Power Balance

Each area has a power balance per time step, where the local demand is met by local production while allowing import and export through the transmission system. The power balance constraint can therefore be seen as the market clearing for each area and time step, where the its dual value is the simulated power price.

We describe the power balance with collective terms, where each of them are explained in more detail in the corresponding technology section.

\[ \textrm{POWBAL}(a,k): \textrm{Production}(a,k) + \textrm{Market}(a,k) + \textrm{Exchange}(a,k) + \textrm{Slack}(a,k) \ \]
\[ = \textrm{Load}(a,k) - \textrm{WindProduction}(a,k) - \textrm{SolarProduction}(a,k) \]

Collective Terms

Where the collective terms on the right-hand side involve variables:

\[ \textrm{Production}(a,k) = \sum_i \mathrm{prod\_hyd} (i,k) + \sum_g \mathrm{prod\_therm}(i,k) \]
\[ \textrm{Market}(a,k) = \sum_m market(m,k) \]
\[\begin{split} \begin{aligned} \textrm{Exchange}(a,k) = \sum_{c \in \textrm{LinesTo}_a} \Bigr(\mathrm{tran\_to}(c,k) - \mathrm{tran\_from}(c,k) \Bigr) \\ -\sum_{c \in \textrm{LinesFrom}_a} \Bigr(\mathrm{tran\_to}(c,k) - \mathrm{tran\_from}(c,k)\Bigr) \end{aligned} \end{split}\]

Slack

In order to ensure that the power balance is always feasible, slack variables are introduced. A \( rationing \) variable is used to reduce demand if power supply is not sufficient to meet the fixed demand. In the opposite case, if must-run power exceeds demand, the model allows possible to \( dump \) power to an artificial demand. Both variables are in GWh.

\[ 0 \leq \textrm{rationing}(a,k) \leq \textrm{RationingMax}(a) \qquad ,\enspace a \in Areas, \ k \in Timesteps \]
\[ 0 \leq \textrm{dump}(a,k) \leq \textrm{DumpMax}(a) \qquad ,\enspace a \in Areas, \ k \in Timesteps \]

The cost of rationing and dumping power is added to the objective:

\[ Cost: += \sum_{a,k} C_{rat}(a,k)*\textrm{rationing}(a,k) \]
\[ Cost: += \sum_{a,k} C_{dump}(a,k)*\textrm{dump}(a,k) \]

Note that the marginal costs for rationing \( C_{rat} \) and dumping \( C_{dump} \) power provides a ‘roof’ and ‘floor’, respectively, for simulated power prices.