Co-optimize Costs and Emissions 

This tutorial will walk through how to co-optimize electricity bill savings and Scope 2 emissions for a simple battery model in either CVXPY or Pyomo by going through the following steps:

Load an electricity tariff spreadsheet and Scope 2 emissions data

Configure an optimization model of the electricity consumer with system constraints

Consider using flexibility metrics to encode system constraints (Flexibility Metrics as Constraints)!

The models are presented step-by-step to demonstrate the model building process, but the complete models are available in the examples folder:

pyomo_battery_model.py

cvxpy_battery_model.py

Create an objective function of electricity costs and emissions with carbon cost

Minimize the electricity costs and emissions of this consumer, given the system constraints, base load consumption, and carbon cost

Display the results to validate that the optimization is correct

CVXPY 

Import dependencies

import datetime
import cvxpy as cp
import pandas as pd
import matplotlib.pyplot as plt
from eeco.units import u
from eeco import emissions
from eeco import costs

Load a electricity tariff and Scope 2 emissions spreadsheet

path_to_emissions_sheet = "eeco/data/emissions.csv"
emission_df = pd.read_csv(path_to_emissions_sheet, sep=",")

# get the carbon intensity
carbon_intensity = emissions.get_carbon_intensity(
    datetime.datetime(2023, 4, 9), datetime.datetime(2023, 4, 11), emission_df, resolution="1m"
)

path_to_tariff_sheet = "eeco/data/tariff.csv"
tariff_df = pd.read_csv(path_to_tariff_sheet, sep=",")

# get the charge dictionary
charge_dict = costs.get_charge_dict(
    datetime.datetime(2023, 4, 9), datetime.datetime(2023, 4, 11), tariff_df, resolution="1m"
)

We are going to evaluate the electricity consumption from only April 9th to April 10th since that is where our synthetic data comes from (https://github.com/we3lab/eeco/blob/main/eeco/data/consumption.csv). You will also see that it is in 1-minute intervals, hence resolution=”1m”.

Configure an optimization model of the electricity consumer with system constraints

# load historical consumption data
load_df = pd.read_csv("eeco/data/consumption.csv", parse_dates=["Datetime"])

# set battery parameters
# create variables for battery total energy, max charge and discharge power, and SOC limits
total_capacity = 10 # kWh
min_soc = 0
max_soc = 1
init_soc = 0.5
fin_soc = 0.5
max_discharge = 5 # kW
max_charge = 5 # kW
T = len(load_df["Datetime"])
delta_t = ((load_df.iloc[-1]["Datetime"] - load_df.iloc[0]["Datetime"]) / T) / datetime.timedelta(hours=1)

# initialize variables
battery_output_kW = cp.Variable(T)
battery_soc = cp.Variable(T+1)
grid_demand_kW = cp.Variable(T)

# set constraints
constraints = [
    battery_output_kW >= -max_discharge,
    battery_output_kW <= max_charge,
    battery_soc >= min_soc,
    battery_soc <= max_soc,
    battery_soc[0] == init_soc,
    battery_soc[T] == fin_soc,
    grid_demand_kW >= 0
]
for t in range(T):
    constraints += [
        battery_soc[t+1] == battery_soc[t] + (battery_output_kW[t] * delta_t) / total_capacity,
        grid_demand_kW[t] == load_df.iloc[t]["Load [kW]"] + battery_output_kW[t]
    ]

This is a standard battery model with energy (i.e., total charge) and power (i.e., discharge/charge rate) constraints. The round-trip efficiency is 1.0 since there is no penalty applied when discharging the battery, but that’s fine for these demonstration purposes.

Create an objective function of electricity costs and Scope 2 emissions with carbon cost

# dollars per kg CO2 - converted from $192/metric ton CO2-eq
cost_of_carbon = 0.192

# NOTE: second entry of the tuple can be ignored since it's for Pyomo
emissions_obj, _ = emissions.calculate_grid_emissions(
    carbon_intensity,
    grid_demand_kW,
    resolution="1m",
    consumption_units=u.kW
)
# requires a consumption dictionary in case there is natural gas in addition to electricity
consumption_data_dict = {"electric": grid_demand_kW}
# NOTE: second entry of the tuple can be ignored since it's for Pyomo
cost_obj, _ = costs.calculate_cost(
    charge_dict,
    {"electric": grid_demand_kW},
    resolution="1m",
    consumption_estimate=load_df["Load [kW]"].sum(),
    desired_utility="electric",
)
obj = cost_obj + emissions_obj * cost_of_carbon

Minimize the costs and emissions of this consumer, given the system constraints, base load consumption, and carbon cost

# solve the CVX problem (objective variable should be named obj)
prob = cp.Problem(cp.Minimize(obj), constraints)
prob.solve()

Display the results to validate that the optimization is correct

Always compute the ex-post cost using numpy due to the convex relaxations that we apply in our optimization code:

# NOTE: second entry of the tuple can be ignored since it's for Pyomo
baseline_electricity_emissions, _ = emissions.calculate_grid_emissions(
    carbon_intensity,
    load_df["Load [kW]"].values,
    resolution="1m",
    consumption_units=u.kW
)
# NOTE: second entry of the tuple can be ignored since it's for Pyomo
optimized_electricity_emissions, _ = emissions.calculate_grid_emissions(
    carbon_intensity,
    grid_demand_kW.value,
    resolution="1m",
    consumption_units=u.kW
)
# NOTE: second entry of the tuple can be ignored since it's for Pyomo
baseline_electricity_cost, _ = costs.calculate_cost(
    charge_dict,
    {"electric": load_df["Load [kW]"].values},
    resolution="1m",
    desired_utility="electric",
)
# NOTE: second entry of the tuple can be ignored since it's for Pyomo
optimized_electricity_cost, _ = costs.calculate_cost(
    charge_dict,
    {"electric": grid_demand_kW.value},
    resolution="1m",
    desired_utility="electric",
)

total_baseline_cost = baseline_electricity_cost + cost_of_carbon * baseline_electricity_emissions.magnitude
total_optimized_cost = optimized_electricity_cost + cost_of_carbon * optimized_electricity_emissions.magnitude

If we print our results, we confirm that the optimal electricity profile has daily emissions of 10.57 kg CO:sub:2-eq, 0.12 kg CO:sub:2-eq less than the baseline emissions of 10.69 kg CO:sub:2-eq. It simultaneously lowers daily costs by $61.48 from $765.29 to $703.81.

>>>print(f"Baseline Scope 2 Emissions: {baseline_electricity_emissions:.2f} kg CO_2-eq")
Baseline Scope 2 Emissions: 10.69 kilogram kg CO_2-eq
>>>print(f"Optimized Scope 2 Emissions: {optimized_electricity_emissions:.2f} kg CO_2-eq")
Optimized Scope 2 Emissions: 10.57 kilogram kg CO_2-eq
>>>print(f"Baseline Electricity Costs: ${baseline_electricity_cost:.2f}")
Baseline Electricity Costs: $765.29
>>>print(f"Optimized Electricity Costs: ${optimized_electricity_cost:.2f}")
Optimized Electricity Costs: $703.81
>>>print(f"Total Baseline Cost w/ $192/ton CO2-eq: ${total_baseline_cost:.2f}")
Total Baseline Cost w/ $192/ton CO2-eq: $767.34
>>>print(f"Total Optimized Cost w/ $192/ton CO2-eq: ${total_optimized_cost:.2f}")
Total Optimized Cost w/ $192/ton CO2-eq: $705.84

We could make similar plots to Optimize Electricity Costs and Optimize Scope 2 Emissions, but we omitted them from these instructions for the sake of space.

Pyomo 

Import dependencies

import datetime
import numpy as np
import pandas as pd
import pyomo.environ as pyo
import matplotlib.pyplot as plt
from eeco.units import u
from eeco import costs
from eeco import emissions
from examples.pyomo_battery_model import BatteryPyomo

Load a electricity tariff and Scope 2 emissions spreadsheet

path_to_emissions_sheet = "eeco/data/emissions.csv"
emission_df = pd.read_csv(path_to_emissions_sheet, sep=",")

# get the carbon intensity
carbon_intensity = emissions.get_carbon_intensity(
    datetime.datetime(2022, 7, 1), datetime.datetime(2022, 8, 1), emission_df, resolution="15m"
)

path_to_tariffsheet = "eeco/data/tariff.csv"
tariff_df = pd.read_csv(path_to_tariffsheet, sep=",")

# get the charge dictionary
charge_dict = costs.get_charge_dict(
    datetime.datetime(2022, 7, 1), datetime.datetime(2022, 8, 1), tariff_df, resolution="15m"
)

We are going to evaluate the electricity consumption for the entire month of July 2022. Below we will create synthetic baseload data for this month with 15-minute resolution, so resolution=”15m”.

Configure an optimization model of the electricity consumer with system constraints

We rely on the virtual battery model in pyomo_battery_model.py. We’re going to stick to the electricity cost calculation details, but we encourage you to go check out the code to better understand the model.

# Define the parameters for the battery model
battery_params = {
    "start_date": "2022-07-01 00:00:00",
    "end_date": "2022-08-01 00:00:00",
    "timestep": 0.25,   # 15 minutes defined in hours
    "rte": 0.86,
    "energycapacity": 100,
    "powercapacity": 50,
    "soc_min": 0.05,
    "soc_max": 0.95,
    "soc_init": 0.5,
}

# Create a sample baseload profile based on a sine wave
baseload = np.sin(np.linspace(0, 4 * np.pi, 96))*100 + 1000 + np.random.normal(0, 10, 96)

# Create an instance of the BatteryOpt class
battery = BatteryPyomo(battery_params, baseload, baseload_repeat=True)

# create the model on the instance battery
battery.create_model()

The above code initializes the battery model with flexibility metrics like round-trip efficiency (RTE), power capacity, and energy capacity.

Create an objective function of electricity costs and Scope 2 emissions with carbon cost

# dollars per kg CO2 - converted from $192/metric ton CO2-eq
cost_of_carbon = 0.192

# compute Scope 2 emissions using Pyomo variable
battery.model.emissions, battery.model = emissions.calculate_grid_emissions(
    carbon_intensity,
    battery.model.net_facility_load,
    resolution="15m",
    consumption_units=u.kW,
    model=battery.model
)

# monthly total consumption - divided by 4 because of 15-min resolution
consumption_estimate = sum(baseload) / 4
# this example tariff only has electric utility types so we do not pass the gas key
consumption_data_dict = {"electric": battery.model.net_facility_load}

# use the built-in helper function to add more terms to the objective
battery.model = costs.build_pyomo_costing(
    charge_dict,
    consumption_data_dict,
    battery.model,
    resolution="15m",
    consumption_estimate=consumption_estimate,
    desired_utility="electric",
    additional_objective_terms=[cost_of_carbon * battery.model.emissions]
)

Minimize the costs and emissions of this consumer, given the system constraints, base load consumption, and carbon cost

# use the glpk solver to solve the model - (any pyomo-supported LP solver will work here)
solver = pyo.SolverFactory("glpk")
results = solver.solve(battery.model, tee=False) # turn tee=True to see solver output

Display the results to validate that the optimization is correct

Always compute the ex-post cost using numpy due to the convex relaxations that we apply in our optimization code:

# retrieve outputs from Pyomo model
net_load = np.array([battery.model.net_facility_load[t].value for t in battery.model.t])
baseload = np.array([battery.model.baseload[t] for t in battery.model.t])

# NOTE: second entry of the tuple can be ignored since it's for Pyomo
baseline_electricity_emissions, _ = emissions.calculate_grid_emissions(
    carbon_intensity,
    baseload,
    resolution="15m",
    consumption_units=u.kW
)
optimized_electricity_emissions = pyo.value(battery.model.emissions)

# NOTE: second entry of the tuple can be ignored since it's for Pyomo
baseline_electricity_cost, _ = costs.calculate_cost(
    charge_dict,
    {"electric": baseload},
    resolution="15m",
    desired_utility="electric",
)
# NOTE: second entry of the tuple can be ignored since it's for Pyomo
optimized_electricity_cost, _ = costs.calculate_cost(
    charge_dict,
    {"electric": net_load},
    resolution="15m",
    desired_utility="electric",
)

total_baseline_cost = baseline_electricity_cost + cost_of_carbon * baseline_electricity_emissions.magnitude: total_optimized_cost = optimized_electricity_cost + cost_of_carbon * optimized_electricity_emissions

If we print our results, we confirm that the optimal electricity profile has monthly emissions of 276,142.32 kg CO:sub:2-eq, 111.82 kg CO:sub:2-eq less than the baseline emissions of 276,254.14 kg CO:sub:2-eq. It simultaneously lowers monthly costs by $2,901.69 from $11,6269.08 to $11,3367.39.

>>>print(f"Baseline Scope 2 Emissions: {baseline_electricity_emissions:.2f} kg CO_2-eq")
Baseline Scope 2 Emissions: 276254.14 kilogram kg CO_2-eq
>>>print(f"Optimized Scope 2 Emissions: {optimized_electricity_emissions:.2f} kg CO_2-eq")
Optimized Scope 2 Emissions: 276142.32 kilogram kg CO_2-eq
>>>print(f"Baseline Electricity Costs: ${baseline_electricity_cost:.2f}")
Baseline Electricity Costs: $116269.08
>>>print(f"Optimized Electricity Costs: ${optimized_electricity_cost:.2f}")
Optimized Electricity Costs: $113367.39
>>>print(f"Total Baseline Cost w/ $192/ton CO2-eq: ${total_baseline_cost:.2f}")
Total Baseline Cost w/ $192/ton CO2-eq: $169309.88
>>>print(f"Total Optimized Cost w/ $192/ton CO2-eq: ${total_optimized_cost:.2f}")
Total Optimized Cost w/ $192/ton CO2-eq: $166386.71

We could make similar plots to Optimize Electricity Costs and Optimize Scope 2 Emissions, but we omitted them from these instructions for the sake of space.

Co-optimize Costs and Emissions

CVXPY

Pyomo

Co-optimize Costs and Emissions 

CVXPY 

Pyomo 