This library provides a layer above brightway2 for defining parametric models and running super fast LCA for monte carlo analysis.

# Introduction

This notebook presents lca-algebraic a small layer above brightay2, designed for the definition of parametric inventories with fast computation of LCA impacts, suitable for monte-carlo analyis.

lca-algebraic provides a set of helper functions for :

• compact & human readable definition of activites :
• search background (tech and biosphere) activities
• create new foreground activites with parametrized amounts
• parametrize / update existing background activities (extending the class Activity)
• definition

# Principles

The main idea of this libray is to move from procedural definition of models (slow and prone to errors) to a declarative / purely functionnal definition of parametric models (models as pure functions).

This enables fast computation of LCA impacts. We leverage the power of symbolic calculus provided by the great libary SymPy.

We define our model in a separate DB, as a nested combination of :

• other foreground activities
• background activities :
• Technical, refering ecoinvent DB
• Biopshere, refering brightway2 biosphere activities

The amounts in exchanges are expressed either as static amounts, or symbolic expressions of pre-defined parameters.

Each activity of our root model is defined as a parametrized combination of the foreground activities, which can themselves be expressed by the background activities.

When computing LCA for foreground models, the library develops the model as a combination of only background activities. It computes once for all the impact of background activities and compiles a fast numpy (vectorial) function for each impact, replacing each background activity by the static value of the corresponding impact.

By providing large vectors of parameter values to those numpy functions, we can compute LCA for thousands of values at a time.

# Compatiblity with brightway2

Under the hood, the activities we define with lca-algebraic are standard brightway2 activities. The amounts of exchanges are stored as float values or serialized as string in the property formula.

Parameters are also stored in the brightay2 projets, making it fully compatible with brightway.

Thus, a model defined with lca-algebraic is stored as a regular bw2 projet. We can use bw2 native support for parametrized dataset for computing LCAs, even if much more slower than the method explain here.

# Doc

The followng notebook explores the main functions. Full doc is available here

# Install extra libraries

We need to install 3 extra libraries beforehand.
Type the following commands into your anaconda shell prompt.

Sympy

conda install -c anaconda sympy


SALib

conda install -c conda-forge salib


slugify

conda install python-slugify


# Imports

Besides external libraries, we import several function from the local file : lca-algebraic.py

# %load_ext autoreload
import pandas as pd
import time
import matplotlib.pyplot as plt
import numpy as np
import brightway2 as bw

# Custom utils defined for inter-acv
from lca_algebraic import *
import lca_algebraic


# Init brightway2 and databases

# Import Ecoinvent DB (if not already done)
# Update the PATH to suit your installation
# importDb(ECOINVENT_DB_NAME, './ecoinvent 3.4_cutoff_ecoSpold02/datasets')

# Setup bw2 and detect version of ecoinvent
initDb('bw2_seminar_2017')

# We use a separate DB for defining our foreground model / activities
USER_DB = 'Geothermal database'

# This is better to cleanup the whole foreground model each time,
# instead of relying on a state or previous run of model.
# Any persistent state is prone to errors.
resetDb(USER_DB)

# Parameters are stored at project level : reset them also
resetParams(USER_DB)


# Introduction to Numpy

Numpy is a python libray for symbolic calculus.

You write Sympy expression as you write standard python expressions, using sympy symbols in them.

The result is then a symbolic expression that can be manipulated, instead of a numeric value.

from sympy import symbols

# create sympy symbol
a = symbols("a")

# Expression are not directly evaluated
f = a * 2 + 4
f

# symbols can be replaced by values afterwards
f.subs(dict(a=3))


In practice, you don't need to care about Sympy. Just remember that :

• The parameters defined below are instances of sympy symbols
• Any valid python expression containing a sympy symbol will create a sympy symbolic expression

# Define parameters

We define the parameters of the model.

The numeric parameters are instances of sympy 'Symbol'.

Thus, any python arithmetic expression composed of parameters will result in a symbolic expression for later evaluation, instead of a numeric result.

# Example of 'float' parameters

a = newFloatParam(
'a',
default=0.5, min=0, max=2,  distrib=DistributionType.TRIANGLE, # Distribution type, linear by default
description="hello world")

b = newFloatParam(
'b',
default=0.5, # Fixed if no min /max provided
description="foo bar")

share_recycled_aluminium = newFloatParam(
'share_recycled_aluminium',
default=0.6, min=0, max=1, std=0.2, distrib=DistributionType.NORMAL, # Normal distrib, with std dev
description="Share of reycled aluminium")

# Yuo can define boolean parameters, taking only discrete values 0 or 1
bool_param = newBoolParam(
'bool_param',
default=1)

# Example 'enum' parameter, acting like a switch between several possibilities
# Enum parameters are not Symbol themselves
# They are a facility to represent many boolean parameters at once '<paramName>_<enumValue>'
# and should be used with the 'newSwitchAct' method
elec_switch_param = newEnumParam(
'elec_switch_param',
values=["us", "eu"],
default="us",
description="Switch on electricty mix")

# Another example enum param
techno_param = newEnumParam(
'techno_param',
values=["technoA", "technoB", "technoC"],
default="technoA",
description="Choice of techonoly")


# Get references to background activities

utils provide two functions for easy and fast (indexed) search of activities in reference databases :

• findBioAct : Search activity in biosphere3 db
• findTechAct : Search activity in ecoinvent db

Those methods are faster and safer than using traditionnal "list-comprehension" search : They will fail with an error if more than one activity matches, preventing the model to be based on a random selection of one activity.

# Biosphere activities
ground_occupuation = findBioAct('Occupation, industrial area')
heat = findBioAct('Heat, waste', categories=['air'])

# Technosphere activities

# You can add an optionnal location selector
alu = findTechAct("aluminium alloy production, AlMg3", loc="RER")
alu_scrap = findTechAct('aluminium scrap, new, Recycled Content cut-off')

# Elec
eu_elec = findTechAct("market group for electricity, medium voltage", 'ENTSO-E')
us_elec = findTechAct("market group for electricity, medium voltage", 'US')


# Define the model

The model is defined as a nested combination of background activities with amounts.

Amounts are defined either as constant float values or algebric formulas implying the parameters defined above.

## Create new activities

# Create a new activity
activity1 = newActivity(USER_DB, # We define foreground activities in our own DB
"first foreground activity", # Name of the activity
"kg", # Unit
exchanges= { # We define exhanges as a dictionarry of 'activity : amount'
ground_occupuation:3 * b, # Amount can be a fixed value
heat: b + 0.2  # Amount can be a Sympy expression (any arithmetic expression of Parameters)
})

# You can create a virtual "switch" activity combining several activities with a switch parameter
elec_mix = newSwitchAct(USER_DB,
"elect mix", # Name
elec_switch_param, # Sith parameter
{ # Dictionnary of enum values / activities
"us" : us_elec,
"eu" : eu_elec})


## Copy and update existing activity

You can copy and update an existing background activity.

Several new helper methods have been added to the class Activity for easy update of exchanges.

alu2 = copyActivity(
USER_DB, # The copy of a background activity is done in our own DB, so that we can safely update it
alu, # Initial activity : won't be altered
"Alu2") # New name

# Update exchanges by their name
alu2.updateExchanges({

# Update amount : the special symbol *old_amount* references the previous amount of this exchange
"aluminium, cast alloy": old_amount * (1 - share_recycled_aluminium),

# Update input activity. Note also that you can use '*' wildcard in exchange name
"electricity*": elec_mix,

# Update both input activity and amount.
# Note that you can use '#' for specifying the location of exchange (useful for duplicate exchange names)
"chromium#GLO" : dict(amount=4.0, input=activity1)
})



## Final model

The final model is just the root foreground activity referencing the others

model = newActivity(USER_DB, "model", "kg", {
activity1 : b * 5 + a + 1, # Reference the activity we just created
alu2: 3 * share_recycled_aluminium,
alu:0.4 * a})


## Display activities

printAct displays the list of all exchanges of an activity.

Note that symbolic expressions have not been evaluated at this stage

# Print_act displays activities as "pandas" tables
printAct(activity1)
printAct(model)

# You can also compute amounts by replacing parameters with a float value
printAct(activity1, b=1.5)

# You print several activities at once to compare them
printAct(alu, alu2)


# Select the impacts to consider

# List of impacts to consider
impacts = [m for m in bw.methods if 'ILCD 1.0.8 2016' in str(m) and 'no LT' in str(m)]


# Compute LCA

We provide two methods for computing LCA :

• multiLCA : It uses brightway2 native parametric support. It is much slower and kept for comparing results.
• multiLCAAlgebric : It computes an algebric expression of the model and computes LCA once for all the background activities. Then it express each impact as a function of the parameters. This expression is then compiled into 'numpy' native code, for fast computation on vectors of samples. This version is 1 million time faster.
# Uses brightway2 parameters
multiLCA(
model,
impacts,

# Parameters of the model
a=1,
b=2,
elec_switch_param="us",
share_recycled_aluminium=0.4)

# Compute with algebric implementation : the values should be the same
multiLCAAlgebric(
model, # The model
impacts, # Impacts

# Parameters of the model
a=1,
b=2,
elec_switch_param="us",
share_recycled_aluminium=0.4)

# Here is what the symbolic model looks like :
expr, _ = actToExpression(model)
expr

# You can compute several LCAs at a time :
multiLCAAlgebric(
[alu, alu2], # The models

impacts, # Impacts

# Parameters of the model
share_recycled_aluminium=0.3,
elec_switch_param="us",
b=4)

# Fast computation for millions of separate samples
multiLCAAlgebric(
model, # The model
impacts, # Impacts

# Parameters of the model
a=list(range(1, 1000000)), # All lists should have the same size
b=list(range(1, 1000000)),
share_recycled_aluminium=1,
elec_switch_param="eu")


# Statistic functions

## One at a time

We provide several functions for computing statistics for local variations of parameters (one at a time).

### oat_matrix(model, impacts)

Shows a matrix of impacts x parameters colored according to the variation of the impact in the bounds of the parameter.

oat_matrix(model, impacts)


### oat_dashboard_matrix

This functions draws a dashboard showing :

• A dropdown list, for choosing a parameter
• Several graphs of evolution of impacts for this parameter
• Full table of data
• A graph of "bars" representing the variation of each impact for this parameter (similar to the information given in oat_matrix)
oat_dashboard_interact(model, impacts)


## Monte-carlo methods & Sobol indices

Here we leverage fast computation of monte-carlo approches.

We compute global sensivity analysis (GSA). Not only local ones.

### Sobol Matrix

Similar to OAT matrix, we compute Sobol indices. they represent the ratio between the variance due to a given parameter and the total variance.

for easier comparison, we translate those relative sobol indices into "deviation / mean" importance :

relative_deviation = sqrt(sobol(param) * total_variance(impact)) / mean(impact)

# Show sobol indices
incer_stochastic_matrix(model, impacts)


### Violin graphs

We provide a dashboard showing violin graphs : the exact probabilistic distribution for each impact. Together with medians of the impacts.

incer_stochastic_violin(model, impacts)


### Full dashboard

A dashboard groups all this information in a single interface with tabs.

It also shows total variation of impacts. This last graph could be improved by showing stacked colored bars with the contribution of each parameter to this variation, according to Sobol indices.

incer_stochastic_dasboard(model, impacts)


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