Basic Component Model

The Basic component model comprises sets of variables, constraints, inter-component constraint-contributions and objective function contributions that apply to all components specified in \(\mathcal{C}\). In this context, the variables and constraints can be divided into either being time-independent or time-dependent.

Component Parameters

Dimensioning of Components

For each component \(c \in C\), capacity variables can be introduced with the parameter \(K_\text{c} \in \{0,1\}\):

\[ \begin{aligned} K_\text{c}= \begin{cases} 1 &\text{, if the component is modeled with a physical capacity, or}\\ 0 &\text{, if the component is modeled without a physical capacity.}\\ \end{cases} \end{aligned} \]

A component which is modeled with physical capacity is for example a gas power plant while an electricity demand does not require one.

The following parameters refer to all components \(\text{c}\in\mathcal{C}\) with \(K_\text{c}=1\):

Parameter	Domain	Description
\(K^\text{min}_\text{c,l}\)	\(\mathbb{R}_0^+\) with \(c \in \mathcal{C}, l \in \mathcal{L}_\text{c}\)	minimum capacity of component c at location l
\(K^\text{max}_\text{c,l}\)	\(\mathbb{R}_0^+\) with \(c \in \mathcal{C}, l \in \mathcal{L}_\text{c}\)	maximum capacity of component c at location l
\(K^\text{fix}_\text{c,l}\)	\(\mathbb{R}_0^+\) with \(c \in \mathcal{C}, l \in \mathcal{L}_\text{c}\)	fixed capacity of component c at location l
\(K^\text{unit}_\text{c}\)	\(\mathbb{R}_0^+\) with \(c \in \mathcal{C}\)	capacity per plant unit of component c
\(B_\text{c}\)	\(\left\{0,1\right\}\) with \(c \in \mathcal{C}\)	introduces decision variable to state if a capacity is built or not
\(B^\text{fix}_\text{c,l}\)	\(\left\{0,1\right\}\) with \(c \in \mathcal{C}, l \in \mathcal{L}_\text{c}\)	fixes decision variable to state if a component c is built or not at location l
\(M_\text{c}\)	\(\mathbb{R}^+\) with \(c \in \mathcal{C}\)	required auxiliary parameter if \(K^\text{bin}_\text{c} = 1\)
\(\text{E}_\text{c,l}\)	\(\left\{0,1\right\}\) with \(c \in \mathcal{C}^\text{node}, l \in \mathcal{L}_\text{c}\)	eligibility of node component c at location l
\(\text{E}_\text{c,a}\)	\(\left\{0,1\right\}\) with \(c \in \mathcal{C}^\text{edge}, a \in \mathcal{A}_\text{c}\)	eligibility of edge component c at arc a

Operation of Components

Parameter	Domain	Description
\(R^\text{min}_{\text{c,l,}\theta}\)	\(\mathbb{R}_0^+\) with \(c \in \mathcal{C}, l \in \mathcal{L}_\text{c}, \theta \in \Theta\)	minimum operation rate of component c at location l and time step t
\(R^\text{max}_{\text{c,l,}\theta}\)	\(\mathbb{R}_0^+\) with \(c \in \mathcal{C}, l \in \mathcal{L}_\text{c}, \theta \in \Theta\)	maximum operation rate of component c at location l and time step t
\(R^\text{fix}_{\text{c,l,}\theta}\)	\(\mathbb{R}_0^+\) with \(c \in \mathcal{C}, l \in \mathcal{L}_\text{c}, \theta \in \Theta\)	fixed operation rate of component c at location l and time step t

Cost Contribution of Components

Parameter	Domain	Description
\(\hat{X}^{\text{capex}_\text{K}}_\text{c,l}\)	\(\mathbb{R}_0^+\) with \(c \in \mathcal{C}, l \in \mathcal{L}_\text{c}\)	expenditures per capacity
\(\hat{X}^{\text{opex}_\text{K}}_\text{c,l}\)	\(\mathbb{R}_0^+\) with \(c \in \mathcal{C}, l \in \mathcal{L}_\text{c}\)	operational expenditures per capacity
\(\hat{X}^{\text{capex}_\text{B}}_\text{c,l}\)	\(\mathbb{R}_0^+\) with \(c \in \mathcal{C}, l \in \mathcal{L}_\text{c}\)	expenditures if capacity is built
\(\hat{X}^{\text{opex}_\text{B}}_\text{c,l}\)	\(\mathbb{R}_0^+\) with \(c \in \mathcal{C}, l \in \mathcal{L}_\text{c}\)	operational expenditures per capacity if capacity is built
\(T^\text{EL}_\text{c}\)	\(\mathbb{Z}_0^+\) with \(c \in \mathcal{C}\)	economic lifetime of component c

Time-Independent Variables and Constraints

A capacity variable \(k_\text{c,l}\in\mathbb{R}^{\geq0}\) is declared for all locations \(\text{l}\in\mathcal{L}_\text{c}\) in the energy system at which the component can appear. Implicitly, this capacity is modeled either as a continuous or discrete value by

\[ \begin{aligned} k_\text{c,l} = \text{K}^\text{unit}_\text{c,l} \cdot n_\text{c,l} \end{aligned} \]

with \(n_\text{c,l} \in \mathbb{R}_0^+\) if the capacity is modeled as a continuous value, and with \(n_\text{c,l} \in \mathbb{N}_0^+\) if the capacity is modeled as discrete value. The variable \(n_\text{c,l}\) describes the number of installed plant units for each component and location, and the parameter \(K^\text{unit}_\text{c}\) describes the capacity per plant unit.

Furthermore, the component can be modeled together with a binary design decision variable \(b_\text{c,l}\in\left\{0,1\right\}\), for all locations \(\text{l}\in\mathcal{L}_\text{c}\), if its boolean parameter \(B_\text{c}\) is set to true (=1). This modeling approach is based on the work of Bemporad and Morari (1999) who give a general description and discussion of this approach in the context of linear integer programming. The optimal value of \(b_\text{c,l}\) states whether a component is built (=1) or not built (=0). The consideration of the binary decision variables is enforced in the model for all \(b_\text{c,l}\) by the constraint

\[ \begin{aligned} \text{M}_\text{c} \cdot b_\text{c,l} ~\geq~ k_\text{c,l}~, \end{aligned} \]

where \(\text{M}_\text{c}\in\mathbb{R}_0^{+}\). The constraint enforces that \(b_\text{c,l} = 1\) if \(k_\text{c,l} > 1\). The parameter \(\text{M}_\text{c}\) has to be chosen large enough such that it does not function as an upper limit on the capacity.

Lower and upper boundaries can be specified for the capacity variables of the component. Lower bounds are enforced, if \(\text{K}^\text{min}_\text{c,l} \in\mathbb{R}^{\geq0}\) is defined for all \(\text{l} \in\mathcal{L}_\text{c}\) of this component, by

\[ \begin{aligned} &&&k_\text{c,l} \geq \begin{cases} K^\text{min}_\text{c,l}\cdot b_\text{c,l} &,~\text{if}~ B_\text{c}=1,\\ K^\text{min}_\text{c,l} &,~\text{if}~ B_\text{c}=0. \end{cases} \end{aligned} \]

Upper bounds are enforced, if \(\text{K}^\text{max}_\text{c,l}\in\mathbb{R}^{\geq0}\) is defined for all \(\text{l} \in\mathcal{L}_\text{c}\), by

\[ \begin{aligned} k_\text{c,l} ~\leq~ \text{K}^\text{max}_\text{c,l}~~. \end{aligned} \]

Moreover, for both the capacity and the binary decision variables, fixed values can be individually specified for a component by

\[ \begin{aligned} k_\text{c,l} &~=~~ && K^\text{fix}_\text{c,l}~~\text{and} \\ k^\text{bin}_\text{c,l} &~=~~ && K^\text{bin,fix}_\text{c,l}~~, \end{aligned} \]

if \(K^\text{fix}_\text{c,l} \in\mathbb{R}^{\geq0},~K^\text{bin,fix}_\text{c,l} \in \left\{0,1\right\}\) are defined for all \(\text{l}\in\mathcal{L}_\text{c}\), respectively.

Basic Time-Dependent Variables and Constraints

Operational variables \(o_{\omega \text{,l,} \theta}\in\mathbb{R}^{\geq0}\) are declared for all operation types of a component \(\omega \in \Omega\), for all locations \(\text{l}\in\mathcal{L}^\text{c}\) and for all periods and time steps \(\theta \in \Theta\). The compound index set \(\Omega\) is individually defined in the respective component extension and describes which modes \(m \in \mathcal{M}\) need to be considered for component \(c \in \mathcal{C}\). The compound index sets are described in Compound Index Sets.

Each operation variable of a component that is modeled with a physical capacity (\(K_\text{c} = 1\)) is limited in one of four ways.

First, the operation variable is limited by

\[ \begin{aligned} o_{\omega \text{, l,} \theta} ~\leq~ \text{T}^\text{hours} \cdot \text{a}_{\omega} \cdot k_\text{c,l} \end{aligned} \]

if the operation of the component is merely limited by its capacity and a time-independent factor \(\text{a}_{\omega}\in\mathbb{R}^{\geq0}\) (default: 1) with \(\omega \in \Omega\).

Second, the operation variable is fixed to

\[ \begin{aligned} o_{\omega \text{,l,} \theta} ~=~ \text{T}^\text{hours} \cdot \text{R}^\text{fix}_{\text{c,l,} \theta} \cdot k_\text{c,l} \end{aligned} \]

if a fixed, relative operation rate \(\text{R}^\text{fix}_{\text{c,l,} \theta}\) is specified for all locations \(\text{l}\in\mathcal{L}_\text{c}\) and for all periods and time steps \(\theta \in \Theta\).

Third, the operation rate is limited by

\[ \begin{aligned} o_{\omega \text{,l,} \theta} ~\leq~ \text{T}^\text{hours} \cdot \text{R}^\text{max}_{\text{c,l,} \theta} \cdot k_\text{c,l} \end{aligned} \]

if a maximum, relative operation rate \(\text{R}^\text{max}_{\text{c,l,} \theta}\) is specified for all locations \(\text{l}\in\mathcal{L}_\text{c}\) and for all periods and time steps \(\theta \in \Theta\).

Lastly, the operation rate is bounded below by

\[ \begin{aligned} o_{\omega \text{,l,} \theta} ~\geq~ \text{T}^\text{hours} \cdot \text{R}^\text{min}_{\text{c,l,} \theta} \cdot k_\text{c,l} \end{aligned} \]

if a minimum, relative operation rate \(\text{R}^\text{min}_{\text{c,l,} \theta}\) is specified for all locations \(\text{l}\in\mathcal{L}_\text{c}\) and for all periods and time steps \(\theta \in \Theta\).

Each operation variable of a component which is modeled without a physical capacity (\(K_\text{c} = 0\)) is limited in one of three ways. The operation variable is fixed to

\[ \begin{aligned} o_{\omega \text{,l,} \theta} ~=~ \text{T}^\text{hours} \cdot \text{R}^\text{fix}_{\text{c,l,} \theta} \end{aligned} \]

if a fixed, relative operation rate \(\text{R}^\text{fix}_{\text{c,l,} \theta}\) is specified for all locations \(\text{l}\in\mathcal{L}_\text{c}\) and for all periods and time steps \(\theta \in \Theta\). This constraint can apply, for example, to the model of an electricity demand.

The operation variable is limited by

\[ \begin{aligned} o_{\omega \text{,l,} \theta} ~\leq~ \text{T}^\text{hours} \cdot \text{R}^\text{max}_{\text{c,l,} \theta} \end{aligned} \]

if a maximum, relative operation rate \(\text{R}^\text{max}_{\text{c,l,} \theta}\) is specified for all locations \(\text{l}\in\mathcal{L}_\text{c}\) and for all periods and time steps \(\theta \in \Theta\). This constraint can apply, for example, to the model of an optional commodity import.

The operation variable is limited by

\[ \begin{aligned} o_{\omega \text{,l,} \theta} ~\geq~ \text{T}^\text{hours} \cdot \text{R}^\text{min}_{\text{c,l,} \theta} \end{aligned} \]

if a minimum, relative operation rate \(\text{R}^\text{min}_{\text{c,l,} \theta}\) is specified for all locations \(\text{l}\in\mathcal{L}_\text{c}\) and for all periods and time steps \(\theta \in \Theta\).

Basic Inter-Component Constraint Contributions

Inter-component constraint contributions are defined to model constraints which affect multiple components. The contribution of a component to a commodity balance equation is labeled \(C_{\text{c,g,l,}\theta}\) and has to be defined for each component which is added to the model.

Moreover, two or more components can compete for a limited capacity potential in an energy system. Components which share a potential in FINE are provided with an identifier. The share of a component on the maximum potential at all locations \(\text{l} \in\mathcal{L}^\text{c}\) is defined by \(k_\text{c,l}/\text{k}^\text{max}_\text{c,l}\).

Basic Objective Function Contribution

The objective function in the framework is defined as the net present value \(NPV\) of all components \(\text{c} \in \mathcal{C}\) and is minimized during optimization. Objective function contributions \(NPV_\text{c}\) [costUnit/a] are specified for each component individually by

\[ \begin{aligned} NPV_\text{c} =& \sum\limits_{\text{l}~\in~\mathcal{L}_\text{c}}\hspace{-3pt} \left( NPV^\text{K}_\text{c,l} ~+~ NPV^\text{B}_\text{c,l} ~+~ NPV^\text{O}_\text{c,l} \right) \end{aligned} \]

The capacity related total annual cost contributions are determined by

\[ \begin{aligned} &NPV^\text{K}_\text{c,l} = \text{F}^\text{K}_\text{c,l} \cdot \left(\frac{\hat{X}^{\text{capex}_\text{K}}_\text{c,l}}{\text{CCF}_\text{c,l}} + \hat{X}^{\text{opex}_\text{K}}_\text{c,l}\right) \cdot k_\text{c,l} \end{aligned} \]

if the component is modeled with a physical capacity. Otherwise, \(NPV^\text{K}_\text{c,l}\) is set to 0. The parameters \(\hat{X}^{\text{capex}_\text{K}}_\text{c,l}\) [costUnit/nominalCapacity] and \(\hat{X}^{\text{opex}_\text{K}}_\text{c,l}\in\mathbb{R}^{\geq0}\) [costUnit/(nominalCapacity\(\cdot\)a)] describe the capital and annual operational expenditures in relation to the capacity. The parameter \(\text{F}^\text{K}_\text{c,l}\) can be defined individually for a component (default: 1).

The total annual cost contributions related to the binary decision variables are determined by

\[ \begin{aligned} &NPV^\text{B}_\text{c,l} = \text{F}^\text{B}_\text{c,l} \cdot \left(\frac{\hat{X}^{\text{capex}_\text{B}}_\text{c,l}}{\text{CCF}_\text{c,l}} + \hat{X}^{\text{opex}_\text{B}}_\text{c,l}\right) \cdot b_\text{c,l} \end{aligned} \]

if the component is modeled with binary decision variables. Otherwise \(NPV^\text{B}_\text{c,l}\) is set to 0. The parameters \(\hat{X}^{\text{capex}_\text{B}}\) [costUnit] and \(\hat{X}^{\text{opex}_\text{B}}\in\mathbb{R}^{\geq0}\) [costUnit/a] describe the capital and annual operational expenditures which arise if the component is built. The parameter \(\text{F}^\text{B}_\text{c,l}\) can be defined individually for a component (default: 1). The factor

\[ \begin{aligned} &\text{CCF}_\text{c,l} = \frac{1}{\text{WACC}_\text{c,l}}-\frac{1}{\left(1+\text{WACC}_\text{c,l}\right)^{\text{T}^\text{EL}_\text{c}}\cdot\text{WACC}_\text{c,l}} \end{aligned} \]

is applied to determine the annuity of the respective invest for one calendar year. Thus, \(\text{WACC}_\text{c,l}\in(0,1]\) is the weighted average cost of capital and \(T^\text{EL}_\text{c}\in\mathbb{Z}_0^{+}\) [a] is the economic lifetime of the component in years.

With the combination of a capacity-dependent and a capacity-independent cost factor, a simplified nonlinear economy-of-scale approach is realized. The operation related total annual cost contributions are determined by

\[ \begin{aligned} &NPV^\text{O}_\text{c,l} = \hspace{-3pt} \sum\limits_{\substack{\theta \\ \in~\Theta}}\hspace{4pt} \sum\limits_{\substack{\text{m} \\ \in~\mathcal{M}_\text{c}}} \text{F}^\text{O}_{\omega \text{,l}}\hspace{-3pt}\cdot o_{\omega\text{,l,} \theta} \cdot \frac{f\left(p\right)}{\text{T}^\text{years}} \end{aligned} \]

where \(F^\text{O}_{\omega \text{,l}}\) [costUnit/(nominalCapacity\(\cdot\)h)] is defined in the individual component model extensions.