Modeling

The ExpansionPlanningModel class implements a modular Generalized Disjunctive Programming (GDP) formulation to produce a generalized model for generation, transmission, and expansion planning (GTEP) for power infrastructure problems. This formulation is represented as a dispatch problem integrated within a unit commitment and an investment problem for a specified representative period. Each of these problems is considered a “stage” in this formulation and is defined using a Pyomo Block component.

Figure 1 shows a graphical representation of these stages, including the relevant model components (variables and constraints), and the terms that connect the stages in the formulation. Note in Figure 1 that the commitment stage serves as the primary link between the stages.

Figure 1. Modular representation of the three stages included in the GTEP model formulation.

In the following section, we provide a more detailed explanation of these stages and how they are defined in the model.

Model Formulation and Construction

To develop the GTEP model in Figure 1, we start by splitting the formulation into three key stages:

1. Investment stage: Estimates relevant investment costs such as installation, expansion (which includes the installation of new selected generation units), operating costs, penalty, and maintenance costs. These costs include representative costs for both thermal and renewable generators as well as transmission equipment (e.g., transmission lines). Additionally, it declares a discrete decision (represented as a disjunction through the GDP formulation) to determine the operational state of the generation and transmission equipment (operational, retired, etc.) that can potentially influence the investment costs.

2. Commitment stage: Determines the operational status of the generators (set as a disjunction) while considering relevant operational costs and constraints to ensure a reliable power generation service. Additionally, it establishes operating limits considering the availability of the equipment from the dispatch stage.

3. Dispatch stage: Defines the status of transmission lines using a discrete decision (set as a disjunction) while considering operational costs, operability limits (such as power flow, ramping, reserves, etc.), and constraints to ensure the effective use of the active power lines.

Additionally, we include a data allocation step (prior to the formulation of the three stages) that assigns all relevant operating and cost parameters of the model. Since we are considering this as a data preprocessing step, not a stage in the formulation, this step is not represented in Figure 1.

Below we outline the steps we followed to construct the GTEP model. Note that the first four steps correspond directly to the stages described above (in some steps, the description is accompanied by example code).

Step 1: Data allocation

1. The initial step to create this model is to import the necessary libraries.

   import json
   import numpy as np
   import math
   from math import ceil
   
   # Import Pyomo components
   from pyomo.environ import *
   from pyomo.environ import units as u
   from pyomo.common.timing import TicTocTimer
   from pyomo.repn.linear import LinearRepnVisitor
   
   from egret.data.model_data import ModelData
   from egret.model_library.transmission.tx_utils import scale_ModelData_to_pu
   from config_options import _get_model_config
   
   # Import IDAES-GTEP functions
   from gtep.gtep_model import model_data_references
   from gtep.gtep_model import model_create_investment_stages
   from gtep.gtep_model import create_objective_function
   

2. Continue the creation of the GTEP model by including a new Python class, which includes a set of relevant parameters for the solution of the model, as arguments of the function.

   class ExpansionPlanningModel:
        """A generalized generation and
           transmission expansion planning model.
        """
   
        def __init__(
            self,
            config=None,
            stages=1,
            formulation=None,
            data=None,
            num_reps=3,
            len_reps=24,
            num_commit=24,
            num_dispatch=4,
        ):
   
            self.stages = stages
            self.formulation = formulation
            self.data = data
            self.num_reps = num_reps
            self.len_reps = len_reps
            self.num_commit = num_commit
            self.num_dispatch = num_dispatch
            self.config = _get_model_config()
            self.timer = TicTocTimer()
   

3. Inside the ExpansionPlanningModel() class, we create a concrete Pyomo model object using the ConcreteModel component. Within this model, we define several time-dependent parameters for each stage, including:

   1. Number of Commitment and Dispatch Periods, which specifies the total number of periods allocated for the commitment and dispatch stages. These are key to determine the operational status of generators over time and how power flows are managed across the transmission network.

   2. Length of Commitment and Dispatch Periods, which defines the duration of each commitment and dispatch periods. These allow us to model the time intervals during which generators can change their status (on, off, etc.) and evaluate operational decisions made during the power dispatch stage.

   def create_model(self):
       """Create concrete Pyomo model object associated with the ExpansionPlanningModel"""
   
       self.timer.tic("Creating GTEP Model")
       m = ConcreteModel()
   
       if self.data is None:
           raise
       elif type(self.data) is list:
           m.data_list = self.data
           m.md = scale_ModelData_to_pu(self.data[0])
   
       model_set_declaration(
           m, self.stages,
           rep_per=[i for i in range(1, self.num_reps + 1)]
       )
       m.representativePeriodLength = Param(
           m.representativePeriods,
           within=PositiveReals,
           default=24,
           units=u.hr
       )
       m.numCommitmentPeriods = Param(
           m.representativePeriods,
           within=PositiveIntegers,
           default=2,
           initialize=self.num_commit,
       )
       m.numDispatchPeriods = Param(
           m.representativePeriods,
           within=PositiveIntegers,
           default=2,
           initialize=self.num_dispatch,
       )
       m.commitmentPeriodLength = Param(
           within=PositiveReals,
           default=1,
           units=u.hr
       )
       m.dispatchPeriodLength = Param(
           within=PositiveReals,
           default=15,
           units=u.min
       )
   

4. Create and label data in the model using a data pre-processing function. This function ties the input data directly to the model by assigning values to all parameters. Look at the Data section for more information.

   model_data_references(m)
   

Step 2: Define Investment Stage

1. We define the investment stage using a Pyomo Block component. This block includes all relevant cost variables and constraints associated with generation and transmission equipment including operating, expansion, maintenance, and penalty costs. Additionally, it incorporates a discrete decision using a Pyomo.GDP Disjunction component to define the operational status of generation and transmission equipment, while considering a series of linking constraints to monitor the operational, installed, and retired states of the equipment. Table 1 presents a list of these key components, including their definition.

   In the GTEP model, we include the pre-defined function below that incorporates all these components.

   model_create_investment_stages(m, self.stages)
   

   During the calculation of the operating costs in this stage, we establish a connection with the commitment and dispatch stages. For this, we define a second Pyomo Block component for the commitment stage. Figure 2 illustrates this interconnection and how these relationships propagate throughout the formulation.

   

   Figure 2. Modular representation of the investment stage.

   Table 1: Model components (parameters, variables, constraints, and disjunctions) in the investment stage

   Component	Type	Definition
   representativePeriods	Set	Representative periods for the investment stage
   commitmentPeriods	Set	Commitment periods in the investment stage
   dispatchPeriods	Set	Dispatch periods in the investment stage
   maxThermalInvestment	Parameter	Maximum investment for thermal generators
   maxRenewableInvestment	Parameter	Maximum investment for renewable generators
   renewableSurplusRepresentative	Variable	Minimum renewable generation requirement per stage
   renewableOperational	Variable	Power provided to the grid by renewable generator at operation
   renewableInstalled	Variable	Power provided to the grid by renewable generator at installation
   renewableRetired	Variable	Power removed from the grid if renewable generator is retired
   renewableExtended	Variable	Power provided to the grid if renewable generator lifetime is extended
   quotaDeficit	Variable	Deficit for investment stage
   expansionCost	Variable	Cost of investment for new installed generators and transmission lines
   renewableCurtailmentInvestment	Variable	Curtailment penalties for investment stage
   operatingCostInvestment	Expression	Operating cost for investment period
   investStatus	Disjunction	Discrete decision to determine the status for generation investment. The alternatives, expressed as disjuncts in the model are: genOperational, genInstalled, genRetired, genDisabled, and genExtended
   branchInvestStatus	Disjunction	Discrete decision to determine the status for the transmission equipment investment. The alternatives (as disjuncts) are: branchOPerational, branchInstalled, branchRetired, branchDisabled, and branchExtended
   gen_stats_link	Constraint	Propagate the thermal generators status considering their previous state
   renewable_stats_link	Constraint	Propagate the renewable generators status considering their previous state
   renewable_retirement	Constraint	Propagate the renewable generation retirement status based on previous state and lifetime of renewable equipment
   consistent_extended	Logical constraint	If a generator is extended at time t, it must stay extended or retired at time t
   consistent_branch_operation	Logical constraint	If a branch is online at time t, it must have been on or installed at time t-1
   consistent_branch_operation_future	Logical constraint	If a branch is online at time t, it must be on, extended, or retired at time t+1
   consistent_branch_extended	Logical constraint	If a branch is extended at time t-1, it must stay extended or be retired at time t
   consistent_operation	Logical constraint	If a generator is online at time t, it should be online or installed at time t-1
   consistent_operation_future	Logical constraint	If a generator is online at time t, it can be online, extended, or retired at time t+1
   consistent_branch_disabled	Logical constraint	If a branch is disabled at time t-1, it must stay disabled or be installed at time t
   full_retirement	Logical constraint	If a generator is retired at time t-1, it should be disable at time t
   full_investment	Logical constraint	Installation in period t-1 implies operational in period t
   full_branch_retirement	Logical constraint	Retirement in period t-1 implies disabled in period t
   full_branch_investment	Logical constraint	Installation in period t-1 implies operational in period t

Step 3: Define Commitment Stage

1. Within the commitment block defined in step 2(a), we incorporate the commitment periods and introduce relevant components to ensure that the operation of transmission and generation equipment aligns with their designated statuses. During this stage, we also introduce cost and operational constraints that are contingent upon the commitment decisions and the outcomes of the dispatch stage. Table 2 provides a detailed overview of the components in the commitment stage of the model formulation.

   Figure 3 illustrates the various components utilized in this stage, highlighting the terms that connect to the dispatch stage.

   

   Figure 3. Modular representation of the commitment stage.

   Table 2: Components in the commitment stage

   Component	Type	Description
   dispatchPeriods	Set	Number of dispatch periods
   carbonTax	Parameter	Carbon tax
   renewableSurplusCommitment	Expression	Total renewable surplus/deficit for commitment
   operatingCostCommitment	Expression	Define total operating costs for commitment
   renewableCurtailmentCommitment	Expression	Define total curtailment for commitment
   genStatus	Disjunction	Discrete decision to set the status of the generation equipment. The alternatives, as disjuncts are: genOn, genStartup, genShutDown, and genOff.
   consistent_commitment_shutdown	Logical constraint
   consistent_commitment_off_after_shutdown	Logical constraint
   consistent_commitment_startup	Logical constraint
   consistent_commitment_on_after_startup	Logical Constraint
   consistent_commitment_uptime	Logical Constraint
   consistent_commitment_shutdown_after_uptime	Logical Constraint
   consistent_commitment_downtime	Logical Constraint
   consistent_commitment_start_after_downtime	Logical Constraint
   commit_active_gens_only	Logical constraint	Generators cannot be committed unless they are operational or just installed

Step 4: Define Dispatch Stage

1. This stage incorporates all key variables and constraints associated with the dispatch problem, including operational limits for generation and curtailment, as well as definitions for load shedding and surplus dispatch. Additionally, this stage outlines the costs associated with these variables. Table 3 and Figure 4 provide a more detailed description of these components and their definitions in the formulation.

   

   Figure 4. Modular representation of the dispatch stage.

   Table 3: Components in dispatch stage

   Component	Type	Description
   thermalGeneration	Variable	Thermal generation power during the dispatch stage
   renewableGeneration	Variable	Renewable generation power during the dispatch stage
   renewableCurtailment	Variable	Amount of curtailed renewable power
   spinningReserve	Variable
   quickstartReserve	Variable
   powerFlow	Variable	Power flow
   renewableGenerationSurplus	Expression	Power surplus per renewable generator
   renewableCurtailmentCost	Expression	Cost of curtailment per renewable generator
   generatorCost	Expression	Thermal generator cost based on the fuel cost
   loadShedCost	Expression	Load shed cost per bus
   branchInUseStatus	Disjunction	Discrete decision to determine the status of the transmission equipment. The alternatives are: branchInUse (which includes the DC OPF equations) and branchNotInUse (which fixes power flow to zero).
   must_use_active_branches	Logical constraint	If a branch is in use, it must be active
   cannot_use_inactive_branches	Logical constraint	If a branch is not in use, it must be inactive

Step 5: Define objective function

Once the data and blocks for the different stages are incorporated, we define the total cost as the objective function using the Pyomo Objective component. The total cost includes not only investment costs, but also operating, expansion, and penalty costs that connect all the stages in the formulation. The objective function is integrated within the following pre-defined function.

create_objective_function(m)

class gtep.gtep_model.ExpansionPlanningModel(config=None, stages=1, formulation=None, data=None, cost_data=None, num_reps=3, len_reps=24, num_commit=24, num_dispatch=4, duration_dispatch=15)

A generalized generation and transmission expansion planning model.

create_model()

Create concrete Pyomo model object associated with the ExpansionPlanningModel

report_large_coefficients(outfile, magnitude_cutoff=100000.0)

Dump very large magnitude (>= 1e5) coefficients to a json file.

Outfile:

should accept filename or open file and write there; (see how we do this in pyomo elsewhere)

Magnitude_cutoff:

magnitude above which to report coefficients

report_model(outfile='pretty_model_output.txt')

Pretty prints Pyomo model to outfile.

Outfile:

(str, optional) _description_. Defaults to “pretty_model_output.txt”.

gtep.gtep_model.add_commitment_constraints(b, comm_per)

Add commitment-associated disjunctions and constraints to representative period block.

gtep.gtep_model.add_commitment_variables(b, commitment_period)

Add variables and disjuncts to commitment period block.

gtep.gtep_model.add_dispatch_constraints(b, disp_per)

Add dispatch-associated inequalities to representative period block.

gtep.gtep_model.add_dispatch_variables(b, dispatch_period)

Add dispatch-associated variables to representative period block.

gtep.gtep_model.add_investment_constraints(b, investment_stage)

Add standard inequalities (i.e., those not involving disjunctions) to investment stage block.

gtep.gtep_model.add_investment_variables(b, investment_stage)

Add variables to investment stage block.

Parameters:

• b – Investment block

• investment_stage – Investment stage index or name

Returns:

None

gtep.gtep_model.add_storage_constraints(m, b, commitment_period)

Battery Discharging Constraints

gtep.gtep_model.commitment_period_rule(b, commitment_period)

Create commitment period block.

Parameters:

• b – commitment period block

• commitment_period – corresponding commitment period label

gtep.gtep_model.create_objective_function(m)

Creates objective function. Total cost is operating cost plus expansion cost plus penalty cost (penalties include generation deficits, renewable quota deficits, and curtailment) :param m: Pyomo GTEP model.

gtep.gtep_model.investment_stage_rule(b, investment_stage)

Creates investment stage block.

B:

Investment block

Investment_stage:

ID for current investment stage

gtep.gtep_model.model_create_investment_stages(m, stages)

Creates investment blocks and linking constraints for GTEP model. Largely manages retirements and links operational units in a given investment stage to operational + installed - retired in the previous investment stage.

M:

Pyomo model object

Stages:

Number of investment stages in planning horizon

gtep.gtep_model.model_data_references(m)

Creates and labels data for GTEP model; ties input data to model directly. :param m: Pyomo model object

gtep.gtep_model.model_set_declaration(m, stages, rep_per=['a', 'b'], com_per=2, dis_per=2)

Creates Pyomo Sets necessary (convenient) for solving the GTEP model.

M:

Pyomo model object

Stages:

Number of stages in investment horizon

gtep.gtep_model.representative_period_rule(b, representative_period)

Create representative period block.

B:

Representative period block

Representative_period:

corresponding representative period label