MDI.TransmissionLine package

Module contents

EELT7030 — Operation and Expansion Planning of Electric Power Systems Federal University of Paraná (UFPR)

Package: TransmissionLine

Author

Augusto Mathias Adams <augusto.adams@ufpr.br>

Description

The TransmissionLine package defines the modular representation of transmission circuits for DC power flow formulations within Pyomo-based optimization frameworks. It encapsulates the data structures, variables, symbolic equations, constraints, and builder routines that describe the physical and operational behavior of power transmission networks.

Modules

TransmissionLineDataTypes: Defines the data classes (TransmissionLineUnit, TransmissionLineData) describing the physical and economic characteristics of each circuit, including susceptance, power limits, and endpoints.
TransmissionLineVars: Declares decision variables associated with transmission lines, such as active power flow for each line and time period.
TransmissionLineEquations: Provides symbolic equations that express DC power flow relations and nodal contributions of line flows to system balance.
TransmissionLineConstraints: Implements physical constraints for transmission lines: - DC flow definition: F_ij,t = b_ij (θ_i,t − θ_j,t) - For transport model power flow is just as is - Symmetric flow limits: −pmax_ij ≤ F_ij,t ≤ pmax_ij
TransmissionLineBuilder: High-level model constructor that integrates variables and constraints, assembling the complete transmission-line subsystem for DC power flow models.

Notes

The package is designed for hierarchical integration with ConnectionBar modules, forming the complete DC power flow network.
The formulation supports both operational (fixed network) and expansion (candidate lines) applications, consistent with the methodologies of DECOMP, DESSEM, and MDI.
Each module can be used independently or through the unified builder build_transmission_lines().

References

[1] CEPEL, DECOMP / DESSEM. Manuais de Metodologia, 2023. [2] Unsihuay Vila, C. Introdução aos Sistemas de Energia Elétrica,

Submodules

MDI.TransmissionLine.TransmissionLineConstraints module

EELT7030 — Operation and Expansion Planning of Electric Power Systems Federal University of Paraná (UFPR)

Module: Transmission Line — Constraints

Author

Augusto Mathias Adams <augusto.adams@ufpr.br>

Description

Defines the physical and operational constraints for transmission lines in DC power flow formulations. These constraints link line flows to phase-angle differences through susceptance and limit the flow magnitude according to each line’s capacity.

Functions

add_transmission_line_flow_constraints(m): Enforce the DC flow definition: F_ij,t = b_ij (θ_i,t − θ_j,t) for all lines and time periods.
add_transmission_line_flow_limits_constraints(m): Apply the flow capacity limits: −pmax_ij ≤ F_ij,t ≤ pmax_ij for all lines and time periods.

Notes

Each line is characterized by: - b_ij : susceptance (1/x_ij) - pmax_ij : maximum active power flow (MW) - endpoints : list of two bars [i, j]
These constraints are suitable for both static (operational) and mixed-integer (expansion) formulations.
Phase angles θ_i,t and θ_j,t must exist in the model before these constraints are declared.

References

[1] CEPEL, DECOMP / DESSEM. Manuais de Metodologia, 2023. [2] Unsihuay Vila, C. Introdução aos Sistemas de Energia Elétrica,

MDI.TransmissionLine.TransmissionLineConstraints.add_transmission_line_flow_constraints(m: ConcreteModel) → ConcreteModel[source]

Add DC flow definition constraints to the model with MILP-compatible linearization.

Enforces, for each line l, time t, and load level p:: F[l,t,p] = b[l] * (θ[i,t,p] − θ[j,t,p]) * x[l,t]

To preserve MILP linearity, this relation is linearized using a Big-M formulation:

where:: M = lines_Fmax[l]

Parameters:: m (pyomo.environ.ConcreteModel) – Model containing: - m.LT : set of transmission lines - m.T : set of time periods - m.P : set of load levels - m.lines_flow[l,t,p] : power flow variable (MW) - m.lines_b[l] : line susceptance (1/x) - m.lines_Fmax[l] : flow capacity (MW) - m.lines_endpoints[l] : tuple/list [i, j] - m.theta[b,t,p] : phase angle (radians) - m.lines_x[l,t] : binary operational state (1 if built)
Returns:: Model with constraint block m.FlowDefinitionConstraint added.
Return type:: pyomo.environ.ConcreteModel

Notes

The formulation ensures linearity by relaxing the DC flow equation when the line is not built (x = 0).
Flow limits (±Fmax) should already be enforced elsewhere.
m.p_base (power base) is optional and defaults to 1 if not present.

MDI.TransmissionLine.TransmissionLineConstraints.add_transmission_line_flow_limits_constraints(m: ConcreteModel) → ConcreteModel[source]

Add flow capacity limit constraints to the model.

For each line l and time period t: −pmax_ij ≤ F_ij,t ≤ pmax_ij

Parameters:: m (pyomo.environ.ConcreteModel) – Pyomo model containing: - m.LT : set of transmission lines - m.T : set of time periods - m.lines_flow[l,t] : power flow variable (MW) - m.lines_pmax[l] : maximum capacity (MW)
Returns:: The same model with constraint block m.LinesFlowMaxLimitConstraint and m.LinesFlowMinLimitConstraint.
Return type:: pyomo.environ.ConcreteModel

Notes

The limit is symmetric (±pmax) and applies to both flow directions.
These constraints should be added after add_transmission_line_flow_constraints().

MDI.TransmissionLine.TransmissionLineConstraints.add_transmission_line_investment_link_constraints(m)[source]

Add the investment linkage constraint for transmission line units.

Defines the relationship between construction decisions and operational availability across time periods. Ensures that the existence variable accumulates investments over the planning horizon.

Parameters:: m (pyomo.environ.ConcreteModel) – Pyomo model instance containing the investment and operational variables.
Returns:: The model with investment linkage and initial-state constraints applied.
Return type:: pyomo.environ.ConcreteModel

MDI.TransmissionLine.TransmissionLineDataTypes module

EELT7030 — Operation and Expansion Planning of Electric Power Systems Federal University of Paraná (UFPR)

Module: Transmission Line — Data Structures

Author

Augusto Mathias Adams <augusto.adams@ufpr.br>

Description

Defines lightweight data classes for representing transmission lines in DC power flow formulations. Each line connects two connection bars (endpoints), has a specified susceptance and maximum power limit.

These structures serve as standardized inputs for Pyomo-based optimization models, supporting both operational (DCOPF) and expansion (MDI) formulations.

Classes

TransmissionLineUnit: Encapsulates the physical andproperties of a single transmission line or circuit, including endpoints and flow limits.
TransmissionLineData: Aggregates multiple line units along with global configuration parameters such as planning horizon and reference data.

Notes

The line susceptance b is defined as the inverse of the series reactance x_ij in per-unit.
The endpoints are represented as a list [bar_i, bar_j].
The state attribute allows distinguishing between existing (1) and candidate (0) lines for expansion studies.

References

[1] CEPEL, DECOMP / DESSEM. Manuais de Metodologia, 2023. [2] Unsihuay Vila, C. Introdução aos Sistemas de Energia Elétrica,

class MDI.TransmissionLine.TransmissionLineDataTypes.TransmissionLineData(horizon: int, units: Dict[str, TransmissionLineUnit])[source]

Bases: object

Aggregated data structure for transmission lines.

Parameters:

horizon (int) – Number of planning periods in the study horizon.
units (dict) – Mapping of line identifiers to their respective TransmissionLineUnit instances.

Notes

This class is typically used to initialize Pyomo sets and parameters.
Each line in units is assumed to have unique endpoints and electrical parameters.

horizon: int

units: Dict[str, TransmissionLineUnit]

class MDI.TransmissionLine.TransmissionLineDataTypes.TransmissionLineUnit(name: str, state: int, c_op: float, c_inv: float, model: str, b: float, pmax: float, endpoints: List[str])[source]

Bases: object

Data container for an individual transmission line or circuit.

Parameters:

name (str) – Unique identifier for the line (e.g., 'LINE_120').
state (int) – Initial state (0 = not installed, 1 = existing).
c_op (float) – Operational cost (per MWh of discharge or equivalent).
c_inv (float) – Investment cost (annualized).
b (float) – Line susceptance (1/x_ij), expressed in per-unit on the system base.
pmax (float) – Maximum allowable active power flow (MW) through the line.
endpoints (list of str) – Names of the two connection bars linked by the line, given as [bar_i, bar_j].

b: float

c_inv: float

c_op: float

endpoints: List[str]

model: str

name: str

pmax: float

state: int

MDI.TransmissionLine.TransmissionLineEquations module

EELT7030 — Operation and Expansion Planning of Electric Power Systems Federal University of Paraná (UFPR)

Module: Transmission Line — Symbolic Equations

Author

Augusto Mathias Adams <augusto.adams@ufpr.br>

Description

Defines symbolic expressions for the active power flow on transmission lines under the DC power flow approximation. These expressions are used as intermediate components for nodal balance equations and other network-level constraints.

Functions

add_transmission_line_flow_expression(m, t, flow_dict): Construct symbolic expressions for each line’s power flow at period t and append them to the supplied dictionary.
add_transmission_line_balance_expression(m, t, balance_dict): Aggregate inflow and outflow terms for each bar at time t to be used in connection-bar power balance constraints.

Notes

The DC power flow model assumes:
F_ij,t = b_ij (θ_i,t − θ_j,t)

where:
- F_ij,t : active power flow (MW)
- b_ij : susceptance (1/x_ij)
- θ_i,t : voltage phase angle at bar i (radians)
Line flows are positive from the first to the second endpoint in the endpoints list, and negative in the opposite direction.
This module produces expressions only; the corresponding constraints are enforced in TransmissionLineConstraints.

References

[1] CEPEL, DECOMP / DESSEM. Manuais de Metodologia, 2023. [2] Unsihuay Vila, C. Introdução aos Sistemas de Energia Elétrica,

MDI.TransmissionLine.TransmissionLineEquations.add_transmission_line_balance_expression(m: ConcreteModel, b: Any, t: Any, p: Any, balance_array: List[Any]) → List[Any][source]

Aggregate inflow and outflow terms for each bar at time t.

This routine computes, for each bar, the algebraic contribution of all connected transmission lines to the nodal power balance.

Mathematically:

Σ_in(F_ij,t) − Σ_out(F_ij,t)

where:

Incoming flows are from neighboring bars j such that line (j,i) exists.
Outgoing flows are from bar i to other bars j.

Parameters:

m (pyomo.environ.ConcreteModel) – Model containing: - m.LT : set of transmission lines - m.lines_endpoints[l] : list of bar identifiers [i, j] - m.lines_b[l] : susceptance (1/x) - m.theta[b,t] : voltage phase angle (radians)
t (int/Any) – Time period index.
p (str/Any) – level index.
balance_dict (Dict[str, List[Any]]) – Dictionary indexed by bar name. Each entry is a list of symbolic expressions contributing to that bar’s power balance.

Returns:

The same dictionary updated with inflow/outflow contributions from all transmission lines.

Return type:

Dict[str, List[Any]]

MDI.TransmissionLine.TransmissionLineEquations.add_transmission_line_cost_expression(m: ConcreteModel, cost_array: List[Any]) → List[Any][source]

Append line transmission cost terms to the total cost expression list.

This function serves as a placeholder for incorporating cost components associated with into the objective function.

Parameters:

m (ConcreteModel) – Pyomo model instance containing hydroelectric generation variables.
cost_array (list of expressions) – List of symbolic expressions used in constructing the total system cost.

Returns:

The input list, optionally extended with hydro-related cost expressions.

Return type:

list of expressions

MDI.TransmissionLine.TransmissionLineBuilder module

EELT7030 — Operation and Expansion Planning of Electric Power Systems Federal University of Paraná (UFPR)

Module: Transmission Line — Builder

Author

Augusto Mathias Adams <augusto.adams@ufpr.br>

Description

Provides the high-level construction routine for assembling the transmission line subsystem within DC power flow formulations. This module integrates sets, variables, and constraints that represent line flows, capacity limits, and their relationship with nodal phase angles.

Functions

build_transmission_lines(m): Create and attach all components of the transmission line subsystem to the given Pyomo model, including variables and constraints.

Notes

This module orchestrates the routines from:
- TransmissionLineVars
- TransmissionLineConstraints
The formulation is consistent with the DC power flow approximation:
F_ij,t = b_ij (θ_i,t − θ_j,t)
The resulting subsystem is compatible with both fixed-network (operational) and mixed-integer (expansion) models.
Phase angles θ[b,t] must be declared before calling this builder.

References

[1] CEPEL, DECOMP / DESSEM. Manuais de Metodologia, 2023. [2] Unsihuay Vila, C. Introdução aos Sistemas de Energia Elétrica,

MDI.TransmissionLine.TransmissionLineBuilder.add_transmission_line_problem(m: ConcreteModel, data: TransmissionLineData, include_objective: bool = False) → ConcreteModel[source]

Attach the transmission-line subsystem to an existing Pyomo model.

This function populates a given ConcreteModel with all sets, parameters, variables, and constraints related to transmission lines under the DC power flow approximation. It forms the network component of the system model, complementing the connection-bar formulation.

Parameters:

m (pyomo.environ.ConcreteModel) –
Pyomo model to which the transmission-line subsystem will be added. The model must contain or be prepared to receive:
- m.LT : set of transmission lines
- m.T : set of time periods
- m.lines_b[l] : susceptance (1/x_ij)
- m.lines_pmax[l] : flow capacity limit (MW)
- m.lines_endpoints[l] : list of connected bars [i, j]
- m.theta[b,t] : phase angle variable (radians)
data (TransmissionLineData) – Structured input containing the system horizon and the physical and operational parameters of each transmission line, including endpoints, susceptance, and capacity.
include_objective (bool, optional) – Reserved argument for interface compatibility (default False). Currently unused, as transmission lines are passive network elements without a direct cost component in the objective.

Returns:

The same model, augmented with transmission-line sets, variables, and constraints.

Return type:

pyomo.environ.ConcreteModel

Notes

Steps performed:
1. Initialize sets and parameters from TransmissionLineData.
2. Declare line flow decision variables (lines_flow[l,t]).
3. Add DC flow definition constraints: F_ij,t = b_ij (θ_i,t − θ_j,t)
4. Add flow capacity limit constraints: −pmax_ij ≤ F_ij,t ≤ pmax_ij.
The order of addition ensures that variables are defined before constraints that depend on them.
The formulation is compatible with both pure DC models and hybrid DC/transport formulations (lines marked with model: "transport" are automatically excluded from the DC flow constraint).

References

[1] CEPEL, DECOMP / DESSEM. Manuais de Metodologia, 2023. [2] Unsihuay Vila, C. Introdução aos Sistemas de Energia Elétrica,

MDI.TransmissionLine.TransmissionLineBuilder.build_transmission_lines(data: TransmissionLineData, include_objective: bool = True) → ConcreteModel[source]

Construct a standalone Pyomo model for the transmission-line subsystem.

This high-level builder creates a new ConcreteModel instance and populates it with all sets, parameters, variables, and constraints required to represent transmission lines in DC power flow formulations. It serves as the main entry point for testing or integrating the transmission-line subsystem independently from other components (e.g., connection bars, generators, or storage).

Parameters:

data (TransmissionLineData) –
Structured input data containing the planning horizon and the collection of transmission lines with their respective physical and economic attributes, including:
- b : susceptance (1/x_ij)
- pmax : maximum flow (MW)
- endpoints : list of connected bars [i, j]
- state : existing (1) or candidate (0)
- model : modeling type (‘dc’ or ‘transport’)
include_objective (bool, optional) – Reserved for future use (default True). Currently, this flag has no effect, as transmission lines are typically passive network elements without an explicit objective term. It is retained for interface consistency with other subsystem builders.

Returns:

A new Pyomo model fully configured with the transmission-line subsystem, including:

Return type:

pyomo.environ.ConcreteModel

Notes

The function internally calls add_transmission_line_problem(), which attaches the subsystem components to the newly created model.
The resulting model can be later coupled with other subsystems via shared variables such as bus angles (θ[b,t]) and power balances.
Both DC and transport models are supported within the same formulation, enabling hybrid operation and expansion studies.

References

[1] CEPEL, DECOMP / DESSEM. Manuais de Metodologia, 2023. [2] Unsihuay Vila, C. Introdução aos Sistemas de Energia Elétrica,

MDI.TransmissionLine.TransmissionLineVars module

EELT7030 — Operation and Expansion Planning of Electric Power Systems Federal University of Paraná (UFPR)

Module: Transmission Line — Sets, Parameters, and Variables

Author

Augusto Mathias Adams <augusto.adams@ufpr.br>

Description

This module defines initialization routines for transmission lines within DC power flow formulations in Pyomo. It provides the sets, parameters, and decision variables associated with transmission circuits, including susceptance, flow limits, and endpoint connectivity between connection bars.

Functions

transmission_line_add_sets_and_params(m, data): Initialize sets and parameters for transmission lines, including susceptance, flow capacity, and bar-to-bar connectivity.
transmission_line_add_variables(m): Declare line-level decision variables representing active power flows for each line and time period.

Notes

Designed for integration with connection-bar models and other subsystems (hydro, thermal, renewable, storage).
The time set m.T is assumed global and will only be created if not already defined.
Each line connects two bars defined by its endpoints attribute.
The parameters b, pmax and endpoints are initialized directly from the data structure TransmissionLineData.

References

[1] CEPEL, DECOMP / DESSEM. Manuais de Metodologia, 2023. [2] Unsihuay Vila, C. Introdução aos Sistemas de Energia Elétrica,

MDI.TransmissionLine.TransmissionLineVars.transmission_line_add_sets_and_params(m: ConcreteModel, data) → ConcreteModel[source]

Initialize sets and parameters for transmission lines.

Adds the transmission line set m.LT and associated dictionaries of electrical parameters: susceptance b, flow limit pmax, and connection endpoints (bar_i, bar_j). The model type and other metadata may also be stored for each line.

Parameters:

m (pyomo.environ.ConcreteModel) – Pyomo model to which transmission-line sets and parameters will be added.
data (TransmissionLineData) –
Data structure containing: - horizon (int): number of time periods. - units (dict[str, TransmissionLineUnit]): mapping of line

identifiers to their attributes (model, b, pmax, endpoints).

Returns:

The same model, augmented with transmission-line sets and parameters.

Return type:

pyomo.environ.ConcreteModel

Notes

If the model already defines m.T, it will not be overwritten.
The time set is defined as RangeSet(1, horizon) if absent.
The parameter mappings are stored as Python dictionaries for efficient access during constraint construction.

MDI.TransmissionLine.TransmissionLineVars.transmission_line_add_variables(m: ConcreteModel) → ConcreteModel[source]

Declare decision variables for transmission lines.

Adds the continuous real variable m.lines_flow[l,t], representing the active power flow (MW) through line l at time period t. The variable may assume positive or negative values, indicating direction of flow between the two connected bars.

Parameters:: m (pyomo.environ.ConcreteModel) – Pyomo model to which transmission-line variables will be added.
Returns:: The same model with line flow variables included.
Return type:: pyomo.environ.ConcreteModel

Notes

The sign convention follows the DC power flow equation: F_ij,t = b_ij * (θ_i,t - θ_j,t).
Line capacity limits (±pmax) are typically imposed in a separate constraint function (see TransmissionLineConstraints).