MDI.ConnectionBar package

Module contents

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

Package: ConnectionBar

Author

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

Description

The ConnectionBar package defines a modular abstraction of electrical connection bars (buses) for DC power flow formulations within Pyomo-based optimization frameworks. It encapsulates data structures, variable definitions, constraint formulations, symbolic equation builders, and model assembly routines used in operation and expansion studies such as DECOMP, DESSEM, and the MDI model.

Modules

ConnectionBarDataTypes

Defines the data classes (ConnectionBarUnit, ConnectionBarData) that describe the electrical and operational attributes of each bar, including demand, state, and slack configuration.

ConnectionBarVars

Declares sets, parameters, and decision variables for connection bars, such as phase angles (θ_b) and deficit variables (D[b,t]), ensuring consistency with system-wide formulations.

ConnectionBarConstraints

Implements the physical and operational constraints of the subsystem: - Slack-bar reference condition (θ = 0) - Angular limits for non-slack bars (−π ≤ θ ≤ π) - Nodal power balance (Kirchhoff’s 1st Law)

ConnectionBarEquations

Provides modular symbolic expression builders used to assemble the nodal power balance equation by integrating hydro, thermal, renewable, storage, and transmission contributions.

ConnectionBarBuilder

High-level model constructor that orchestrates all components, producing a complete Pyomo ConcreteModel configured with the relevant sets, variables, and constraints for each connection bar.

Notes

  • The package is designed for hierarchical integration with higher-level systems involving generation and transmission models.

  • It serves as the foundation for hybrid and multi-bar DC formulations in short-term (DESSEM), medium-term (DECOMP), and long-term (MDI) optimization contexts.

  • Each module can be used independently or assembled through build_connection_bars().

References

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

Lecture Notes, EELT7030/UFPR, 2023.

Submodules

MDI.ConnectionBar.ConnectionBarConstraints module

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

Module: Connection Bar — Balance Constraints

Author

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

Description

This module defines the nodal (per-bar) power balance constraints for Pyomo-based operation and expansion models. It enforces Kirchhoff’s Current Law (1st Law) at each connection bar and time period, ensuring that total injections (generation, inflows) and withdrawals (demand, outflows, and storage) are balanced.

Functions

add_connection_bar_balance_constraint(m)

Assemble and add the nodal power balance constraints for all bars and time periods using the component expressions provided by ConnectionBarEquations.

Notes

  • The routine calls modular expression builders such as add_hydraulic_balance_expression(), add_thermal_balance_expression(), add_renewable_balance_expression(), add_storage_balance_expression(), and add_transmission_line_balance_expression().

  • Each time period t generates a set of symbolic expressions by bar, which are then converted into Pyomo constraints and appended to m.Balance (a pyomo.core.base.constraint.ConstraintList).

  • This formulation supports hybrid systems (hydro, thermal, renewable, storage) with multiple connection bars.

  • The deficit variable m.D[bar,t] ensures model feasibility whenever the available generation and imports cannot fully meet local demand.

  • For single-bus systems, the resulting formulation is equivalent to the traditional system-wide balance equation.

References

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

Lecture Notes, EELT7030/UFPR, 2023.

MDI.ConnectionBar.ConnectionBarConstraints.add_connection_bar_angle_limits_constraints(m: ConcreteModel) → ConcreteModel[source]

Add angular bounds for non-slack connection bars.

Enforces -π ≤ θ_b ≤ π for all bars that are not marked as slack. This constraint bounds the nodal phase angles in DC power flow formulations, preventing unrealistic or numerically unstable solutions.

Parameters:

m (pyomo.environ.ConcreteModel) – Pyomo model instance containing: - m.BAR : set of all bars - m.SB : set of slack bars - m.theta[b] : voltage phase angle variable (radians)

Returns:

The same model with constraint block m.AngleLimits.

Return type:

pyomo.environ.ConcreteModel

Notes

  • The bounds are symbolic (±π radians) and apply only to non-slack bars.

  • The restriction helps numerical stability when solving large MILP or MINLP systems involving binary investment variables.

  • This function complements add_connection_bar_slack_constraints().

References

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

Lecture Notes, EELT7030/UFPR, 2023.

MDI.ConnectionBar.ConnectionBarConstraints.add_connection_bar_balance_constraints(m: ConcreteModel) → ConcreteModel[source]

Add nodal power balance constraints for all connection bars and time periods.

For each period t in the planning horizon, this function assembles the power balance expressions by connection bar (node) using add_connection_bar_balance_expression(), and converts them into Pyomo constraints stored in m.Balance.

Mathematically, for each bar b and period t:

\[\sum_{s \in S_b} P^{\text{gen}}_{s,t} + \sum_{i:(i,b)\in\mathcal{L}} F_{i b,t} - \sum_{j:(b,j)\in\mathcal{L}} F_{b j,t} + D_{b,t} = d_{b,t}\]
where:

  • P^{gen} represents generation injections (hydro, thermal, renewable, storage),

  • F_{ij,t} are line flows between bars i and j,

  • D_{b,t} is the local deficit variable,

  • d_{b,t} is the active demand at bar b.

Parameters:

m (pyomo.environ.ConcreteModel) – Pyomo model instance containing: - Sets: m.BAR (connection bars), m.T (time periods) - Variables: m.D[bar,t] (deficit), generation and flow variables - Parameters: m.d[bar,t] (demand per bar and time)

Returns:

The same model with all bar-level power balance constraints added to m.Balance.

Return type:

pyomo.environ.ConcreteModel

MDI.ConnectionBar.ConnectionBarConstraints.add_connection_bar_capacity_constraints(m: ConcreteModel) → ConcreteModel[source]

Add system adequacy (generation capacity) constraints to the model.

This function enforces that the total available generation and storage capacity across all bars, time periods, and load levels must be at least equal to the total system demand. It represents the resource adequacy condition of the model and defines, in its dual form, the Custo Marginal de Expansão (CME) or Expansion Marginal Cost.

Mathematically

For each time period t and load level p:

\[\sum_{g \in G} P^{max}_g \; x_{g,t} \; + \; \sum_{s \in S} \frac{P^{max}_{dis,s,p}}{h_p} \; x_{s,t} \; \geq \; \sum_{b \in B} D_{b,t,p}\]
where:

  • \(x_{g,t}\) is the investment or operational existence variable for generator g,

  • \(x_{s,t}\) is the existence variable for storage unit s,

  • \(P^{max}_g\) is the maximum generation capacity of unit g (MW),

  • \(P^{max}_{dis,s,p}\) is the maximum discharge power of storage unit s (MWh/patamar),

  • \(h_p\) is the number of hours in load level p,

  • \(D_{b,t,p}\) is the demand at bus b, time t, and level p.

param m:

The Pyomo model containing the sets, parameters, and variables for generation, storage, and demand. It must include: - GU : set of generators, - SU : set of storage units, - CB : set of connection bars, - T : set of time periods, - P : set of load levels.

type m:

pyomo.environ.ConcreteModel

returns:

The same model instance, augmented with the adequacy constraints.

rtype:

pyomo.environ.ConcreteModel

MDI.ConnectionBar.ConnectionBarConstraints.add_connection_bar_slack_constraints(m: ConcreteModel) → ConcreteModel[source]

Add the slack-bar reference constraints to fix phase angles.

Enforces θ_b = 0 for all bars marked as slack in the system data, establishing the angular reference for the DC power flow model.

Parameters:

m (pyomo.environ.ConcreteModel) – Pyomo model instance containing: - m.theta[b] : voltage phase angle variable (radians) - m.SB : set of bars designated as reference

Returns:

The same model with constraint block m.SlackConstraint.

Return type:

pyomo.environ.ConcreteModel

Notes

  • The DC power flow equations determine only angle differences, so a reference (slack) bar must be fixed to zero to avoid singularity in the system Jacobian.

  • In multi-slack configurations (rare but possible), all selected bars are fixed to θ = 0.

  • This constraint should be added after variable declaration and set initialization (i.e., after ConnectionBarVariables).

MDI.ConnectionBar.ConnectionBarDataTypes module

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

Module: Connection Bar Data Structures

Author

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

Description

This module defines lightweight data classes for representing connection bars (buses) in power system operation and expansion models. Each connection bar encapsulates its electrical state, demand profile, and configuration parameters. The aggregated system-level data structure (ConnectionBarData) groups all bars with the planning horizon and deficit penalty cost.

Classes

ConnectionBarUnit

Represents a single connection bar (bus) with demand profile, slack flag, and operational attributes.

ConnectionBarData

Aggregates multiple connection bars with common parameters such as planning horizon, system-level demand reference, and deficit cost.

Notes

  • The demand profile of each bar is expressed as a deterministic time series aligned with the planning horizon.

  • These structures serve as standardized inputs for Pyomo-based formulations of DECOMP, DESSEM, or the abstracted MDI model.

  • Extend as needed to include locational data, voltage levels, or stochastic demand scenarios.

References

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

Lecture Notes, EELT7030/UFPR, 2023.

class MDI.ConnectionBar.ConnectionBarDataTypes.ConnectionBarData(horizon: int, units: Dict[str, ConnectionBarUnit])[source]

Bases: object

System-wide data structure for connection bars.

Parameters:

Notes

  • The attribute units aggregates all bars of the system.

  • Cdef is used to penalize deficit variables D[b,t] in the objective.

  • In hierarchical models, this class provides the bar-level data consumed by the module ConnectionBarVariables.

horizon: int
units: Dict[str, ConnectionBarUnit]
class MDI.ConnectionBar.ConnectionBarDataTypes.ConnectionBarUnit(name: str, slack: bool, Cdef: float, c_pmax: float, demand: Dict[str, List[float]])[source]

Bases: object

Data container for an electrical connection bar (bus).

Parameters:

  • name (str) – Unique identifier for the bar (e.g., 'BAR_1').

  • slack (bool) – True if the bar is the system reference (angle fixed at 0 rad).

  • Cdef (float, optional) – Penalty cost for unmet demand (R$/MWh). Default is 500.0.

  • c_pmax (float, optional) – Max power of the deficit (MW)

  • demand (dict of {str: list of float}) – Nested dictionary containing demand data per load level and time period. Example structure: {"Ponta": [500, 550, 600], "Fora": [300, 320, 350]}

Notes

  • The demand list length must equal the planning horizon.

  • The attribute slack is used to define the voltage angle reference in DC power flow formulations.

  • The fields c_op and c_inv are placeholders for compatibility with expansion models (MDI), though typically unused in operation models.

Cdef: float
c_pmax: float
demand: Dict[str, List[float]]
name: str
slack: bool

MDI.ConnectionBar.ConnectionBarEquations module

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

Module: Connection Bar — Symbolic Expressions for Pyomo Models

Author

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

Description

This module defines symbolic expressions for connection bars (buses) in the context of integrated operation and expansion planning models (MDI). It provides functions to build cost and balance expressions that integrate generation, storage, and transmission subsystems within a unified mathematical formulation.

The functions are designed for modular assembly of Pyomo-based models, allowing each subsystem to contribute its respective terms to the global objective function and nodal power balance constraints.

Each function performs internal verification of required components, ensuring flexibility when operating within partially defined or staged model configurations (e.g., single-node, multi-bar, or hybrid systems).

Intended Use

  • To assemble total system cost expressions including generation, storage, deficit, and network effects.

  • To construct nodal (bus-level) power balance equations that ensure Kirchhoff’s law compliance across the network.

Examples

>>> cost_terms = []
>>> add_connection_bar_cost_expression(model, cost_terms)
>>> model.TotalCost = Objective(expr=sum(cost_terms), sense=minimize)

>>> balance_dict = {}
>>> add_connection_bar_balance_expression(model, t=1, p="Ponta", balance_array=balance_dict)
>>> for bar, expr in balance_dict.items():
...     model.PowerBalance.add(expr)

Notes

  • This module assumes that the model contains: CB (connection bars), T (time set), P (load levels), and subsystem modules with balance functions for generation, storage, and transmission.

  • Cost expressions include deficit penalties (Cdef * D[b,t,p]) weighted by discount factors and load duration.

  • The resulting symbolic expressions are compatible with both ConstraintList and indexed Constraint formulations in Pyomo.

References

[1] CEPEL, DECOMP / DESSEM — Metodologia de Modelagem Hidrotérmica, 2023 [2] Unsihuay Vila, C. (2023). Introdução aos Sistemas de Energia Elétrica, Lecture Notes, EELT7030/UFPR.

MDI.ConnectionBar.ConnectionBarEquations.add_connection_bar_balance_expression(m: ConcreteModel, b: Any, t: Any, p: str, balance_array: List[Any]) → List[Any][source]

Construct the nodal power balance expressions for each connection bar.

Aggregates all inflows and outflows of power at time t and load level p for each connection bar. Includes generation, storage, transmission, and deficit components.

Mathematically, for each bar b:

\[\sum_{g \in G_b} P_{g,t,p} + \sum_{s \in S_b} (dis_s - ch_s) + \sum_{ℓ \in Ω_b^{in}} F_{ℓ,t} - \sum_{ℓ \in Ω_b^{out}} F_{ℓ,t} + D_{b,t,p} = D_{b,t,p}^{dem}\]
Parameters:

  • m (ConcreteModel) – Pyomo model instance containing subsystems and bars.

  • t (int) – Time period index.

  • p (str) – Load level (e.g., “Ponta” or “Fora”).

  • balance_array (dict) – Dictionary mapping each bar to a list of symbolic expressions contributing to its power balance.

Returns:

Updated dictionary of symbolic expressions for each bar.

Return type:

dict

Notes

  • The resulting expressions can be added as constraints via:

    >>> for bar, expr in balance_dict.items():
>>>     model.PowerBalance.add(expr)

  • Each bar’s balance includes the local deficit term D[b,t,p].

  • Transmission flows contribute with positive sign for inflow and negative for outflow.

MDI.ConnectionBar.ConnectionBarEquations.add_connection_bar_capacity_expression(m: ConcreteModel, t: Any, p: str, capacity_array: List[Any]) → List[Any][source]
MDI.ConnectionBar.ConnectionBarEquations.add_connection_bar_cost_expression(m: ConcreteModel, cost_array: List[Any]) → List[Any][source]

Append deficit penalty terms to the total cost expression.

Constructs and appends the total cost expression associated with unmet demand (deficit) across all bars, time periods, and load levels. The cost is discounted over time and weighted by load duration.

Parameters:

  • m (ConcreteModel) – Pyomo model containing bars, time periods, and load levels.

  • cost_array (list of Pyomo expressions) – List of symbolic expressions to which the total deficit cost term will be appended.

Returns:

The same list, extended with the deficit cost expression.

Return type:

list of Pyomo expressions

Notes

  • The expression added is of the form:

    \[C_{def} = \sum_{b,t,p} (discount_t \cdot h_p \cdot Cdef_b \cdot D_{b,t,p}) \]

  • The cost reflects the economic penalty of unserved energy and integrates directly into the objective function of the MDI model.

MDI.ConnectionBar.ConnectionBarBuilder module

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

Module: Connection Bar — Model Builder

Author

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

Description

This module provides high-level routines for constructing and assembling Pyomo models representing electrical connection bars (buses) within DC power flow formulations. It integrates the components defined in submodules:

  • ConnectionBarDataTypes

  • ConnectionBarVars

  • ConnectionBarConstraints

The builder functions create or extend a ConcreteModel with sets, parameters, variables, and constraints associated with each connection bar. The formulation supports hybrid systems (hydro, thermal, renewable, and storage) as well as networked multi-bar expansion studies (e.g., MDI, DECOMP, DESSEM).

Functions

build_connection_bars(data, include_objective=True)

Create and return a complete Pyomo model representing the connection-bar subsystem.

add_connection_bar_problem(m, data, include_objective=False)

Attach all sets, parameters, variables, and constraints related to connection bars to an existing Pyomo model.

Notes

  • This module is the main interface between the connection-bar data structures and their corresponding Pyomo model elements.

  • The resulting model includes:

    • Sets and parameters (bars, demands, and slack identification)

    • Variables (phase angles and deficits)

    • Constraints:

      • Slack-bar reference (θ = 0)

      • Angular limits (-π ≤ θ ≤ π)

      • Power balance per bar and time period

  • The functions are designed for integration with higher-level system builders that include generators and transmission lines.

  • The optional argument include_objective is reserved for compatibility with renewable-only formulations and may be deprecated in future versions.

References

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

Lecture Notes, EELT7030/UFPR, 2023.

MDI.ConnectionBar.ConnectionBarBuilder.add_connection_bar_problem(m: ConcreteModel, data: ConnectionBarData, include_objective: bool = False) → ConcreteModel[source]

Attach connection-bar components and constraints to a Pyomo model.

Populates an existing ConcreteModel with all connection-bar-related elements, including sets, parameters, variables, and the fundamental constraints for DC power flow consistency.

Parameters:

  • m (pyomo.environ.ConcreteModel) – Pyomo model instance to which connection-bar elements will be added.

  • data (ConnectionBarData) –

    Input data containing:

    • Planning horizon (horizon)

    • Bar-level demand profiles

    • Slack-bar configuration

    • Deficit cost (Cdef)

  • include_objective (bool, optional) – Reserved parameter for compatibility with legacy renewable-only builders. Currently unused (default False).

Returns:

The updated model containing all connection-bar components.

Return type:

pyomo.environ.ConcreteModel

Notes

  • Functions called:

    • connection_bar_add_sets_and_params()

    • connection_bar_add_variables()

    • add_connection_bar_slack_constraints()

    • add_connection_bar_angle_limits_constraints()

  • The sequence ensures that:

    1. Sets and parameters are initialized.

    2. Variables are declared with appropriate bounds.

    3. The slack bar is fixed (θ = 0).

    4. Non-slack bars are bounded between ±π.

  • The routine can be reused within multi-bar or multi-region models, such as those in DECOMP and MDI formulations.

MDI.ConnectionBar.ConnectionBarBuilder.build_connection_bars(data: ConnectionBarData, include_objective: bool = True) → ConcreteModel[source]

Build a complete Pyomo model representing the connection-bar subsystem.

Creates a new ConcreteModel and populates it with all components associated with connection bars (sets, parameters, variables, and constraints). It serves as the main entry point for standalone tests or integration into larger system models.

Parameters:

  • data (ConnectionBarData) – Structured data containing the planning horizon, bar demands, slack-bar configuration, and deficit penalty cost.

  • include_objective (bool, optional) – Reserved argument for backward compatibility with renewable-only dispatch models (default is True). Currently unused.

Returns:

A fully initialized Pyomo model containing connection-bar elements.

Return type:

pyomo.environ.ConcreteModel

Notes

  • The created model includes:

    • Time set m.T and bar set m.BAR

    • Demand and deficit penalty parameters

    • Phase-angle (θ_b) and deficit (D[b,t]) variables

    • Slack-bar, angular-limit, and balance constraints

  • Intended for modular integration with generation and transmission builders in network expansion formulations.

MDI.ConnectionBar.ConnectionBarVars module

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

Module: Connection Bar — Sets, Parameters, and Variables

Author

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

Description

This module defines initialization routines for connection bars (nodes) within power system operation and expansion models in Pyomo. It provides the sets, parameters, and decision variables associated with network buses, including demand, deficit, and phase angles for DC power flow formulations.

Functions

connection_bar_add_sets_and_params(m, data)

Initialize sets and parameters for connection bars, including demand profiles and bar-specific attributes (state, slack status, etc.).

connection_bar_add_variables(m)

Declare bar-level decision variables: deficit (unserved demand) and phase angle (for DC models).

Notes

  • Designed for integration with hydrothermal and renewable subsystems.

  • The time set m.T is assumed global and will only be created if not already defined.

  • Each bar (bus) has a time-varying demand demand[b,t] and, optionally, a deficit variable D[b,t] penalized by m.Cdef in the objective.

  • The slack bar is identified by bar.slack = True and its phase angle is fixed at zero.

References

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

Lecture Notes, EELT7030/UFPR, 2023.

MDI.ConnectionBar.ConnectionBarVars.connection_bar_add_sets_and_params(m: ConcreteModel, data) → ConcreteModel[source]

Initialize sets and parameters for connection bars.

Adds the connection bar set m.BAR, the demand parameter m.demand[b,t], and the deficit penalty m.Cdef. It also records which bars are slack (reference) for the DC model.

Parameters:

  • m (pyomo.environ.ConcreteModel) – Pyomo model to which the bar sets and parameters will be added.

  • data (ConnectionBarData) –

    Data structure containing: - horizon (int): number of time periods. - demand (dict[str, list[float]]): demand time series per bar. - Cdef (float): deficit cost. - units (dict[str, ConnectionBar]): each bar object has

    attributes state, slack, demand.

Returns:

The same model, augmented with bar-related sets and parameters.

Return type:

pyomo.environ.ConcreteModel

Notes

  • If the model already defines m.T or m.Cdef, they are not overwritten.

  • The time set is defined as RangeSet(1, horizon) if absent.

  • Each bar’s demand is initialized using its demand array with 0-based indexing, so t-1 is used during initialization.

MDI.ConnectionBar.ConnectionBarVars.connection_bar_add_variables(m: ConcreteModel) → ConcreteModel[source]

Declare decision variables associated with connection bars.

Adds the following variables: - m.D[b,t,p] : non-negative deficit (unserved demand) per bar and time and level. - m.theta[b,t, p] : phase angle of the bus voltage (radians), continuous real.

For DC power flow models, the reference (slack) bar has its angle fixed to zero through a constraint or by setting bounds.

Parameters:

m (pyomo.environ.ConcreteModel) – Pyomo model to which bar variables will be added.

Returns:

The same model with bar variables included.

Return type:

pyomo.environ.ConcreteModel

Notes

  • m.D[b,t] is created even if a global deficit variable exists; each bar has its own local deficit component.

  • The variable m.theta[b] represents the voltage angle used in linearized DC power flow equations.

  • The fixing of the slack bar angle (theta = 0) should be performed externally by a constraint after this routine.