# User’s guide¶

Solph is an oemof-package, designed to create and solve linear or mixed-integer linear optimization problems. The package is based on pyomo. To create an energy system model generic and specific components are available. To get started with solph, checkout the examples in the Examples section.

This User’s guide provides a user-friendly introduction into oemof-solph, which includes small examples and nice illustrations. However, the functionalities of oemof-solph go beyond the content of this User’s guide section. So, if you want to know all details of a certain component or a function, please go the API Reference. There, you will find a detailed and complete description of all oemof-solph modules.

## How can I use solph?¶

To use solph you have to install oemof.solph and at least one solver (see Installation), which can be used together with pyomo (e.g. CBC, GLPK, Gurobi, Cplex). See the pyomo installation guide for all supported solvers. You can test it by executing one of the existing examples (see Examples). Be aware that the examples require the CBC solver but you can change the solver name in the example files to your solver.

Once the examples work you are close to your first energy model.

### Handling of Warnings¶

The solph library is designed to be as generic as possible to make it possible to use it in different use cases. This concept makes it difficult to raise Errors or Warnings because sometimes untypical combinations of parameters are allowed even though they might be wrong in over 99% of the use cases.

Therefore, a SuspiciousUsageWarning was introduced. This warning will warn you if you do something untypical. If you are sure that you know what you are doing you can switch the warning off.

See the debugging module of oemof-tools for more information.

### Set up an energy system¶

In most cases an EnergySystem object is defined when we start to build up an energy system model. The EnergySystem object will be the main container for the model’s elements.

The model time is defined by the number of intervals and the length of intervals. The length of each interval does not have to be the same. This can be defined in two ways:

Define the length of each interval in an array/Series where the number of the elements is the number of intervals.

Define a pandas.DatetimeIndex with all time steps that encloses an interval. Be aware that you have to define n+1 time points to get n intervals. For non-leap year with hourly values that means 8761 time points to get 8760 interval e.g. 2018-01-01 00:00 to 2019-01-01 00:00.

The index will also be used for the results. For a numeric index the resulting time series will indexed with a numeric index starting with 0.

One can use the function
`create_time_index()`

to create an equidistant datetime index. By default the function creates an hourly index for one year, so online the year has to be passed to the function. But it is also possible to change the length of the interval to quarter hours etc. The default number of intervals is the number needed to cover the given year but the value can be overwritten by the user.

It is also possible to define the datetime index using pandas. See pandas date_range guide for more information.

Both code blocks will create an hourly datetime index for 2011:

```
from oemof.solph import create_time_index
my_index = create_time_index(2011)
```

```
import pandas as pd
my_index = pd.date_range('1/1/2011', periods=8761, freq='H')
```

This index can be used to define the EnergySystem:

```
import oemof.solph as solph
my_energysystem = solph.EnergySystem(timeindex=my_index)
```

Now you can start to add the components of the network.

### Add components to the energy system¶

After defining an instance of the EnergySystem class, in the following you have to add all nodes you define to your EnergySystem.

Basically, there are two types of *nodes* - *components* and *buses*. Every Component has to be connected with one or more *buses*. The connection between a *component* and a *bus* is the *flow*.

All solph *components* can be used to set up an energy system model but you should read the documentation of each *component* to learn about usage and restrictions. For example it is not possible to combine every *component* with every *flow*. Furthermore, you can add your own *components* in your application (see below) but we would be pleased to integrate them into solph if they are of general interest (see Feature requests and feedback).

An example of a simple energy system shows the usage of the nodes for real world representations:

The figure shows a simple energy system using the four basic network classes and the Bus class. If you remove the transmission line (transport 1 and transport 2) you get two systems but they are still one energy system in terms of solph and will be optimised at once.

There are different ways to add components to an *energy system*. The following line adds a *bus* object to the *energy system* defined above.

```
my_energysystem.add(solph.buses.Bus())
```

It is also possible to assign the bus to a variable and add it afterwards. In that case it is easy to add as many objects as you like.

```
my_bus1 = solph.buses.Bus()
my_bus2 = solph.buses.Bus()
my_energysystem.add(my_bus1, my_bus2)
```

Therefore it is also possible to add lists or dictionaries with components but you have to dissolve them.

```
# add a list
my_energysystem.add(*my_list)
# add a dictionary
my_energysystem.add(*my_dictionary.values())
```

#### Bus¶

All flows into and out of a *bus* are balanced (by default). Therefore an instance of the Bus class represents a grid or network without losses. To define an instance of a Bus only a unique label is necessary. If you do not set a label a random label is used but this makes it difficult to get the results later on.

To make it easier to connect the bus to a component you can optionally assign a variable for later use.

```
solph.buses.Bus(label='natural_gas')
electricity_bus = solph.buses.Bus(label='electricity')
```

Note

See the `Bus`

class for all parameters and the mathematical background.

#### Flow¶

The flow class has to be used to connect nodes and buses. An instance of the Flow class is normally used in combination with the definition of a component.
A Flow can be limited by upper and lower bounds (constant or time-dependent) or by summarised limits.
For all parameters see the API documentation of the `Flow`

class or the examples of the nodes below. A basic flow can be defined without any parameter.

```
solph.flows.Flow()
```

oemof.solph has different types of *flows* but you should be aware that you cannot connect every *flow* type with every *component*.

Note

See the `Flow`

class for all parameters and the mathematical background.

#### Components¶

Components are divided in two categories. Well-tested components (solph.components) and experimental components (solph.components.experimental). The experimental section was created to lower the entry barrier for new components. Be aware that these components might not be properly documented or even sometimes do not even work as intended. Let us know if you have successfully used and tested these components. This is the first step to move them to the regular components section.

See Solph components for a list of all components.

### Optimise your energy system¶

The typical optimisation of an energy system in solph is the dispatch optimisation, which means that the use of the sources is optimised to satisfy the demand at least costs. Therefore, variable cost can be defined for all components. The cost for gas should be defined in the gas source while the variable costs of the gas power plant are caused by operating material. The actual fuel cost in turn is calculated in the framework itself considering the efficiency of the power plant. You can deviate from this scheme but you should keep it consistent to make it understandable for others.

Costs do not have to be monetary costs but could be emissions or other variable units.

Furthermore, it is possible to optimise the capacity of different components using the investment mode (see Investment optimisation).

Since v0.5.1, there also is the possibility to have multi-period (i.e. dynamic) investments over longer-time horizon which is in experimental state (see Multi-period (investment) mode (experimental)).

```
# set up a simple least cost optimisation
om = solph.Model(my_energysystem)
# solve the energy model using the CBC solver
om.solve(solver='cbc', solve_kwargs={'tee': True})
```

If you want to analyse the lp-file to see all equations and bounds you can write the file to you disc. In that case you should reduce the timesteps to 3. This will increase the readability of the file.

```
# set up a simple least cost optimisation
om = solph.Model(my_energysystem)
# write the lp file for debugging or other reasons
om.write('path/my_model.lp', io_options={'symbolic_solver_labels': True})
```

### Analysing your results¶

If you want to analyse your results, you should first dump your EnergySystem instance to permanently store results. Otherwise you would have to run the simulation again.

```
my_energysystem.results = processing.results(om)
my_energysystem.dump('my_path', 'my_dump.oemof')
```

If you need the meta results of the solver you can do the following:

```
my_energysystem.results['main'] = processing.results(om)
my_energysystem.results['meta'] = processing.meta_results(om)
my_energysystem.dump('my_path', 'my_dump.oemof')
```

To restore the dump you can simply create an EnergySystem instance and restore your dump into it.

```
import oemof.solph as solph
my_energysystem = solph.EnergySystem()
my_energysystem.restore('my_path', 'my_dump.oemof')
results = my_energysystem.results
# If you use meta results do the following instead of the previous line.
results = my_energysystem.results['main']
meta = my_energysystem.results['meta']
```

If you call dump/restore without any parameters, the dump will be stored as *‘es_dump.oemof’* into the *‘.oemof/dumps/’* folder created in your HOME directory.

See Handling Results to learn how to process, plot and analyse the results.

## Solph components¶

### Sink (basic)¶

A sink is normally used to define the demand within an energy model but it can also be used to detect excesses.

The example shows the electricity demand of the electricity_bus defined above.
The *‘my_demand_series’* should be sequence of normalised valueswhile the *‘nominal_value’* is the maximum demand the normalised sequence is multiplied with.
Giving *‘my_demand_series’* as parameter *‘fix’* means that the demand cannot be changed by the solver.

```
solph.components.Sink(label='electricity_demand', inputs={electricity_bus: solph.flows.Flow(
fix=my_demand_series, nominal_value=nominal_demand)})
```

In contrast to the demand sink the excess sink has normally less restrictions but is open to take the whole excess.

```
solph.components.Sink(label='electricity_excess', inputs={electricity_bus: solph.flows.Flow()})
```

Note

The Sink class is only a plug and provides no additional constraints or variables.

### Source (basic)¶

A source can represent a pv-system, a wind power plant, an import of natural gas or a slack variable to avoid creating an in-feasible model.

While a wind power plant will have as feed-in depending on the weather conditions the natural_gas import might be restricted by maximum value (*nominal_value*) and an annual limit (*full_load_time_max*).
As we do have to pay for imported gas we should set variable costs.
Comparable to the demand series an *fix* is used to define a fixed the normalised output of a wind power plant.
Alternatively, you might use *max* to allow for easy curtailment.
The *nominal_value* sets the installed capacity.

```
solph.components.Source(
label='import_natural_gas',
outputs={my_energysystem.groups['natural_gas']: solph.flows.Flow(
nominal_value=1000, full_load_time_max=1000000, variable_costs=50)})
solph.components.Source(label='wind', outputs={electricity_bus: solph.flows.Flow(
fix=wind_power_feedin_series, nominal_value=1000000)})
```

Note

The Source class is only a plug and provides no additional constraints or variables.

### Converter (basic)¶

An instance of the Converter class can represent a node with multiple input and output flows such as a power plant, a transport line or any kind of a transforming process as electrolysis, a cooling device or a heat pump. The efficiency has to be constant within one time step to get a linear transformation. You can define a different efficiency for every time step (e.g. the thermal powerplant efficiency according to the ambient temperature) but this series has to be predefined and cannot be changed within the optimisation.

A condensing power plant can be defined by a converter with one input (fuel) and one output (electricity).

```
b_gas = solph.buses.Bus(label='natural_gas')
b_el = solph.buses.Bus(label='electricity')
solph.components.Converter(
label="pp_gas",
inputs={bgas: solph.flows.Flow()},
outputs={b_el: solph.flows.Flow(nominal_value=10e10)},
conversion_factors={electricity_bus: 0.58})
```

A CHP power plant would be defined in the same manner but with two outputs:

```
b_gas = solph.buses.Bus(label='natural_gas')
b_el = solph.buses.Bus(label='electricity')
b_th = solph.buses.Bus(label='heat')
solph.components.Converter(
label='pp_chp',
inputs={b_gas: Flow()},
outputs={b_el: Flow(nominal_value=30),
b_th: Flow(nominal_value=40)},
conversion_factors={b_el: 0.3, b_th: 0.4})
```

A CHP power plant with 70% coal and 30% natural gas can be defined with two inputs and two outputs:

```
b_gas = solph.buses.Bus(label='natural_gas')
b_coal = solph.buses.Bus(label='hard_coal')
b_el = solph.buses.Bus(label='electricity')
b_th = solph.buses.Bus(label='heat')
solph.components.Converter(
label='pp_chp',
inputs={b_gas: Flow(), b_coal: Flow()},
outputs={b_el: Flow(nominal_value=30),
b_th: Flow(nominal_value=40)},
conversion_factors={b_el: 0.3, b_th: 0.4,
b_coal: 0.7, b_gas: 0.3})
```

A heat pump would be defined in the same manner. New buses are defined to make the code cleaner:

```
b_el = solph.buses.Bus(label='electricity')
b_th_low = solph.buses.Bus(label='low_temp_heat')
b_th_high = solph.buses.Bus(label='high_temp_heat')
# The cop (coefficient of performance) of the heat pump can be defined as
# a scalar or a sequence.
cop = 3
solph.components.Converter(
label='heat_pump',
inputs={b_el: Flow(), b_th_low: Flow()},
outputs={b_th_high: Flow()},
conversion_factors={b_el: 1/cop,
b_th_low: (cop-1)/cop})
```

If the low-temperature reservoir is nearly infinite (ambient air heat pump) the low temperature bus is not needed and, therefore, a Converter with one input is sufficient.

Note

See the `Converter`

class for all parameters and the mathematical background.

### ExtractionTurbineCHP (component)¶

The `ExtractionTurbineCHP`

inherits from the Converter (basic) class. Like the name indicates,
the application example for the component is a flexible combined heat and power
(chp) plant. Of course, an instance of this class can represent also another
component with one input and two output flows and a flexible ratio between
these flows, with the following constraints:

\[\begin{split}& (1)\dot H_{Fuel}(t) = \frac{P_{el}(t) + \dot Q_{th}(t) \cdot \beta(t)} {\eta_{el,woExtr}(t)} \\ & (2)P_{el}(t) \geq \dot Q_{th}(t) \cdot C_b\end{split}\]where:

\[\beta(t) = \frac{\eta_{el,woExtr}(t) - \eta_{el,maxExtr}(t)}{\eta_{th,maxExtr}(t)}\]and:

\[C_b = \frac{\eta_{el,maxExtr}(t)}{\eta_{th,maxExtr}(t)}\]The first equation is the result of the relation between the input flow and the two output flows, the second equation stems from how the two output flows relate to each other, and the symbols used are defined as follows (with Variables (V) and Parameters (P)):

symbol

attribute

type

explanation

\(\dot H_{Fuel}\)

flow[i, n, t]

V

fuel input flow

\(P_{el}\)

flow[n, main_output, t]

V

electric power

\(\dot Q_{th}\)

flow[n, tapped_output, t]

V

thermal output

\(\beta\)

main_flow_loss_index[n, t]

P

power loss index

\(\eta_{el,woExtr}\)

conversion_factor_full_condensation[n, t]

P

- electric efficiency
without heat extraction

\(\eta_{el,maxExtr}\)

conversion_factors[main_output][n, t]

P

- electric efficiency
with max heat extraction

\(\eta_{th,maxExtr}\)

conversion_factors[tapped_output][n, t]

P

- thermal efficiency with
maximal heat extraction

These constraints are applied in addition to those of a standard
`Converter`

. The constraints limit the range of
the possible operation points, like the following picture shows. For a certain
flow of fuel, there is a line of operation points, whose slope is defined by
the power loss factor \(\beta\) (in some contexts also referred to as
\(C_v\)). The second constraint limits the decrease of electrical power and
incorporates the backpressure coefficient \(C_b\).

For now, `ExtractionTurbineCHP`

instances must
have one input and two output flows. The class allows the definition
of a different efficiency for every time step that can be passed as a series
of parameters that are fixed before the optimisation. In contrast to the
`Converter`

, a main flow and a tapped flow is
defined. For the main flow you can define a separate conversion factor that
applies when the second flow is zero (*`conversion_factor_full_condensation`*).

```
solph.components._extractionTurbineCHP(
label='variable_chp_gas',
inputs={b_gas: solph.flows.Flow(nominal_value=10e10)},
outputs={b_el: solph.flows.Flow(), b_th: solph.flows.Flow()},
conversion_factors={b_el: 0.3, b_th: 0.5},
conversion_factor_full_condensation={b_el: 0.5})
```

The key of the parameter *‘conversion_factor_full_condensation’* defines which
of the two flows is the main flow. In the example above, the flow to the Bus
*‘b_el’* is the main flow and the flow to the Bus *‘b_th’* is the tapped flow.
The following plot shows how the variable chp (right) schedules it’s electrical
and thermal power production in contrast to a fixed chp (left). The plot is the
output of an example in the example directory.

Note

See the `ExtractionTurbineCHP`

class for all parameters and the mathematical background.

### GenericCHP (component)¶

With the GenericCHP class it is possible to model different types of CHP plants (combined cycle extraction turbines, back pressure turbines and motoric CHP), which use different ranges of operation, as shown in the figure below.

Combined cycle extraction turbines: The minimal and maximal electric power without district heating (red dots in the figure) define maximum load and minimum load of the plant. Beta defines electrical power loss through heat extraction. The minimal thermal condenser load to cooling water and the share of flue gas losses at maximal heat extraction determine the right boundary of the operation range.

```
solph.components.GenericCHP(
label='combined_cycle_extraction_turbine',
fuel_input={bgas: solph.flows.Flow(
H_L_FG_share_max=[0.19 for p in range(0, periods)])},
electrical_output={bel: solph.flows.Flow(
P_max_woDH=[200 for p in range(0, periods)],
P_min_woDH=[80 for p in range(0, periods)],
Eta_el_max_woDH=[0.53 for p in range(0, periods)],
Eta_el_min_woDH=[0.43 for p in range(0, periods)])},
heat_output={bth: solph.flows.Flow(
Q_CW_min=[30 for p in range(0, periods)])},
Beta=[0.19 for p in range(0, periods)],
back_pressure=False)
```

For modeling a back pressure CHP, the attribute back_pressure has to be set to True. The ratio of power and heat production in a back pressure plant is fixed, therefore the operation range is just a line (see figure). Again, the P_min_woDH and P_max_woDH, the efficiencies at these points and the share of flue gas losses at maximal heat extraction have to be specified. In this case “without district heating” is not to be taken literally since an operation without heat production is not possible. It is advised to set Beta to zero, so the minimal and maximal electric power without district heating are the same as in the operation point (see figure). The minimal thermal condenser load to cooling water has to be zero, because there is no condenser besides the district heating unit.

```
solph.components.GenericCHP(
label='back_pressure_turbine',
fuel_input={bgas: solph.flows.Flow(
H_L_FG_share_max=[0.19 for p in range(0, periods)])},
electrical_output={bel: solph.flows.Flow(
P_max_woDH=[200 for p in range(0, periods)],
P_min_woDH=[80 for p in range(0, periods)],
Eta_el_max_woDH=[0.53 for p in range(0, periods)],
Eta_el_min_woDH=[0.43 for p in range(0, periods)])},
heat_output={bth: solph.flows.Flow(
Q_CW_min=[0 for p in range(0, periods)])},
Beta=[0 for p in range(0, periods)],
back_pressure=True)
```

A motoric chp has no condenser, so Q_CW_min is zero. Electrical power does not depend on the amount of heat used so Beta is zero. The minimal and maximal electric power (without district heating) and the efficiencies at these points are needed, whereas the use of electrical power without using thermal energy is not possible. With Beta=0 there is no difference between these points and the electrical output in the operation range. As a consequence of the functionality of a motoric CHP, share of flue gas losses at maximal heat extraction but also at minimal heat extraction have to be specified.

```
solph.components.GenericCHP(
label='motoric_chp',
fuel_input={bgas: solph.flows.Flow(
H_L_FG_share_max=[0.18 for p in range(0, periods)],
H_L_FG_share_min=[0.41 for p in range(0, periods)])},
electrical_output={bel: solph.flows.Flow(
P_max_woDH=[200 for p in range(0, periods)],
P_min_woDH=[100 for p in range(0, periods)],
Eta_el_max_woDH=[0.44 for p in range(0, periods)],
Eta_el_min_woDH=[0.40 for p in range(0, periods)])},
heat_output={bth: solph.flows.Flow(
Q_CW_min=[0 for p in range(0, periods)])},
Beta=[0 for p in range(0, periods)],
back_pressure=False)
```

Modeling different types of plants means telling the component to use different constraints. Constraint 1 to 9 are active in all three cases. Constraint 10 depends on the attribute back_pressure. If true, the constraint is an equality, if not it is a less or equal. Constraint 11 is only needed for modeling motoric CHP which is done by setting the attribute H_L_FG_share_min.

\[\begin{split}& (1)\qquad \dot{H}_F(t) = fuel\ input \\ & (2)\qquad \dot{Q}(t) = heat\ output \\ & (3)\qquad P_{el}(t) = power\ output\\ & (4)\qquad \dot{H}_F(t) = \alpha_0(t) \cdot Y(t) + \alpha_1(t) \cdot P_{el,woDH}(t)\\ & (5)\qquad \dot{H}_F(t) = \alpha_0(t) \cdot Y(t) + \alpha_1(t) \cdot ( P_{el}(t) + \beta \cdot \dot{Q}(t) )\\ & (6)\qquad \dot{H}_F(t) \leq Y(t) \cdot \frac{P_{el, max, woDH}(t)}{\eta_{el,max,woDH}(t)}\\ & (7)\qquad \dot{H}_F(t) \geq Y(t) \cdot \frac{P_{el, min, woDH}(t)}{\eta_{el,min,woDH}(t)}\\ & (8)\qquad \dot{H}_{L,FG,max}(t) = \dot{H}_F(t) \cdot \dot{H}_{L,FG,sharemax}(t)\\ & (9)\qquad \dot{H}_{L,FG,min}(t) = \dot{H}_F(t) \cdot \dot{H}_{L,FG,sharemin}(t)\\ & (10)\qquad P_{el}(t) + \dot{Q}(t) + \dot{H}_{L,FG,max}(t) + \dot{Q}_{CW, min}(t) \cdot Y(t) = / \leq \dot{H}_F(t)\\\end{split}\]where \(= / \leq\) depends on the CHP being back pressure or not.

The coefficients \(\alpha_0\) and \(\alpha_1\) can be determined given the efficiencies maximal/minimal load:

\[\begin{split}& \eta_{el,max,woDH}(t) = \frac{P_{el,max,woDH}(t)}{\alpha_0(t) \cdot Y(t) + \alpha_1(t) \cdot P_{el,max,woDH}(t)}\\ & \eta_{el,min,woDH}(t) = \frac{P_{el,min,woDH}(t)}{\alpha_0(t) \cdot Y(t) + \alpha_1(t) \cdot P_{el,min,woDH}(t)}\\\end{split}\]

If \(\dot{H}_{L,FG,min}\) is given, e.g. for a motoric CHP:

\[\begin{split}& (11)\qquad P_{el}(t) + \dot{Q}(t) + \dot{H}_{L,FG,min}(t) + \dot{Q}_{CW, min}(t) \cdot Y(t) \geq \dot{H}_F(t)\\[10pt]\end{split}\]The symbols used are defined as follows (with Variables (V) and Parameters (P)):

math. symbol

attribute

type

explanation

\(\dot{H}_{F}\)

H_F[n,t]

V

input of enthalpy through fuel input

\(P_{el}\)

P[n,t]

V

provided electric power

\(P_{el,woDH}\)

P_woDH[n,t]

V

electric power without district heating

\(P_{el,min,woDH}\)

P_min_woDH[n,t]

P

min. electric power without district heating

\(P_{el,max,woDH}\)

P_max_woDH[n,t]

P

max. electric power without district heating

\(\dot{Q}\)

Q[n,t]

V

provided heat

\(\dot{Q}_{CW, min}\)

Q_CW_min[n,t]

P

minimal therm. condenser load to cooling water

\(\dot{H}_{L,FG,min}\)

H_L_FG_min[n,t]

V

flue gas enthalpy loss at min heat extraction

\(\dot{H}_{L,FG,max}\)

H_L_FG_max[n,t]

V

flue gas enthalpy loss at max heat extraction

\(\dot{H}_{L,FG,sharemin}\)

H_L_FG_share_min[n,t]

P

share of flue gas loss at min heat extraction

\(\dot{H}_{L,FG,sharemax}\)

H_L_FG_share_max[n,t]

P

share of flue gas loss at max heat extraction

\(Y\)

Y[n,t]

V

status variable on/off

\(\alpha_0\)

n.alphas[0][n,t]

P

coefficient describing efficiency

\(\alpha_1\)

n.alphas[1][n,t]

P

coefficient describing efficiency

\(\beta\)

beta[n,t]

P

power loss index

\(\eta_{el,min,woDH}\)

Eta_el_min_woDH[n,t]

P

el. eff. at min. fuel flow w/o distr. heating

\(\eta_{el,max,woDH}\)

Eta_el_max_woDH[n,t]

P

el. eff. at max. fuel flow w/o distr. heating

Note

See the `GenericCHP`

class for all parameters and the mathematical background.

### GenericStorage (component)¶

A component to model a storage with its basic characteristics. The
GenericStorage is designed for one input and one output.
The `nominal_storage_capacity`

of the storage signifies the storage capacity. You can either set it to the net capacity or to the gross capacity and limit it using the min/max attribute.
To limit the input and output flows, you can define the `nominal_value`

in the Flow objects.
Furthermore, an efficiency for loading, unloading and a loss rate can be defined.

```
solph.components.GenericStorage(
label='storage',
inputs={b_el: solph.flows.Flow(nominal_value=9, variable_costs=10)},
outputs={b_el: solph.flows.Flow(nominal_value=25, variable_costs=10)},
loss_rate=0.001, nominal_storage_capacity=50,
inflow_conversion_factor=0.98, outflow_conversion_factor=0.8)
```

For initialising the state of charge before the first time step (time step zero) the parameter `initial_storage_level`

(default value: `None`

) can be set by a numeric value as fraction of the storage capacity.
Additionally the parameter `balanced`

(default value: `True`

) sets the relation of the state of charge of time step zero and the last time step.
If `balanced=True`

, the state of charge in the last time step is equal to initial value in time step zero.
Use `balanced=False`

with caution as energy might be added to or taken from the energy system due to different states of charge in time step zero and the last time step.
Generally, with these two parameters four configurations are possible, which might result in different solutions of the same optimization model:

`initial_storage_level=None`

,`balanced=True`

(default setting): The state of charge in time step zero is a result of the optimization. The state of charge of the last time step is equal to time step zero. Thus, the storage is not violating the energy conservation by adding or taking energy from the system due to different states of charge at the beginning and at the end of the optimization period.

`initial_storage_level=0.5`

,`balanced=True`

: The state of charge in time step zero is fixed to 0.5 (50 % charged). The state of charge in the last time step is also constrained by 0.5 due to the coupling parameter`balanced`

set to`True`

.

`initial_storage_level=None`

,`balanced=False`

: Both, the state of charge in time step zero and the last time step are a result of the optimization and not coupled.

`initial_storage_level=0.5`

,`balanced=False`

: The state of charge in time step zero is constrained by a given value. The state of charge of the last time step is a result of the optimization.

The following code block shows an example of the storage parametrization for the second configuration:

```
solph.components.GenericStorage(
label='storage',
inputs={b_el: solph.flows.Flow(nominal_value=9, variable_costs=10)},
outputs={b_el: solph.flows.Flow(nominal_value=25, variable_costs=10)},
loss_rate=0.001, nominal_storage_capacity=50,
initial_storage_level=0.5, balanced=True,
inflow_conversion_factor=0.98, outflow_conversion_factor=0.8)
```

If you want to view the temporal course of the state of charge of your storage
after the optimisation, you need to check the `storage_content`

in the results:

```
from oemof.solph import processing, views
results = processing.results(om)
column_name = (('your_storage_label', 'None'), 'storage_content')
SC = views.node(results, 'your_storage_label')['sequences'][column_name]
```

The `storage_content`

is the absolute value of the current stored energy.
By calling:

```
views.node(results, 'your_storage_label')['scalars']
```

you get the results of the scalar values of your storage, e.g. the initial
storage content before time step zero (`init_content`

).

For more information see the definition of the `GenericStorage`

class or check the Examples.

#### Using an investment object with the GenericStorage component¶

Based on the GenericStorage object the GenericInvestmentStorageBlock adds two main investment possibilities.

Invest into the flow parameters e.g. a turbine or a pump

Invest into capacity of the storage e.g. a basin or a battery cell

Investment in this context refers to the value of the variable for the ‘nominal_value’ (installed capacity) in the investment mode.

As an addition to other flow-investments, the storage class implements the possibility to couple or decouple the flows with the capacity of the storage. Three parameters are responsible for connecting the flows and the capacity of the storage:

`invest_relation_input_capacity`

fixes the input flow investment to the capacity investment. A ratio of 1 means that the storage can be filled within one time-period.

`invest_relation_output_capacity`

fixes the output flow investment to the capacity investment. A ratio of 1 means that the storage can be emptied within one period.

`invest_relation_input_output`

fixes the input flow investment to the output flow investment. For values <1, the input will be smaller and for values >1 the input flow will be larger.

You should not set all 3 parameters at the same time, since it will lead to overdetermination.

The following example pictures a Pumped Hydroelectric Energy Storage (PHES). Both flows and the storage itself (representing: pump, turbine, basin) are free in their investment. You can set the parameters to None or delete them as None is the default value.

```
solph.components.GenericStorage(
label='PHES',
inputs={b_el: solph.flows.Flow(nominal_value=solph.Investment(ep_costs=500))},
outputs={b_el: solph.flows.Flow(nominal_value=solph.Investment(ep_costs=500)},
loss_rate=0.001,
inflow_conversion_factor=0.98, outflow_conversion_factor=0.8),
investment = solph.Investment(ep_costs=40))
```

The following example describes a battery with flows coupled to the capacity of the storage.

```
solph.components.GenericStorage(
label='battery',
inputs={b_el: solph.flows.Flow()},
outputs={b_el: solph.flows.Flow()},
loss_rate=0.001,
inflow_conversion_factor=0.98,
outflow_conversion_factor=0.8,
invest_relation_input_capacity = 1/6,
invest_relation_output_capacity = 1/6,
investment = solph.Investment(ep_costs=400))
```

Note

See the `GenericStorage`

class for all parameters and the mathematical background.

### Link¶

The Link allows to model connections between two busses, e.g. modeling the transshipment of electric energy between two regions.

Note

See the `Link`

class for all parameters and the mathematical background.

### OffsetConverter (component)¶

The OffsetConverter object makes it possible to create a Converter with different efficiencies in part load condition. For this object it is necessary to define the inflow as a nonconvex flow and to set a minimum load. The following example illustrates how to define an OffsetConverter for given information for the output:

```
eta_min = 0.5 # efficiency at minimal operation point
eta_max = 0.8 # efficiency at nominal operation point
P_out_min = 20 # absolute minimal output power
P_out_max = 100 # absolute nominal output power
# calculate limits of input power flow
P_in_min = P_out_min / eta_min
P_in_max = P_out_max / eta_max
# calculate coefficients of input-output line equation
c1 = (P_out_max-P_out_min)/(P_in_max-P_in_min)
c0 = P_out_max - c1*P_in_max
# define OffsetConverter
solph.components.OffsetConverter(
label='boiler',
inputs={bfuel: solph.flows.Flow(
nominal_value=P_in_max,
max=1,
min=P_in_min/P_in_max,
nonconvex=solph.NonConvex())},
outputs={bth: solph.flows.Flow()},
coefficients = [c0, c1])
```

This example represents a boiler, which is supplied by fuel and generates heat. It is assumed that the nominal thermal power of the boiler (output power) is 100 (kW) and the efficiency at nominal power is 80 %. The boiler cannot operate under 20 % of nominal power, in this case 20 (kW) and the efficiency at that part load is 50 %. Note that the nonconvex flow has to be defined for the input flow. By using the OffsetConverter a linear relation of in- and output power with a power dependent efficiency is generated. The following figures illustrate the relations:

Now, it becomes clear, why this object has been named OffsetConverter. The linear equation of in- and outflow does not hit the origin, but is offset. By multiplying the Offset \(C_{0}\) with the binary status variable of the nonconvex flow, the origin (0, 0) becomes part of the solution space and the boiler is allowed to switch off:

\[\begin{split}& P_{in}(p, t) = C_1(t) \cdot P_{out}(p, t) + C_0(t) \cdot P_max(p) \cdot Y(t) \\\end{split}\]The symbols used are defined as follows (with Variables (V) and Parameters (P)):

symbol

attribute

type

explanation

\(P_{out}(t)\)

flow[n,o,p,t]

V

Outflow of converter

\(P_{in}(t)\)

flow[i,n,p,t]

V

Inflow of converter

\(Y(t)\)

V

Binary status variable of nonconvex inflow

\(P_{max}(t)\)

V

Maximum Outflow of converter

\(C_1(t)\)

coefficients[1][n,t]

P

Linear coefficient 1 (slope)

\(C_0(t)\)

coefficients[0][n,t]

P

Linear coefficient 0 (y-intersection)

Note that \(P_{max}(t) \cdot Y(t)\) is merged into one variable, called status_nominal[n, o, p, t].

The following figures shows the efficiency dependent on the output power, which results in a nonlinear relation:

The parameters \(C_{0}\) and \(C_{1}\) can be given by scalars or by series in order to define a different efficiency equation for every timestep.

Note

See the `OffsetConverter`

class for all parameters and the mathematical background.

### ElectricalLine (experimental)¶

Electrical line.

Note

See the `ElectricalLine`

class for all parameters and the mathematical background.

### GenericCAES (experimental)¶

Compressed Air Energy Storage (CAES). The following constraints describe the CAES:

\[\begin{split}& (1) \qquad P_{cmp}(t) = electrical\_input (t) \quad \forall t \in T \\ & (2) \qquad P_{cmp\_max}(t) = m_{cmp\_max} \cdot CAS_{fil}(t-1) + b_{cmp\_max} \quad \forall t \in\left[1, t_{max}\right] \\ & (3) \qquad P_{cmp\_max}(t) = b_{cmp\_max} \quad \forall t \notin\left[1, t_{max}\right] \\ & (4) \qquad P_{cmp}(t) \leq P_{cmp\_max}(t) \quad \forall t \in T \\ & (5) \qquad P_{cmp}(t) \geq P_{cmp\_min} \cdot ST_{cmp}(t) \quad \forall t \in T \\ & (6) \qquad P_{cmp}(t) = m_{cmp\_max} \cdot CAS_{fil\_max} + b_{cmp\_max} \cdot ST_{cmp}(t) \quad \forall t \in T \\ & (7) \qquad \dot{Q}_{cmp}(t) = m_{cmp\_q} \cdot P_{cmp}(t) + b_{cmp\_q} \cdot ST_{cmp}(t) \quad \forall t \in T \\ & (8) \qquad \dot{Q}_{cmp}(t) = \dot{Q}_{cmp_out}(t) + \dot{Q}_{tes\_in}(t) \quad \forall t \in T \\ & (9) \qquad r_{cmp\_tes} \cdot\dot{Q}_{cmp\_out}(t) = \left(1-r_{cmp\_tes}\right) \dot{Q}_{tes\_in}(t) \quad \forall t \in T \\ & (10) \quad\; P_{exp}(t) = electrical\_output (t) \quad \forall t \in T \\ & (11) \quad\; P_{exp\_max}(t) = m_{exp\_max} CAS_{fil}(t-1) + b_{exp\_max} \quad \forall t \in\left[1, t_{\max }\right] \\ & (12) \quad\; P_{exp\_max}(t) = b_{exp\_max} \quad \forall t \notin\left[1, t_{\max }\right] \\ & (13) \quad\; P_{exp}(t) \leq P_{exp\_max}(t) \quad \forall t \in T \\ & (14) \quad\; P_{exp}(t) \geq P_{exp\_min}(t) \cdot ST_{exp}(t) \quad \forall t \in T \\ & (15) \quad\; P_{exp}(t) \leq m_{exp\_max} \cdot CAS_{fil\_max} + b_{exp\_max} \cdot ST_{exp}(t) \quad \forall t \in T \\ & (16) \quad\; \dot{Q}_{exp}(t) = m_{exp\_q} \cdot P_{exp}(t) + b_{cxp\_q} \cdot ST_{cxp}(t) \quad \forall t \in T \\ & (17) \quad\; \dot{Q}_{exp\_in}(t) = fuel\_input(t) \quad \forall t \in T \\ & (18) \quad\; \dot{Q}_{exp}(t) = \dot{Q}_{exp\_in}(t) + \dot{Q}_{tes\_out}(t)+\dot{Q}_{cxp\_add}(t) \quad \forall t \in T \\ & (19) \quad\; r_{exp\_tes} \cdot \dot{Q}_{exp\_in}(t) = (1 - r_{exp\_tes})(\dot{Q}_{tes\_out}(t) + \dot{Q}_{exp\_add}(t)) \quad \forall t \in T \\ & (20) \quad\; \dot{E}_{cas\_in}(t) = m_{cas\_in}\cdot P_{cmp}(t) + b_{cas\_in}\cdot ST_{cmp}(t) \quad \forall t \in T \\ & (21) \quad\; \dot{E}_{cas\_out}(t) = m_{cas\_out}\cdot P_{cmp}(t) + b_{cas\_out}\cdot ST_{cmp}(t) \quad \forall t \in T \\ & (22) \quad\; \eta_{cas\_tmp} \cdot CAS_{fil}(t) = CAS_{fil}(t-1) + \tau\left(\dot{E}_{cas\_in}(t) - \dot{E}_{cas\_out}(t)\right) \quad \forall t \in\left[1, t_{max}\right] \\ & (23) \quad\; \eta_{cas\_tmp} \cdot CAS_{fil}(t) = \tau\left(\dot{E}_{cas\_in}(t) - \dot{E}_{cas\_out}(t)\right) \quad \forall t \notin\left[1, t_{max}\right] \\ & (24) \quad\; CAS_{fil}(t) \leq CAS_{fil\_max} \quad \forall t \in T \\ & (25) \quad\; TES_{fil}(t) = TES_{fil}(t-1) + \tau\left(\dot{Q}_{tes\_in}(t) - \dot{Q}_{tes\_out}(t)\right) \quad \forall t \in\left[1, t_{max}\right] \\ & (26) \quad\; TES_{fil}(t) = \tau\left(\dot{Q}_{tes\_in}(t) - \dot{Q}_{tes\_out}(t)\right) \quad \forall t \notin\left[1, t_{max}\right] \\ & (27) \quad\; TES_{fil}(t) \leq TES_{fil\_max} \quad \forall t \in T \\ &\end{split}\]

Table: Symbols and attribute names of variables and parameters

¶ symbol

attribute

type

explanation

\(ST_{cmp}\)

cmp_st[n,t]

V

Status of compression

\({P}_{cmp}\)

cmp_p[n,t]

V

Compression power

\({P}_{cmp\_max}\)

cmp_p_max[n,t]

V

Max. compression power

\(\dot{Q}_{cmp}\)

cmp_q_out_sum[n,t]

V

Summed heat flow in compression

\(\dot{Q}_{cmp\_out}\)

cmp_q_waste[n,t]

V

Waste heat flow from compression

\(ST_{exp}(t)\)

exp_st[n,t]

V

Status of expansion (binary)

\(P_{exp}(t)\)

exp_p[n,t]

V

Expansion power

\(P_{exp\_max}(t)\)

exp_p_max[n,t]

V

Max. expansion power

\(\dot{Q}_{exp}(t)\)

exp_q_in_sum[n,t]

V

Summed heat flow in expansion

\(\dot{Q}_{exp\_in}(t)\)

exp_q_fuel_in[n,t]

V

Heat (external) flow into expansion

\(\dot{Q}_{exp\_add}(t)\)

exp_q_add_in[n,t]

V

Additional heat flow into expansion

\(CAV_{fil}(t)\)

cav_level[n,t]

V

Filling level if CAE

\(\dot{E}_{cas\_in}(t)\)

cav_e_in[n,t]

V

Exergy flow into CAS

\(\dot{E}_{cas\_out}(t)\)

cav_e_out[n,t]

V

Exergy flow from CAS

\(TES_{fil}(t)\)

tes_level[n,t]

V

Filling level of Thermal Energy Storage (TES)

\(\dot{Q}_{tes\_in}(t)\)

tes_e_in[n,t]

V

Heat flow into TES

\(\dot{Q}_{tes\_out}(t)\)

tes_e_out[n,t]

V

Heat flow from TES

\(b_{cmp\_max}\)

cmp_p_max_b[n,t]

P

Specific y-intersection

\(b_{cmp\_q}\)

cmp_q_out_b[n,t]

P

Specific y-intersection

\(b_{exp\_max}\)

exp_p_max_b[n,t]

P

Specific y-intersection

\(b_{exp\_q}\)

exp_q_in_b[n,t]

P

Specific y-intersection

\(b_{cas\_in}\)

cav_e_in_b[n,t]

P

Specific y-intersection

\(b_{cas\_out}\)

cav_e_out_b[n,t]

P

Specific y-intersection

\(m_{cmp\_max}\)

cmp_p_max_m[n,t]

P

Specific slope

\(m_{cmp\_q}\)

cmp_q_out_m[n,t]

P

Specific slope

\(m_{exp\_max}\)

exp_p_max_m[n,t]

P

Specific slope

\(m_{exp\_q}\)

exp_q_in_m[n,t]

P

Specific slope

\(m_{cas\_in}\)

cav_e_in_m[n,t]

P

Specific slope

\(m_{cas\_out}\)

cav_e_out_m[n,t]

P

Specific slope

\(P_{cmp\_min}\)

cmp_p_min[n,t]

P

Min. compression power

\(r_{cmp\_tes}\)

cmp_q_tes_share[n,t]

P

Ratio between waste heat flow and heat flow into TES

\(r_{exp\_tes}\)

exp_q_tes_share[n,t]

P

Ratio between external heat flow into expansionand heat flows from TES and additional source\(\tau\)

m.timeincrement[n,t]

P

Time interval length

\(TES_{fil\_max}\)

tes_level_max[n,t]

P

Max. filling level of TES

\(CAS_{fil\_max}\)

cav_level_max[n,t]

P

Max. filling level of TES

\(\tau\)

cav_eta_tmp[n,t]

P

Temporal efficiency(loss factor to take intertemporal losses into account)\(electrical\_input\)

flow[list(n.electrical_input.keys())[0], p, n, t]

P

Electr. power input into compression

\(electrical\_output\)

flow[n, list(n.electrical_output.keys())[0], p, t]

P

Electr. power output of expansion

\(fuel\_input\)

flow[list(n.fuel_input.keys())[0], n, p, t]

P

Heat input (external) into Expansion

Note

See the `GenericCAES`

class for all parameters and the mathematical background.

### SinkDSM (experimental)¶

`SinkDSM`

can used to represent flexibility in a demand time series.
It can represent both, load shifting or load shedding.
For load shifting, elasticity of the demand is described by upper (~oemof.solph.custom.sink_dsm.SinkDSM.capacity_up) and lower (~oemof.solph.custom.SinkDSM.capacity_down) bounds where within the demand is allowed to vary.
Upwards shifted demand is then balanced with downwards shifted demand.
For load shedding, shedding capability is described by ~oemof.solph.custom.SinkDSM.capacity_down.
It both, load shifting and load shedding are allowed, ~oemof.solph.custom.SinkDSM.capacity_down limits the sum of both downshift categories.

`SinkDSM`

provides three approaches how the Demand-Side Management (DSM) flexibility is represented in constraints
It can be used for both, dispatch and investments modeling.

“DLR”: Implementation of the DSM modeling approach from by Gils (2015): Balancing of Intermittent Renewable Power Generation by Demand Response and Thermal Energy Storage, Stuttgart,, Details:

`SinkDSMDLRBlock`

and`SinkDSMDLRInvestmentBlock`

“DIW”: Implementation of the DSM modeling approach by Zerrahn & Schill (2015): On the representation of demand-side management in power system models, in: Energy (84), pp. 840-845, 10.1016/j.energy.2015.03.037. Details:

`SinkDSMDIWBlock`

and`SinkDSMDIWInvestmentBlock`

“oemof”: Is a fairly simple approach. Within a defined windows of time steps, demand can be shifted within the defined bounds of elasticity. The window sequentially moves forwards. Details:

`SinkDSMOemofBlock`

and`SinkDSMOemofInvestmentBlock`

Cost can be associated to either demand up shifts or demand down shifts or both.

This small example of PV, grid and SinkDSM shows how to use the component

```
# Create some data
pv_day = [(-(1 / 6 * x ** 2) + 6) / 6 for x in range(-6, 7)]
pv_ts = [0] * 6 + pv_day + [0] * 6
data_dict = {"demand_el": [3] * len(pv_ts),
"pv": pv_ts,
"Cap_up": [0.5] * len(pv_ts),
"Cap_do": [0.5] * len(pv_ts)}
data = pd.DataFrame.from_dict(data_dict)
# Do timestamp stuff
datetimeindex = pd.date_range(start='1/1/2013', periods=len(data.index), freq='H')
data['timestamp'] = datetimeindex
data.set_index('timestamp', inplace=True)
# Create Energy System
es = solph.EnergySystem(timeindex=datetimeindex)
# Create bus representing electricity grid
b_elec = solph.buses.Bus(label='Electricity bus')
es.add(b_elec)
# Create a back supply
grid = solph.components.Source(label='Grid',
outputs={
b_elec: solph.flows.Flow(
nominal_value=10000,
variable_costs=50)}
)
es.add(grid)
# PV supply from time series
s_wind = solph.components.Source(label='wind',
outputs={
b_elec: solph.flows.Flow(
fix=data['pv'],
nominal_value=3.5)}
)
es.add(s_wind)
# Create DSM Sink
demand_dsm = solph.custom.SinkDSM(label="DSM",
inputs={b_elec: solph.flows.Flow()},
demand=data['demand_el'],
capacity_up=data["Cap_up"],
capacity_down=data["Cap_do"],
delay_time=6,
max_demand=1,
max_capacity_up=1,
max_capacity_down=1,
approach="DIW",
cost_dsm_down=5)
es.add(demand_dsm)
```

Yielding the following results

Note

Keyword argument method from v0.4.1 has been renamed to approach in v0.4.2 and methods have been renamed.

The parameters demand, capacity_up and capacity_down have been normalized to allow investments modeling. To retreive the original dispatch behaviour from v0.4.1, set max_demand=1, max_capacity_up=1, max_capacity_down=1.

This component is a candidate component. It’s implemented as a custom component for users that like to use and test the component at early stage. Please report issues to improve the component.

See the

`SinkDSM`

class for all parameters and the mathematical background.

## Investment optimisation¶

As described in Optimise your energy system the typical way to optimise an energy system is the dispatch optimisation based on marginal costs. Solph also provides a combined dispatch and investment optimisation. This standard investment mode is limited to one period where all investments happen at the start of the optimization time frame. If you want to optimize longer-term horizons and allow investments at the beginning of each of multiple periods, also taking into account units lifetimes, you can try the Multi-period (investment) mode (experimental). Please be aware that the multi-period feature is experimental. If you experience any bugs or unexpected behaviour, please report them.

In the standard investment mode, based on investment costs you can compare the usage of existing components against building up new capacity. The annual savings by building up new capacity must therefore compensate the annuity of the investment costs (the time period does not have to be one year, but depends on your Datetime index).

See the API of the `Investment`

class to see all possible parameters.

Basically, an instance of the Investment class can be added to a Flow, a
Storage or a DSM Sink. All parameters that usually refer to the *nominal_value/capacity* will
now refer to the investment variables and existing capacity. It is also
possible to set a maximum limit for the capacity that can be build.
If existing capacity is considered for a component with investment mode enabled,
the *ep_costs* still apply only to the newly built capacity, i.e. the existing capacity
comes at no costs.

The investment object can be used in Flows and some components. See the Solph components section for detailed information of each component. Besides the flows, it can be invested into

For example if you want to find out what would be the optimal capacity of a wind
power plant to decrease the costs of an existing energy system, you can define
this model and add an investment source.
The *wind_power_time_series* has to be a normalised feed-in time series of you
wind power plant. The maximum value might be caused by limited space for wind
turbines.

```
solph.components.Source(label='new_wind_pp', outputs={electricity: solph.flows.Flow(
fix=wind_power_time_series,
nominal_value=solph.Investment(ep_costs=epc, maximum=50000))})
```

Let’s slightly alter the case and consider for already existing wind power capacity of 20,000 kW. We’re still expecting the total wind power capacity, thus we allow for 30,000 kW of new installations and formulate as follows.

```
solph.components.Source(label='new_wind_pp', outputs={electricity: solph.flows.Flow(
fix=wind_power_time_series,
nominal_value=solph.Investment(ep_costs=epc,
maximum=30000,
existing=20000))})
```

The periodical costs (*ep_costs*) are typically calculated as annuities, i.e. as follows:

```
capex = 1000 # investment cost
lifetime = 20 # life expectancy
wacc = 0.05 # weighted average of capital cost
epc = capex * (wacc * (1 + wacc) ** lifetime) / ((1 + wacc) ** lifetime - 1)
```

This also implemented in the annuity function of the economics module in the oemof.tools package. The code above would look like this:

```
from oemof.tools import economics
epc = economics.annuity(1000, 20, 0.05)
```

So far, the investment costs and the installed capacity are mathematically a
line through origin. But what if there is a minimum threshold for doing an
investment, e.g. you cannot buy gas turbines lower than a certain
nominal power, or, the marginal costs of bigger plants
decrease.
Therefore, you can use the parameter *nonconvex* and *offset* of the
investment class. Both, work with investment in flows and storages. Here is an
example of a converter:

```
trafo = solph.components.Converter(
label='converter_nonconvex',
inputs={bus_0: solph.flows.Flow()},
outputs={bus_1: solph.flows.Flow(
nominal_value=solph.Investment(
ep_costs=4,
maximum=100,
minimum=20,
nonconvex=True,
offset=400))},
conversion_factors={bus_1: 0.9})
```

In this examples, it is assumed, that independent of the size of the converter, there are always fix investment costs of 400 (€). The minimum investment size is 20 (kW) and the costs per installed unit are 4 (€/kW). With this option, you could theoretically approximate every cost function you want. But be aware that for every nonconvex investment flow or storage you are using, an additional binary variable is created. This might boost your computing time into the limitless.

The following figures illustrates the use of the nonconvex investment flow.
Here, \(c_{invest,fix}\) is the *offset* value and \(c_{invest,var}\) is
the *ep_costs* value:

In case of a convex investment (which is the default setting
nonconvex=False), the *minimum* attribute leads to a forced investment,
whereas in the nonconvex case, the investment can become zero as well.

The calculation of the specific costs per kilowatt installed capacity results in the following relation for convex and nonconvex investments:

See `InvestmentFlow`

and
`GenericInvestmentStorageBlock`

for all the
mathematical background, like variables and constraints, which are used.

Note

At the moment the investment class is not compatible with the MIP classes `NonConvex`

.

## Multi-period (investment) mode (experimental)¶

Sometimes you might be interested in how energy systems could evolve in the longer-term, e.g. until 2045 or 2050 to meet some carbon neutrality and climate protection or RES and energy efficiency targets.

While in principle, you could try to model this in oemof.solph using the standard investment mode described above (see Investment optimisation), you would make the implicit assumption that your entire system is built at the start of your optimization and doesn’t change over time. To address this shortcoming, the multi-period (investment) feature has been introduced. Be aware that it is still experimental. So feel free to report any bugs or unexpected behaviour if you come across them.

While in principle, you can define a dispatch-only multi-period system, this doesn’t make much sense. The power of the multi-period feature only unfolds if you look at long-term investments. Let’s see how.

First, you start by defining your energy system as you might have done before, but you

choose a longer-term time horizon (spanning multiple years, i.e. multiple periods) and

explicitly define the periods attribute of your energy system which lists the time steps for each period.

```
import pandas as pd
import oemof.solph as solph
my_index = pd.date_range('1/1/2013', periods=17520, freq='H')
periods = [
pd.date_range('1/1/2013', periods=8760, freq='H'),
pd.date_range('1/1/2014', periods=8760, freq='H'),
]
my_energysystem = solph.EnergySystem(timeindex=my_index, periods=periods)
```

If you want to use a multi-period model you have define periods of your energy system explicitly. This way, you are forced to critically think, e.g. about handling leap years, and take some design decisions. It is possible to define periods with different lengths, but remember that decommissioning of components is possible only at the beginning of each period. This means that if the life of a component is just a little longer, it will remain for the entire next period. This can have a particularly large impact the longer your periods are.

To assist you, here is a plain python snippet that includes leap years which you can just copy and adjust to your needs:

```
def determine_periods(datetimeindex):
"""Explicitly define and return periods of the energy system
Leap years have 8784 hourly time steps, regular years 8760.
Parameters
----------
datetimeindex : pd.date_range
DatetimeIndex of the model comprising all time steps
Returns
-------
periods : list
periods for the optimization run
"""
years = sorted(list(set(getattr(datetimeindex, "year"))))
periods = []
filter_series = datetimeindex.to_series()
for number, year in enumerate(years):
start = filter_series.loc[filter_series.index.year == year].min()
end = filter_series.loc[filter_series.index.year == year].max()
periods.append(pd.date_range(start, end, freq=datetimeindex.freq))
return periods
```

So if you want to use this, the above would simplify to:

```
import pandas as pd
import oemof.solph as solph
# Define your method (or import it from somewhere else)
def determine_periods(datetimeindex):
...
my_index = pd.date_range('1/1/2013', periods=17520, freq='H')
periods = determine_periods(my_index) # Make use of method
my_energysystem = solph.EnergySystem(timeindex=my_index, periods=periods)
```

Then you add all the *components* and *buses* to your energy system, just as you are used to with, but with few additions.

```
hydrogen_bus = solph.buses.Bus(label="hydrogen")
coal_bus = solph.buses.Bus(label="coal")
electricity_bus = solph.buses.Bus(label="electricity")
hydrogen_source = solph.components.Source(
label="green_hydrogen",
outputs={
hydrogen_bus: solph.flows.Flow(
variable_costs=[25] * 8760 + [30] * 8760
)
},
)
coal_source = solph.components.Source(
label="hardcoal",
outputs={
coal_bus: solph.flows.Flow(variable_costs=[20] * 8760 + [24] * 8760)
},
)
electrical_sink = solph.components.Sink(
label="electricity_demand",
inputs={
electricity_bus: solph.flows.Flow(
nominal_value=1000, fix=[0.8] * len(my_index)
)
},
)
```

So defining buses is the same as for standard models. Also defining components that do not have any investments associated with them or any lifetime limitations is the same.

Now if you want to have components that can be invested into, you use the investment option, just as in Investment optimisation, but with a few minor additions and modifications in the investment object itself which you specify by additional attributes:

You have to specify a lifetime attribute. This is the components assumed technical lifetime in years. If it is 20 years, the model invests into it and your simulation has a 30 years horizon, the plant will be decommissioned. Now the model is free to reinvest or choose another option to fill up the missing capacity.

You can define an initial age if you have existing capacity. If you do not specify anything, the default value 0 will be used, meaning your existing capacity has just been newly invested.

You can define an interest_rate that the investor you model has, i.e. the return he desires expressed as the weighted average osts of capital (wacc) and used for calculating annuities in the model itself.

You also can define fixed_costs, i.e. costs that occur every period independent of the plants usage.

Here is an example

```
hydrogen_power_plant = solph.components.Converter(
label="hydrogen_pp",
inputs={hydrogen_bus: solph.flows.Flow()},
outputs={
electricity_bus: solph.flows.Flow(
nominal_value=solph.Investment(
maximum=1000,
ep_costs=1e6,
lifetime=30,
interest_rate=0.06,
fixed_costs=100,
),
variable_costs=3,
)
},
conversion_factors={electricity_bus: 0.6},
)
```

Warning

The ep_costs attribute for investments is used in a different way in a multi-period model. Instead of periodical costs, it depicts (nominal or real) investment expenses, so actual Euros you have to pay per kW or MW (or whatever power or energy unit) installed. Also, you can depict a change in investment expenses over time, so instead of providing a scalar value, you could define a list with investment expenses with one value for each period modelled.

Annuities are calculated within the model. You do not have to do that. Also the model takes care of discounting future expenses / cashflows.

Below is what it would look like if you altered ep_costs and fixed_costs per period. This can be done by simply providing a list. Note that the length of the list must equal the number of periods of your model. This would mean that for investments in the particular period, these values would be the one that are applied over their lifetime.

```
hydrogen_power_plant = solph.components.Converter(
label="hydrogen_pp",
inputs={hydrogen_bus: solph.flows.Flow()},
outputs={
electricity_bus: solph.flows.Flow(
nominal_value=solph.Investment(
maximum=1000,
ep_costs=[1e6, 1.1e6],
lifetime=30,
interest_rate=0.06,
fixed_costs=[100, 110],
),
variable_costs=3,
)
},
conversion_factors={electricity_bus: 0.6},
)
```

For components that is not invested into, you also can specify some additional attributes for their inflows and outflows:

You can specify a lifetime attribute. This can be used to depict existing plants going offline when reaching their lifetime.

You can define an initial age. Also, this can be used for existing plants.

You also can define fixed_costs, i.e. costs that occur every period independent of the plants usage. How they are handled depends on whether the flow has a limited or an unlimited lifetime.

```
coal_power_plant = solph.components.Converter(
label="existing_coal_pp",
inputs={coal_bus: solph.flows.Flow()},
outputs={
electricity_bus: solph.flows.Flow(
nominal_value=600,
max=1,
min=0.4,
lifetime=50,
age=46,
fixed_costs=100,
variable_costs=3,
)
},
conversion_factors={electricity_bus: 0.36},
)
```

To solve our model and retrieve results, you basically perform the same operations as for standard models. So it works like this:

```
my_energysystem.add(
hydrogen_bus,
coal_bus,
electricity_bus,
hydrogen_source,
coal_source,
electrical_sink,
hydrogen_power_plant,
coal_power_plant,
)
om = solph.Model(my_energysystem)
om.solve(solver="cbc", solve_kwargs={"tee": True})
# Obtain results
results = solph.processing.results(om)
hydrogen_results = solph.views.node(results, "hydrogen_pp")
# Show investment plan for hydrogen power plants
print(hydrogen_results["period_scalars"])
```

The keys in the results dict in a multi-period model are “sequences” and “period_scalars”. So for sequences, it is all the same, while for scalar values, we now have values for each period.

Besides the invest variable, new variables are introduced as well. These are:

total: The total capacity installed, i.e. how much is actually there in a given period.

old: (Overall) capacity to be decommissioned in a given period.

old_end: Endogenous capacity to be decommissioned in a given period. This is capacity that has been invested into in the model itself.

old_exo: Exogenous capacity to be decommissioned in a given period. This is capacity that was already existing and given by the existing attribute.

Note

You can specify a discount_rate for the model. If you do not do so, 0.02 will be used as a default, corresponding to sort of a social discount rate. If you work with costs in real terms, discounting is obsolete, so define discount_rate = 0 in that case.

You can specify an interest_rate for every investment object. If you do not do so, it will be chosen the same as the model’s discount_rate. You could use this default to model a perfect competition administered by some sort of social planner, but even in a social planner setting, you might want to deviate from the discount_rate value and/or discriminate among technologies with different risk profiles and hence different interest requirements.

For storage units, the initial_content is not allowed combined with multi-period investments. The storage inflow and outflow are forced to zero until the storage unit is invested into.

You can specify periods of different lengths, but the frequency of your timeindex needs to be consistent. Also, you could use the timeincrement attribute of the energy system to model different weightings. Be aware that this has not yet been tested.

For now, both, the timeindex as well as the timeincrement of an energy system have to be defined since they have to be of the same length for a multi-period model.

You can choose whether to re-evaluate assets at the end of the optimization horizon. If you set attribute use_remaining_value of the energy system to True (defaults to False), this leads to the model evaluating the difference in the asset value at the end of the optimization horizon vs. at the time the investment was made. The difference in value is added to or subtracted from the respective investment costs increment, assuming assets are to be liquidated / re-evaluated at the end of the optimization horizon.

Also please be aware, that periods correspond to years by default. You could also choose monthly periods, but you would need to be very careful in parameterizing your energy system and your model and also, this would mean monthly discounting (if applicable) as well as specifying your plants lifetimes in months.

## Modelling cellular energy systems and modularizing energy system models¶

The cellular approach is a concept proposed by the [VDE-ETG](https://shop.vde.com/en/vde-study-the-cellular-approach). It is related to smart-grids and multi-microgrid systems but extends both. The idea is to group the components of an energy system into a hierarchically aggregating structure of cells. For example, the sources, sinks, storages and converters of a household could be a single cell. Then a group of physically neighboring households could form another cell, consisting of household-cells. This behaviour can be scaled up. The real game-changer in the cellular approach is the way the cells are operated, which will not be covered here. Here, we focus on the way such cellular energy systems can be modeled.

So far, the implementation in solph is just a neat way to group different parts of a larger energy system into cells. However, the implementation can also be regarded as a precursor for further functionality. Decomposition techniques such as [Benders](https://en.wikipedia.org/wiki/Benders_decomposition) or [Dantzig-Wolfe](https://en.wikipedia.org/wiki/Dantzig%E2%80%93Wolfe_decomposition) could be implemented in solph. These methods are dependent on a special constraint matrix structure, which the cellular modelling approach presented here is helping to obtain.

### Modelling procedure¶

Similar to the creation of regular energy systems, the creation of energy cells is the first step in model creation. Essentially,
each energy cell is just an energy system, therefore we use the class `oemof.solph.EnergySystem`

to create energy cells.

```
from oemof.solph import EnergySystem
es = EnergySystem(
label="es", timeindex=timeindex, infer_last_interval=False
)
ec_1 = EnergySystem(
label="ec_1", timeindex=timeindex, infer_last_interval=False
)
ec_2 = EnergySystem(
label="ec_2", timeindex=timeindex, inver_last_interval=False
)
```

Now we can go on and add components to the energy cells just like we do with regular energy systems.

```
from oemof import solph
bus_el_es = solph.buses.Bus(label="bus_el_es")
es.add(bus_el_es)
bus_el_ec_1 = solph.buses.Bus(label="bus_el_ec_1")
sink_el_ec_1 = solph.components.Sink(
label="sink_el_ec_1",
inputs={bus_el_ec_1: flows.Flow(fix=10, nominal_value=1)},
)
source_el_ec_1 = solph.components.Source(
label="source_el_ec_1",
outputs={
bus_el_ec_1: flows.Flow(
max=30, nominal_value=1, variable_costs=10,
),
},
)
ec_1.add(bus_el_ec_1, sink_el_ec_1, source_el_ec_1)
```

Note

This is just an exemplary piece of code. A (little bit more interesting) working example can be found in the examples.

The next step would be to model the connections between cells. Here, we resort to the class
`oemof.solph.components.Link`

. Each connection Link has two inputs (from the
“parent cell” and the “child cell”) and two outputs (to the “parent cell” and the “child
cell”). A connection between the “parent cell” es and the “child cell” ec_1 could look
like this:

```
connector_el_ec_1 = solph.buses.Bus(
label="connector_el_ec_1",
inputs={
bus_el_es: flows.Flow(),
bus_el_ec_1: flows.Flow(),
},
outputs={
bus_el_es: flows.Flow(),
bus_el_ec_1: flows.Flow(),
},
conversion_factors={
(bus_el_es, bus_el_ec_1): 0.85,
(bus_el_ec_1, bus_el_es): 0.85
}
)
es.add(connector_el_ec_1)
```

The conversion_factors can be used to model transmission losses. Here, a symmetrical loss of 15% is assumed. All connection Links are added to the upmost energy cell.

Note

Note that we do not add the energy cells as components to their parent cells! Instead, the hierarchical structure is flattened and all connections between the cells are created as depicted above.

The last step is to create (and solve) the model. Again, this is fairly similar to the
regular model creation. However, instead of passing just one instance of
`oemof.solph.EnergySystem`

, a list of energy systems is passed.

Warning

By convention the first element of the list is assumed to be the upmost energy cell. The ordering afterwards does not play a role.

Note

The resulting model is monolithic. This means that all components of all energy
cells are actually grouped into one pyomo model. It would, therefore, also be possible
to model all the components in one `oemof.solph.EnergySystem`

instance and
the results would be identical.

```
cmodel = Model(
energysystem=[es, ec_1, ec_2, ...]
)
cmodel.solve()
```

As pointed out above, the resulting model is monolithic. Nonetheless, this modelling approach holds some benefits:

Better overview through segmentation of the energy system

(Facilitated) opportunity to model cellular energy systems where the energy exchanged between cells is of interest

Segmentation of the energy system is a necessary precursor for distributed optimization via Dantzig-Wolfe

## Mixed Integer (Linear) Problems¶

Solph also allows you to model components with respect to more technical details, such as minimum power production. This can be done in both possible combinations, as dispatch optimization with fixed capacities or combined dispatch and investment optimization.

### Dispatch Optimization¶

In dispatch optimization, it is assumed that the capacities of the assets are already known,
but the optimal dispatch strategy must be obtained.
For this purpose, the class `NonConvex`

should be used, as seen in the following example.

Note that this flow class’s usage is incompatible with the `Investment`

option. This means that,
as stated before, the optimal capacity of the converter cannot be obtained using the `NonConvexFlow`

class, and only the optimal dispatch strategy of an existing asset with a given capacity can be optimized here.

```
b_gas = solph.buses.Bus(label='natural_gas')
b_el = solph.buses.Bus(label='electricity')
b_th = solph.buses.Bus(label='heat')
solph.components.Converter(
label='pp_chp',
inputs={b_gas: solph.flows.Flow()},
outputs={b_el: solph.flows.Flow(
nonconvex=solph.NonConvex(),
nominal_value=30,
min=0.5),
b_th: solph.flows.Flow(nominal_value=40)},
conversion_factors={b_el: 0.3, b_th: 0.4})
```

The class `NonConvex`

for the electrical output of the created Converter (i.e., CHP)
will create a ‘status’ variable for the flow.
This will be used to model, for example, minimal/maximal power production constraints if the
attributes min/max of the flow are set. It will also be used to include start-up constraints and costs
if corresponding attributes of the class are provided. For more information, see the API of the
`NonConvexFlow`

class.

Note

The usage of this class can sometimes be tricky as there are many interdenpendencies. So check out the examples and do not hesitate to ask the developers if your model does not work as expected.

### Combination of Dispatch and Investment Optimisation¶

Since version ‘v0.5’, it is also possilbe to combine the investment and nonconvex option.
Therefore, a new constraint block for flows, called `InvestNonConvexFlowBlock`

has been developed,
which combines both `Investment`

and `NonConvex`

classes.
The new class offers the possibility to perform the investment optimization of an asset considering min/max values of the flow
as fractions of the optimal capacity. Moreover, it obtains the optimal ‘status’ of the flow during the simulation period.

It must be noted that in a straighforward implementation, a binary variable
representing the ‘status’ of the flow at each time is multiplied by the ‘invest’ parameter,
which is a continuous variable representing the capacity of the asset being optimized (i.e., \(status \times invest\)).
This nonlinearity is linearised in the
`InvestNonConvexFlowBlock`

```
b_diesel = solph.buses.Bus(label='diesel')
b_el = solph.buses.Bus(label='electricity')
solph.components.Converter(
label='diesel_genset',
inputs={b_diesel: solph.flows.Flow()},
outputs={
b_el: solph.flows.Flow(
variable_costs=0.04,
min=0.2,
max=1,
nonconvex=solph.NonConvex(),
nominal_value=solph.Investment(
ep_costs=90,
maximum=150, # required for the linearization
),
)
},
conversion_factors={b_el: 0.3})
```

The following diagram shows the duration curve of a typical diesel genset in a hybrid mini-grid system consisting of a diesel genset,
PV cells, battery, inverter, and rectifier. By using the `InvestNonConvexFlowBlock`

class,
it is possible to obtain the optimal capacity of this component and simultaneously limit its operation between min and max loads.

Without using the new `InvestNonConvexFlowBlock`

class, if the same system is optimized again, but this
time using the `InvestmentFlowBlock`

, the corresponding duration curve would be similar to the following
figure. However, assuming that the diesel genset has a minimum operation load of 20% (as seen in the figure), the
`InvestmentFlowBlock`

cannot prevent operations at lower loads than 20%, and it would result in
an infeasible operation of this device for around 50% of its annual operation.

Moreover, using the `InvestmentFlowBlock`

class in the given case study would result in a significantly
oversized diesel genset, which has a 30% larger capacity compared with the optimal capacity obtained from the
`InvestNonConvexFlowBlock`

class.

Solving such an optimisation problem considering min/max loads without the `InvestNonConvexFlowBlock`

class, the only possibility is first to obtain the optimal capacity using the
`InvestmentFlowBlock`

and then implement the min/max loads using the
`NonConvexFlowBlock`

class. The following duration curve would be obtained by applying
this method to the same diesel genset.

Because of the oversized diesel genset obtained from this approach, the capacity of the PV and battery in the given case study
would be 13% and 43% smaller than the capacities obtained using the `NonConvexInvestmentFlow`

class.
This results in a 15% reduction in the share of renewable energy sources to cover the given demand and a higher levelized
cost of electricity. Last but not least, apart from the nonreliable results, using `Investment`

and `NonConvex`

classes for the dispatch and investment optimization of the given case study
increases the computation time by more than 9 times compared to the
`NonConvexInvestmentFlow`

class.

## Adding additional constraints¶

You can add additional constraints to your `Model`

.
See Custom constraints to learn how to do it.

Some predefined additional constraints can be found in the
`constraints`

module.

Emission limit for the model ->

`emission_limit()`

Generic integral limit (general form of emission limit) ->

`generic_integral_limit()`

Coupling of two variables e.g. investment variables) with a factor ->

`equate_variables()`

Overall investment limit ->

`investment_limit()`

Generic investment limit ->

`additional_investment_flow_limit()`

Limit active flow count ->

`limit_active_flow_count()`

Limit active flow count by keyword ->

`limit_active_flow_count_by_keyword()`

## The Grouping module (Sets)¶

To construct constraints,
variables and objective expressions inside all Block classes
and the `models`

modules, so called groups are used. Consequently,
certain constraints are created for all elements of a specific group. Thus,
mathematically the groups depict sets of elements inside the model.

The grouping is handled by the solph grouping module `groupings`

which is based on the groupings module functionality of oemof network. You
do not need to understand how the underlying functionality works. Instead, checkout
how the solph grouping module is used to create groups.

The simplest form is a function that looks at every node of the energy system and returns a key for the group depending e.g. on node attributes:

```
def constraint_grouping(node):
if isinstance(node, Bus) and node.balanced:
return blocks.Bus
if isinstance(node, Converter):
return blocks.Converter
GROUPINGS = [constraint_grouping]
```

This function can be passed in a list to groupings of
`oemof.solph.network.energy_system.EnergySystem`

. So that we end up with two groups,
one with all Converters and one with all Buses that are balanced. These
groups are simply stored in a dictionary. There are some advanced functionalities
to group two connected nodes with their connecting flow and others
(see for example: FlowsWithNodes class in the oemof.network package).

## Using the Excel (csv) reader¶

Alternatively to a manual creation of energy system component objects as describe above, can also be created from a excel sheet (libreoffice, gnumeric…).

The idea is to create different sheets within one spreadsheet file for different components. Afterwards you can loop over the rows with the attributes in the columns. The name of the columns may differ from the name of the attribute. You may even create two sheets for the GenericStorage class with attributes such as C-rate for batteries or capacity of turbine for a PHES.

Once you have create your specific excel reader you can lower the entry barrier for other users. It is some sort of a GUI in form of platform independent spreadsheet software and to make data and models exchangeable in one archive.

See Spreadsheet (Excel) Reader for an excel reader example.

## Handling Results¶

The main purpose of the processing module is to collect and organise results. The views module will provide some typical representations of the results. Plots are not part of solph, because plots are highly individual. However, the provided pandas.DataFrames are a good start for plots. Some basic functions for plotting of optimisation results can be found in the separate repository oemof_visio.

The `processing.results`

function gives back the results as a python
dictionary holding pandas Series for scalar values and pandas DataFrames for
all nodes and flows between them. This way we can make use of the full power
of the pandas package available to process the results.

See the pandas documentation to learn how to visualise, read or write or how to access parts of the DataFrame to process them.

The results chapter consists of three parts:

The first step is the processing of the results (Collecting results) This is followed by basic examples of the general analysis of the results (General approach) and finally the use of functionality already included in solph for providing a quick access to your results (Easy access). Especially for larger energy systems the general approach will help you to write your own results processing functions.

### Collecting results¶

Collecting results can be done with the help of the processing module. A solved model is needed:

```
[...]
model.solve(solver=solver)
results = solph.processing.results(model)
```

The scalars and sequences describe nodes (with keys like (node, None)) and flows between nodes (with keys like (node_1, node_2)). You can directly extract the data in the dictionary by using these keys, where “node” is the name of the object you want to address. Processing the results is the prerequisite for the examples in the following sections.

### General approach¶

As stated above, after processing you will get a dictionary with all result data. If you want to access your results directly via labels, you can continue with Easy access. For a systematic analysis list comprehensions are the easiest way of filtering and analysing your results.

The keys of the results dictionary are tuples containing two nodes. Since flows have a starting node and an ending node, you get a list of all flows by filtering the results using the following expression:

```
flows = [x for x in results.keys() if x[1] is not None]
```

On the same way you can get a list of all nodes by applying:

```
nodes = [x for x in results.keys() if x[1] is None]
```

Probably you will just get storages as nodes, if you have some in your energy system. Note, that just nodes containing decision variables are listed, e.g. a Source or a Converter object does not have decision variables. These are in the flows from or to the nodes.

All items within the results dictionary are dictionaries and have two items with ‘scalars’ and ‘sequences’ as keys:

```
for flow in flows:
print(flow)
print(results[flow]['scalars'])
print(results[flow]['sequences'])
```

There many options of filtering the flows and nodes as you prefer. The following will give you all flows which are outputs of converter:

```
flows_from_converter = [x for x in flows if isinstance(
x[0], solph.components.Converter)]
```

You can filter your flows, if the label of in- or output contains a given string, e.g.:

```
flows_to_elec = [x for x in results.keys() if 'elec' in x[1].label]
```

Getting all labels of the starting node of your investment flows:

```
flows_invest = [x[0].label for x in flows if hasattr(
results[x]['scalars'], 'invest')]
```

### Easy access¶

The solph package provides some functions which will help you to access your results directly via labels, which is helpful especially for small energy systems. So, if you want to address objects by their label, you can convert the results dictionary such that the keys are changed to strings given by the labels:

```
views.convert_keys_to_strings(results)
print(results[('wind', 'bus_electricity')]['sequences']
```

Another option is to access data belonging to a grouping by the name of the grouping (note also this section on groupings. Given the label of an object, e.g. ‘wind’ you can access the grouping by its label and use this to extract data from the results dictionary.

```
node_wind = energysystem.groups['wind']
print(results[(node_wind, bus_electricity)])
```

However, in many situations it might be convenient to use the views module to collect information on a specific node. You can request all data related to a specific node by using either the node’s variable name or its label:

```
data_wind = solph.views.node(results, 'wind')
```

A function for collecting and printing meta results, i.e. information on the objective function, the problem and the solver, is provided as well:

```
meta_results = solph.processing.meta_results(om)
pp.pprint(meta_results)
```