DC Power Flow

JuliaGrid employs standard network components and the Unified Branch Model to obtain the DC power flow solution. To begin, let us generate the PowerSystem composite type, as illustrated by the following example:

@power(MW, MVAr, MVA)
@voltage(pu, deg, V)

system = powerSystem()

addBus!(system; label = 1, type = 3)
addBus!(system; label = 2, type = 1, active = 21.7)
addBus!(system; label = 3, type = 2, conductance = 0.07)

addBranch!(system; from = 1, to = 2, reactance = 0.26)
addBranch!(system; from = 1, to = 3, reactance = 0.38)
addBranch!(system; from = 2, to = 3, reactance = 0.17, turnsRatio = 0.97)

addGenerator!(system; bus = 1, active = 2.0)
addGenerator!(system; bus = 1, active = 4.0)
addGenerator!(system; bus = 3, active = 5.0)

To review, we can conceptualize the bus/branch model as the graph denoted by $\mathcal{G} = (\mathcal{N}, \mathcal{E})$, where we have the set of buses $\mathcal{N} = \{1, \dots, n\}$, and the set of branches $\mathcal{E} \subseteq \mathcal{N} \times \mathcal{N}$ within the power system:

julia> 𝒩 = collect(keys(system.bus.label))3-element Vector{String}:
 "1"
 "2"
 "3"
julia> ℰ = [𝒩[system.branch.layout.from] 𝒩[system.branch.layout.to]]3×2 Matrix{String}:
 "1"  "2"
 "1"  "3"
 "2"  "3"

Power Flow Solution

As discussed in section DC Model, the DC power flow problem can be represented by a set of linear equations:

\[ \mathbf {P} = \mathbf{B} \bm {\Theta} + \mathbf{P_\text{tr}} + \mathbf{P}_\text{sh}.\]

Implementation

JuliaGrid offers a set of functions to solve the DC power flow problem and obtain the bus voltage angles. Firstly, the power system is loaded and the DC model is built using the following code sequence:

dcModel!(system)

The DC power flow solution is obtained through a non-iterative approach by solving the system of linear equations:

\[ \bm {\Theta} = \mathbf{B}^{-1}(\mathbf {P} - \mathbf{P_\text{tr}} - \mathbf{P}_\text{sh}).\]

JuliaGrid begins the process by establishing the DC power flow framework:

analysis = dcPowerFlow(system)

The subsequent step involves performing the LU factorization of the nodal matrix $\mathbf{B} = \mathbf{L}\mathbf{U}$ and computing the bus voltage angles using:

solve!(system, analysis)

The factorization of the nodal matrix can be accessed using:

julia> 𝐋 = analysis.method.factorization.L3×3 SparseArrays.SparseMatrixCSC{Float64, Int64} with 4 stored entries:
 1.0    ⋅         ⋅
  ⋅    1.0        ⋅
  ⋅   -0.662261  1.0
julia> 𝐔 = analysis.method.factorization.U3×3 SparseArrays.SparseMatrixCSC{Float64, Int64} with 4 stored entries:
 1.0   ⋅          ⋅
  ⋅   0.620383  -0.379617
  ⋅    ⋅         0.337739

It is important to note that the slack bus voltage angle is excluded from the vector $\bm{\Theta}$ only during the computation step. As a analysis, the corresponding elements in the vectors $\mathbf {P}$, $\mathbf{P_\text{tr}}$, $\mathbf{P}_\text{sh}$, and the corresponding row and column of the matrix $\mathbf{B}$ are removed. It is worth mentioning that this process is handled internally, and the stored elements remain unchanged.

Finally, the resulting bus voltage angles are saved in the vector as follows:

julia> 𝚯 = analysis.voltage.angle3-element Vector{Float64}:
  0.0
 -0.03214364268853274
 -0.016746829916759843

Power Analysis

After obtaining the solution from the DC power flow, we can calculate powers related to buses, branches, and generators using the power! function:

power!(system, analysis)

Power Injections

Active power injections are stored as the vector $\mathbf{P} = [P_i]$, and can be retrieved using the following commands:

julia> 𝐏 = analysis.power.injection.active3-element Vector{Float64}:
  0.1677
 -0.217
  0.05

Generator Power Injections

The active power supplied by generators to the buses can be calculated by summing the given generator active powers in the input data, except for the slack bus, which can be determined as:

\[ P_{\text{p}i} = P_i + P_{\text{d}i},\;\;\; i \in \mathcal{N}_{\text{sb}},\]

where $P_{\text{d}i}$ represents the active power demanded by consumers at the slack bus. The vector of active power injected by generators to the buses, denoted by $\mathbf{P}_{\text{p}} = [P_{\text{p}i}]$, can be obtained using the following command:

julia> 𝐏ₚ = analysis.power.supply.active3-element Vector{Float64}:
 0.1677
 0.0
 0.05

Power Flows

The resulting active power flows are stored as the vector $\mathbf{P}_{\text{i}} = [P_{ij}]$, which can be retrieved using:

julia> 𝐏ᵢ = analysis.power.from.active3-element Vector{Float64}:
  0.12362939495589513
  0.04407060504410485
 -0.09337060504410487

Similarly, the resulting active power flows are stored as the vector $\mathbf{P}_{\text{j}} = [P_{ji}]$, which can be retrieved using:

julia> 𝐏ⱼ = analysis.power.to.active3-element Vector{Float64}:
 -0.12362939495589513
 -0.04407060504410485
  0.09337060504410487

Generators Power Outputs

The output active power of each generator located at bus $i \in \mathcal{N}_{\text{pv}} \cup \mathcal{N}_{\text{pq}}$ is equal to the active power specified in the input data. If there are multiple generators, their output active powers are also equal to the active powers specified in the input data. However, the output active power of a generator located at the slack bus will be:

\[ P_{\text{g}i} = P_i + P_{\text{d}i},\;\;\; i \in \mathcal{N}_{\text{sb}}.\]

In the case of multiple generators connected to the slack bus, the first generator in the input data is assigned the obtained value of $P_{\text{g}i}$. Then, this amount of power is reduced by the output active power of the other generators.

To retrieve the vector of active power outputs of generators, denoted as $\mathbf{P}_{\text{g}} = [P_{\text{g}i}]$, $i \in \mathcal{S}$, where the set $\mathcal{S}$ represents the set of generators, users can utilize the following command:

julia> 𝐏ₒ = analysis.power.generator.active3-element Vector{Float64}:
 0.12769999999999998
 0.04
 0.05