Intertemporal Optimization Model¶

Intertemporal models are a more general type of model than the minimal case. For such models a second time domain is introduced to capture the behavior of the system over a timeframe of many years, thus rendering a modeling of the system development, rather than the optimal system configuration, possible. To keep the model as small as possible while still capturing most of the intertemporal behavior, the second time slice is approximated by a number of support timeframes (years) \(Y=(y_1,...,y_n)\), which is in general smaller than the total model horizon. Each modeled timeframe is then essentially a minimal (or multinode-) model in its own right. The basic approximative assumptions linking the modeled timeframes are then given by:

Each modeled year is repeated \(k\) times if the next modeled year is \(k\) years later. The last year is repeated a user specified number of times.
The depreciation period is assumed to be also the operational period of any unit built. There is no operation in an economically fully depreciated state.
A unit can only be operated in a given modeled year when it is operational for the entire period that year represents, i.e., until the next modeled year.
All payments are exponentially discounted with a discount rate \(j\) that is set once for the entire modeling horizon.

The variable vector is as a first step only changed in so far, as the second time domain is entering the index. It now reads:

\[x^{\text{T}}=(\zeta, \underbrace{\rho_{yct}}_{\text{commodity variables}}, \underbrace{\kappa_{yp}, \widehat{\kappa}_{yp}, \tau_{ypt}, \epsilon^{\text{in}}_{ycpt}, \epsilon^{\text{out}}_{ycpt}}_{\text{process variables}}).\]

Here, \(\zeta\) represents the total discounted system costs over the entire modeling horizon, \(\rho_{yct}\) the amount of commodities \(c\) taken from a virtual, infinite stock in year \(y\) at time \(t\), \(\kappa_{yp}\) and \(\widehat{\kappa}_{yp}\) the total and the newly installed process capacities in year \(y\) of processes \(p\), \(\tau_{ypt}\) the operational state in year \(y\) of processes \(p\) at time \(t\) and \(\epsilon^{\text{in}}_{ycpt}\) and \(\epsilon^{\text{out}}_{ycpt}\) the total inputs and outputs in year \(y\) of commodities \(c\) to and from process \(p\) at time \(t\), respectively.

All dispatch constraint equations for commodities and processes described in the minimal model section, as well as all such constraints for transmissions, storages, DSM described in their respective dedicated sections, remain structurally the same in an intertemporal model. The only modification there is, that the modeled year shows up as an additional index.

The parts that change in a more meaningful way are the costs and the unit expansion constraints.

Costs¶

As in the minimal model the total cost variable can be split into the following sum:

\[\zeta = \zeta_{\text{inv}} + \zeta_{\text{fix}} + \zeta_{\text{var}} + \zeta_{\text{fuel}} + \zeta_{\text{env}},\]

where \(\zeta_{\text{inv}}\) are the discounted invest costs accumulated over the entire modeled period, \(\zeta_{\text{fix}}\) the discounted, accumulated fixed costs, \(\zeta_{\text{var}}\) the discounted, sum over the modeled years of all variable costs accumulated over each year, \(\zeta_{\text{fuel}}\) the discounted sum over the modeled years of fuel costs accumulated over each year and \(\zeta_{\text{env}}\) the discounted total penalty for environmental pollution.

All costs are discounted by the same exponent \(j\) for the entire modeling horizon on a yearly basis. This means that any payment \(x\) that has to be made in a year \(k\) will be discounted for the cost function \(\zeta\) by:

\[x_{\text{discounted}}=(1+j)^{-k}\cdot x\]

Since all non-modeled years are just treated as exact copies of the last modeled year before them, the discounted sum of fix, variable, fuel and environmental costs can simply be taken as the costs of the representative modeled year \(m\) multiplied by a factor \(D_m\). If the distance to the next modeled year is \(k\), it can be calculated via:

\[\begin{split}D_m&=\sum_{l=m}^{m+k-1}(1+j)^{-l}=(1+j)^{-m}\sum_{l=0}^{k-1}(1+j)^{-l}= (1+j)^{-m}\frac{1-(1+j)^{-k}}{1-(1+j)^{-1}}=\\\\ &=(1+j)^{1-m}\frac{1-(1+j)^{-k}}{j}.\end{split}\]

So for example the variable costs for modeled year \(m\) and its \(k\) identical, non-modeled copies \(\zeta_{\text{var}}^{\{m,m+1,..,m+k-1\}}\) are given by:

\[\zeta_{\text{var}}^{\{m,m+1,..,m+k-1\}}=D_m\cdot\zeta_{\text{var}}^{m},\]

if \(\zeta_{\text{var}}^m\) is the sum of all variable costs accumulated by the use of units in the year \(m\) alone by the model.

Intertemporal Calculation of Invest Costs¶

In the intertemporal model, invest costs are calculated using the annuity method. This directly entails that there are no rest values of any units built by the model that have to be considered for the cost function. It is then possible to multiply the annuity payments \(fC\) for a unit with investment costs \(C\) built in year \(m\) simply with the factor \(D_{m}\). The only difference is, that the investment annuity payments are not restricted to the modeled years but have to be paid for the entire depreciation period provided that it is within the modeled time horizon. When the depreciation period is \(n\) and \(k\) is the number of payments that fall in the modeled time horizon, the total costs \(C_{\text{total}}\) of an investment of size \(C\) made in year \(m\) is given by:

\[\begin{split}C^{\text{total}}_{\text{m}}&=D_{m}\cdot f \cdot C = (1+j)^{1-m}\frac{1-(1+j)^{-k}}{j} \cdot \frac{(1+i)^n\cdot i}{(1+i)^n-1} \cdot C=\\\\ &=\underbrace{(1+j)^{1-m}\cdot \frac{i}{j}\cdot \left(\frac{1+i}{1+j}\right)^n\cdot \frac{(1+j)^n-(1+j)^{n-k}}{(1+i)^n-1}}_{=:I_{\text{m}}}\cdot C\end{split}\]

For either \(i=0\) or \(j=0\) a distinction has to be made, which takes the following form:

\(i=0,~j=0\):

\[C^{\text{total}}_{\text{m}}=\underbrace{\frac{k}{n}}_{=:I_{\text{m}}}\cdot C\]
\(i\neq0,~j=0\):

\[C^{\text{total}}_{\text{m}}=k\cdot f\cdot C=\underbrace{k\cdot \frac{(1+i)^n\cdot i}{(1+i)^n-1}}_{=:I_{\text{m}}}\cdot C\]
\(i=0,~j\neq0\):

\[C^{\text{total}}_{\text{m}}=\frac 1n \cdot (1+j)^{-m} \sum_{l=0}^{k-1}(1+j)^{-l} \cdot C=\underbrace{\frac 1n \cdot (1+j)^{1-m} \cdot \frac{(1+j)^k-1}{(1+j)^k\cdot j}}_{=:I_{\text{m}}}\cdot C\]

In any case the total invest costs are then given by:

\[\begin{split}\zeta_{\text{inv}}=\sum_{y\in Y\\p\in P}C^{\text{total}}_{\text{m}}= \sum_{y\in Y\\p\in P}I_{\text{y}}k^{\text{inv}}_{yp} \widehat{\kappa}_{yp}\end{split}\]

Unit Expansion Constraints¶

Apart from the costs there are also changes in the unit expansion constraints for an intertemporal model. These changes mostly concern how the amount of installed units is found. This becomes an issue since units built in an earlier modeled year or already installed in the first modeled year, may or may not be operational in a given modeled year \(m\) and through \(m+k-1\). Here, \(k\) is the distance to the next modeled year or the end of the modeled horizon in case of \(m\) being the last modeled year. To properly calculate the capacity of a process in a year \(y\) the following two sets are necessary:

\[\begin{split}O&:=\{(p,y_i,y_j)|p\in P,~\{y_i,y_j\}\in Y,~y_i\leq y_j,~ y_i + L_p \geq\ y_{j+1}\}\\\\ O_{\text{inst}}&:=\{(p, y_j)|p\in P_0,~y\in Y,~y_0+T_p\geq y_{j+1}\},\end{split}\]

where \(L_p\) is the lifetime of processes \(p\), \(P_0\) the subset of processes that are already installed in the first modeled year \(y_0\) and \(T_{p}\) the rest lifetime of already installed processes. If \(y_j\) is the last modeled year, \(y_{j+1}\) stands for the end of the model horizon. The set \(O_{\text{inst}}\) can be found in the model implementation as Initially Installed Units. This is the set of units, that are already installed in the beginning of the model. The set \(O\) describes the set of those units, that are installed during the model and its implementation can be found at Installation in Earlier Support Timeframe.

With these two sets the installed process capacity in a given year is then given by:

\[\begin{split}\kappa_{yp}&=\sum_{y^{\prime}\in Y\\(p,y^{\prime},y)\in O} \widehat{\kappa}_{y^{\prime}p} + K_{p} ~,~~\text{if}~(p,y)\in O_{\text{inst}}\\\\ \kappa_{yp}&=\sum_{y^{\prime}\in Y\\(p,y^{\prime},y)\in O} \widehat{\kappa}_{y^{\prime}p}~,~~\text{else}\end{split}\]

where \(K_{p}\) is the installed capacity of process \(p\) at the beginning of the modeling horizon. Since for each modeled year still the capacity constraint

\[\begin{split}&\forall y\in Y,~ p \in P:\\ &\underline{K}_{yp}\leq\kappa_{yp}\leq\overline{K}_{yp}\end{split}\]

is valid, the set constraints can have effects across years and especially the modeller has to be careful not to set infeasible constraints.

Commodity Dispatch Constraints¶

While in an intertemporal model all commodity constraints within one modeled year remain valid one addition is possible concerning CO2 emissions. Here, a budget can be given, which is valid over the entire modeling horizon:

\[\begin{split}-w\sum_{y\in Y\\t\in T_{m}}\text{CB}(y,\text{CO}_2,t)\leq \overline{\overline{L}}_{\text{CO}_2}\end{split}\]

Here, \(\overline{\overline{L}}_c\) is the global budget for the emission of the environmental commodity. Currently this is hard coded for CO2 only.

This rule concludes the model additions introduced by intertemporal modeling.