# Optimal Power Flow¶

See the module `pypsa.opf`

and `pypsa.linopf`

. Optimisation with the linearised power flow equations for (mixed) AC
and DC networks is fully supported. Note that optimisation with the full non-linear power flow equations is not yet supported.

All constraints and variables are listed below.

## Overview¶

The linear OPF module can optimise the dispatch of generation and storage and the capacities of generation, storage and transmission infrastructure.

It is assumed that the load is inelastic and must be met in every snapshot (this will be relaxed in future versions).

The optimisation currently uses continuous variables for most functionality; unit commitment with binary variables is also implemented for generators.

The objective function is the total system cost for the snapshots optimised.

Each snapshot can be given a weighting \(w_t\) to represent e.g. multiple hours.

This set-up can also be used for stochastic optimisation, if you interpret the weighting as a probability.

Each transmission asset has a capital cost.

Each generation and storage asset has a capital cost and a marginal cost.

Execute:

```
network.lopf(snapshots, solver_name="glpk", solver_io=None,
extra_functionality=None, solver_options={},
keep_files=False, formulation="angles",
extra_postprocessing=None)
```

where `snapshots`

is an iterable of snapshots, `solver_name`

is a
string, e.g. “gurobi” or “glpk”, `solver_io`

is a string,
`extra_functionality`

is a function of network and snapshots that is
called before the solver (see below), `extra_postprocessing`

is a
function of network, snapshots and duals that is called after solving
(see below), `solver_options`

is a dictionary of flags to pass to
the solver, `keep_files`

means that the `.lp`

file is saved and
`formulation`

is a string in
`["angles","cycles","kirchhoff","ptdf"]`

(see Passive branch flow formulations
for more details).

See `pypsa.Network.lopf()`

for the documentation.

Important

Since version v0.16.0, PyPSA enables optimisation without the use of pyomo by setting `pyomo=False`

. This make the `lopf`

function much more efficient in terms of memory usage and time. For this purpose two new module were introduced, `pypsa.linopf`

and `pypsa.linopt`

which mainly reflect the functionality of `pypsa.opf`

and `pypsa.opt`

but without using pyomo.
Note that when setting pyomo to False, the `extra_functionality`

has to be adapted to the appropriate syntax (see guidelines below). Some unit commitment functionality is not yet implemented without pyomo.

Warning

If the transmission capacity is changed in passive networks, then the impedance will also change (i.e. if parallel lines are installed). This is NOT reflected in the ordinary LOPF, however `pypsa.linopf.ilopf`

covers this through an iterative process as done in here.

## Optimising dispatch only - a market model¶

Capacity optimisation can be turned off so that only the dispatch is optimised, like a short-run electricity market model.

For simplified transmission representation using Net Transfer Capacities (NTCs), there is a Link component which does controllable power flow like a transport model (and can also represent a point-to-point HVDC link).

## Optimising total annual system costs¶

To minimise long-run annual system costs for meeting an inelastic electrical load, capital costs for transmission and generation should be set to the annualised investment costs in e.g. EUR/MW/a, marginal costs for dispatch to e.g. EUR/MWh and the weightings (now with units hours per annum, h/a) are chosen such that

In this case the objective function gives total system cost in EUR/a to meet the total load.

## Stochastic optimisation¶

For the very simplest stochastic optimisation you can use the
weightings `w_t`

as probabilities for the snapshots, which can
represent different load/weather conditions. More sophisticated
functionality is planned.

## Variables and notation summary¶

\(n \in N = \{0,\dots |N|-1\}\) |
label the buses |

\(t \in T = \{0,\dots |T|-1\}\) |
label the snapshots |

\(l \in L = \{0,\dots |L|-1\}\) |
label the branches |

\(s \in S = \{0,\dots |S|-1\}\) |
label the different generator/storage types at each bus |

\(w_t\) |
weighting of time \(t\) in the objective function |

\(g_{n,s,t}\) |
dispatch of generator \(s\) at bus \(n\) at time \(t\) |

\(\bar{g}_{n,s}\) |
nominal power of generator \(s\) at bus \(n\) |

\(\bar{g}_{n,s,t}\) |
availability of generator \(s\) at bus \(n\) at time \(t\) per unit of nominal power |

\(u_{n,s,t}\) |
binary status variable for generator with unit commitment |

\(suc_{n,s,t}\) |
start-up cost if generator with unit commitment is started at time \(t\) |

\(sdc_{n,s,t}\) |
shut-down cost if generator with unit commitment is shut down at time \(t\) |

\(c_{n,s}\) |
capital cost of extending generator nominal power by one MW |

\(o_{n,s}\) |
marginal cost of dispatch generator for one MWh |

\(f_{l,t}\) |
flow of power in branch \(l\) at time \(t\) |

\(F_{l}\) |
capacity of branch \(l\) |

\(\eta_{n,s}\) |
efficiency of generator \(s\) at bus \(n\) |

\(\eta_{l}\) |
efficiency of controllable link \(l\) |

\(e_s\) |
CO2-equivalent-tonne-per-MWh of the fuel carrier \(s\) |

Further definitions are given below.

## Objective function¶

See `pypsa.opf.define_linear_objective(network,snapshots)`

.

The objective function is composed of capital costs \(c\) for each component and operation costs \(o\) for generators

Additional variables which do not appear in the objective function are the storage uptake variable, the state of charge and the voltage angle for each bus.

## Generator constraints¶

These are defined in `pypsa.opf.define_generator_variables_constraints(network,snapshots)`

.

Generator nominal power and generator dispatch for each snapshot may be optimised.

Each generator has a dispatch variable \(g_{n,s,t}\) where \(n\) labels the bus, \(s\) labels the particular generator at the bus (e.g. it can represent wind/gas/coal generators at the same bus in an aggregated network) and \(t\) labels the time.

It obeys the constraints:

where \(\bar{g}_{n,s}\) is the nominal power (`generator.p_nom`

)
and \(\tilde{g}_{n,s,t}\) and \(\bar{g}_{n,s,t}\) are
time-dependent restrictions on the dispatch (per unit of nominal
power) due to e.g. wind availability or power plant de-rating.

For generators with time-varying `p_max_pu`

in `network.generators_t`

the per unit
availability \(\bar{g}_{n,s,t}\) is a time series.

For generators with static `p_max_pu`

in `network.generators`

the per unit
availability is a constant.

If the generator’s nominal power \(\bar{g}_{n,s}\) is also the
subject of optimisation (`generator.p_nom_extendable == True`

) then
limits `generator.p_nom_min`

and `generator.p_nom_max`

on the
installable nominal power may also be introduced, e.g.

## Generator unit commitment constraints¶

These are defined in `pypsa.opf.define_generator_variables_constraints(network,snapshots)`

.

Important

Unit commitment constraints will only be build fully if pyomo is set to True. If pyomo is set to False a simplified version of the unit commitment is calculated by ignoring the parameters min_up_time, min_down_time, start_up_cost, shut_down_cost, up_time_before and down_time_before.

The implementation is a complete implementation of the unit commitment constraints defined in Chapter 4.3 of Convex Optimization of Power Systems by Joshua Adam Taylor (CUP, 2015).

Unit commitment can be turned on for any generator by setting `committable`

to be `True`

. This introduces a
times series of new binary status variables \(u_{n,s,t} \in \{0,1\}\), saved in `network.generators_t.status`

,
which indicates whether the generator is running (1) or not (0) in
period \(t\). The restrictions on generator output now become:

so that if \(u_{n,s,t} = 0\) then also \(g_{n,s,t} = 0\).

Note that a generator cannot be both extendable (`generator.p_nom_extendable == True`

) and committable (`generator.committable == True`

) because of the coupling of the variables \(u_{n,s,t}\)
and \(\bar{g}_{n,s}\) here.

If the minimum up time \(T_{\textrm{min_up}}\) (`generator.min_up_time`

) is set then we have for generic times

i.e. if the generator has just started up at time \(t\) then \(u_{n,s,t-1} = 0\), \(u_{n,s,t} = 1\) and \(u_{n,s,t} - u_{n,s,t-1} = 1\), so that it has to run for at least \(T_{\textrm{min_up}}\) periods.

The generator may have been up for some periods before the `snapshots`

simulation period. If the up-time before `snapshots`

starts is less than the minimum up-time, then the generator is forced to be up for the difference at the start of `snapshots`

. If the start of `snapshots`

is the start of `network.snapshots`

, then the up-time before the simulation is read from the input variable `generator.up_time_before`

. If `snapshots`

falls in the middle of `network.snapshots`

, then PyPSA assumes the statuses for hours before `snapshots`

have been set by previous simulations, and reads back the previous up-time by examining the previous statuses. If the start of `snapshots`

is very close to the start of `network.snapshots`

, it will also take account of `generator.up_time_before`

as well as the statuses in between.

At the end of `snapshots`

the minimum up-time in the constraint is only enforced for the remaining snapshots, if the number of remaining snapshots is less than \(T_{\textrm{min_up}}\).

Similarly if the minimum down time \(T_{\textrm{min_down}}\) (`generator.min_up_time`

) is set then we have

You can also defined `generator.down_time_before`

for periods before `network.snapshots`

, analogous to the up time.

For non-zero start up costs \(suc_{n,s}\) a new variable \(suc_{n,s,t} \geq 0\) is introduced for each time period \(t\) and added to the objective function. The variable satisfies

so that it is only non-zero if \(u_{n,s,t} - u_{n,s,t-1} = 1\), i.e. the generator has just started, in which case the inequality is saturated \(suc_{n,s,t} = suc_{n,s}\). Similarly for the shut down costs \(sdc_{n,s,t} \geq 0\) we have

## Generator ramping constraints¶

These are defined in `pypsa.opf.define_generator_variables_constraints(network,snapshots)`

.

The implementation follows Chapter 4.3 of Convex Optimization of Power Systems by Joshua Adam Taylor (CUP, 2015).

Ramp rate limits can be defined for increasing power output \(ru_{n,s}\) and decreasing power output \(rd_{n,s}\). By default these are null and ignored. They should be given per unit of the generator nominal power. The generator dispatch then obeys

for \(t \in \{1,\dots |T|-1\}\).

For generators with unit commitment you can also specify ramp limits at start-up \(rusu_{n,s}\) and shut-down \(rdsd_{n,s}\)

## Storage Unit constraints¶

These are defined in `pypsa.opf.define_storage_variables_constraints(network,snapshots)`

.

Storage nominal power and dispatch for each snapshot may be optimised.

With a storage unit the maximum state of charge may not be independently optimised from the maximum power output (they’re linked by the maximum hours variable) and the maximum power output is linked to the maximum power input. To optimise these capacities independently, build a storage unit out of the more fundamental `Store`

and `Link`

components.

The storage nominal power is given by \(\bar{h}_{n,s}\).

In contrast to the generator, which has one time-dependent variable, each storage unit has three:

The storage dispatch \(h_{n,s,t}\) (when it depletes the state of charge):

The storage uptake \(f_{n,s,t}\) (when it increases the state of charge):

and the state of charge itself:

where \(r_{n,s}\) is the number of hours at nominal power that fill the state of charge.

The variables are related by

\(\eta_{\textrm{stand};n,s}\) is the standing losses dues to e.g. thermal losses for thermal storage. \(\eta_{\textrm{store};n,s}\) and \(\eta_{\textrm{dispatch};n,s}\) are the efficiency losses for power going into and out of the storage unit.

There are two options for specifying the initial state of charge \(soc_{n,s,t=-1}\): you can set
`storage_unit.cyclic_state_of_charge = False`

(the default) and the value of
`storage_unit.state_of_charge_initial`

in MWh; or you can set
`storage_unit.cyclic_state_of_charge = True`

and then
the optimisation assumes \(soc_{n,s,t=-1} = soc_{n,s,t=|T|-1}\).

If in the time series `storage_unit_t.state_of_charge_set`

there are
values which are not NaNs, then it will be assumed that these are
fixed state of charges desired for that time \(t\) and these will
be added as extra constraints. (A possible usage case would be a
storage unit where the state of charge must empty every day.)

## Store constraints¶

These are defined in `pypsa.opf.define_store_variables_constraints(network,snapshots)`

.

Store nominal energy and dispatch for each snapshot may be optimised.

The store nominal energy is given by \(\bar{e}_{n,s}\).

The store has two time-dependent variables:

The store dispatch \(h_{n,s,t}\):

and the energy:

The variables are related by

\(\eta_{\textrm{stand};n,s}\) is the standing losses dues to e.g. thermal losses for thermal storage.

There are two options for specifying the initial energy
\(e_{n,s,t=-1}\): you can set
`store.e_cyclic = False`

(the default) and the
value of `store.e_initial`

in MWh; or you can
set `store.e_cyclic = True`

and then the
optimisation assumes \(e_{n,s,t=-1} = e_{n,s,t=|T|-1}\).

## Passive branch flows: lines and transformers¶

See `pypsa.opf.define_passive_branch_flows(network,snapshots)`

and
`pypsa.opf.define_passive_branch_constraints(network,snapshots)`

and `pypsa.opf.define_branch_extension_variables(network,snapshots)`

.

For lines and transformers, whose power flows according to impedances, the power flow \(f_{l,t}\) in AC networks is given by the difference in voltage angles \(\theta_{n,t}\) at bus0 and \(\theta_{m,t}\) at bus1 divided by the series reactance \(x_l\)

(For DC networks, replace the voltage angles by the difference in voltage magnitude \(\delta V_{n,t}\) and the series reactance by the series resistance \(r_l\).)

This flow is the limited by the capacity :math:`F_l`

of the line

Note

If \(F_l\) is also subject to optimisation
(`branch.s_nom_extendable -- True`

), then the impedance \(x\) of
the line is NOT automatically changed with the capacity (to represent
e.g. parallel lines being added).

There are two choices here:

Iterate the LOPF again with the updated impedances, see e.g. http://www.sciencedirect.com/science/article/pii/S0360544214000322#, like done by

`pypsa.linopf.ilopf`

João Gorenstein Dedecca has also implemented a MILP version of the transmission expansion, see https://github.com/jdedecca/MILP_PyPSA, which properly takes account of the impedance with a disjunctive relaxation. This will be pulled into the main PyPSA code base soon.

## Passive branch flow formulations¶

PyPSA implements four formulations of the linear power flow equations that are mathematically equivalent, but may have different solving times. These different formulations are described and benchmarked in the arXiv preprint paper Linear Optimal Power Flow Using Cycle Flows.

You can choose the formulation by passing `network.lopf`

the
argument `formulation`

, which must be in
`["angles","cycles","kirchhoff","ptdf"]`

.

`angles`

is the standard formulations based on voltage angles described above, used for the linear power flow and found in textbooks.`ptdf`

uses the Power Transfer Distribution Factor (PTDF) formulation, found for example in http://www.sciencedirect.com/science/article/pii/S0360544214000322#.`kirchhoff`

and`cycles`

are two new formulations based on a graph-theoretic decomposition of the network flows into a spanning tree and closed cycles.

Based on the benchmarking in Linear Optimal Power Flow Using Cycle
Flows for standard networks,
`kirchhoff`

almost always solves fastest, averaging 3 times faster
than the `angles`

formulation and up to 20 times faster in specific
cases. The speedup is higher for larger networks with dispatchable
generators at most nodes.

## Controllable branch flows: links¶

See `pypsa.opf.define_controllable_branch_flows(network,snapshots)`

and `pypsa.opf.define_branch_extension_variables(network,snapshots)`

.

For links, whose power flow is controllable, there is simply an optimisation variable for each component which satisfies

If the link flow is positive \(f_{l,t} > 0\) then it withdraws
\(f_{l,t}\) from `bus0`

and feeds in \(\eta_l f_{l,t}\) to
`bus1`

, where \(\eta_l\) is the link efficiency.

If additional output buses `busi`

for \(i=2,3,\dots\) are
defined (i.e. `bus2`

, `bus3`

, etc) and their associated
efficiencies `efficiencyi`

, i.e. \(\eta_{i,l}\), then at
`busi`

the feed-in is \(\eta_{i,l} f_{l,t}\). See also
Link with multiple outputs or inputs.

## Nodal power balances¶

See `pypsa.opf.define_nodal_balances(network,snapshots)`

.

This is the most important equation, which guarantees that the power balances at each bus \(n\) for each time \(t\).

Where \(d_{n,s,t}\) is the exogenous load at each node (`load.p_set`

) and the incidence matrix \(K_{nl}\) for the graph takes values in \(\{-1,0,1\}\) depending on whether the branch \(l\) ends or starts at the bus. \(\lambda_{n,t}\) is the shadow price of the constraint, i.e. the locational marginal price, stored in `network.buses_t.marginal_price`

.

The bus’s role is to enforce energy conservation for all elements feeding in and out of it (i.e. like Kirchhoff’s Current Law).

## Global constraints¶

See `pypsa.opf.define_global_constraints(network,snapshots)`

.

Global constraints apply to more than one component.

Currently only “primary energy” constraints are defined. They depend on the power plant efficiency and carrier-specific attributes such as specific CO2 emissions.

Suppose there is a global constraint defined for CO2 emissions with
sense `<=`

and constant `\textrm{CAP}_{CO2}`

. Emissions can come
from generators whose energy carriers have CO2 emissions and from
stores and storage units whose storage medium releases or absorbs CO2
when it is converted. Only stores and storage units with non-cyclic
state of charge that is different at the start and end of the
simulation can contribute.

If the specific emissions of energy carrier \(s\) is \(e_s\)
(`carrier.co2_emissions`

) CO2-equivalent-tonne-per-MWh and the
generator with carrier \(s\) at node \(n\) has efficiency
\(\eta_{n,s}\) then the CO2 constraint is

The first sum is over generators; the second sum is over stores and
storage units. \(\mu\) is the shadow price of the constraint,
i.e. the CO2 price in this case. \(\mu\) is an output of the
optimisation stored in `network.global_constraints.mu`

.

## Optimising investment and operation over multiple investment periods¶

In general, there are two different methods of pathway optimisation with perfect foresight. These differ in the way of accounting the investment costs:

In the first case (type I), the complete overnight investment costs are applied.

In the second case (type II), the investment costs are annualised over the years, in which an asset is active (depending on the build year and lifetime).

Method II is used in PyPSA since it allows a separation of the discounting over different years and the end-of-horizon effects are smaller compared to method I. For a more detailed comparison of the two methods and a reference to other energy system models see https://nworbmot.org/energy/multihorizon.pdf.

Note

Be aware, that the attribute `capital_cost`

represents the annualised investment costs
NOT the overnight investment costs for the multi-investment.

Multi-year investment instead of investing a single time is currently only
implemented without pyomo. It can be passed by setting the argument
`multi_investment_periods`

when calling the
`network.lopf(multi_investment_periods=True, pyomo=False)`

. For the pathway
optimisation `snapshots`

have to be a pandas.MultiIndex, with the first level
as a subset of the investment periods.

The investment periods are defined in the component `investment_periods`

.
They have to be integer and increasing (e.g. [2020, 2030, 2040, 2050]).
The investment periods can be weighted both in time called `years`

(e.g. for global constraints such as CO2 emissions) and in the objective function
`objective`

(e.g. for a social discount rate) using the
`investment_period_weightings`

.

The objective function is then expressed by

Where \(A\) are the investment periods, \(w^o_a\) the objective weighting of the investment period, \(b_s\) is the build year of an asset \(s\) with lifetime \(L_s\), \(c_{s,a}\) the annualised investment costs, \(o_{s,a, t}\) the operational cots and \(w^\tau_{s,a}\) the temporal weightings (including snapshot objective weightings and investment period temporal weightings).

Example jupyter notebook for multi-investment and python
script `examples/multi-decade-example.py`

.

## Custom constraints and other functionality¶

Since PyPSA v0.16.0, the lopf function is provided by two different modules. The ordinary implementation based on the `pypsa.opf`

module uses
pyomo to set up the linear optimisation problem and passing it to the solver. The implementation without pyomo, based on the module `pypsa.linopf`

, uses a straight-forward approach to write out the `.lp`

file directly and explicitly running it from a solver’s interface. Therefore the application of custom constraints depends on whether pyomo is activated or not.

In general for a custom constraint, pass the function `network.lopf`

a
function `extra_functionality`

as an argument. This function must
take two arguments `extra_functionality(network,snapshots)`

and is
called after the model building is complete, but before it is sent to
the solver. It allows the user to add, change or remove constraints
and alter the objective function.

### 1. pyomo is set to True¶

You can easily extend the optimisation problem constructed by PyPSA using the usual pyomo syntax.

The CHP example and the
example that replaces generators and storage units with fundamental links
and stores
both pass an `extra_functionality`

argument to the LOPF to add
functionality.

The function `extra_postprocessing`

is called after the model has
solved and the results are extracted. This function must take three
arguments extra_postprocessing(network,snapshots,duals). It allows
the user to extract further information about the solution, such as
additional shadow prices for constraints.

### 2. pyomo is set to False¶

In general when pyomo is disabled, all variable and constraint references are stored in the network object itself. Thus every variable and constraint is attached to a component, e.g. the dispatch variable of network.generators.p is attached to the component ‘Generator’ and can be easily accessed by

```
>>> get_var(n, 'Generator', 'p')
```

An additional constraint can easily be implemented by using the functions

`pypsa.linopt.get_var()`

for getting the variables which should be included in the constraint`pypsa.linopt.linexpr()`

for creating linear expressions for the left hand side (lhs) of the constraint. Note that only the lhs includes all terms with variables, the rhs is a constant.`pypsa.linopt.define_constraints()`

for defining a network constraint.

The function `extra_postprocessing`

is not necessary when pyomo is deactivated. For retrieving additional shadow prices, just pass the name of the constraint, to which the constraint is attached, to the `keep_shadowprices`

parameter of the `lopf`

function.

## Inputs¶

For the linear optimal power flow, the following data for each component are used. For almost all values, defaults are assumed if not explicitly set. For the defaults and units, see Components.

network.{snapshot_weightings}

bus.{v_nom, carrier}

load.{p_set}

generator.{p_nom, p_nom_extendable, p_nom_min, p_nom_max, p_min_pu, p_max_pu, marginal_cost, capital_cost, efficiency, carrier}

storage_unit.{p_nom, p_nom_extendable, p_nom_min, p_nom_max, p_min_pu, p_max_pu, marginal_cost, capital_cost, efficiency*, standing_loss, inflow, state_of_charge_set, max_hours, state_of_charge_initial, cyclic_state_of_charge}

store.{e_nom, e_nom_extendable, e_nom_min, e_nom_max, e_min_pu, e_max_pu, e_cyclic, e_initial, capital_cost, marginal_cost, standing_loss}

line.{x, s_nom, s_nom_extendable, s_nom_min, s_nom_max, capital_cost}

transformer.{x, s_nom, s_nom_extendable, s_nom_min, s_nom_max, capital_cost}

link.{p_min_pu, p_max_pu, p_nom, p_nom_extendable, p_nom_min, p_nom_max, capital_cost}

carrier.{carrier_attribute}

global_constraint.{type, carrier_attribute, sense, constant}

Note

Note that for lines and transformers you MUST make sure that \(x\) is non-zero, otherwise the bus admittance matrix will be singular.

## Outputs¶

bus.{v_mag_pu, v_ang, p, marginal_price}

load.{p}

generator.{p, p_nom_opt}

storage_unit.{p, p_nom_opt, state_of_charge, spill}

store.{p, e_nom_opt, e}

line.{p0, p1, s_nom_opt, mu_lower, mu_upper}

transformer.{p0, p1, s_nom_opt, mu_lower, mu_upper}

link.{p0, p1, p_nom_opt, mu_lower, mu_upper}

global_constraint.{mu}