# Source code for temoa_rules

"""
Tools for Energy Model Optimization and Analysis (Temoa):
An open source framework for energy systems optimization modeling

Copyright (C) 2015,  NC State University

This program is free software; you can redistribute it and/or modify
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

A complete copy of the GNU General Public License v2 (GPLv2) is available
in LICENSE.txt.  Users uncompressing this from an archive may not have
"""

# Import below required in Python 2.7 to avoid integer division
# (e.g., 1/2 = 0 instead of 0.5)
from __future__ import division

from temoa_initialize import *

# ---------------------------------------------------------------
# Define the derived variables used in the objective function
# and constraints below.
# ---------------------------------------------------------------

[docs]def Activity_Constraint(M, p, s, d, t, v): r""" The Activity constraint defines the Activity convenience variable. The Activity variable is mainly used in the objective function to calculate the cost associated with use of a technology. This constraint sums the :math:\textbf{FO}_{p,s,d,i,t,v,o} over all input and output commodities. There is one caveat to keep in mind in regards to the Activity variable: if there is more than one output, there is currently no attempt by Temoa to convert to a common unit of measurement. For example, common measurements for heat include mass of steam at a given temperature, or total BTUs, while electricity is generally measured in a variant of watt-hours. Reconciling these units of measurement, as for example with a cogeneration plant, is currently left as an accounting exercise for the modeler. .. math:: :label: Activity \textbf{ACT}_{p, s, d, t, v} = \sum_{I, O} \textbf{FO}_{p,s,d,i,t,v,o} \\ \forall \{p, s, d, t, v\} \in \Theta_{\text{Activity}} """ return sum( \ M.V_FlowOut[p, s, d, S_i, t, v, S_o] \ for S_i in M.processInputs[p, t, v] \ for S_o in M.ProcessOutputsByInput[p, t, v, S_i] \ ) \ == M.V_Activity[p, s, d, t, v]
[docs]def Capacity_Constraint(M, p, s, d, t, v): r""" This constraint ensures that the capacity of a given process is sufficient to support its activity across all time periods and time slices. The calculation on the left hand side of the equality is the maximum amount of energy a process can produce in the timeslice <s,\ d>. Note that the curtailment variable shown below only applies to technologies are members of the curtailment set. Curtailment is necessary to track explicitly in scenarios that include a high renewable target. Without it, the model can generate more activity than is used to meet demand, and have all activity (including the portion curtailed) count towards the target. Tracking activity and curtailment separately prevents this possibility. .. math:: :label: Capacity \left ( \text{CFP}_{t, v} \cdot \text{C2A}_{t} \cdot \text{SEG}_{s, d} \cdot \text{TLF}_{p, t, v} \right ) \cdot \textbf{CAP}_{t, v} = \textbf{ACT}_{p, s, d, t, v} + \sum_{I, O} \textbf{CUR}_{p,s,d,i,t,v,o} \\ \forall \{p, s, d, t, v\} \in \Theta_{\text{Activity}} """ if t in M.tech_storage: return Constraint.Skip # The expressions below are defined in-line to minimize the amount of # expression cloning taking place with Pyomo. if t in M.tech_curtailment: # If technologies are present in the curtailment set, then enough # capacity must be available to cover both activity and curtailment. return value(M.CapacityFactorProcess[s, d, t, v]) \ * value(M.CapacityToActivity[t]) * value(M.SegFrac[s, d]) \ * value(M.ProcessLifeFrac[p, t, v]) \ * M.V_Capacity[t, v] == M.V_Activity[p, s, d, t, v] + sum( \ M.V_Curtailment[p, s, d, S_i, t, v, S_o] \ for S_i in M.processInputs[p, t, v] \ for S_o in M.ProcessOutputsByInput[p, t, v, S_i]) else: return value(M.CapacityFactorProcess[s, d, t, v]) \ * value(M.CapacityToActivity[t]) \ * value(M.SegFrac[s, d]) \ * value(M.ProcessLifeFrac[p, t, v]) \ * M.V_Capacity[t, v] >= M.V_Activity[p, s, d, t, v]
def ActivityByPeriodAndProcess_Constraint(M, p, t, v): r""" This constraint creates a derived variable in which the activity variable is summed over the season and time-of-day time slices: .. math:: :label: ActivityByPeriodAndProcess \textbf{ACT}_{p, t, v} = \sum_{I, O} \textbf{ACT}_{p,s,d,t,v} \\ \forall \{p, s, d, t, v\} \in \Theta_{\text{activity}} """ if p < v or v not in M.processVintages[p, t]: return Constraint.Skip activity = sum( M.V_Activity[p, S_s, S_d, t, v] for S_s in M.time_season for S_d in M.time_of_day ) if int is type(activity): return Constraint.Skip expr = M.V_ActivityByPeriodAndProcess[p, t, v] == activity return expr # This is required for MGA objective function def ActivityByTech_Constraint(M, t): r""" This constraint is utilized by the MGA objective function and sums activity by each technology over all time elements and vintages: .. math:: :label: ActivityByTech \textbf{ACT}_{t} = \sum_{I, O} \textbf{ACT}_{p,s,d,t,v} \\ \forall \{p, s, d, t, v\} \in \Theta_{\text{activity}} """ activity = sum( M.V_Activity[S_p, S_s, S_d, t, S_v] for S_p, S_v in M.processTechs[t] for S_s in M.time_season for S_d in M.time_of_day ) if int is type(activity): return Constraint.Skip expr = M.V_ActivityByTech[t] == activity return expr
[docs]def CapacityAvailableByPeriodAndTech_Constraint(M, p, t): r""" The :math:\textbf{CAPAVL} variable is nominally for reporting solution values, but is also used in the Max and Min constraint calculations. For any process with an end-of-life (EOL) on a period boundary, all of its capacity is available for use in all periods in which it is active (the process' TLF is 1). However, for any process with an EOL that falls between periods, Temoa makes the simplifying assumption that the available capacity from the expiring technology is available through the whole period, but only as much percentage as its lifespan through the period. For example, if a process expires 3 years into an 8 year period, then only :math:\frac{3}{8} of the installed capacity is available for use throughout the period. .. math:: :label: CapacityAvailable \textbf{CAPAVL}_{p, t} = \sum_{V} {TLF}_{p, t, v} \cdot \textbf{CAP} \\ \forall p \in \text{P}^o, t \in T """ cap_avail = sum( value(M.ProcessLifeFrac[p, t, S_v]) * M.V_Capacity[t, S_v] for S_v in M.processVintages[p, t] ) expr = M.V_CapacityAvailableByPeriodAndTech[p, t] == cap_avail return expr
[docs]def ExistingCapacity_Constraint(M, t, v): r""" Temoa treats existing capacity installed prior to the beginning of the model's optimization horizon as regular processes that require the same parameter specification as do new vintage technologies, except for the :code:CostInvest parameter. This constraint sets the capacity of processes for model periods that exist prior to the optimization horizon to user-specified values. .. math:: :label: ExistingCapacity \textbf{CAP}_{t, v} = ECAP_{t, v} \forall \{t, v\} \in \Theta_{\text{ExistingCapacity}} """ expr = M.V_Capacity[t, v] == M.ExistingCapacity[t, v] return expr
def EmissionActivityByPeriodAndTech_Constraint(M, e, p, t): r""" This constraint creates a derived variable that tracks the total emissions by pollutant, model time period, and technology. """ emission_total = sum( M.V_FlowOut[p, S_s, S_d, S_i, t, S_v, S_o] * M.EmissionActivity[e, S_i, t, S_v, S_o] for tmp_e, S_i, S_t, S_v, S_o in M.EmissionActivity.sparse_iterkeys() if tmp_e == e and S_t == t if (p, S_t, S_v) in M.processInputs.keys() for S_s in M.time_season for S_d in M.time_of_day ) if type(emission_total) is int: return Constraint.Skip expr = M.V_EmissionActivityByPeriodAndTech[e, p, t] == emission_total return expr # --------------------------------------------------------------- # Define the Objective Function # ---------------------------------------------------------------
[docs]def TotalCost_rule(M): r""" Using the :code:Activity and :code:Capacity variables, the Temoa objective function calculates the cost of energy supply, under the assumption that capital costs are paid through loans. This implementation sums up all the costs incurred, and is defined as :math:C_{tot} = C_{loans} + C_{fixed} + C_{variable}. Each term on the right-hand side represents the cost incurred over the model time horizon and discounted to the initial year in the horizon (:math:{P}_0). The calculation of each term is given below. .. math:: :label: obj_loan C_{loans} = \sum_{t, v \in \Theta_{IC}} \left ( \left [ IC_{t, v} \cdot LA_{t, v} \cdot \frac{(1 + GDR)^{P_0 - v +1} \cdot (1 - (1 + GDR)^{-{LLN}_{t, v}})}{GDR} \cdot \frac{ 1-(1+GDR)^{-LPA_{t,v}} }{ 1-(1+GDR)^{-LP_{t,v}} } \right ] \cdot \textbf{CAP}_{t, v} \right ) Note that capital costs (:math:{IC}_{t,v}) are handled in several steps. First, each capital cost is amortized using the loan rate (i.e., technology-specific discount rate) and loan period. Second, the annual stream of payments is converted into a lump sum using the global discount rate and loan period. Third, the new lump sum is amortized at the global discount rate and technology lifetime. Fourth, loan payments beyond the model time horizon are removed and the lump sum recalculated. The terms used in Steps 3-4 are :math:\frac{ GDR }{ 1-(1+GDR)^{-LP_{t,v} } }\cdot \frac{ 1-(1+GDR)^{-LPA_{t,v}} }{ GDR }. The product simplifies to :math:\frac{ 1-(1+GDR)^{-LPA_{t,v}} }{ 1-(1+GDR)^{-LP_{t,v}} }, where :math:LPA_{t,v} represents the active lifetime of a process :math:(t,v) before the end of the model horizon, and :math:LP_{t,v} represents the full lifetime of a process :math:(t,v). Fifth, the lump sum is discounted back to the beginning of the horizon (:math:P_0) using the global discount rate. While an explicit salvage term is not included, this approach properly captures the capital costs incurred within the model time horizon, accounting for technology-specific loan rates and periods. .. math:: :label: obj_fixed C_{fixed} = \sum_{p, t, v \in \Theta_{FC}} \left ( \left [ FC_{p, t, v} \cdot \frac{(1 + GDR)^{P_0 - p +1} \cdot (1 - (1 + GDR)^{-{MPL}_{t, v}})}{GDR} \right ] \cdot \textbf{CAP}_{t, v} \right ) .. math:: :label: obj_variable C_{variable} = \sum_{p, t, v \in \Theta_{VC}} \left ( MC_{p, t, v} \cdot \frac{ (1 + GDR)^{P_0 - p + 1} \cdot (1 - (1 + GDR)^{-{MPL}_{p,t, v}}) }{ GDR } \cdot \textbf{ACT}_{t, v} \right ) """ return sum(PeriodCost_rule(M, p) for p in M.time_optimize)
def PeriodCost_rule(M, p): P_0 = min(M.time_optimize) P_e = M.time_future.last() # End point of modeled horizon GDR = value(M.GlobalDiscountRate) MLL = M.ModelLoanLife MPL = M.ModelProcessLife x = 1 + GDR # convenience variable, nothing more. loan_costs = sum( M.V_Capacity[S_t, S_v] * ( value(M.CostInvest[S_t, S_v]) * value(M.LoanAnnualize[S_t, S_v]) * ( value(M.LifetimeLoanProcess[S_t, S_v]) if not GDR else ( x ** (P_0 - S_v + 1) * (1 - x ** (-value(M.LifetimeLoanProcess[S_t, S_v]))) / GDR ) ) ) * ( (1 - x ** (-min(value(M.LifetimeProcess[S_t, S_v]), P_e - S_v))) / (1 - x ** (-value(M.LifetimeProcess[S_t, S_v]))) ) for S_t, S_v in M.CostInvest.sparse_iterkeys() if S_v == p ) fixed_costs = sum( M.V_Capacity[S_t, S_v] * ( value(M.CostFixed[p, S_t, S_v]) * ( value(MPL[p, S_t, S_v]) if not GDR else (x ** (P_0 - p + 1) * (1 - x ** (-value(MPL[p, S_t, S_v]))) / GDR) ) ) for S_p, S_t, S_v in M.CostFixed.sparse_iterkeys() if S_p == p ) variable_costs = sum( M.V_ActivityByPeriodAndProcess[p, S_t, S_v] * ( value(M.CostVariable[p, S_t, S_v]) * ( value(MPL[p, S_t, S_v]) if not GDR else (x ** (P_0 - p + 1) * (1 - x ** (-value(MPL[p, S_t, S_v]))) / GDR) ) ) for S_p, S_t, S_v in M.CostVariable.sparse_iterkeys() if S_p == p ) period_costs = loan_costs + fixed_costs + variable_costs return period_costs # --------------------------------------------------------------- # Define the Model Constraints. # The order of constraint definitions follows the same order as the # declarations in temoa_model.py. # ---------------------------------------------------------------
[docs]def Demand_Constraint(M, p, s, d, dem): r""" The Demand constraint drives the model. This constraint ensures that supply at least meets the demand specified by the Demand parameter in all periods and slices, by ensuring that the sum of all the demand output commodity (:math:c) generated by :math:\textbf{FO} must meet the modeler-specified demand, in each time slice. .. math:: :label: Demand \sum_{I, T, V} \textbf{FO}_{p, s, d, i, t, v, dem} = {DEM}_{p, dem} \cdot {DSD}_{s, d, dem} \\ \forall \{p, s, d, dem\} \in \Theta_{\text{Demand}} Note that the validity of this constraint relies on the fact that the :math:C^d set is distinct from both :math:C^e and :math:C^p. In other words, an end-use demand must only be an end-use demand. Note that if an output could satisfy both an end-use and internal system demand, then the output from :math:\textbf{FO} would be double counted. """ if (s,d,dem) not in M.DemandSpecificDistribution.sparse_keys(): return Constraint.Skip supply = sum( M.V_FlowOut[p, s, d, S_i, S_t, S_v, dem] for S_t, S_v in M.commodityUStreamProcess[p, dem] for S_i in M.ProcessInputsByOutput[p, S_t, S_v, dem] ) DemandConstraintErrorCheck(supply, p, s, d, dem) expr = supply == M.Demand[p, dem] * M.DemandSpecificDistribution[s, d, dem] return expr
[docs]def DemandActivity_Constraint(M, p, s, d, t, v, dem, s_0, d_0): r""" For end-use demands, it is unreasonable to let the optimizer only allow use in a single time slice. For instance, if household A buys a natural gas furnace while household B buys an electric furnace, then both units should be used throughout the year. Without this constraint, the model might choose to only use the electric furnace during the day, and the natural gas furnace during the night. This constraint ensures that the ratio of a process activity to demand is constant for all time slices. Note that if a demand is not specified in a given time slice, or is zero, then this constraint will not be considered for that slice and demand. This is transparently handled by the :math:\Theta superset. .. math:: :label: DemandActivity DEM_{p, s, d, dem} \cdot \sum_{I} \textbf{FO}_{p, s_0, d_0, i, t, v, dem} = DEM_{p, s_0, d_0, dem} \cdot \sum_{I} \textbf{FO}_{p, s, d, i, t, v, dem} \\ \forall \{p, s, d, t, v, dem, s_0, d_0\} \in \Theta_{\text{DemandActivity}} """ if (s,d,dem) not in M.DemandSpecificDistribution.sparse_keys(): return Constraint.Skip DSD = M.DemandSpecificDistribution # lazy programmer act_a = sum( M.V_FlowOut[p, s_0, d_0, S_i, t, v, dem] for S_i in M.ProcessInputsByOutput[p, t, v, dem] ) act_b = sum( M.V_FlowOut[p, s, d, S_i, t, v, dem] for S_i in M.ProcessInputsByOutput[p, t, v, dem] ) expr = act_a * DSD[s, d, dem] == act_b * DSD[s_0, d_0, dem] return expr
[docs]def CommodityBalance_Constraint(M, p, s, d, c): r""" Where the Demand constraint :eq:Demand ensures that end-use demands are met, the CommodityBalance constraint ensures that the endogenous system demands are met. This constraint requires the total production of a given commodity to equal the amount consumed, thus ensuring an energy balance at the system level. .. math:: :label: CommodityBalance \sum_{I, T, V} \textbf{FO}_{p, s, d, i, t, v, c} = \sum_{T, V, O} \textbf{FI}_{p, s, d, c, t, v, o} \\ \forall \{p, s, d, c\} \in \Theta_{\text{CommodityBalance}} """ if c in M.commodity_demand: return Constraint.Skip vflow_in = sum( M.V_FlowIn[p, s, d, c, S_t, S_v, S_o] for S_t, S_v in M.commodityDStreamProcess[p, c] for S_o in M.ProcessOutputsByInput[p, S_t, S_v, c] ) vflow_out = sum( M.V_FlowOut[p, s, d, S_i, S_t, S_v, c] for S_t, S_v in M.commodityUStreamProcess[p, c] for S_i in M.ProcessInputsByOutput[p, S_t, S_v, c] ) CommodityBalanceConstraintErrorCheck(vflow_out, vflow_in, p, s, d, c) expr = vflow_out == vflow_in return expr
[docs]def ProcessBalance_Constraint(M, p, s, d, i, t, v, o): r""" The ProcessBalance constraint is one of the most fundamental constraints in Temoa. It defines the basic relationship between the energy entering a process (:math:\textbf{FI}) and the energy leaving a process (:math:\textbf{FO}). This constraint sets the :code:FlowOut variable, upon which all other constraints rely. This constraint requires that the output energy of a given process is equal to the product of its input energy and conversion efficiency. Note that the curtailment variable shown below only applies to technologies are members of the curtailment set. As noted in the :code:Capacity_Constraint, curtailment is necessary to track explicitly in scenarios that include a high renewable target. .. math:: :label: ProcessBalance \textbf{FO}_{p, s, d, i, t, v, o} + \textbf{CUR}_{p, s, d, i, t, v, o} = EFF_{i, t, v, o} \cdot \textbf{FI}_{p, s, d, i, t, v, o} \\ \forall \{p, s, d, i, t, v, o\} \in \Theta_{\text{ProcessBalance}} """ if t in M.tech_curtailment: return M.V_FlowOut[p, s, d, i, t, v, o] + \ M.V_Curtailment[p, s, d, i, t, v, o] == \ M.V_FlowIn[p, s, d, i, t, v, o] * \ value(M.Efficiency[i, t, v, o]) else: return M.V_FlowOut[p, s, d, i, t, v, o] == \ M.V_FlowIn[p, s, d, i, t, v, o] * \ value(M.Efficiency[i, t, v, o])
[docs]def ResourceExtraction_Constraint(M, p, r): r""" The ResourceExtraction constraint allows a modeler to specify an annual limit on the amount of a particular resource Temoa may use in a period. .. math:: :label: ResourceExtraction \sum_{S, D, I, t \in T^r, V} \textbf{FO}_{p, s, d, i, t, v, c} \le RSC_{p, c} \forall \{p, c\} \in \Theta_{\text{ResourceExtraction}} """ collected = sum( M.V_FlowOut[p, S_s, S_d, S_i, S_t, S_v, r] for S_i, S_t, S_v in M.ProcessByPeriodAndOutput.keys() for S_s in M.time_season for S_d in M.time_of_day ) expr = collected <= M.ResourceBound[p, r] return expr
[docs]def BaseloadDiurnal_Constraint(M, p, s, d, t, v): r""" Some electric generators cannot ramp output over a short period of time (e.g., hourly or daily). Temoa models this behavior by forcing technologies in the :code:tech_baseload set to maintain a constant output across all times-of-day within the same season. Note that the output of a baseload process can vary between seasons. Ideally, this constraint would not be necessary, and baseload processes would simply not have a :math:d index. However, implementing the more efficient functionality is currently on the Temoa TODO list. .. math:: :label: BaseloadDaily SEG_{s, D_0} \cdot \textbf{ACT}_{p, s, d, t, v} = SEG_{s, d} \cdot \textbf{ACT}_{p, s, D_0, t, v} \\ \forall \{p, s, d, t, v\} \in \Theta_{\text{BaseloadDiurnal}} """ # Question: How to set the different times of day equal to each other? # Step 1: Acquire a "canonical" representation of the times of day l_times = sorted(M.time_of_day) # i.e. a sorted Python list. # This is the commonality between invocations of this method. index = l_times.index(d) if 0 == index: # When index is 0, it means that we've reached the beginning of the array # For the algorithm, this is a terminating condition: do not create # an effectively useless constraint return Constraint.Skip # Step 2: Set the rest of the times of day equal in output to the first. # i.e. create a set of constraints that look something like: # tod[ 2 ] == tod[ 1 ] # tod[ 3 ] == tod[ 1 ] # tod[ 4 ] == tod[ 1 ] # and so on ... d_0 = l_times[0] # Step 3: the actual expression. For baseload, must compute the /average/ # activity over the segment. By definition, average is # (segment activity) / (segment length) # So: (ActA / SegA) == (ActB / SegB) # computationally, however, multiplication is cheaper than division, so: # (ActA * SegB) == (ActB * SegA) expr = ( M.V_Activity[p, s, d, t, v] * M.SegFrac[s, d_0] == M.V_Activity[p, s, d_0, t, v] * M.SegFrac[s, d] ) return expr
[docs]def StorageEnergy_Constraint(M, p, s, d, t, v): r""" This constraint tracks the amount of storage assuming ordered time slices. The storage unit is initialized at a user-specified charge level (0-100%) in the first time slice of each period, and then the charge level is updated each time slice based on the amount of energy stored or discharged. At the end of the last time slice associated with each period, the charge level must be zeroed out. """ # This is the sum of all input=i sent TO storage tech t of vintage v with # output=o in p,s,d charge = sum( M.V_FlowIn[p, s, d, S_i, t, v, S_o] * M.Efficiency[S_i, t, v, S_o] for S_i in M.processInputs[p, t, v] for S_o in M.ProcessOutputsByInput[p, t, v, S_i] ) # This is the sum of all output=o withdrawn FROM storage tech t of vintage v # with input=i in p,s,d discharge = sum( M.V_FlowOut[p, s, d, S_i, t, v, S_o] for S_o in M.processOutputs[p, t, v] for S_i in M.ProcessInputsByOutput[p, t, v, S_o] ) stored_energy = charge - discharge # This storage formulation allows stored energy to carry over through # time of day and seasons, but must be zeroed out at the end of each period, i.e., # the last time slice of the last season must zero out if d == M.time_of_day.last() and s == M.time_season.last(): d_prev = M.time_of_day.prev(d) expr = M.V_StorageLevel[p, s, d_prev, t, v] + stored_energy == 0 # First time slice of the first season (i.e., start of period), starts at full charge elif d == M.time_of_day.first() and s == M.time_season.first(): initial_storage = ( M.StorageInit[t] * M.V_Capacity[t, v] * M.StorageDuration[t] * M.CapacityToActivity[t] * value(M.ProcessLifeFrac[p, t, v]) ) expr = M.V_StorageLevel[p, s, d, t, v] == initial_storage + stored_energy # First time slice of any season that is NOT the first season elif d == M.time_of_day.first(): d_last = M.time_of_day.last() s_prev = M.time_season.prev(s) expr = ( M.V_StorageLevel[p, s, d, t, v] == M.V_StorageLevel[p, s_prev, d_last, t, v] + stored_energy ) # Any time slice that is NOT covered above (i.e., not the time slice ending # the period, or the first time slice of any season) else: d_prev = M.time_of_day.prev(d) expr = ( M.V_StorageLevel[p, s, d, t, v] == M.V_StorageLevel[p, s, d_prev, t, v] + stored_energy ) return expr
[docs]def StorageEnergyUpperBound_Constraint(M, p, s, d, t, v): r""" This constraint ensures that the amount of energy stored does not exceed the upper bound set by the energy capacity of the storage device. """ energy_capacity = ( M.V_Capacity[t, v] * M.StorageDuration[t] * M.CapacityToActivity[t] * value(M.ProcessLifeFrac[p, t, v]) ) expr = M.V_StorageLevel[p, s, d, t, v] <= energy_capacity return expr
[docs]def StorageChargeRate_Constraint(M, p, s, d, t, v): r""" This constraint ensures that the charge rate of the storage unit is limited by the power capacity (typically GW) of the storage unit. """ # Calculate energy charge in each time slice slice_charge = sum( M.V_FlowIn[p, s, d, S_i, t, v, S_o] * M.Efficiency[S_i, t, v, S_o] for S_i in M.processInputs[p, t, v] for S_o in M.ProcessOutputsByInput[p, t, v, S_i] ) # Maximum energy charge in each time slice max_charge = ( M.V_Capacity[t, v] * M.CapacityToActivity[t] * M.SegFrac[s, d] * value(M.ProcessLifeFrac[p, t, v]) ) # Energy charge cannot exceed the power capacity of the storage unit expr = slice_charge <= max_charge return expr
[docs]def StorageDischargeRate_Constraint(M, p, s, d, t, v): r""" This constraint ensures that the discharge rate of the storage unit is limited by the power capacity (typically GW) of the storage unit. """ # Calculate energy discharge in each time slice slice_discharge = sum( M.V_FlowOut[p, s, d, S_i, t, v, S_o] for S_o in M.processOutputs[p, t, v] for S_i in M.ProcessInputsByOutput[p, t, v, S_o] ) # Maximum energy discharge in each time slice max_discharge = ( M.V_Capacity[t, v] * M.CapacityToActivity[t] * M.SegFrac[s, d] * value(M.ProcessLifeFrac[p, t, v]) ) # Energy discharge cannot exceed the capacity of the storage unit expr = slice_discharge <= max_discharge return expr
[docs]def StorageThroughput_Constraint(M, p, s, d, t, v): r""" It is not enough to only limit the charge and discharge rate separately. We also need to ensure that the maximum throughput (charge + discharge) does not exceed the capacity (typically GW) of the storage unit. """ discharge = sum( M.V_FlowOut[p, s, d, S_i, t, v, S_o] for S_o in M.processOutputs[p, t, v] for S_i in M.ProcessInputsByOutput[p, t, v, S_o] ) charge = sum( M.V_FlowIn[p, s, d, S_i, t, v, S_o] * M.Efficiency[S_i, t, v, S_o] for S_i in M.processInputs[p, t, v] for S_o in M.ProcessOutputsByInput[p, t, v, S_i] ) throughput = charge + discharge max_throughput = ( M.V_Capacity[t, v] * M.CapacityToActivity[t] * M.SegFrac[s, d] * value(M.ProcessLifeFrac[p, t, v]) ) expr = throughput <= max_throughput return expr
[docs]def RampUpDay_Constraint(M, p, s, d, t, v): # M.time_of_day is a sorted set, and M.time_of_day.first() returns the first # element in the set, similarly, M.time_of_day.last() returns the last element. # M.time_of_day.prev(d) function will return the previous element before s, and # M.time_of_day.next(d) function will return the next element after s. r""" The ramp rate constraint is utilized to limit the rate of electricity generation increase and decrease between two adjacent time slices in order to account for physical limits associated with thermal power plants. Note that this constraint only applies to technologies with ramp capability, which is defined in the set :math:\textbf{T}^{ramp}. We assume for simplicity the rate limits for both ramp up and down are equal and they do not vary with technology vintage. The ramp rate limits (:math:r_t) for technology :math:t should be expressed in percentage of its rated capacity. Note that when :math:d_{nd} is the last time-of-day, :math:d_{nd + 1} \not \in \textbf{D}, i.e., if one time slice is the last time-of-day in a season and the other time slice is the first time-of-day in the next season, the ramp rate limits between these two time slices can not be expressed by :eq:ramp_up_day. Therefore, the ramp rate constraints between two adjacent seasons are represented in :eq:ramp_up_season. In Equation :eq:ramp_up_day and :eq:ramp_up_season, we assume :math:\textbf{S} = \{s_i, i = 1, 2, \cdots, ns\} and :math:\textbf{D} = \{d_i, i=1, 2, \cdots, nd\}. .. math:: \frac{ \textbf{ACT}_{p, s, d_{i + 1}, t, v} }{ SEG_{s, d_{i + 1}} \cdot C2A_t } - \frac{ \textbf{ACT}_{p, s, d_i, t, v} }{ SEG_{s, d_i} \cdot C2A_t } \leq r_t \cdot \textbf{CAPAVL}_{p,t} \\ \forall p \in \textbf{P}^o, s \in \textbf{S}, d_i, d_{i + 1} \in \textbf{D}, t \in \textbf{T}^{ramp}, v \in \textbf{V} :label: ramp_up_day """ if d != M.time_of_day.first(): d_prev = M.time_of_day.prev(d) expr_left = ( M.V_Activity[p, s, d, t, v] / value(M.SegFrac[s, d]) - M.V_Activity[p, s, d_prev, t, v] / value(M.SegFrac[s, d_prev]) ) / value(M.CapacityToActivity[t]) expr_right = M.V_Capacity[t, v] * value(M.RampUp[t]) expr = expr_left <= expr_right else: return Constraint.Skip return expr
[docs]def RampDownDay_Constraint(M, p, s, d, t, v): r""" Similar to Equation :eq:ramp_up_day, we use Equation :eq:ramp_down_day to limit ramp down rates between any two adjacent time slices. .. math:: \frac{ \textbf{ACT}_{p, s, d_{i + 1}, t, v} }{ SEG_{s, d_{i + 1}} \cdot C2A_t } - \frac{ \textbf{ACT}_{p, s, d_i, t, v} }{ SEG_{s, d_i} \cdot C2A_t } \geq -r_t \cdot \textbf{CAPAVL}_{p,t} \\ \forall p \in \textbf{P}^o, s \in \textbf{S}, d_i, d_{i + 1} \in \textbf{D}, t \in \textbf{T}^{ramp}, v \in \textbf{V} :label: ramp_down_day """ if d != M.time_of_day.first(): d_prev = M.time_of_day.prev(d) expr_left = ( M.V_Activity[p, s, d, t, v] / value(M.SegFrac[s, d]) - M.V_Activity[p, s, d_prev, t, v] / value(M.SegFrac[s, d_prev]) ) / value(M.CapacityToActivity[t]) expr_right = -(M.V_Capacity[t, v] * value(M.RampDown[t])) expr = expr_left >= expr_right else: return Constraint.Skip return expr
[docs]def RampUpSeason_Constraint(M, p, s, t, v): r""" Note that :math:d_1 and :math:d_{nd} represent the first and last time-of-day, respectively. .. math:: \frac{ \textbf{ACT}_{p, s_{i + 1}, d_1, t, v} }{ SEG_{s_{i + 1}, d_1} \cdot C2A_t } - \frac{ \textbf{ACT}_{p, s_i, d_{nd}, t, v} }{ SEG_{s_i, d_{nd}} \cdot C2A_t } \leq r_t \cdot \textbf{CAPAVL}_{p,t} \\ \forall p \in \textbf{P}^o, s_i, s_{i + 1} \in \textbf{S}, d_1, d_{nd} \in \textbf{D}, t \in \textbf{T}^{ramp}, v \in \textbf{V} :label: ramp_up_season """ if s != M.time_season.first(): s_prev = M.time_season.prev(s) d_first = M.time_of_day.first() d_last = M.time_of_day.last() expr_left = ( M.V_Activity[p, s, d_first, t, v] / M.SegFrac[s, d_first] - M.V_Activity[p, s_prev, d_last, t, v] / M.SegFrac[s_prev, d_last] ) / value(M.CapacityToActivity[t]) expr_right = M.V_Capacity[t, v] * value(M.RampUp[t]) expr = expr_left <= expr_right else: return Constraint.Skip return expr
[docs]def RampDownSeason_Constraint(M, p, s, t, v): r""" Similar to Equation :eq:ramp_up_season, we use Equation :eq:ramp_down_season to limit ramp down rates between any two adjacent seasons. .. math:: \frac{ \textbf{ACT}_{p, s_{i + 1}, d_1, t, v} }{ SEG_{s_{i + 1}, d_1} \cdot C2A_t } - \frac{ \textbf{ACT}_{p, s_i, d_{nd}, t, v} }{ SEG_{s_i, d_{nd}} \cdot C2A_t } \geq -r_t \cdot \textbf{CAPAVL}_{p,t} \\ \forall p \in \textbf{P}^o, s_i, s_{i + 1} \in \textbf{S}, d_1, d_{nd} \in \textbf{D}, t \in \textbf{T}^{ramp}, v \in \textbf{V} :label: ramp_down_season """ if s != M.time_season.first(): s_prev = M.time_season.prev(s) d_first = M.time_of_day.first() d_last = M.time_of_day.last() expr_left = ( M.V_Activity[p, s, d_first, t, v] / value(M.SegFrac[s, d_first]) - M.V_Activity[p, s_prev, d_last, t, v] / value(M.SegFrac[s_prev, d_last]) ) / value(M.CapacityToActivity[t]) expr_right = -(M.V_Capacity[t, v] * value(M.RampDown[t])) expr = expr_left >= expr_right else: return Constraint.Skip return expr
def RampUpPeriod_Constraint(M, p, t, v): # if p != M.time_future.first(): # p_prev = M.time_future.prev(p) # s_first = M.time_season.first() # s_last = M.time_season.last() # d_first = M.time_of_day.first() # d_last = M.time_of_day.last() # expr_left = ( # M.V_Activity[ p, s_first, d_first, t, v ] - # M.V_Activity[ p_prev, s_last, d_last, t, v ] # ) # expr_right = ( # M.V_Capacity[t, v]* # value( M.RampUp[t] )* # value( M.CapacityToActivity[ t ] )* # value( M.SegFrac[s, d]) # ) # expr = (expr_left <= expr_right) # else: # return Constraint.Skip # return expr return Constraint.Skip # We don't need inter-period ramp up/down constraint. def RampDownPeriod_Constraint(M, p, t, v): # if p != M.time_future.first(): # p_prev = M.time_future.prev(p) # s_first = M.time_season.first() # s_last = M.time_season.last() # d_first = M.time_of_day.first() # d_last = M.time_of_day.last() # expr_left = ( # M.V_Activity[ p, s_first, d_first, t, v ] - # M.V_Activity[ p_prev, s_last, d_last, t, v ] # ) # expr_right = ( # -1* # M.V_Capacity[t, v]* # value( M.RampDown[t] )* # value( M.CapacityToActivity[ t ] )* # value( M.SegFrac[s, d]) # ) # expr = (expr_left >= expr_right) # else: # return Constraint.Skip # return expr return Constraint.Skip # We don't need inter-period ramp up/down constraint.
[docs]def ReserveMargin_Constraint(M, p, s, d): r""" During each period :math:p, the sum of the available capacity of all reserve technologies :math:\sum_{t \in T^{res}} \textbf{CAPAVL}_{p,t}, which are defined in the set :math:\textbf{T}^{res}, should exceed the peak load by :math:RES_z, the regional reserve margin. Note that the reserve margin is expressed in percentage of the peak load. Generally speaking, in a database we may not know the peak demand before running the model, therefore, we write this equation for all the time-slices defined in the database in each region. .. math:: \sum_{t \in T^{res}} { CC_t \cdot \textbf{CAPAVL}_{p,t} \cdot SEG_{s^*,d^*} \cdot C2A_t } \geq \sum_{t \in T^{res}} { \sum_{t \in v^{vintage}} \textbf{ACT}_{p, s, d, t, v}} \cdot (1 + RES_z) \\ \forall p \in \textbf{P}^o, z \in \textbf{C}^{zone} :label: reserve_margin """ if not M.tech_reserve: # If reserve set empty, skip the constraint return Constraint.Skip cap_avail = sum( value(M.CapacityCredit[p, t]) * M.V_CapacityAvailableByPeriodAndTech[p, t] * value(M.CapacityToActivity[t]) * value(M.SegFrac[s, d]) for t in M.tech_reserve # Make sure (p,t) combinations are defined if (p,t) in M.activeCapacityAvailable_pt ) # In most Temoa input databases, demand is endogenous, so we use electricity # generation instead. total_generation = sum( M.V_Activity[p, s, d, t, S_v] for (t,S_v) in M.processReservePeriods[p] ) cap_target = total_generation * (1 + value(M.PlanningReserveMargin)) return cap_avail >= cap_target
[docs]def EmissionLimit_Constraint(M, p, e): r""" A modeler can track emissions through use of the :code:commodity_emissions set and :code:EmissionActivity parameter. The :math:EAC parameter is analogous to the efficiency table, tying emissions to a unit of activity. The EmissionLimit constraint allows the modeler to assign an upper bound per period to each emission commodity. .. math:: :label: EmissionLimit \sum_{I,T,V,O|{e,i,t,v,o} \in EAC_{ind}} \left ( EAC_{e, i, t, v, o} \cdot \textbf{FO}_{p, s, d, i, t, v, o} \right ) \le ELM_{p, e} \\ \forall \{p, e\} \in \Theta_{\text{EmissionLimit}} """ emission_limit = M.EmissionLimit[p, e] actual_emissions = sum( M.V_FlowOut[p, S_s, S_d, S_i, S_t, S_v, S_o] * M.EmissionActivity[e, S_i, S_t, S_v, S_o] for tmp_e, S_i, S_t, S_v, S_o in M.EmissionActivity.sparse_iterkeys() if tmp_e == e # EmissionsActivity not indexed by p, so make sure (p,t,v) combos valid if (p, S_t, S_v) in M.processInputs.keys() for S_s in M.time_season for S_d in M.time_of_day ) if int is type(actual_emissions): msg = ( "Warning: No technology produces emission '%s', though limit was " "specified as %s.\n" ) SE.write(msg % (e, emission_limit)) return Constraint.Skip expr = actual_emissions <= emission_limit return expr
[docs]def GrowthRateConstraint_rule(M, p, t): r""" This constraint sets an upper bound growth rate on technology-specific capacity. .. math:: :label: GrowthRate CAPAVL_{p_{i},t} \le GRM \cdot CAPAVL_{p_{i-1},t} + GRS, \\ \forall \{p, t\} \in \Theta_{\text{GrowthRate}} where :math:GRM is the maximum growth rate, and should be specified as :math:(1+r) and :math:GRS is the growth rate seed, which has units of capacity. Without the seed, any technology with zero capacity in the first time period would be restricted to zero capacity for the remainder of the time horizon. """ GRS = value(M.GrowthRateSeed[t]) GRM = value(M.GrowthRateMax[t]) CapPT = M.V_CapacityAvailableByPeriodAndTech periods = sorted(set(p_ for p_, t_ in CapPT if t_ == t)) if p not in periods: return Constraint.Skip if p == periods[0]: expr = CapPT[p, t] <= GRS else: p_prev = periods.index(p) p_prev = periods[p_prev - 1] expr = CapPT[p, t] <= GRM * CapPT[p_prev, t] + GRS return expr
[docs]def MaxActivity_Constraint(M, p, t): r""" The MaxActivity sets an upper bound on the activity from a specific technology. Note that the indices for these constraints are period and tech, not tech and vintage. .. math:: :label: MaxActivity \sum_{S,D,V} \textbf{ACT}_{p,s,d,t,v} \le MAXACT_{p, t} \forall \{p, t\} \in \Theta_{\text{MaxActivity}} """ activity_pt = sum( M.V_Activity[p, S_s, S_d, t, S_v] for S_s in M.time_season for S_d in M.time_of_day for S_v in M.processVintages[p, t] ) max_act = value(M.MaxActivity[p, t]) expr = activity_pt <= max_act return expr
[docs]def MinActivity_Constraint(M, p, t): r""" The MinActivity sets a lower bound on the activity from a specific technology. Note that the indices for these constraints are period and tech, not tech and vintage. .. math:: :label: MinActivity \sum_{S,D,V} \textbf{ACT}_{p,s,d,t,v} \ge MINACT_{p, t} \forall \{p, t\} \in \Theta_{\text{MinActivity}} """ activity_pt = sum( M.V_Activity[p, S_s, S_d, t, S_v] for S_s in M.time_season for S_d in M.time_of_day for S_v in M.processVintages[p, t] ) min_act = value(M.MinActivity[p, t]) expr = activity_pt >= min_act return expr
[docs]def MinActivityGroup_Constraint(M, p, g): r""" The MinActivityGroup constraint sets a minimum activity limit for a user-defined technology group. Each technology within each group is multiplied by a weighting function, which determines what technology activity share can count towards the constraint. .. math:: :label: MinActivityGroup \sum_{S,D,T,V} \textbf{ACT}_{p,s,d,t,v} \cdot \ge MGGT_{p, g} \forall \{p, g\} \in \Theta_{\text{MinActivityGroup}} where :math:g represents the assigned technology group and :math:MGGT refers to the :code:MinGenGroupTarget parameter. """ activity_p = sum( M.V_Activity[p, S_s, S_d, S_t, S_v] * M.MinGenGroupWeight[S_t, g] for S_t in M.tech_groups for S_s in M.time_season for S_d in M.time_of_day for S_v in M.processVintages[p, S_t] ) min_act = value(M.MinGenGroupTarget[p, g]) expr = activity_p >= min_act return expr
[docs]def MaxCapacity_Constraint(M, p, t): r""" The MaxCapacity constraint sets a limit on the maximum available capacity of a given technology. Note that the indices for these constraints are period and tech, not tech and vintage. .. math:: :label: MaxCapacity \textbf{CAPAVL}_{p, t} \le MAX_{p, t} \forall \{p, t\} \in \Theta_{\text{MaxCapacity}} """ max_cap = value(M.MaxCapacity[p, t]) expr = M.V_CapacityAvailableByPeriodAndTech[p, t] <= max_cap return expr
[docs]def MaxCapacitySet_Constraint(M, p): r""" See MaxCapacity_Constraint """ max_cap = value(M.MaxCapacitySum[p]) aggcap = sum( M.V_CapacityAvailableByPeriodAndTech[p, t] for t in M.tech_capacity_max ) expr = aggcap <= max_cap return expr
[docs]def MinCapacity_Constraint(M, p, t): r""" The MinCapacity constraint sets a limit on the minimum available capacity of a given technology. Note that the indices for these constraints are period and tech, not tech and vintage. .. math:: :label: MinCapacityCapacityAvailableByPeriodAndTech \textbf{CAPAVL}_{p, t} \ge MIN_{p, t} \forall \{p, t\} \in \Theta_{\text{MinCapacity}} """ min_cap = value(M.MinCapacity[p, t]) expr = M.V_CapacityAvailableByPeriodAndTech[p, t] >= min_cap return expr
def MinCapacitySet_Constraint(M, p): r""" See MinCapacity_Constraint """ min_cap = value(M.MinCapacitySum[p]) aggcap = sum( M.V_CapacityAvailableByPeriodAndTech[p, t] for t in M.tech_capacity_min ) expr = aggcap >= min_cap return expr
[docs]def TechInputSplit_Constraint(M, p, s, d, i, t, v): r""" Allows users to specify fixed or minimum shares of commodity inputs to a process producing a single output. These shares can vary by model time period. See TechOutputSplit_Constraint for an analogous explanation. """ inp = sum( M.V_FlowIn[p, s, d, i, t, v, S_o] for S_o in M.ProcessOutputsByInput[p, t, v, i] ) total_inp = sum( M.V_FlowIn[p, s, d, S_i, t, v, S_o] for S_i in M.processInputs[p, t, v] for S_o in M.ProcessOutputsByInput[p, t, v, i] ) expr = inp >= M.TechInputSplit[p, i, t] * total_inp return expr
[docs]def TechOutputSplit_Constraint(M, p, s, d, t, v, o): r""" Some processes take a single input and make multiple outputs, and the user would like to specify either a constant or time-varying ratio of outputs per unit input. The most canonical example is an oil refinery. Crude oil is used to produce many different refined products. In many cases, the modeler would like to specify a minimum share of each refined product produced by the refinery. For example, a hypothetical (and highly simplified) refinery might have a crude oil input that produces 4 parts diesel, 3 parts gasoline, and 2 parts kerosene. The relative ratios to the output then are: .. math:: d = \tfrac{4}{9} \cdot \text{total output}, \qquad g = \tfrac{3}{9} \cdot \text{total output}, \qquad k = \tfrac{2}{9} \cdot \text{total output} Note that it is possible to specify output shares that sum to less than unity. In such cases, the model optimizes the remaining share. In addition, it is possible to change the specified shares by model time period. The constraint is formulated as follows: .. math:: :label: TechOutputSplit \sum_{I} \textbf{FO}_{p, s, d, i, t, v, o} \geq SPL_{p, t, o} \cdot \textbf{ACT}_{p, s, d, t, v} \forall \{p, s, d, t, v, o\} \in \Theta_{\text{TechOutputSplit}} """ out = sum( M.V_FlowOut[p, s, d, S_i, t, v, o] for S_i in M.ProcessInputsByOutput[p, t, v, o] ) expr = out >= M.TechOutputSplit[p, t, o] * M.V_Activity[p, s, d, t, v] return expr
# --------------------------------------------------------------- # Define rule-based parameters # --------------------------------------------------------------- def ParamModelLoanLife_rule(M, t, v): loan_length = value(M.LifetimeLoanProcess[t, v]) mll = min(loan_length, max(M.time_future) - v) return mll def ParamModelProcessLife_rule(M, p, t, v): life_length = value(M.LifetimeProcess[t, v]) tpl = min(v + life_length - p, value(M.PeriodLength[p])) return tpl def ParamPeriodLength(M, p): # This specifically does not use time_optimize because this function is # called /over/ time_optimize. periods = sorted(M.time_future) i = periods.index(p) # The +1 won't fail, because this rule is called over time_optimize, which # lacks the last period in time_future. length = periods[i + 1] - periods[i] return length def ParamPeriodRate(M, p): """\ The "Period Rate" is a multiplier against the costs incurred within a period to bring the time-value back to the base year. The parameter PeriodRate is not directly specified by the modeler, but is a convenience calculation based on the GlobalDiscountRate and the length of each period. One may refer to this (pseudo) parameter via M.PeriodRate[ a_period ] """ rate_multiplier = sum( (1 + M.GlobalDiscountRate) ** (M.time_optimize.first() - p - y) for y in range(0, M.PeriodLength[p]) ) return value(rate_multiplier) def ParamProcessLifeFraction_rule(M, p, t, v): """\ Calculate the fraction of period p that process :math:<t, v> operates. For most processes and periods, this will likely be one, but for any process that will cease operation (rust out, be decommissioned, etc.) between periods, calculate the fraction of the period that the technology is able to create useful output. """ eol_year = v + value(M.LifetimeProcess[t, v]) frac = eol_year - p period_length = value(M.PeriodLength[p]) if frac >= period_length: # try to avoid floating point round-off errors for the common case. return 1 # number of years into final period loan is complete frac /= float(period_length) return frac def ParamLoanAnnualize_rule(M, t, v): dr = value(M.DiscountRate[t, v]) lln = value(M.LifetimeLoanProcess[t, v]) if not dr: return 1.0 / lln annualized_rate = dr / (1.0 - (1.0 + dr) ** (-lln)) return annualized_rate