Power System Optimization#
Important
Since version v0.22.0, PyPSA enables the optimisation with Linopy through the optimization module. The core function is available via Network.optimize which per default follows the behaviour of the lopf function and includes all its feature. The new optimization module should provide flexibility as well as performance. For an introduction to the new implementation have a look at the example notebook for a migration guide of extra functionalities have a look at our migration guide. The following section will be adapted in the future to the new optimization interface.
See the modules 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#
PyPSA 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={},
formulation="angles", extra_postprocessing=None, pyomo=True)
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, formulation is a string in
["angles","cycles","kirchhoff","ptdf"] (see Passive branch flow formulations for more
details) and pyomo is a boolean to switch between formulating the
optimisation problem using pyomo or PyPSA’s custom optimisation framework.
See pypsa.Network.lopf() for the full documentation.
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 \(s\) |
Objective function#
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#
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#
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#
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#
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#
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).
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"].
anglesis the standard formulations based on voltage angles described above, used for the linear power flow and found in textbooks.ptdfuses the Power Transfer Distribution Factor (PTDF) formulation, found for example in http://www.sciencedirect.com/science/article/pii/S0360544214000322#.kirchhoffandcyclesare 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#
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#
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#
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#
The pypsa.Network.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 PyPSA’s own internal optimisation framework that writes
out the .lp file directly and explicitly runs it from a solver’s interface.
Therefore the application of custom constraints depends on whether pyomo is
activated or not. Pyomo is activated by default, but to switch to the internal
optimisation framework run pypsa.lopf(pyomo=False).
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#
To use PyPSA’s own internal optimisation framework linopt run network.lopf(pyomo=False). The linopt framework uses considerably less memory and time than pyomo, however it is slightly harder to customise.
Several customisations with linopt are demonstrated in the example Optimization without pyomo.
linopt works by assigning an integer to each variable and constraint. Constraints are then built as strings by adding variables with coefficients.
All variable and constraint references are stored in the network object itself, attached to the relevant component. By accessing these references inside an extra_functionality(network, snapshots) function passed to network.lopf, you can select variables and build constraints using the following 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.join_exprs()for summing linear expressions.pypsa.linopt.define_constraints()for defining a network constraint.
Once the problem has been built, all names of variable sets are stored in n.variables and all names of constraint sets in n.constraints.
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.
get_var#
The function linopt.get_var is used to access the variables attached to a component. To find out which variables are available, look inside n.variables once the n.lopf(pyomo=False) has run.
For example, to access the the dispatch variable of network.generators_t.p attached to the component Generator use
>>> get_var(n, 'Generator', 'p')
This will return a pd.DataFrame with index of network.snapshots and columns of network.generators.index with the variable references in each entry.
To access the capacities of extendable generators use
>>> get_var(n, 'Generator', 'p_nom')
This will return a pd.Series with index of network.generators.index with the variable references in each entry.
linexpr#
The function linopt.linexpr is used to build linear combinations of variables.
It takes a tuple of twoples, where the first entry is the coefficients and the second entry is the variables.
Beware that the indices and columns of the pd.DataFrame or pd.Series you combine must have aligned indices and columns. This applies both to coefficients and variables.
For example, to subtract the extendable generator capacities from their dispatch for each snapshot do
>>> ext_i = n.get_extendable_i('Generator')
>>> p = get_var(n, 'Generator', 'p')[ext_i]
>>> p_nom = get_var(n, 'Generator', 'p_nom')
>>> linexpr((1, p), (-1, p_nom))
This will return a pd.DataFrame with index of network.snapshots and columns of the extendable generators ext_i with the constraint strings in each entry.
To add the dispatch weighted by the generator efficiency do
>>> ext_i = n.get_extendable_i('Generator')
>>> p = get_var(n, 'Generator', 'p')[ext_i]
>>> p_nom = get_var(n, 'Generator', 'p_nom')
>>> efficiency = n.generators.efficiency[ext_i]
>>> linexpr((efficiency, p), (-1, p_nom))
To add the dispatch weighted by the snapshot weightings do
>>> ext_i = n.get_extendable_i('Generator')
>>> p = get_var(n, 'Generator', 'p')[ext_i]
>>> p_nom = get_var(n, 'Generator', 'p_nom')
>>> weightings = pd.DataFrame({gen: n.snapshot_weightings.generators for gen in ext_i})
>>> linexpr((weightings, p), (-1, p_nom))
You may need to rename indices if you’re adding components with different names. Consider this example subtracting battery discharging from charging capacities
>>> chargers = n.links.index[n.links.index.str.contains('charger')]
>>> dischargers = n.links.index[n.links.index.str.contains('discharger')]
>>> linexpr((1, get_var(n, 'Link', 'p_nom')[chargers]), (-1, get_var(n, 'Link', 'p_nom')[dischargers].rename(lambda name: name.replace("discharger","charger"))))
join_exprs#
The function linopt.join_exprs is used to sum up variables along different axes.
For example, to sum up all dispatch variables over all generators and times do
>>> join_exprs(linexpr((1,get_var(n, "Generator", "p"))))
This returns a string.
To sum up only over the index n.snapshots, i.e. to get for each generator its total dispatch over the period, do
>>> linexpr((1,get_var(n, "Generator", "p"))).apply(join_exprs)
This returns a pd.Series of strings indexed by n.generators.index, where each string is a sum over time.
To sum up only over the columns n.generators.index, i.e. to get for each time the sum of generator dispatch, do
>>> linexpr((1,get_var(n, "Generator", "p"))).apply(join_exprs, axis=1)
This returns a pd.Series of strings indexed by n.snapshots, where each string is a sum over generators.
define_constraints#
The function linopt.define_constraints is used to add constraints to the model.
It typically has the form
>>> define_constraints(n, lhs, "=", rhs, 'Link', 'charger_ratio')
where lhs is a linear expression linexpr, the sense follows
(one of =, <= or >=), rhs is a constant or linear
expression linexpr, the next argument tells on which component to
store the constraints and then the name of the constraints.
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}
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}