the Newton-Raphson or fast Newton-Raphson methods in JuliaGrid, the mismatch! function must be included inside the iteration loop. For instance:

for iteration = 1:100
    mismatch!(system, analysis)
    solve!(system, analysis)
end

Upon completion of the AC power flow analysis, the solution is conveyed through the bus voltage magnitudes and angles. Here are the values corresponding to the buses:

julia> print(system.bus.label, analysis.voltage.magnitude, analysis.voltage.angle)Bus 1: 1.1, 0.0
Bus 2: 1.1312984943545519, 0.022034582385283743
Bus 3: 1.123144964293639, 0.005894335154140862

In contrast, the iterative loop of the Gauss-Seidel method does not require the mismatch! function:

analysis = gaussSeidel(system)
for iteration = 1:100
    solve!(system, analysis)
end

In these examples, the algorithms run until the specified number of iterations is reached.

Breaking the Iterative Process

We can terminate the iterative process using the mismatch! function. The following code shows an example of how to use the function to break out of the iteration loop:

analysis = newtonRaphson(system)
for iteration = 1:100
    stopping = mismatch!(system, analysis)
    if all(stopping .< 1e-8)
        println("Solution Found.")
        break
    end
    solve!(system, analysis)
end

Solution Found.

The mismatch! function returns the maximum absolute values of active and reactive power injection mismatches, which are commonly used as a convergence criterion in iterative AC power flow algorithms. Note that the function can also be used to terminate the loop when using the Gauss-Seidel method, even though it is not required.

Combining Methods

The PowerSystem type, once created, can be shared among different methods, offering several advantages.

For instance, while the Gauss-Seidel method is commonly used to swiftly derive an approximate solution, the Newton-Raphson method is favored for obtaining precise final solutions. Hence, a strategy involves employing the Gauss-Seidel method for a limited number of iterations, followed by initializing the Newton-Raphson method with the voltages obtained from the Gauss-Seidel method, leveraging it as a starting point for further refinement:

gs = gaussSeidel(system)
for iteration = 1:5
    solve!(system, gs)
end

Next, we can initialize the Newton-Raphson method with the voltages obtained from the Gauss-Seidel method and start the algorithm from that point:

analysis = newtonRaphson(system)

for i = 1:system.bus.number
    analysis.voltage.magnitude[i] = gs.voltage.magnitude[i]
    analysis.voltage.angle[i] = gs.voltage.angle[i]
end

for iteration = 1:100
    stopping = mismatch!(system, analysis)
    if all(stopping .< 1e-8)
        break
    end
    solve!(system, analysis)
end

Print Results in the REPL

Users have the option to print the results in the REPL using any units that have been configured, such as:

@voltage(pu, deg, V)
printBusData(system, analysis)

|----------------------------|
| Bus Data                   |
|----------------------------|
| Label |      Voltage       |
|       |                    |
|   Bus | Magnitude |  Angle |
|       |      [pu] |  [deg] |
|-------|-----------|--------|
| Bus 1 |    1.1000 | 0.0000 |
| Bus 2 |    1.1313 | 1.2625 |
| Bus 3 |    1.1231 | 0.3377 |
|----------------------------|

Next, users can easily customize the print results for specific buses, for example:

printBusData(system, analysis; label = "Bus 1", header = true)
printBusData(system, analysis; label = "Bus 2")
printBusData(system, analysis; label = "Bus 3", footer = true)

Save Results to a File

Users can also redirect print output to a file. For example, data can be saved in a text file as follows:

open("bus.txt", "w") do file
    printBusData(system, analysis, file)
end

Save Results to a CSV File

For CSV output, users should first generate a simple table with style = false, and then save it to a CSV file:

using CSV

io = IOBuffer()
printBusData(system, analysis, io; style = false)
CSV.write("bus.csv", CSV.File(take!(io); delim = "|"))

Power System Update

After establishing the PowerSystem type using the powerSystem function and configuring the AC model with acModel!, users gain the capability to incorporate new branches and generators. Furthermore, they can adjust buses, branches, and generators.

Once updates are done, users can progress towards generating the ACPowerFlow type using the newtonRaphson, fastNewtonRaphsonBX, fastNewtonRaphsonXB, or gaussSeidel function. Ultimately, resolving the AC power flow is achieved through the utilization of the mismatch! and solve! functions:

system = powerSystem() # <- Initialize the PowerSystem instance

addBus!(system; label = "Bus 1", type = 3, active = 0.5, magnitude = 0.9, angle = 0.0)
addBus!(system; label = "Bus 2", type = 1, reactive = 0.05, magnitude = 1.1, angle = -0.1)

@branch(resistance = 0.02, conductance = 1e-4, susceptance = 0.04)
addBranch!(system; label = "Branch 1", from = "Bus 1", to = "Bus 2", reactance = 0.05)

addGenerator!(system; label = "Generator 1", bus = "Bus 1", magnitude = 1.1, active = 3.2)

acModel!(system)
analysis = newtonRaphson(system) # <- Build ACPowerFlow for the defined power system
for iteration = 1:100
    mismatch!(system, analysis)
    solve!(system, analysis)
end

updateBus!(system; label = "Bus 2", active = 0.2)

addBranch!(system; label = "Branch 2", from = "Bus 1", to = "Bus 2", reactance = 1)
updateBranch!(system; label = "Branch 1", status = 0)

addGenerator!(system; label = "Generator 2", bus = "Bus 1", active = 0.2)
updateGenerator!(system; label = "Generator 1", active = 0.3)

analysis = newtonRaphson(system) # <- Build ACPowerFlow for the updated power system
for iteration = 1:100
    mismatch!(system, analysis)
    solve!(system, analysis)
end

Power Flow Update

An advanced methodology involves users establishing the ACPowerFlow type using newtonRaphson, fastNewtonRaphsonBX, fastNewtonRaphsonXB, or gaussSeidel just once. After this initial setup, users can integrate new branches and generators, and also have the capability to modify buses, branches, and generators, all without the need to recreate the ACPowerFlow type.

This advancement extends beyond the previous scenario where recreating the PowerSystem and AC model was unnecessary, to now include the scenario where ACPowerFlow also does not need to be recreated. Such efficiency proves particularly beneficial in cases where JuliaGrid can reuse established Jacobian matrices or even factorizations, especially when users choose the fast Newton-Raphson method.

By modifying the previous example, we observe that we now create the ACPowerFlow type only once using the newtonRaphson function. This approach allows us to circumvent the need for reinitializing the Jacobian matrix, enabling us to proceed directly with iterations:

system = powerSystem() # <- Initialize the PowerSystem instance

addBus!(system; label = "Bus 1", type = 3, active = 0.5, magnitude = 0.9, angle = 0.0)
addBus!(system; label = "Bus 2", type = 1, reactive = 0.05, magnitude = 1.1, angle = -0.1)

@branch(resistance = 0.02, conductance = 1e-4, susceptance = 0.04)
addBranch!(system; label = "Branch 1", from = "Bus 1", to = "Bus 2", reactance = 0.05)

addGenerator!(system; label = "Generator 1", bus = "Bus 1", magnitude = 1.1, active = 3.2)

acModel!(system)
analysis = newtonRaphson(system) # <- Build ACPowerFlow for the defined power system
for iteration = 1:100
    mismatch!(system, analysis)
    solve!(system, analysis)
end

updateBus!(system, analysis; label = "Bus 2", active = 0.2)

addBranch!(system, analysis; label = "Branch 2", from = "Bus 1", to = "Bus 2", reactance = 1)
updateBranch!(system, analysis; label = "Branch 1", status = 0)

addGenerator!(system, analysis; label = "Generator 2", bus = "Bus 1", active = 0.2)
updateGenerator!(system, analysis; label = "Generator 1", active = 0.3)

# <- No need for re-build; we have already updated the existing ACPowerFlow instance
for iteration = 1:100
    mismatch!(system, analysis)
    solve!(system, analysis)
end

Fast Newton-Raphson: Reusing Jacobian Matrices Factorizations

An intriguing scenario unfolds when employing the fast Newton-Raphson method. Continuing from the previous example, let us now initialize the fast Newton-Raphson method and proceed with iterations as outlined below:

analysis = fastNewtonRaphsonBX(system)
for iteration = 1:100
    mismatch!(system, analysis)
    solve!(system, analysis)
end

Throughout this process, JuliaGrid will factorize the constant Jacobian matrices that govern the fast Newton-Raphson method.

Now, let us make changes to the power system and proceed directly to the iteration step:

updateBus!(system, analysis; label = "Bus 2", reactive = 0.02)
updateGenerator!(system, analysis; label = "Generator 1", reactive = 0.1)

for iteration = 1:100
    mismatch!(system, analysis)
    solve!(system, analysis)
end

Warm Start

In these scenarios, users leverage the previously created PowerSystem type with the AC model and also reuse the ACPowerFlow type, proceeding directly to the iterations. This approach offers the advantage of a "warm start", wherein the initial voltages for the subsequent iteration step align with the solution from the previous iteration step. This alignment facilitates an efficient continuation of the power flow analysis.

Let us now make another alteration to the power system:

updateBus!(system, analysis; label = "Bus 1", active = 0.1, magnitude = 0.95, angle = -0.07)
updateGenerator!(system, analysis; label = "Generator 2", reactive = 0.2, magnitude = 1.1)

With these modifications we are not only altering the power system, but also starting voltages. For the next uses of one of the methods, these values now are:

julia> print(system.bus.label, analysis.voltage.magnitude, analysis.voltage.angle)Bus 1: 1.1, -0.07
Bus 2: 1.0839300986879146, -0.16864600045398928

Therefore, users possess the flexibility to adjust these initial values as needed by employing the magnitude and angle keywords within the updateBus! and updateGenerator! functions.

If users prefer to set starting voltages according to the typical scenario, they can accomplish this through the startingVoltage! function:

startingVoltage!(system, analysis)

Now, we have starting voltages defined exclusively according to the PowerSystem. These values are exactly the same as if we executed the newtonRaphson, fastNewtonRaphsonBX, fastNewtonRaphsonXB, or gaussSeidel function after all the updates we performed:

julia> print(system.bus.label, analysis.voltage.magnitude, analysis.voltage.angle)Bus 1: 1.1, -0.07
Bus 2: 1.1, -0.1

Limitations

The newtonRaphson, fastNewtonRaphsonBX, fastNewtonRaphsonXB, or gaussSeidel function oversees bus type validations, as outlined in the Bus Type Modification section. Consequently, attempting to change bus types or leaving generator buses without a generator and then proceeding directly to the iteration process is not viable.

In such scenarios, JuliaGrid will raise an error:

julia> updateBus!(system, analysis; label = "Bus 2", type = 2)ERROR: The power flow model cannot be reused due to required bus type conversion.

In this scenario, the user must execute the newtonRaphson, fastNewtonRaphsonBX, fastNewtonRaphsonXB, or gaussSeidel function instead of trying to reuse them, for example:

updateBus!(system; label = "Bus 2", type = 2)

analysis = fastNewtonRaphsonBX(system)
for iteration = 1:100
    mismatch!(system, analysis)
    solve!(system, analysis)
end

Power and Current Analysis

After obtaining the solution from the AC power flow, we can calculate various electrical quantities related to buses, branches, and generators using the power! and current! functions. For instance, let us consider the power system for which we obtained the AC power flow solution:

system = powerSystem()

addBus!(system; label = "Bus 1", type = 3, active = 0.6)
addBus!(system; label = "Bus 2", type = 1, reactive = 0.1, susceptance = 0.03)
addBus!(system; label = "Bus 3", type = 1, conductance = 0.02)

@branch(resistance = 0.02, conductance = 1e-4, susceptance = 0.04)
addBranch!(system; label = "Branch 1", from = "Bus 1", to = "Bus 2", reactance = 0.5)
addBranch!(system; label = "Branch 2", from = "Bus 1", to = "Bus 2", reactance = 0.1)
addBranch!(system; label = "Branch 3", from = "Bus 2", to = "Bus 3", reactance = 0.4)

addGenerator!(system; label = "Generator 1", bus = "Bus 1", active = 0.2)
addGenerator!(system; label = "Generator 2", bus = "Bus 2", active = 1.0, reactive = 0.2)

analysis = newtonRaphson(system)
for iteration = 1:100
    stopping = mismatch!(system, analysis)
    if all(stopping .< 1e-8)
        break
    end
    solve!(system, analysis)
end

We can now utilize the provided functions to compute powers and currents:

power!(system, analysis)
current!(system, analysis)

For instance, if we want to show the active power injections and the to-bus current angles, we can employ the following code:

julia> print(system.bus.label, analysis.power.injection.active)Bus 1: -0.9645033643807599
Bus 2: 1.0000000022772795
Bus 3: -4.359533802980213e-10
julia> print(system.branch.label, analysis.current.to.angle)Branch 1: -0.15017765586673948
Branch 2: -0.08939059735692804
Branch 3: -3.073480808526896

Print Results in the REPL

Users can utilize any of the print functions outlined in the Print Power System Data or Print Power System Summary. For example, to create a bus summary with the desired units, users can use the following function:

@voltage(pu, deg, V)
@power(MW, MVAr, pu)
printBusSummary(system, analysis)

|----------------------------------------------------------------------------|
| Bus Summary                                                                |
|----------------------------------------------------------------------------|
|       Type        |     Minimum      |     Maximum      | In-Use |  Total  |
|                   |                  |                  |        |         |
|                   | Label |    Value | Label |    Value |        |         |
|-------------------|-------|----------|-------|----------|--------|---------|
| Voltage           |       |          |       |          |      3 |         |
|  Magnitude [pu]   | Bus 1 |   1.0000 | Bus 3 |   1.0362 |        |         |
|  Angle [deg]      | Bus 1 |   0.0000 | Bus 2 |   4.3886 |        |         |
|-------------------|-------|----------|-------|----------|--------|---------|
| Power Generation  |       |          |       |          |      1 |         |
|  Active [MW]      | Bus 3 |   0.0000 | Bus 3 |   0.0000 |        |  0.0000 |
|  Reactive [MVAr]  | Bus 3 |   0.0000 | Bus 3 |   0.0000 |        |  0.0000 |
|-------------------|-------|----------|-------|----------|--------|---------|
| Power Demand      |       |          |       |          |      2 |         |
|  Active [MW]      | Bus 2 |   0.0000 | Bus 1 |  60.0000 |        | 60.0000 |
|  Reactive [MVAr]  | Bus 1 |   0.0000 | Bus 2 |  10.0000 |        | 10.0000 |
|-------------------|-------|----------|-------|----------|--------|---------|
| Power Injection   |       |          |       |          |      3 |         |
|  Active [MW]      | Bus 1 | -96.4503 | Bus 2 | 100.0000 |        |  3.5497 |
|  Reactive [MVAr]  | Bus 1 | -17.6931 | Bus 2 |  10.0000 |        | -7.6931 |
|-------------------|-------|----------|-------|----------|--------|---------|
| Shunt Power       |       |          |       |          |      2 |         |
|  Active [MW]      | Bus 2 |   0.0000 | Bus 3 |   2.1475 |        |  2.1475 |
|  Reactive [MVAr]  | Bus 2 |  -3.1727 | Bus 3 |  -0.0000 |        | -3.1727 |
|-------------------|-------|----------|-------|----------|--------|---------|
| Current Injection |       |          |       |          |      3 |         |
|  Magnitude [pu]   | Bus 3 |   0.0000 | Bus 1 |   0.9806 |        |         |
|  Angle [rad]      | Bus 3 |  -2.7799 | Bus 1 |   2.9602 |        |         |
|----------------------------------------------------------------------------|

Active and Reactive Power Injection

To calculate the active and reactive power injection associated with a specific bus, the function can be used:

julia> active, reactive = injectionPower(system, analysis; label = "Bus 1")(-0.9645033643807599, -0.17693071258892878)

Active and Reactive Power Injection from Generators

To calculate the active and reactive power injection from the generators at a specific bus, the function can be used:

julia> active, reactive = supplyPower(system, analysis; label = "Bus 1")(-0.3645033643807599, -0.17693071258892878)

Active and Reactive Power at Shunt Element

To calculate the active and reactive power associated with shunt element at a specific bus, the function can be used:

julia> active, reactive = shuntPower(system, analysis; label = "Bus 3")(0.021474952575336232, -0.0)

Active and Reactive Power Flow

Similarly, we can compute the active and reactive power flow at both the from-bus and to-bus ends of the specific branch by utilizing the functions provided below:

julia> active, reactive = fromPower(system, analysis; label = "Branch 2")(-0.8053928587014492, -0.11256618205670144)
julia> active, reactive = toPower(system, analysis; label = "Branch 2")(0.8186418709823436, 0.13714551291797428)

Active and Reactive Power at Charging Admittances

To calculate the active and reactive power linked with branch charging admittances of the particular branch, the function can be used:

julia> active, reactive = chargingPower(system, analysis; label = "Branch 1")(0.00010287834702025066, -0.04115133880810026)

Active powers indicate active losses within the branch's charging admittances. Moreover, charging admittances injected reactive powers into the power system due to their capacitive nature.

Active and Reactive Power at Series Impedance

To calculate the active and reactive power across the series impedance of the branch, the function can be used:

julia> active, reactive = seriesPower(system, analysis; label = "Branch 2")(0.013146133933874347, 0.06573066966937174)

The active power also considers active losses originating from the series resistance of the branch, while the reactive power represents reactive losses resulting from the impedance's inductive characteristics.

Generator Active and Reactive Power Output

We can compute the active and reactive power output of a particular generator using the function:

julia> active, reactive = generatorPower(system, analysis; label = "Generator 1")(-0.3645033643807599, -0.17693071258892878)

Current Injection

To calculate the current injection associated with a specific bus, the function can be used:

julia> magnitude, angle = injectionCurrent(system, analysis; label = "Bus 1")(0.9805973776015471, 2.9601674608926722)

Current Flow

We can compute the current flow at both the from-bus and to-bus ends of the specific branch by utilizing the provided functions below:

julia> magnitude, angle = fromCurrent(system, analysis; label = "Branch 2")(0.8132212504540907, 3.0027266550776193)
julia> magnitude, angle = toCurrent(system, analysis; label = "Branch 2")(0.807142930926126, -0.08939059735692804)

Current Through Series Impedance

To calculate the current passing through the series impedance of the branch in the direction from the from-bus end to the to-bus end, we can use the following function:

julia> magnitude, angle = seriesCurrent(system, analysis; label = "Branch 2")(0.8107445323242811, 3.027168837917323)

Generator Reactive Power Limits

The function reactiveLimit! can be used to check if the generators' output of reactive power is within the defined limits after obtaining the solution from the AC power flow analysis:

system = powerSystem()

addBus!(system; label = "Bus 1", type = 3)
addBus!(system; label = "Bus 2", type = 1, active = 0.5)
addBus!(system; label = "Bus 3", type = 2, reactive = 0.05)
addBus!(system; label = "Bus 4", type = 2, reactive = 0.05)

@branch(resistance = 0.015)
addBranch!(system; from = "Bus 1", to = "Bus 2", reactance = 0.05)
addBranch!(system; from = "Bus 1", to = "Bus 3", reactance = 0.01)
addBranch!(system; from = "Bus 2", to = "Bus 3", reactance = 0.04)
addBranch!(system; from = "Bus 2", to = "Bus 4", reactance = 0.004)

@generator(minReactive = -0.4, maxReactive = 0.1)
addGenerator!(system; label = "Generator 1", bus = "Bus 1")
addGenerator!(system; label = "Generator 2", bus = "Bus 3", reactive = 0.8)
addGenerator!(system; label = "Generator 3", bus = "Bus 4", reactive = 0.9)

analysis = newtonRaphson(system)
for iteration = 1:100
    stopping = mismatch!(system, analysis)
    if all(stopping .< 1e-8)
        break
    end
    solve!(system, analysis)
end

violate = reactiveLimit!(system, analysis)

The output reactive power of the observed generators is subject to limits which are defined as follows:

julia> [system.generator.capability.minReactive system.generator.capability.maxReactive]3×2 Matrix{Float64}:
 -0.4  0.1
 -0.4  0.1
 -0.4  0.1

After obtaining the solution of the AC power flow analysis, the reactiveLimit! function is used to internally calculate the output powers of the generators and verify if these values exceed the defined limits. Consequently, the variable violate indicates whether there is a violation of limits.

In the provided example, it can be observed that the Generator 2 and Generator 3 violate the maximum limit:

julia> print(system.generator.label, violate)Generator 1: 0
Generator 2: 1
Generator 3: 1

Due to these violations of limits, the PowerSystem type undergoes modifications, and the output reactive power at the limit-violating generators is adjusted as follows:

julia> print(system.generator.label, system.generator.output.reactive)Generator 1: 0.0
Generator 2: 0.1
Generator 3: 0.1

To ensure that these values stay within the limits, the bus type must be changed from the generator bus (type = 2) to the demand bus (type = 1), as shown below:

julia> print(system.bus.label, system.bus.layout.type)Bus 1: 3
Bus 2: 1
Bus 3: 1
Bus 4: 1

After modifying the PowerSystem type as described earlier, we can run the simulation again with the following code:

analysis = newtonRaphson(system)
for iteration = 1:100
    stopping = mismatch!(system, analysis)
    if all(stopping .< 1e-8)
        break
    end
    solve!(system, analysis)
end

Once the simulation is complete, we can verify that all generator reactive power outputs now satisfy the limits by checking the violate variable again:

julia> violate = reactiveLimit!(system, analysis)3-element Vector{Int64}:
 0
 0
 0

New Slack Bus

Looking at the following code example, we can see that the output limits of the generator are set only for Generator 1 that is connected to the slack bus:

system = powerSystem()

addBus!(system; label = "Bus 1", type = 3, active = 0.5, reactive = 0.05)
addBus!(system; label = "Bus 2", type = 1, active = 0.5)
addBus!(system; label = "Bus 3", type = 2)
addBus!(system; label = "Bus 4", type = 2)

@branch(resistance = 0.01)
addBranch!(system; from = "Bus 1", to = "Bus 2", reactance = 0.05)
addBranch!(system; from = "Bus 1", to = "Bus 3", reactance = 0.01)
addBranch!(system; from = "Bus 2", to = "Bus 3", reactance = 0.04)
addBranch!(system; from = "Bus 2", to = "Bus 4", reactance = 0.004)

addGenerator!(system; label = "Generator 1", bus = "Bus 1", maxReactive = 0.2)
addGenerator!(system; label = "Generator 2", bus = "Bus 4", reactive = 0.3)

analysis = newtonRaphson(system)
for iteration = 1:100
    stopping = mismatch!(system, analysis)
    if all(stopping .< 1e-8)
        break
    end
    solve!(system, analysis)
end

Upon checking the limits, we can observe that the slack bus has been transformed by executing the following code:

julia> violate = reactiveLimit!(system, analysis)[ Info: The slack bus labeled Bus 1 is converted to generator bus. The bus labeled Bus 4 is the new slack bus.
2-element Vector{Int64}:
 -1
  0

Here, the generator connected to the slack bus is violating the minimum reactive power limit, which indicates the need to convert the slack bus. It is important to note that the new slack bus can be created only from the generator bus (type = 2). We will now perform another AC power flow analysis on the modified system using the following:

analysis = newtonRaphson(system)
for iteration = 1:100
    stopping = mismatch!(system, analysis)
    if all(stopping .< 1e-8)
        break
    end
    solve!(system, analysis)
end

After examining the bus voltages, we will focus on the angles:

julia> print(system.bus.label, analysis.voltage.angle)Bus 1: 0.013581064245677444
Bus 2: 0.0005651921742472857
Bus 3: 0.01069570666159161
Bus 4: 0.0

We can observe that the angles have been calculated based on the new slack bus. JuliaGrid offers the function to adjust these angles to match the original slack bus as follows:

adjustAngle!(system, analysis; slack = "Bus 1")

After executing the above code, the updated results can be viewed:

julia> print(system.bus.label, analysis.voltage.angle)Bus 1: 0.0
Bus 2: -0.013015872071430158
Bus 3: -0.0028853575840858334
Bus 4: -0.013581064245677444