Optimal Power Flow Module

This module provides functions for AC/DC hybrid optimal power flow analysis [1].

functions are found in pyflow_acdc.ACDC_OPF and pyflow_acdc.ACDC_OPF_NL_model

AC/DC Hybrid Optimal Power Flow

Running the OPF

This function runs the AC/DC hybrid optimal power flow calculation. It creates the model, chooses an objective function, and solves the model.

Optimal_PF(grid, ObjRule=None, PV_set=False, OnlyGen=True, Price_Zones=False)

Performs AC/DC hybrid optimal power flow calculation.

Parameter	Type	Description	Default
grid	Grid	Grid to optimize	Required
ObjRule	dict	Objective function weights	None
PV_set	bool	Sets PV and Slack voltage as fixed variables	False
OnlyGen	bool	Only generators are considered in the cost function	True
Price_Zones	bool	Enable price zone constraints	False

Example

model, model_res , timing_info, solver_stats =pyf.Optimal_PF(grid, ObjRule=None, PV_set=False, OnlyGen=True, Price_Zones=False)

Creating the OPF model

OPF_create_NLModel_ACDC(model, grid, PV_set, Price_Zones, TEP=False, limit_flow_rate=True)

Creates the OPF model.

Parameter	Type	Description
model	Model	Model to create
grid	Grid	Grid to optimize
PV_set	bool	Sets PV and Slack voltage as fixed variables
Price_Zones	bool	Enable price zone constraints
TEP	bool	Enable TEP investment variables (lines/generators)
limit_flow_rate	bool	Enable flow rate limits

Variables

The optimization model includes variables for:

• AC node voltages and angles

• DC node voltages

• Generator active/reactive power

• Renewable generation and curtailment

• Line flows

• Converter power flows

• Price zone variables

Constraints

The model enforces constraints for:

• AC power flow equations

• DC power flow equations

• Generator limits

• AC branch thermal limits

• DC branch thermal limits

• Voltage and angle limits

• Converter operation limits

• Price zone balancing

For more details on the constraints, please refer to the System Modelling page.

Example

from pyflow_acdc.ACDC_OPF_NL_model import OPF_create_NLModel_ACDC
model = OPF_create_NLModel_ACDC(model, grid, PV_set=False, Price_Zones=False)

Objective Functions

The user can define the objective by setting the weight of each sub objective. The objective function is defined as:

OPF_obj(model, grid, ObjRule, OnlyGen, OnlyAC=False)

This function creates a weighted sum of the different sub objectives.

\[\min \frac{\sum_{i \in O} \left( w_i \cdot f_i \right)}{\sum_{i \in O} w_i}\]

where \(f_i\) is the sub objective and \(w_i\) is the weight.

The following table shows the pre-built objective functions as defined in [1] :

Weight	Description	Formula
Ext_Gen	External generation minimization or maximum export	\(\sum_{g \in G} \cdot P_{g}\)
Energy_cost	Energy cost	\(\sum_{g \in \mathcal{G}_{ac}} \left(P_{g}^2 \cdot \alpha_g + P_{g} \cdot \beta_g \right)\)
Curtailment_Red	Renewable curtailment reduction	\(\sum_{rg \in \mathcal{RG}_{ac}}\left((1-\gamma_rg)P_{rg}\cdot \rho_{rg} \sigma_{rg}\right)\)
AC_losses	AC transmission losses	\(\sum_{j \in \mathcal{B}_{ac}} \left( P_{j,\text{from}} +P_{j,\text{to}} \right)\)
DC_losses	DC transmission losses	\(\sum_{e \in \mathcal{B}_{dc}} \left( P_{e,\text{from}} +P_{e,\text{to}} \right)\)
Converter_Losses	Converter losses	\(\sum_{cn \in \mathcal{C}_{n}} \left( P_{loss_{cn}} + |\left(P_{c_{cn}}-P_{s_{cn}}\right)| \right)\)
General_Losses	Generation minus demand	\(\left(\sum_{g \in \mathcal{G}} P_{g}+\sum_{rg \in \mathcal{RG}} P_{rg}*\gamma_{rg}- \sum_{l \in \mathcal{L}} P_{L} \right)\)

The following table shows the pre-built objective functions as defined in [2]:

Weight	Description	Formula
PZ_cost_of_generation	Price zone generation cost	\(\sum_{m \in \mathcal{M}} CG(P_N)_m\)

The following table shows the pre-built objective functions in development:

Weight	Description	Formula
Renewable_profit	Renewable generation profit	\(- \left(\sum_{rg \in \mathcal{RG}} P_{rg}*\gamma_{rg} + \sum_{cn \in \mathcal{C}} \left(P_{loss,cn} + P_{AC,loss,cn}\right)\right)\)
Gen_set_dev	Generator setpoint deviation	\(\sum_{g \in G} \left(P_g -P_{g,set}\right)^2\)

Example

weights_def = {
'Ext_Gen': {'w': 0},
'Energy_cost': {'w': 0},
'Curtailment_Red': {'w': 0},
'AC_losses': {'w': 0},
'DC_losses': {'w': 0},
'Converter_Losses': {'w': 0},
'PZ_cost_of_generation': {'w': 0},
'Renewable_profit': {'w': 0},
'Gen_set_dev': {'w': 0}
}

Solvers

The OPF module supports pyomo solvers.

To see the available solvers, use the following command:

pyomo help --solvers

Tested with:

• IPOPT

• Bonmin

pyomo_model_solve(model, grid, solver='ipopt')

Solves the OPF model using the specified solver.

Parameters:

• model – The optimization model

• grid – The grid to optimize

• solver – The solver to use

Example

results, solver_stats =pyf.pyomo_model_solve(model,grid)

Result Translation Helpers

OPF_line_res(model, grid)

Extracts AC/DC line OPF results from a solved model into pandas-friendly structures.

OPF_price_priceZone(model, grid)

Extracts price-zone results from a solved OPF model.

Translate_pyf_OPF(grid, Price_Zones=False)

Translates solved OPF variables into grid result containers for plotting/export.

References

[1] (1,2)

B.C. Valerio, V. A. Lacerda, M. Cheah-Mane, P. Gebraad and O. Gomis-Bellmunt, “An optimal power flow tool for AC/DC systems, applied to the analysis of the North Sea Grid for offshore wind integration” in IEEE Transactions on Power Systems, doi: 10.1109/TPWRS.2023.3533889.

[2]

B. C. Valerio, V. A. Lacerda, M. Cheah-Mañe, P. Gebraad, and O. Gomis-Bellmunt, “Optimizing offshore wind integration through multi-terminal DC grids: a market-based OPF framework for the North Sea interconnectors,” IET Conference Proceedings, vol. 2025, no. 6, pp. 150–155, 2025. doi: 10.1049/icp.2025.1198