Relationships between investments

The additional constraints from the package do not allow for relationships between individual investments. However, it can sometimes be beneficial to model relationships between individual technologies, e.g., for modelling a maximum budget for investments in a set of technologies or couplings between different technology investments. To this end, EnergyModelsInvestments provides the user with several functions which must be called after the creation of the JuMP model instance.

If not stated differently, the constraints are added for all investment periods, i.e., $\forall t_{inv} \in T^{Inv}$.

Maximum budget

The function max_budget can be utilized to provide a maximum budget for investments in a given set of technologies. This budget can be either for all investment periods within the model or alternatively, if you e.g., utilize a TwoLevelTree and want to limit the budget within an individual scenario or only have a budget in a subset of the investment periods, for a specified set of investment periods.

Given a set of investments specified as a Vector of Tuples, corresponding to the element as first entry and the capacity prefix as second entry, it limits the sum of the undiscounted capital expenditures into these technologies based on the specified limit.

An example would be given by the following code snippet

# Included technologies in the budgets
investments = [
    (tech, :cap),
    (storage, :stor_charge),
    (pipeline, :trans_cap),
]

# Maximum budget
limit = 1e7

# Addition of the constraints
max_budget(m, limit, investments, 𝒯)

In this situation, we have an overall budget of 10 000 000 € to invest into three technologies, a technology tech in which the capacity variables use the prefix :cap, a storage technology storage in which the capacity variables use the prefix :stor_charge (the charging capacity), and a transmission mode pipeline in which the capacity variables use the prefix :trans_cap. This budget applies to all investment periods, given as $T^{Inv}$, of the time structure.

This corresponds to the following mathematical description

\[\begin{aligned} 1e7 \geq \sum_{t_{inv} \in T^{Inv}} & \texttt{cap\_add}[tech, t_{inv}] + {} \\ & \texttt{stor\_charge\_add}[storage, t_{inv}] + {} \\ & \texttt{trans\_cap\_add}[pipeline, t_{inv}] \\ \end{aligned}\]

or generally speaking to

\[limit \geq \sum_{t_{inv} \in sps, (element, prefix) \in investments}{\texttt{prefix\_add}[element, t_{inv}]}\]

where $sps$ corresponds either to all investment periods $T^{Inv}$ as in the example above or to the specified investment periods through the keyword arguments $sps\_spec$.

Relations between investment actions

There are several functions for creating relations between investment actions in technologies. These relations link binary activations, not necessarily positive capacity additions. This implies that the capacity additions of the elements are not directly affected.

Maximum and minimum number of investments

The functions max_investments and min_investments provide bounds to the number of investment actions for a given set of technologies.

Considering again the example from above,

# Included technologies in the budgets
investments = [
    (tech, :cap),
    (storage, :stor_charge),
    (pipeline, :trans_cap),
]

# Maximum number of investment actions
max_limit = 5
max_investments(m, max_limit, investments, 𝒯)

# Minimum number of investment actions
min_limit = 3
min_investments(m, min_limit, investments, 𝒯)

the addition of the constraints implies that we need to have at least ***three*** investment actions and at maximum ***five***. This limit applies to all investment periods, given as $T^{Inv}$, of the time structure.

This corresponds to the following mathematical description

\[\begin{aligned} 3 \leq \sum_{t_{inv} \in T^{Inv}} & \texttt{cap\_invest\_b}[tech, t_{inv}] + {} \\ & \texttt{stor\_charge\_invest\_b}[storage, t_{inv}] + {} \\ & \texttt{trans\_cap\_invest\_b}[pipeline, t_{inv}] \\ \leq {} & {} 5 \end{aligned}\]

or generally speaking to

\[\sum_{t_{inv} \in sps, (element, prefix) \in investments}{\texttt{prefix\_invest\_b}[element, t_{inv}]} \leq limit\]

or

\[\sum_{t_{inv} \in sps, (element, prefix) \in investments}{\texttt{prefix\_invest\_b}[element, t_{inv}]} \geq limit\]

where $sps$ corresponds either to all investment periods $T^{Inv}$ as in the example above or to the specified investment periods through the keyword arguments $sps\_spec$.

Required investment actions

Required investment actions, introduced through the function require_investment, imply that you must invest in the prerequisite element if you want to invest in the dependent element. However, it is possible to invest in the prerequisite element without investing into the dependent element

Consider the following example:

# Dependent elements
elements_dep = [
    (tech_1, :cap),
    (tech_2, :cap),
]

# Prerequisite element
element_pre = (tech_3, :cap)

# Add the constraints
require_investment(m, elements_dep, element_pre, 𝒯)

This corresponds to the following mathematical description

\[\texttt{cap\_invest\_b}[tech\_1, t_{inv}] \leq \texttt{cap\_invest\_b}[tech\_3, t_{inv}]\]

or generally speaking $\forall (element\_dep, prefix\_dep) \in elements\_dep$ to

\[\texttt{prefix\_dep\_invest\_b}[element\_dep, t_{inv}] \leq \texttt{prefix\_pre\_invest\_b}[element\_pre, t_{inv}]\]

Couple investment actions

Coupling investment actions, introduced through the function couple_investments, behave similar to require_investment, but the constraint is replaced through an equality constraint requiring the investment in both elements or none.

Consider the following example:

# Investments to exclude
elements = [
    (tech_1, :cap),
    (tech_2, :cap),
]

# Add the constraints
couple_investments(m, elements, 𝒯)

This corresponds to the following mathematical description

\[\texttt{cap\_invest\_b}[tech\_1, t_{inv}] = \texttt{cap\_invest\_b}[tech\_2, t_{inv}]\]

or generally speaking $\forall (element, prefix) \in elements[2:end]$ to

\[\texttt{elements[1][2]\_invest\_b}[elements[1][1], t_{inv}] = \texttt{prefix\_invest\_b}[element, t_{inv}]\]

Exclude investment actions

Exclude investments, introduced through the function exclude_investments, results in constraints that you can in each strategic period $t_{inv}$ only invest in one of the elements.

Consider the following example:

# Investments to exclude
elements = [
    (tech_1, :cap),
    (tech_2, :cap),
]

# Add the constraints
exclude_investments(m, elements, 𝒯)

This corresponds to the following mathematical description

\[\texttt{cap\_invest\_b}[tech\_1, t_{inv}] + \texttt{cap\_invest\_b}[tech\_2, t_{inv}] \leq 1\]

or generally speaking to

\[\sum_{(element, prefix) \in elements} \texttt{prefix\_invest\_b}[element, t_{inv}] \leq 1\]

Relations between installed capacities

There are several functions for creating relations between the installed capacity of two elements.

Coupled capacities

The installed capacities (specified by prefix_1 and prefix_2) of two elements (specified by element_1 and element_2) can be coupled through the function couple_capacity.

Consider the following example:

# Investments to couple
element_1 = (tech_1, :cap)
element_2 = (tech_2, :cap)

# Ratio between the capacity
capacity_ratio = 2

# Addition of the constraints
couple_capacity(m, element_1, element_2, 𝒯; capacity_ratio)

This implies that the capacity of $tech\_1$ must be twice the capacity of $tech\_2$. The following mathematical description is hence given

\[\texttt{cap\_current}[tech\_1, t_{inv}] = 2 \times \texttt{cap\_current}[tech\_2, t_{inv}]\]

or generally speaking to

\[\begin{aligned} \texttt{element\_1[2]\_current}&[element\_1[1], t_{inv}] = \\ &capacity\_ratio \times \texttt{element\_2[2]\_current}[element\_2[1], t_{inv}] \end{aligned}\]

Required and preceded capacities

Capacity requirements can be introduced through the functions require_capacity and precede_capacity. These functions connect the capacities of dependent elements, specified as (element_dep, prefix_dep) tuples, with the capacity of a prerequisite element specified as an (element_pre, prefix_pre) tuple. The two capacities are linked through the keyword argument capacity_ratio.

Consider the following example:

# Dependent elements
elements_dep = [
    (tech_1, :cap),
    (tech_2, :cap),
]

# Prerequisite element
element_pre = (tech_3, :cap)

# Ratio between the capacity
capacity_ratio = 2

# Addition of the constraints
require_capacity(m, elements_dep, element_pre, 𝒯; capacity_ratio)

This implies that we can invest in at most twice the capacity of each dependent technology relative to the capacity of $tech\_3$. The following mathematical description is hence given

\[\begin{aligned} \texttt{cap\_current}[tech\_1, t_{inv}] & {} \leq 2 \times \texttt{cap\_current}[tech\_3, t_{inv}] \\ \texttt{cap\_current}[tech\_2, t_{inv}] & {} \leq 2 \times \texttt{cap\_current}[tech\_3, t_{inv}] \end{aligned}\]

or generally speaking to

\[\begin{aligned} \texttt{prefix\_dep\_current}&[element\_dep, t_{inv}] \leq \\ & capacity\_ratio \times \texttt{prefix\_pre\_current}[element\_pre, t_{inv}] \\ \forall (element\_dep, & prefix\_dep) \in elements\_dep \end{aligned}\]

precede_capacity works similarly but requires the capacity to be available at the beginning of the investment period, that is, any capacity additions within the investment period do not count towards the requirement.

Given the example from above, the following call adds the preceded-capacity constraints:

precede_capacity(m, elements_dep, element_pre, 𝒯; capacity_ratio)

\[\begin{aligned} \texttt{cap\_current}[tech\_1, t_{inv}] & {} \leq 2 \times (\texttt{cap\_current}[tech\_3, t_{inv}] - \texttt{cap\_add}[tech\_3, t_{inv}]) \\ \texttt{cap\_current}[tech\_2, t_{inv}] & {} \leq 2 \times (\texttt{cap\_current}[tech\_3, t_{inv}] - \texttt{cap\_add}[tech\_3, t_{inv}]) \end{aligned}\]

or generally speaking to

\[\begin{aligned} \texttt{prefix\_dep\_current}[element\_dep, t_{inv}] & \leq capacity\_ratio \times {} \\ (\texttt{prefix\_pre\_current}&[element\_pre, t_{inv}] - \texttt{prefix\_pre\_add}[element\_pre, t_{inv}]) \\ \forall (element\_dep, prefix\_dep) & \in elements\_dep \end{aligned}\]

Retirement of capacities

In some cases, it is required that the capacity of a technology is retired before it is possible to invest in the capacity of a second technology. This is achieved through the function retire_capacity, which requires the retirement of the capacity of the prerequisite technology before it is possible to invest in the capacity of the dependent technology.

Consider the following example:

# Dependent elements
elements_dep = [
    (tech_1, :cap, investment_data(tech_1)),
    (tech_2, :cap, investment_data(tech_2)),
]

# Prerequisite element
element_pre = (tech_3, :cap, investment_data(tech_3))

# Add the constraints
retire_capacity(m, elements_dep, element_pre, 𝒯)

The function introduces first a number of binary variables similar to the number of investment periods $T^{Inv}$. These variables are not registered with a name. Hence, they are called $\texttt{var\_bin}$ below. The binary variables correspond to the retirement of the prerequisite element, i.e., a value of 1 implies that the installed capacity of the prerequisite element has an upper bound of 0.

The binary variables of subsequent investment periods ($t_{inv,pre}$ and $t_{inv}$) are coupled by

\[\texttt{var\_bin}[t_{inv}] \geq \texttt{var\_bin}[t_{inv,pre}]\]

This implies that once the prerequisite technology is retired, it cannot be invested in any longer.

The main constraints are then given by

\[\begin{aligned} \texttt{cap\_current}[tech\_1, t_{inv}] & {} \leq {} \texttt{var\_bin}[t_{inv}] \times max\_installed(inv\_data\_dep, t_{inv}) {} \\ \texttt{cap\_current}[tech\_2, t_{inv}] & {} \leq {} \texttt{var\_bin}[t_{inv}] \times max\_installed(inv\_data\_dep, t_{inv}) {} \\ \texttt{cap\_current}[tech\_3, t_{inv}] & {} \leq (1-\texttt{var\_bin}[t_{inv}]) \times max\_installed(inv\_data\_pre, t_{inv}) {} \\ \end{aligned}\]

or generally speaking $\forall (element\_dep, prefix\_dep, inv\_data\_dep) \in elements\_dep$ to

\[\begin{aligned} \texttt{prefix\_dep\_current} & {} [element\_dep, t_{inv}] \leq \\ & {} \texttt{var\_bin}[t_{inv}] \times max\_installed(inv\_data\_dep, t_{inv}) \end{aligned}\]

and for the prerequisite element to

\[\begin{aligned} \texttt{prefix\_pre\_current} & {} [element\_pre, t_{inv}] \leq \\ & {} (1-\texttt{var\_bin}[t_{inv}]) \times max\_installed(inv\_data\_pre, t_{inv}) \end{aligned}\]