StorageDispatchWithReserves Mathematical Model

This is the full mathematical model underlying the StorageDispatchWithReserves formulation:

Sets

\[\begin{align*} &\mathcal{P}^{\text{as}_\text{up}} & \text{Up Ancillary Service Products Set}\\ &\mathcal{P}^{\text{as}_\text{dn}} & \text{Down Ancillary Service Products Set}\\ &\mathcal{P}^{\text{as}} := \bigcup\left\{ \mathcal{P}^{\text{as}_\text{up}}, \mathcal{P}^{\text{as}_\text{dn}}\right\} & \text{Ancillary Service Products Set}\\ &\mathcal{T} := \{1,\dots,T\} & \text{Time steps} \\ \end{align*}\]

Parameters

Operational Parameters

\[\begin{align*} &P^{max,ch}_{st} &\text{Max Charge Power Storage [MW]}\\ &P^{max,ds}_{st} &\text{Max Discharge Power Storage [MW]}\\ &\eta^{ch}_{st} &\text{Charge Efficiency Storage [\%/hr]}\\ &\eta^{ds}_{st} &\text{Discharge Efficiency Storage [\%/hr]}\\ &R^{*}_{p, t} &\text{Ancillary Service deployment Forecast at time $t$ for service $p \in \mathcal{P}^{\text{as}}$ [\$/MW]}\\ &E^{max}_{st} &\text{Max Energy Storage Capacity [MWh]}\\ &E^{st}_{0} &\text{Storage initial energy [MWh]}\\ &E^{st}_{T} &\text{Storage Energy Target at the end of the horizon, i.e., time-step $T$ [MWh]}\\ &\Delta t &\text{Timestep length}\\ &C_{st} & \text{Maximum number of cycles over the horizon.} \\ && \text{For DA the value is fixed to 3 and in RT the value depends on the DA allocation of cycles} \\ &N_{p} & \text{Number of periods of compliance to supply an AS.}\\ && \text{For example Spinning reserve has 12 for 1 hour of compliance when $\Delta_t$ is 5-minutes.} \end{align*}\]

Cost Parameters

\[\begin{align*} &\text{VOM} &\text{Storage Variable Operation and Maintenance Cost [\%/MWh]}\\ &\rho^{e+} &\text{Storage Surplus penalty at end of target cost [\$/MWh]. Used when \texttt{use\_slacks = true}}\\ &\rho^{e-} &\text{Storage Shortage penalty at end of target cost [\$/MWh]. Used when \texttt{use\_slacks = true}}\\ &\rho^{c} &\text{Storage Cycling Penalty [\$/MWh]. Used when \texttt{use\_slacks = true}}\\ &\rho^{z} &\text{Regularization Terms Penalty. Used when \texttt{"regularization" => true}}\\ \end{align*}\]

Variables

\[\begin{align*} &p^{st, ch}_{t} & \in [0, P^{max,ch}_{st}] &\quad\text{Expected Storage charging power}\\ &p^{st, ds}_{t} & \in [0, P^{max,ds}_{st}] &\quad\text{Expected Storage discharging power}\\ &e^{st}_{t} & \in [0, E^{max}_{st}] &\quad \text{Expected Storage Energy}\\ &\text{ss}^{st}_{t} & \in \{ 0, 1 \} &\quad \text{Charge/Discharge status Storage. Used when \texttt{"reservation" => true}}\\ &sb_{stc,p,t} & \in [0, P^{max,ch}_{st}] & \quad \text{Ancillary service fraction assigned to Storage Charging}\\ &sb_{std,p,t} & \in [0, P^{max,ds}_{st}] & \quad \text{Ancillary service fraction assigned to Storage Discharging}\\ &e^{st+} & \in [0, E^{max}_{st}] &\quad \text{Storage Energy Surplus above target. Used when \texttt{use\_slacks = true}}\\ &e^{st-} & \in [0, E^{max}_{st}] &\quad \text{Storage Energy Shortage below target. Used when \texttt{use\_slacks = true}}\\ &c^{ch-} & \in [0, T C_{st}] &\quad \text{Charging Cycling Shortage. Used when \texttt{use\_slacks = true}}\\ &c^{ds-} & \in [0, T C_{st}] &\quad \text{Discharging Cycling Shortage. Used when \texttt{use\_slacks = true}}\\ &z^{st, ch}_{t} & \in [0, P^{max,ch}_{st}] &\quad \text{Regularization charge variable. Used when \texttt{"regularization" => true}}\\ &z^{st, ds}_{t} & \in [0, P^{max,ds}_{st}] &\quad \text{Regularization discharge variable. Used when \texttt{"regularization" => true}}\\ \end{align*}\]

Model

\[\begin{align*} \min_{\substack{\boldsymbol{p}^{st, ch}, \boldsymbol{p}^{st, ds}, \boldsymbol{e}^{st}, \\ e^{st+}, e^{st-}, c^{ch-} + c^{ds-}}} & \rho^{e+} e^{st+} + \rho^{e-} e^{st-} + \rho^{c} \left(c^{ch-} + c^{ds-} \right) + \rho^{z} \left(\frac{z^{ch}}{P^{max,ch}_{st}} + \frac{z^{ds}}{P^{max,ds}_{st}} \right)\\ & +\Delta t \sum_{t \in \mathcal{T}} \text{VOM}_{st} \left ( \left(\sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,t} sb_{stc,p,t} + p^{st,ch}_{t} \right) + \left(\sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,t} sb_{std,p,t} + p^{st,ds}_{t}\right) \right) & \end{align*}\]

\[\begin{align*} \text{s.t.} & &\\ &\text{Power Limit Constraints.}&\\ &p^{st, ch}_{t} + \sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} sb_{stc,p,t} \leq (1 - \text{ss}^{st}_{t})P^{max,ch}_{st} & \quad \forall t \in \mathcal{T} \\ & p^{st, ch}_{t} - \sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} sb_{stc,p,t} \geq 0 & \quad \forall t \in \mathcal{T}\\ & p^{st, ds}_{t} + \sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} sb_{std,p,t} \leq \text{ss}^{st}_{t}P^{max,ds}_{st} & \forall t \in \mathcal{T}\\ & p^{st, ds}_{t} - \sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} sb_{std,p,t} \geq 0 & \forall t \in \mathcal{T}\\ &\text{Energy Storage Limit Constraints}&\\ &e^{st}_{t} \leq E^{max}_{st} & \forall t \in \mathcal{T}\\ & e^{st}_{t} \geq E^{min}_{st} & \forall t \in \mathcal{T}\\ &\text{Energy Bookkeeping Constraints}&\\ & E^{st}_{0} + \Delta t \left(\sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,1} sb_{stc,p,1} + p^{st,ch}_{1} - \sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,1} sb_{stc,p,1}\right)\eta^{ch}_{st}&\\ &-\Delta t\left(\sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,1} sb_{std,p,1} + p^{st,ds}_{1} - \sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,t} sb_{std,p,1}\right)\frac{1}{\eta^{ds}_{st}}=e^{st}_{1}\\ &e^{st}_{t-1} + \Delta t \left(\sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,t} sb_{stc,p,t} + p^{st,ch}_{t} - \sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,t} sb_{stc,p,t}\right)\eta^{ch}_{st}&\\ &-\Delta t\left(\sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,t} sb_{std,p,t} + p^{st,ds}_{t} - \sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,t} sb_{std,p,t}\right)\frac{1}{\eta^{ds}_{st}} =e^{st}_{t} & \forall t \in \mathcal{T} \setminus {1}\\ &\text{End of period energy target constraint. Used when \texttt{"energy\_target" => true}}&\\ &e^{st}_{T} + e^{st+} - e^{st-} = E^{st}_{T}&\\ &\text{Storage Cycling Limits Constraints. Used when \texttt{"cycling\_limits" => true}}&\\ & \sum_{t \in \mathcal{T}} \left(\sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,t} sb_{std,p,t} + p^{st,ds}_{t}\right)\frac{1}{\eta^{ds}_{st}} \Delta t - c^{ds-} \leq C_{st} E^{max}_{st} &\\ & \sum_{t \in \mathcal{T}} \left(\sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,t} sb_{stc,p,t} + p^{st,ch}_{t} \right)\eta^{ch}_{st} \Delta t - c^{ch-} \leq C_{st} E^{max}_{st} \\ &\text{Single Ancillary Services Energy Coverage}&\\ & sb_{stc,p,t} \eta^{ch}_{st} N_{p} \Delta t \le E_{st}^{max} - e^{st}_{t} & \forall p \in \mathcal{P}^{as_{dn}} \ \forall t \in \mathcal{T}\\ & sb_{std,p,t} \frac{1}{\eta^{ds}_{st}} N_{p} \Delta t \leq e^{st}_{t}- E^{min}_{st} & \forall p \in \mathcal{P}^{as_{up}}, \ \forall t \in \mathcal{T}\\ & sb_{stc,p,1} \eta^{ch}_{st} N_{p} \Delta t \le E_{st}^{max} - e^{st}_0 & \forall p \in \mathcal{P}^{as_{dn}}\\ & sb_{stc,p,t} \eta^{ch}_{st} N_{p} \Delta t \le E_{st}^{max} - e^{st}_{t-1} & \forall p \in \mathcal{P}^{as_{dn}} \ \forall t \in \mathcal{T} \setminus 1\\ &sb_{std,p,1} \frac{1}{\eta^{ds}_{st}} N_{p} \Delta t \leq e^{st}_0 - E^{min}_{st} & \forall p \in \mathcal{P}^{as_{up}}\\ & sb_{std,p,t} \frac{1}{\eta^{ds}_{st}} N_{p} \Delta t \leq e^{st}_{t-1} - E^{min}_{st} & \forall p \in \mathcal{P}^{as_{up}}, \ \forall t \in \mathcal{T} \setminus 1 \\ &\text{Complete Ancillary Services Energy Coverage. Used when \texttt{"complete\_coverage" => true}}&\\ & \sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} sb_{stc,p,t} \eta^{ch}_{st} N_{p} \Delta t \le E_{st}^{max} - e^{st}_{t} & \forall t \in \mathcal{T}\\ & \sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} sb_{std,p,t} \frac{1}{\eta^{ds}_{st}} N_{p} \Delta t \leq e^{st}_{t}- E^{min}_{st} & \forall t \in \mathcal{T}\\ & \sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} sb_{stc,p,1} \eta^{ch}_{st} N_{p} \Delta t \le E_{st}^{max} - e^{st}_0 &\\ &\sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} sb_{stc,p,t} \eta^{ch}_{st} N_{p} \Delta t \le E_{st}^{max} - e^{st}_{t-1} & \forall t \in \mathcal{T} \setminus 1\\ &\sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} sb_{std,p,1} \frac{1}{\eta^{ds}_{st}} N_{p} \Delta t \leq e^{st}_0- E^{min}_{st} & \\ & \sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} sb_{std,p,t} \frac{1}{\eta^{ds}_{st}} N_{p} \Delta t \leq e^{st}_{t-1}- E^{min}_{st} & \forall t \in \mathcal{T} \setminus 1\\ &\text{Regularization Constraints. Used when \texttt{"regularization" => true}}&\\ & \left(\sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,t-1} sb_{stc,p,t-1} + p^{st,ch}_{t-1} - \sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,t-1} sb_{stc,p,t-1}\right) &\\ & - \left(\sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,t} sb_{stc,p,t} + p^{st,ch}_{t} - \sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,t} sb_{stc,p,t}\right) \le z^{st, ch}_{t} & \forall t \in \mathcal{T} \setminus 1\\ & \left(\sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,t-1} sb_{stc,p,t-1} + p^{st,ch}_{t-1} - \sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,t-1} sb_{stc,p,t-1}\right) &\\ & - \left(\sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,t} sb_{stc,p,t} + p^{st,ch}_{t} - \sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,t} sb_{stc,p,t}\right) \ge -z^{st, ch}_{t} & \forall t \in \mathcal{T} \setminus 1\\ &\left(\sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,t-1} sb_{std,p,t-1} + p^{st,ds}_{t-1} - \sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,t-1} sb_{std,p,t-1}\right) &\\ &-\left(\sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,t} sb_{std,p,t-1} + p^{st,ds}_{t} - \sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,t} sb_{std,p,t}\right) \le z^{st, ds}_{t} & \forall t \in \mathcal{T} \setminus 1\\ &\left(\sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,t-1} sb_{std,p,t-1} + p^{st,ds}_{t-1} - \sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,t-1} sb_{std,p,t-1}\right) &\\ &-\left(\sum_{p \in \mathcal{P}^{\text{as}_\text{up}}} R^*_{p,t} sb_{std,p,t} + p^{st,ds}_{t} - \sum_{p \in \mathcal{P}^{\text{as}_\text{dn}}} R^*_{p,t} sb_{std,p,t}\right) \ge -z^{st, ds}_{t} & \forall t \in \mathcal{T} \setminus 1 \end{align*}\]