# Reserve requirements

In the following, we will present the main equations involved in modelling reserve requirements for thermal and hydro plants in Trident. The term "power plant" will henceforth be used to refer to either a thermal or hydro plant.

## Mathematical description
We begin by defining the running state of the power plant
$$
    P_{\mathrm{cap}}(i,k)* P_{\mathrm{min}}(i)* u_{\mathrm{run}}(i,k) \leq p(i,k) \leq P_{\mathrm{cap}}(i,k)*u_{\mathrm{run}}(i,k)\qquad(1)
$$
where $\textrm{P}_{cap}(i, k)$ is the installed/available capacity for a power plant, $\textrm{P}_{min}(i)$ the minimum production capacity for a power plant [in percentage], $p$ is the plant production and $u_{run}\in [0,1]$ the running state of the power plant. $i$ is an index for the thermal unit and $k$ denotes the time step. The constraint is already defined in connection with start-up costs for thermal/hydro plants, and we refer to the documentation for thermal/hydro plants for a more detailed explanation. 

If start up costs are not defined for a reserve unit, the above equation and the following ones still hold with $u_{\mathrm{run}}=1$.  

We define the upward reserve provision, $r_{up}$, from a power plant as follows
$$
    0 \le r_{up}(i, k) \le \textrm{P}_{cap}(i, k)*u_{run}(i, k) - p(i, k)\qquad(2)
$$
which is connected to the upward reserve requirement per area, $a$,
$$
    \sum_{i \in \mathbb{I}(a)}r_{up}(i, k) + v_{up}(a, k) \ge \textrm{R}_{up}(a, k) \qquad(3)
$$
where $\mathbb{I}(a)$ is an index set containing indices of all generating units (both thermal and hydro) with reserve requirements in area $a$ and where we have introduced a penalty variable, $v_{up}$, for not exceeding the minimum upward reserve capacity, $R_{up}$. 

Similarly, we define the downward reserve provision, $r_{down}$, from a power plant as
$$
    0 \le r_{down}(i, k) \le p(i, k) - \textrm{P}_{cap}(i, k)*\textrm{P}_{min}(i)*u_{run}(i,k)\qquad(4)
$$
and the downward reserve requirement per area
$$
    \sum_{i \in \mathbb{I}(a)}r_{down}(i, k) + v_{down}(a, k) \ge \textrm{R}_{down}(a, k) \qquad(5)
$$
with a penalty variable, $v_{down}$, for not exceeding the minimum downward reserve capacity, $R_{down}$.

The penalty variables enter into the objective function as
$$
V_{R} * \sum_{k\in \mathrm{time\;steps}, a \in A}(v_{up}(a, k) + v_{down}(a, k)) \qquad(6)
$$
with an associated cost, $V_R$.