Piecewise linearisation¶

If you have non-linear relationships between two decision variables, you may want to represent them as a piecewise linear function. Common non-linear relationships found in energy systems include:

marginal cost curves: Where the marginal cost of generating a unit of energy changes as the quantity of energy generated changes.
capital cost curves: Where the capital cost of investing in the next unit of nominal capacity changes as the quantity nominal capacity changes.
part-load efficiency curves: Where the efficiency of energy generation / conversion changes as the load rate (\(\frac{\text{generation}}{\text{nominal capacity}}\)) changes.

Defining piecewise data¶

A limited set of component attributes can be defined using piecewise curves:

component	x	y	aux_variable	allow_extendable
Generator	p_pu	marginal_cost	marginal_cost_piecewise	False
Generator	p_nom	capital_cost	capital_cost_piecewise	True
Generator	p_pu	efficiency	p_primary_piecewise	False
Line	s_nom	capital_cost	capital_cost_piecewise	True
Link	p_pu	marginal_cost	marginal_cost_piecewise	False
Link	p_nom	capital_cost	capital_cost_piecewise	True
Link	p_pu	efficiency	p{port}_piecewise	False
Process	p_pu	marginal_cost	marginal_cost_piecewise	False
Process	p_nom	capital_cost	capital_cost_piecewise	True
Process	p_pu	rate	p{port}_piecewise	False
StorageUnit	p_dispatch_pu	marginal_cost	marginal_cost_piecewise	False
StorageUnit	p_nom	capital_cost	capital_cost_piecewise	True
Store	p_pu	marginal_cost	marginal_cost_piecewise	False
Store	e_nom	capital_cost	capital_cost_piecewise	True
Transformer	s_nom	capital_cost	capital_cost_piecewise	True

For each, they can be defined when using pypsa.Network.add by either providing an appropriately formatted dictionary or pandas.DataFrame. If a dictionary, keys should be x-values and values should be y-values at each of the breakpoints of the curve being described. If a DataFrame, the index should be breakpoint number, the columns should be the x- and y-axis names (e.g. p_pu and marginal_cost), and the values should be the x-/y-axis values at each breakpoint.

What is a breakpoint?

A breakpoint is a point along the piecewise curve where two straight lines of different gradients meet. We define breakpoints in terms of integer values (breakpoint 0, 1, 2, etc.).

Inspecting piecewise data¶

Piecewise data is stored in a component piecewise dictionary-like object, much like dynamic data. For instance, to access marginal_cost piecewise data for generators, you can call n.c.generators.piecewise.marginal_cost. The returned pandas.DataFrame will be of the same form as when defining an input attribute as a DataFrame, with an added column level with the names of the components for which you've defined piecewise data.

Piecewise formulations¶

When applying the piecewise constraints to the optimisation problem, each will be formulated by linopy in the format deemed to be "cheapest" for solving the problem. As of writing, this could be in the form of:

Each formulation will create a piecewise auxiliary decision variable of the form <component>-<aux_variable> in the optimisation problem (see the table above for all possible component and aux_variable combinations). For instance, a piecewise capital cost constraint applied to a generator will yield the variable Generator-capital_cost_piecewise, accessible at n.model.variables["Generator-capital_cost_piecewise"].

Depending on the formulation used, linopy will also create additional auxiliary variables and constraints. For instance, the incremental (delta) formulation will lead to the auxiliary variables <component>-<aux_variable>_delta and <component>-<aux_variable>_binary and the constraints:

<component>-<aux_variable>_delta_bound
<component>-<aux_variable>_fill_order
<component>-<aux_variable>_binary_order
<component>-<aux_variable>_link

These additional constraints and auxiliary variables will not find their way into the PyPSA network results or statistics. You will, however, have access to the results capital_cost_piecewise_opt (static) and marginal_cost_piecewise_opt (dynamic) if applying piecewise costs. Piecewise efficiencies will be applied directly to the appropriate dynamic result attribute (e.g. p2 for a Link if the piecewise curve is applied to efficiency2).

You can override the default linopy formulation by configuring your own piecewise options.

Configuring piecewise formulations¶

Our assumptions, and the linopy defaults, can be overridden when calling n.optimize by using the piecewise_options list argument. Each dictionary in the list defines a formulation variation. The options that you can override are defined in pypsa.optimization.piecewise.PiecewiseOptions.

Example

By default, we do not pin piecewise costs to the curve, instead we allow costs to increase above those defined by the curve. With some piecewise curve shapes, this approach allows for a fully linear piecewise formulation. This should be fine in most cases; the optimisation is cost-minimising, so you will not actually get costs that deviate from the piecewise curve.

If you want to pin piecewise costs to the curve, you need to use an equality sign. Assuming we are applying this to marginal costs of generators:

n.optimize(piecewise_options=[{"component": "Generator", "attribute": "marginal_cost", "sign": "=="}])

Cumulative vs direct curves¶

Piecewise curves can be used in two different contexts in energy system modelling. The choice is configurable but you should understand the difference before considering overriding defaults.

Info

The default curve type can be overwritten using the boolean piecewise option cumulative_attr (True = cumulative, False = direct). See pypsa.optimization.piecewise.PiecewiseOptions for more information.

Cumulative curves¶

Here, the y-value at each point on the curve represents an incremental contribution, so the total operating point \(X\) is the integral of the curve up to \(X\). We assume that piecewise marginal operating and capital cost curves are cumulative.

Warning

Curves must start at \(x = 0\), and the y-value there prices a zero-width increment and is ignored.

Example

For an increasing marginal cost curve, the increments are dispatched in cost order. A PyPSA CCGT generator might represent many individual assets, each with different marginal costs. At a 50% load rate, the cheapest half run at full output and the rest stay off, since that is the least-cost operating schedule. The value at \(X\) is then the marginal cost of the generator at 50% load rate, the cost of the last increment. The total operating cost is the integral along the curve up to \(X\).

Direct curves¶

Here, the curve is read at the operating point and no integration takes place. The value at the operating rate \(X\) is the realised value directly, evaluated point by point rather than accumulated from the rates below it.

We assume that part-load efficiency curves are direct. A cost curve can also be direct where its values are meant to be read at the operating point rather than accumulated. In such a case, the total cost is the curve value at \(X\) multiplied by \(𝑋\).

Example

Consider a component with a part-load efficiency curve: at a 50% load rate the asset operates at 50% of its nameplate capacity and the efficiency is the curve value at 50%, regardless of its efficiency at other load rates.