Get Python 3¶

Friendly Sam is developed in Python 3 (at the time of this writing, Python 3.4). Download and install it now, if you haven’t already.

Use a virtual environment¶

It is highly recommended that you use a virtual environment. It’s not strictly necessary, but if you choose not to, there is a risk that you will have conflicts between different versions of the packages that Friendly Sam and other Python packages depend on. Google for python virtualenv if you want to learn more. If not, you can also do it the way I do, using vex.

• If you are on Windows

  1. Open a command prompt.
  2. Make sure you have the latest setuptools by running pip install setuptools --upgrade
  3. Install vex by running pip install --user vex
  4. Create a virtual environment named my_project_name and enter it by running vex -m --python C:\Python34\python.exe my_project_name cmd

  Now, whenever you want to use your virtual environment, open a command prompt and run vex my_project_name cmd.

• If you are on Linux

  Basically, you follow the instructions for Windows above but exchange C:\Python34\python.exe for something more suitable, and then do vex my_project_name bash instead. Also see the docs for vex if you have problems.

Install Friendly Sam¶

Assuming you have entered/activated your Python virtual environment, or wherever you want to install it, open a command prompt/shell and run the command:

pip install friendlysam

pip install friendlysam[pandas]
pip install friendlysam[pickling]
pip install friendlysam[pandas,pickling]

Install in developer mode¶

If you are developing the source code of Friendly Sam, you probably want to install it in “develop” mode instead. This has two benefits. First, you get some extra dependencies such as nose (testing package), sphinx (documentation package) and twine and wheel (used for releasing), etc. Second, you won’t have to reinstall the package into your Python site-packages directory every time you change something.

1. Get Python 3. (Note: If you are on Windows it might be convenient to use a ready-made distribution like WinPython and skip step 5 below, but we can’t guarantee it will work.)

2. Download the source code

   • Alternative 1: Download a zip file: https://github.com/sp-etx/friendlysam/archive/master.zip

   • Alternative 2: If you know git, clone into the repository:

     git clone https://github.com/sp-etx/friendlysam.git
     

3. You probably want to install Friendly Sam in a virtual environment. Create one and activate it before you take the next step.

4. Now, to install Friendly Sam in develop mode, do this:

   pip install -r develop.txt
   

Make Sphinx documentation¶

The documentation for residues is made with Sphinx and hosted with Read the Docs. To parse nice, human-readable docstrings, we use Napoleon.

• If you want to make a very minor change to the documentation, you can actually just edit the source, push to the github repository and magically, the docs will update at readthedocs.org.

• However, if you want to edit the docs a lot, you probably want to make test builds on your own machine. In that case, you need to learn about Sphinx. To build the docs, open a command prompt, go to friendlysam\docs and run the command:

  make html
  

The resulting HTML can be previewed under friendlysam\docs\_build\index.html.

Run tests¶

Please run the tests before pushing to the master branch.

To run all the tests, including doctests in the source code and doctests in this documentation, go to the project root directory and run:

nosetests --with-doctest --doctest-options=+ELLIPSIS

Releasing Friendly Sam¶

Why build another tool?¶

There are a lot of different tools for optimization-based modeling. Why in the world do we need another one?

The short answer is this: Friendly Sam is a domain specific toolbox. For the type of models we work with, the model code is shorter, more readable and easier to debug than it would be with many other tools. Furthermore, Friendly Sam makes data handling and analysis easier. Because Friendly Sam is implemented in Python, we get access to all our favorite Python tools for scientific computing and visualization, including Pandas, NumPy, SciPy, matplotlib, etc. This is a strong advantage because the majority of our modeling work is preparing input data and analyzing results.

In the coming paragraphs we’ll explain more about what Friendly Sam is. And at end we’ll also say a few things Friendly Sam is not.

Data handling is easier with Python¶

Friendly Sam was designed to simplify our work with optimization-based models of energy systems, so-called dispatch models. This is a common type of model in research and in applied analysis of energy systems, based on the thought that the operator(s) of an energy system always act so as to minimize the cost of delivering energy to customers, or maybe (in a parallel universe) to minimize the carbon emissions, or some other objective function. A dispatch model is usually formulated as a minimization problem: “Minimize the operation cost of this system in this time period, subject to the technical and legal constraints of the system.”

There are a zillion different variants of such models, but many of them have in common that there is a lot of data going in and out. Some examples of possible input data are prices for different forms of energy, demand profiles, technical constraints, etc. The output data could be operation decisions, system costs, greenhouse gas emissions, and many other things. Therefore, a large part of our modeling work is data handling: Reading and wrangling data files, transforming and resampling input and output data, visualizing results, making statistical tests, etc.

Many optimization-based models are implemented using a generic optimization modeling language like GAMS, AMPL, AIMMS or CMPL. These languages can be wonderful to work with when formulating models because they are made specifically for optimization, and they are efficient in transforming your human-readable code into something that can be understood by almost any optimization solver. However, the infrastructure for handling input and output data in GAMS and AMPL is sub-optimal (pun intended). Anyone who implemented a large, complicated model in one of those languages knows it’s not an easy ride to keep track of all the data going in and out, especially not if you want to make a lot of similar runs with different parameter sets. I know several people who wrote their own tools for getting inputs and outputs back and forth between GAMS and their favorite data crunching tool (Excel, Python, MATLAB, R, etc).

When we started writing what would later become Friendly Sam, we chose Python because of the great ecosystem of open source tools that come with it. We have paid specific attention to numpy, pandas, and matplotlib when developing Friendly Sam. It’s not necessary to use these tools with Friendly Sam, but there is a great chance they will make your life easier. What about optimization then? To formulate and solve the actual optimization problems, we first used the Python API of the Gurobi optimizer. Gurobi’s Python API exposes a Variable class with overloaded operators for addition, multiplication, etc, so you can make algebraic expressions for the optimization objective and all the constraints in Python code. The Gurobi backend then translates these expression objects into a well-formed optimization problem, solves the problem and delivers the solution back through the Python API so you never have to leave Python. In Friendly Sam 1.0 we have created an abstraction layer to reduce the dependence on a certain solver backend. We are now using PuLP to interact with the Gurobi and CBC solvers, but you never have to interact directly with the backend, and it is not too hard to switch to another backend if we want to.

Domain specific toolbox¶

Friendly Sam is a Python library for formulating, running, and analyzing optimization-based models of energy systems.

In fact, it’s not only suitable for modeling energy systems, but also for other systems where you want to optimize flow networks of physical or abstract quantities, be it energy carriers, money, solid waste, cargo deliveries, virtual water or something else.

In principle, you are not even restricted to modeling systems with flow networks, because the optimization engine behind Friendly Sam is exposed so you can formulate a large class of optimization problems. But if you want a generic tool for formulating optimization problems you should probably check out other tools instead. In Python it’s worth to look at CyLP, cvxpy, PuLP, and Pyomo. If you want a pure optimization language, look at GAMS, AMPL, AIMMS or CMPL.

So although Friendly Sam can be used as a rather generic optimization modeling tool, it is domain specific in the sense that it has vocabulary for energy systems and similar systems. We developed it specifically to help us formulate dispatch models. In our energy system models, there are almost always balance equations for energy or materials, so Friendly Sam contains definitions of things like FlowNetwork, Node and Cluster to simplify the formulation of such constraints. And the Node class is a perfect starting point for modeling things like power plants, energy storages, and other things you typically find in an energy system. Friendly Sam also has a simple formulation of a myopic dispatch model of the type we often encounter in the academic literature on energy system modeling. If you use these building blocks, you will have to think less about sign errors in balance equations and instead concentrate on what your model really means.

Friendly Sam code is meant to be readable. For example, in a district heating model we can have instances of Node subclasses, one named LinearCHP, another named HeatPump, etc. This makes perfect sense to us, because the code is naturally structured similar to how we think about the energy system we are modeling. When the underlying optimization problem is solved, we can query the state of the model objects with code like heat_pump.consumption['power'](time).

The code can also be easier to debug. When you have a bewildering error somewhere, it can be helpful to just eyeball the constraints of your optimization problem, to see if you can spot the error. Friendly Sam makes this easier by automatically naming constraints after their “owner”, for example the HeatPump instance we just mentioned. You can also name variables and add descriptions to constraints. These features help you understand where things come from when you are looking at a long list of constraints.

What Friendly Sam is not¶

First, we want to clarify that Friendly Sam is not “a model”. It is a toolbox we use to build models.

Second, Friendly Sam is not fool proof. It is entirely possible to make models that are stupid or wrong with Friendly Sam. We have tried to design Friendly Sam to produce readable, understandable, debuggable models, and to make idioms and conventions that help to avoid common errors. But having this tool is not an alternative to knowing and understanding the optimization problems you are creating. Friendly Sam is a tool to help us focus on what is important, rather than chasing indexing errors and how to formulate piecewise affine functions using special ordered sets.

Third, Friendly Sam is not primarily optimized for speed. If you want to solve a really big model fast, you are probably better off with something like AMPL or GAMS, or maybe writing your own code in a compiled language. However, if your model is moderately big you might get the job done faster with Friendly Sam because debugging, data handling, analysis and visualization will be so much faster. In our experience, the development phase often consumes more time and money than the computation phase, so development convenience is often more important than execution speed.

OK, let’s get started!¶

You are now ready to read about Variables and expressions.

Names don’t mean anything¶

In the example above, we named the Variable object 'x'. This is nothing more than a string attached to the object, and it does not say anything about the identity of the variable. In principle you can have several Variable objects with the same name, but that’s really confusing and should not be necessary.

>>> my_var = Variable('y')
>>> my_other_var = Variable('y')
>>> my_var == my_other_var
False
>>> print(my_var + my_other_var)
y + y

It is often a good idea to give your variables names you can recognize, because that simplifies debugging when you want to inspect the expressions you have made with the variables. But if you don’t want to name variables you don’t have to. The variables are then named automatically.

>>> Variable()
<friendlysam.opt.Variable at 0x...: x1>
>>> Variable()
<friendlysam.opt.Variable at 0x...: x2>

VariableCollection is like an indexed Variable¶

There is also a convenient class called VariableCollection. It is a sort of lazy dictionary, which creates variables when you ask for them:

>>> from friendlysam import VariableCollection
>>> z = VariableCollection('z')
>>> z
<friendlysam.opt.VariableCollection at 0x...: z>
>>> z(1)
<friendlysam.opt.Variable at 0x...: z(1)>
>>> z((1, 'a'))
<friendlysam.opt.Variable at 0x...: z((1, 'a'))>
>>> z(None)
<friendlysam.opt.Variable at 0x...: z(None)>

You can think of VariableCollection as an indexed variable, but all it really does is to create variables when you call it, and then remember them.

The index must be hashable. For example, tuples are valid indices, but not lists:

>>> z((3, 1, 4))
<friendlysam.opt.Variable at 0x...: z((3, 1, 4))>
>>> z([3, 1, 4])
Traceback (most recent call last):
...
TypeError: unhashable type: 'list'

Variables can be named in a namespace, like this:

>>> from friendlysam import namespace
>>> with namespace('cheese'):
...     cheese1 = Variable('gorgonzola')
...     cheese2 = VariableCollection('ricotta')
...
>>> cheese1
<friendlysam.opt.Variable at 0x...: cheese.gorgonzola>
>>> cheese2
<friendlysam.opt.VariableCollection at 0x...: cheese.ricotta>

The namespace doesn’t affect the function of a variable in any way. It only prepends a string representation of whatever object to the variable name, so you can also do things like this:

>>> with namespace(dict()):
...     Variable('x')
...
<friendlysam.opt.Variable at 0x...: {}.x>

Variables can have values¶

You can assign a value to a variable. The variable will still work in expressions:

>>> x = Variable('x')
>>> x.value = 39
>>> expression = x + 3
>>> expression
<friendlysam.opt.Add at 0x...>
>>> print(expression)
x + 3

The difference is that you can now evaluate expressions. But note that the expression object is unchanged.

>>> float(expression)
42.0
>>> print(expression)
x + 3

You can change or delete the value:

>>> x.value = 0.5
>>> int(expression)
3
>>> float(expression)
3.5
>>> expression.value
3.5
>>> del x.value
>>> float(expression)
Traceback (most recent call last):
...
friendlysam.opt.NoValueError: cannot get a numeric value: x + 3 evaluates to x + 3

And it works for relations, too:

>>> x.value = 10
>>> (x <= 12).value
True

Expressions are immutable¶

Expressions are hashed by structure: If they do the same thing, they hash and compare equal. This also means they are considered equal e.g. as dict keys.

>>> expr1 = x * 2
>>> expr2 = x * 2
>>> expr1 is expr2 # Different objects!
False
>>> expr1 == expr2 # But similar
True
>>> d = dict()
>>> d[expr1] = 'some value'
>>> d[expr2]
'some value'

Expressions are immutable, meaning that their state can never be changed. In the example above, expr1 == expr2 and that will always be true. Two expressions are interchangeable if (and only if) they compare equal. For any purpose, in any situation, expr1 will always do the same thing as expr2.

However, as you saw above, the result of float(expr1) may vary depending on whether variables in the expression have values. Let’s look a little bit closer:

>>> x.value = 3
>>> expression = x + 39
>>> float(expression)
42.0
>>> x.value = 100
>>> another_expression = x + 39
>>> expression == another_expression
True
>>> float(expression)
139.0
>>> float(another_expression)
139.0

This is pretty much analogous to a tuple of mutable objects. The tuple itself may never change, but its contents may:

>>> a = [1, 2, 3]
>>> my_tuple = (a, 'something')
>>> my_tuple
([1, 2, 3], 'something')
>>> a[:] = ['changed'] # Only changing the contents of the list
>>> another_tuple = (a, 'something')
>>> my_tuple == another_tuple
True
>>> my_tuple
(['changed'], 'something')
>>> another_tuple
(['changed'], 'something')

Behind value is evaluate()¶

You might want to know what is happening behind the scenes when you ask for expression.value or float(expression). In that case, check out the method evaluate().

Creating a problem¶

We use Friendly Sam to formulate MILP problems. The optimization library could be extended to allow other types of problems, too, but this is what is supported today.

Now, let’s begin with a full example of an optimization problem.

>>> import friendlysam as fs
>>>
>>> # Create the problem
>>> x = fs.VariableCollection('x')
>>> prob = fs.Problem()
>>> prob.objective = fs.Maximize(x(1) + x(2))
>>> prob.add(8 * x(1) + 4 * x(2) <= 11)
>>> prob.add(2 * x(1) + 4 * x(2) <= 5)
>>>
>>> # Get a solver and solve the problem
>>> solver = fs.get_solver()
>>> solution = solver.solve(prob)
>>> type(solution)
<class 'dict'>
>>> solution[x(1)]
1.0
>>> solution[x(2)]
0.75

The solver does not in any way affect the problem or the variables. It just reads the problem, solves it and handles back a dict with your Variable objects as keys and their solutions as values.

If you set the value of some variables, those will be inserted into the problem before solving it:

>>> x(1).value = 0
>>> solution = solver.solve(prob)
>>> solution
{<friendlysam.opt.Variable at 0x...: x(2)>: 1.25}
>>> x(1) in solution
False

x(1) is not in the solution, because you already set its value, so it was handled like a number by the solver.

Debugging constraints¶

Now let’s add another constraint:

>>> x(1).value = 0
>>> prob.add(1 <= x(1))
>>> solver.solve(prob)
Traceback (most recent call last):
...
friendlysam.opt.ConstraintError: The expression in <Constraint: Ad hoc constraint> evaluates to False, so the problem is infeasible.

In this case it’s obvious why the problem could not be solved. But for argument’s sake, let’s say we didn’t know which constraint was causing a problem. The error message was not too helpful, but the ConstraintError luckily also contains a reference to the constraint that failed, so we can pick it out like this:

>>> try:
...     solver.solve(prob)
... except fs.ConstraintError as e:
...     failed_constraint = e.constraint
...     print(repr(failed_constraint))
...     print(repr(failed_constraint.expr))
...     print(failed_constraint.expr)
...     print(failed_constraint.desc)
...     print(failed_constraint.origin)
...
<friendlysam.opt.Constraint at 0x...>
<friendlysam.opt.LessEqual at 0x...>
1 <= x(1)
Ad hoc constraint
None

OK, that’s helpful! We got the problematic constraint out. And there are a few things you should note.

1. The type of the failed constraint is friendlysam.opt.Constraint. It was automatically created when we added a friendlysam.opt.LessEqual constraint to the problem, and its sole purpose is to wrap the inequality 1 <= x(1) and to add some metadata.
2. The Constraint object contains the LessEqual object that we added to the problem.
3. The Constraint object contains also a description desc and a variable called origin which is supposed to say something about where the constraint comes from.

>>> print(failed_constraint.long_description)
<friendlysam.opt.Constraint at 0x...>
Description: Ad hoc constraint
Origin: None

If you want to make your model easier to debug, you can use Constraint instances with custom description and/or origin, like in this stupid example:

>>> from friendlysam import Constraint
>>> def constr(var, parameter):
...     return var / 42 >= parameter
>>> for i in range(5):
...     expr = constr(x(i), i)
...     origin = (constr, x(i), i)
...     prob += Constraint(expr, desc='Some description', origin=origin)
...

Different ways to add constraints¶

>>> prob.add(8 * x(1) + 4 * x(2) <= 11)
>>> prob += Constraint(expr, desc='Some description', origin=origin)

>>> prob += ([constr(x(i), i), constr(x(i+1), i)] for i in range(5))

Special ordered sets¶

Friendly Sam also supports special ordered sets. You specify them as a sort of constraint: Check out SOS1 and SOS2.

Interconnected parts¶

Friendly Sam is made for optimization-based modeling of interconnected parts, producing and consuming various resources. For example, an urban energy system may have grids for district heating, electric power, fossil gas, etc. Consumers, producers, storages and other parts are connected to each other through these grids. To describe these relations, Friendly Sam models make heavy use of the Part class and its subclasses like Node, FlowNetwork, Cluster, Storage, etc. We will introduce these in due time, but first a few general things about Part.

Parts have indexed constraints¶

A Part typically represents something physical, like a heat consumer or a power grid. You can attach constraint functions to parts. Constraint functions are probably easiest to explain with a concrete example:

>>> from friendlysam import Part, namespace, VariableCollection
>>> class ChocolateFactory(Part):
...     def __init__(self):
...         with namespace(self):
...             self.output = VariableCollection('output')
...         self.constraints += self.more_than_last
...
...     def more_than_last(self, time):
...         last = self.step_time(time, -1)
...         return self.output(last) <= self.output(time)
...

OK, what happens above is the following: We define ChocolateFactory as a subclass of Part. Upon setup, in __init__(), we add a constraint function called more_than_last, which defines the (admittedly bizarre) rule that the factory may never decrease its output from one time step to the next.

In Friendly Sam’s vocabulary, the time argument in the example above is called an index. Our typical use case for indexing is a discrete time model, where each hour, day, year, or whatever time period, is an index of the model, and each constraint “belongs” to a time step just like in the silly example above.

Going back to the example, we can get the constraints out by making a ChocolateFactory instance and calling constraints.make() with an index:

>>> chocolate_factory = ChocolateFactory() # Create an instance
>>> constraints = chocolate_factory.constraints.make(47)
>>> constraints
{<friendlysam.opt.Constraint at 0x...>}
>>> for c in constraints:
...     print(c.expr)
...
ChocolateFactory0001.output(46) <= ChocolateFactory0001.output(47)

The result of constraints.make(47) is a set with one single constraint in it, saying that output at “time” 47 must be greater than or equal to output at “time” 46.

>>> from pandas import Timestamp, Timedelta
>>> chocolate_factory.time_unit = Timedelta('6h')
>>> constraints = chocolate_factory.constraints.make(Timestamp('2015-06-10 18:00'))
>>> for c in constraints:
...     print(c.expr)
...
ChocolateFactory0001.output(2015-06-10 12:00:00) <= ChocolateFactory0001.output(2015-06-10 18:00:00)

Advanced indexing¶

Indexing is really just a way to organize constraints in groups that belong together. When we have a whole bunch of parts, to get all the constraints that belong together, we can do things like this:

>>> from itertools import chain
>>> parts = Part(), Part(), Part() # Put something more useful here...
>>> some_index = 'could be anything'
>>> constraints = set.union(*(p.constraints.make(some_index) for p in parts))

Indexing is typically used to represent time, but it is really up to you to decide what an index means, and what to use as indices. We call it “index” rather than “time” because it is something more general than a representation of time. In fact, any hashable object can be used as an index, so you can do all sorts of complicated things. Examples of indexing can be found in the docs for step_time(). Also check out times() and times_between().

Friendly Sam currently has no mechanism for using constraint functions without indices. If you want to make a static model and really don’t need indexing, then just use some common index like None or 0 for everything. (Or come up with a better solution and discuss it with us on GitHub.)

In the next section Flow networks: Nodes and resources you will also see how indexing is used to represent time in flow networks.

Nodes and balance constraints¶

>>> from friendlysam import Node, VariableCollection, namespace
>>> class PowerPlant(Node):
...     def __init__(self):
...         with namespace(self):
...             x = VariableCollection('output')
...         self.production['power'] = x
...
>>> class Consumer(Node):
...     def __init__(self, demand):
...         self.consumption['power'] = lambda time: demand[time]
...

>>> power_plant = PowerPlant()
>>> power_plant.production['power'](3)
<friendlysam.opt.Variable at 0x...: PowerPlant0001.output(3)>

>>> power_demand = [25, 30, 33, 29, 27]
>>> consumer = Consumer(power_demand)
>>> consumer.consumption['power'](3)
29

>>> from friendlysam import FlowNetwork
>>> power_grid = FlowNetwork('power', name='Power grid')
>>> power_grid.connect(power_plant, consumer)
>>> power_grid.children == {power_plant, consumer}
True

>>> for part in [consumer, power_plant, power_grid]:
...     for constraint in part.constraints.make(3):
...         print(constraint.long_description)
...         print(constraint.expr)
...         print()
...
<friendlysam.opt.Constraint at 0x...>
Description: Balance constraint (resource=power)
Origin: CallTo(func=<bound method Consumer.balance_constraints of <Consumer at 0x...: Consumer0001>>, index=3, owner=<Consumer at 0x...: Consumer0001>)
Power grid.flow(PowerPlant0001-->Consumer0001)(3) == 29

<friendlysam.opt.Constraint at 0x...>
Description: Balance constraint (resource=power)
Origin: CallTo(func=<bound method PowerPlant.balance_constraints of ...>, index=3, owner=<PowerPlant at 0x...: PowerPlant0001>)
PowerPlant0001.output(3) == Power grid.flow(PowerPlant0001-->Consumer0001)(3)

>>> class CHPPlant(Node):
...     def __init__(self, name=None):
...         if name:
...             self.name = name
...         ...
>>> chp_plant = CHPPlant(name='Rya KVV')
>>> chp_plant.name == str(chp_plant) == 'Rya KVV'
True

FlowNetwork¶

A FlowNetwork essentially does two things: It creates the variable collections representing flows in the network, and it modifies the inflows and outflows of nodes when you call connect().

>>> power_grid.connect(power_plant, consumer, bidirectional=True)

>>> flow = power_grid.get_flow(power_plant, consumer)
>>> flow
<friendlysam.opt.VariableCollection at 0x...: Power grid.flow(PowerPlant0001-->Consumer0001)>
>>> flow.ub = 40

Clusters and multi-area models¶

>>> from friendlysam import Cluster
>>> power_plant = PowerPlant()
>>> consumer = Consumer(power_demand)
>>> power_cluster = Cluster(power_plant, consumer, resource='power', name='Power cluster')
>>> for part in power_cluster.descendants_and_self:
...     for constraint in part.constraints.make(2):
...         print(constraint.long_description)
...         print(constraint.expr)
...
<friendlysam.opt.Constraint at 0x...>
Description: Balance constraint (resource=power)
Origin: CallTo(func=<bound method Cluster.balance_constraints ...>, index=2, owner=<Cluster at 0x...: Power cluster>)
PowerPlant0002.output(2) == 33

area_A = Cluster(*nodes_in_area_A, resource='heat')
area_B = Cluster(*nodes_in_area_B, resource='heat')
area_C = Cluster(*nodes_in_area_C, resource='heat')

heat_grid = FlowNetwork('heat')
heat_grid.connect(area_A, area_B, bidirectional=True, capacity=ab)
heat_grid.connect(area_A, area_C, bidirectional=True, capacity=ac)
heat_grid.connect(area_B, area_C, bidirectional=True, capacity=bc)

Time in flow networks¶

It is natural to think of indices like time periods: All the expressions for flows, production and consumption must add up, for each index (time period). As shown in the examples above, the balance constraints for an index is called by passing the index to production, consumption, outflows and inflows.

There is another dictionary which is always used in balance constraints: accumulation. It works just like the dictionaries production and consumption. To learn more, read the API docs for Storage, and look at this example:

>>> from friendlysam import Storage
>>> from pandas import Timestamp, Timedelta
>>> battery = Storage('power', name='Battery')
>>> battery.time_unit = Timedelta('3h')
>>> t = Timestamp('2015-06-10 18:00')
>>> print(battery.accumulation['power'](t))
Battery.volume(2015-06-10 21:00:00) - Battery.volume(2015-06-10 18:00:00)

Variables¶

Expressions¶

Constraints and optimization¶

Models¶

Utilities¶

Exceptions¶