Optimal Power Flow (OPF)

An optimal power flow constitutes the core of power system operations. It is an optimization extension of the power flow analysis, where in addition to feasibility, an optimal dispatch is also desired. The objective of power system operation depends on the what kind of problem is being solved. Usually it is to serve the customers at the minimum cost, taking into consideration the physical constraints of the system. Optimal power flow can be expressed as an optimization problem and when security constraints are added it is referred to as security-constrained optimal power flow (SCOPF). In addition to power system operations, a solution to the optimal power flow is also essential for power system planning.

In an OPF problem, the most important equations are about the representation of transmission branches. There are different approaches to model a transmission branch in a power system. The most commonly used representation of the transmission branch is called the π-model, which is what have adopted too.

Transmission Model

The figure above shows a general representation of a π-model of a transmission branch, where $y_{ij}$ is the branch series admittance, $y^s_{ij}$ is the branch shunt admittance, $y^M_{ij}$ is the branch mutual admittance, and $τ_{ij}$ is the branch complex tap ratio. If a branch is only a transmission line then $τ^i_{ij}=1$ and $y^M_{ij}=0$. If a branch is only a transformer, then all the line parameters are zero. Thus, the figure represents a general model that can represent any type of transmission branch by simply adjusting the values of the parameters shown in the figure above.

Let $y_{ij}^{net} = y^M_{ij}+(y_{ij}+y^s_{ij}) \cdot \left\lvert \tau^i_{ij}\right\lvert^2$, $y_{ji}^{net} = y_{ij}+y^s_{ij}$, and $2\mathcal{K} = \{ \mathcal{K} \} \bigcup \{ -\mathcal{K} \}$. The admittance matrix can be written as shown below:

$\begin{subequations}\begin{align*}&Y_{ii}=\sum_{\substack{k=1 \\ k \neq i}}^{\mathcal{K}_i^f} y^{net}_{ik}+\sum_{\substack{k=1 \\ k \neq i}}^{\mathcal{K}_i^t}y^{net}_{ki}, \\&Y_{ij}=-y_{ij} \cdot \tau^{i*}_{ij} \\&Y_{ji}=-y_{ij} \cdot \tau^i_{ij}\end{align*}\end{subequations}$

An ACOPF problem can be formulated as an optimization problem as shown below:

$\begin{subequations}\begin{align*}& min& & \sum_{\substack{g \in \mathcal{G}}} c_{2g} \cdot (\Re[s_g])^2 + c_{1g} \cdot \Re[s_g] + c_{0g} \\& \text{s.t.}& & s_{ij}=v_i \cdot (y^{net}_{ij} \cdot v_i+Y_{ij} \cdot v_j)^*, &\forall \{i,j\} \in 2\mathcal{K}.\\& & & \sum_{\substack{g \in \mathcal{G}_n}}s_g - \sum_{\substack{d \in \mathcal{D}_n}}s_d=\sum_{\substack{k \in \mathcal{K}_n^f}}s_{nk}+\sum_{\substack{k \in \mathcal{K}_n^t}}s_{kn} +Y^L_{n}\cdot |v_n|^2, &\forall n \in \mathcal{N}.\\& & & V_n^{Min} \leq |v_n| \leq V_n^{Max}, &\forall n \in \mathcal{N}. \\& & & |s_{ij}| \leq I^{Max}_{ij} \cdot |v_n|, &\forall \{i,j\} \in 2\mathcal{K}. \\& & & s_g^{Min} \leq s_g \leq s_g^{Max}, &\forall g \in \mathcal{G}.\end{align*}\end{subequations}$

# Load packages
using Powersense, Ipopt

# Build Powersense Data model. 
# Path is the address where PSSE or MATPOWER file types are located
Data = create_PowersenseData(path)

run_opf!(Data, solver = Ipopt.Optimizer);

OPF Formulations

An ACOPF problem in its complex form can be expanded into different solvable optimization formulations as shown in the figure below. While these formulations represent the same original ACOPF problem, they have different optimization structures that can have different convergence patterns.

Formulations Chart

1. Power Branch-Flow Rectangular Admittance Polar Voltage (PBRAPV):

Definition of the formulation. The syntax for this formulation is PBRAPVmodel.

# Solving optimal power flow using PBRAPV formulation
run_opf!(Data, solver = Ipopt.Optimizer, formulation = PBRAPVmodel);

2. Power Branch-Flow Rectangular Admittance Rectangular Voltage (PBRARV):

Definition of the formulation. The syntax for this formulation is PBRARVmodel.

# Solving optimal power flow using PBRARV formulation
run_opf!(Data, solver = Ipopt.Optimizer, formulation = PBRARVmodel);

3. Current Branch-Flow Rectangular Admittance Rectangular Voltage (CBRARV):

Definition of the formulation. The syntax for this formulation is CBRARVmodel.

# Solving optimal power flow using CBRARV formulation
run_opf!(Data, solver = Ipopt.Optimizer, formulation = CBRARVmodel);

4. Current Branch-Flow Rectangular Admittance W-matrix Voltage (CBRAWV):

Definition of the formulation. The syntax for this formulation is CBRAWVmodel.

# Solving optimal power flow using CBRAWV formulation
run_opf!(Data, solver = Ipopt.Optimizer, formulation = CBRAWVmodel);

5. Power Nodal-Injection Polar Admittance Polar Voltage (PNPAPV):

Definition of the formulation. The syntax for this formulation is PNPAPVmodel.

# Solving optimal power flow using PNPAPV formulation
run_opf!(Data, solver = Ipopt.Optimizer, formulation = PNPAPVmodel);

6. Power Nodal-Injection Rectangular Admittance Polar Voltage (PNRAPV):

Definition of the formulation. The syntax for this formulation is PNRAPVmodel.

# Solving optimal power flow using PNRAPV formulation
run_opf!(Data, solver = Ipopt.Optimizer, formulation = PNRAPVmodel);

7. Power Nodal-Injection Rectangular Admittance Rectangular Voltage (PNRARV):

Definition of the formulation. The syntax for this formulation is PNRARVmodel.

# Solving optimal power flow using PNRARV formulation
run_opf!(Data, solver = Ipopt.Optimizer, formulation = PNRARVmodel);

If you find this work useful, we kindly request that you cite the following papers:

@article{9562928,
    author={Sadat, Sayed Abdullah},
    booktitle={2021 IEEE International Smart Cities Conference (ISC2)}, 
    title={Evaluating the Performance of Various ACOPF Formulations Using Nonlinear Interior-Point Method}, 
    year={2021},
    pages={1-7},
    doi={10.1109/ISC253183.2021.9562928}
}

@article{9562929,
    author={Sayed Abdullah Sadat and Kibaek Kim},
    booktitle={2021 Australasian Universities Power Engineering Conference (AUPEC)}, 
    title={Numerical Performance of Different Formulations for Alternating Current Optimal Power Flow}, 
    year={2021},
    pages={1-6}
}

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