| Notation | Description |
|---|
| $D$ | Set of demand, index $d$ |
| $G$ | Set of all types of generating units, index $g$ |
| $H$ | Set of hours, index $h$ |
| $K$ | Set of technology types, index $k$ |
| $T$ | Set of time periods (e.g., representative days of seasons), index $t$ |
| $S$ | Set of storage units, index $s$ |
| $I,J$ | Set of zones, index $i,j$ |
| $L$ | Set of transmission corridors, index $l$ |
| $W$ | Set of states, index $w/w’$ |
| Notation | Description |
|---|
| $D_{i}$ | Set of demand connected to zone $i$, a subset of $D$ |
| $G^{PV}$, $G^{W}$, $G^{F}$ | Set of solar, wind, and dispatchable generators, respectively, subsets of $G$ |
| $G^{RPS}$ | Set of generators could provide RPS credits, subsets of $G$ |
| $G^{L}_{l}$ | Set of generators linked to line $i$, subset of $G$ |
| $G_{i}$ | Set of generating units connected to zone $i$, subset of $G$ |
| $G^{E}/G^{+}$ | Set of existing/candidate generation units, index $g$, subset of $G$ |
| $H_{t}$ | Set of hours in time period (day) $t$, index $h$, subset of $H$ |
| $S^{E}/S^{+}$ | Set of existing/candidate storage units, subset of $S$ |
| $S_{i}$ | Set of storage units connected to zone $i$, subset of $S$ |
| $L^{E}/L^{+}$ | Set of existing/candidate transmission corridors |
| $LS_{l}/LR_{l}$ | Set of sending/receiving corridors for zone $i$, subset of $L$ |
| $WIR_{w}$ | Set of states that state w can import renewable credits from (includes $w$ itself), subset of $W$ |
| $WER_{w}$ | Set of states that state w can export renewable credits to (excludes $w$ itself), subset of $W$ |
| Notation | Description |
|---|
| $ALW_{t,w}$ | Total carbon allowance in time period $t$ in state $w$, ton |
| $AFRE_{g,h,i}$ | Availability factor of renewable energy source $g$ in hour $h$ in zone $i$, $g \in G^{PV} \bigcup G^{W}$ |
| $CC_{g/s}$ | Capacity credit of resource $g/s$, unitless |
| $CP_{g}$ | Carbon price of generation $g \in\ G^{F}$, M/t |
| $DR_{i,t,h}^{ref}$ | Reference demand of demand response aggregator in time-period $t$ in hour $h$, MW |
| $DR_{i}^{MAX}$ | Maximum capacity limit for demand consumption of DR aggregator in zone $i$, MW |
| $DRC$ | Cost of demand response, unitless |
| $EF_{g}$ | Carbon emission factor of generator $g$, t/MWh |
| $ELMT_{w}$ | Carbon emission limits at state $w, t$ |
| $F^{max}_{l}$ | Maximum capacity of transmission corridor/line $l$, MW |
| $\tilde{I}_{g}$ | Investment cost of candidate generator $g$, M$ |
| $\tilde{I}_{l}$ | Investment cost of transmission line $l$, M$ |
| $\tilde{I}_{s}$ | Investment cost of storage unit $s$, M$ |
| $IBG$ | Total investment budget for generators |
| $IBL$ | Total investment budget for transmission lines |
| $IBS$ | Total investment budget for storages |
| $N_{t}$ | Number of time periods (days) represented by time period (day) $t$ per year, /sum{t /in T} N{t} |
| $NI_{i.h}$ | Net interchange in zone $i$ in hour $h, MWh |
| $P_{d,h}$ | Active power demand, MW |
| $PK$ | Peak power demand, MW |
| $PT^{rps}$ | RPS volitation penalty, /MWh |
| $PT^{emis}$ | Carbon emission volitation penalty, /t |
| $P_{g}^{min}/P_{g}^{max}$ | Minimum/Maximum power generation of unit $g$, MW |
| $RPS_{w}$ | Renewable portfolio standard in state $w$, %, unitless |
| $RM$ | Planning reserve margin, unitless |
| $SCAP_{s}$ | Maximum capacity of storage unit $s$, MW |
| $SECAP_{s}$ | Maximum energy capacity of storage unit $s$, MWh |
| $SC_{s}/SD_{s}$ | The maximum rates of charging/discharging, unitless |
| $VCG_{g}$ | Variable cost of generation unit $g$, / MWh |
| $VCS_{g}$ | Variable (degradation) cost of storage unit $s$, / MWh |
| $VOLL_{d}$ | Value of loss of load $d$, /MWh |
| $\epsilon_{ch}$ | Charging efficiency of storage unit $s$, unitless |
| $\epsilon_{dis}$ | Discharging efficiency of storage unit $s$, unitless |
| Notation | Description |
|---|
| $a_{g,t}$ | Bidding carbon allowance of unit $g$ in time period $t$, ton |
| $b_{g,t}$ | Banking of allowance of g in time period $t$, ton |
| $dr_{i,t,h}^{UP/DN}$ | Upwards/downwards demand change relative to reference demand during $h$ in time period $t$ in zone $i$, MW |
| $dr_{i,t,h}^{DR}$ | Demand from DR aggregator during $h$ in time period $t$ in zone $i$, MW |
| $p_{g,t,h}$ | Active power generation of unit $g$ in time period $t$ hour $h$, MW |
| $pw_{g,w}$ | Total renewable generation of unit $g$ in state $w$, MWh |
| $p^{LS}_{d,t,h}$ | Load shedding of demand $d$ in time period $t$ in hour $h$, MW |
| $pt^{rps}_{w}$ | Amount of active power violated RPS policy in state $w$, MW |
| $pwi_{g,w,w'}$ | State $w$ imported renewable credits of from state $w'$ annually, MWh |
| $f_{l,t,h}$ | Active power of generator $g$ through transmission corridor/line $l$ in time period $t$ and hour $h$, MW |
| $em^{emis}_{w}$ | Carbon emission violated emission limit in state $w$, ton |
| $x_{g}$ | Decision variable for candidate generator $g$, binary |
| $y_{l}$ | Decision variable for candidate line $l$, binary |
| $z_{s}$ | Decision variable for candidate storage $s$, binary |
| $soc_{s,t,h}$ | State of charge level of storage $s$ in time period $t$ in hour $h$, MWh |
| $c_{s,t,h}$ | Charging power of storage $s$ from grid in time period $t$ in hour $h$, MW |
| $dc_{s,t,h}$ | Discharging power of storage $s$ from grid in time period $t$ in hour $h$, MW |