diff --git a/docs/src/notation.md b/docs/src/notation.md index 59b6926..e48de9d 100644 --- a/docs/src/notation.md +++ b/docs/src/notation.md @@ -9,149 +9,87 @@ CurrentModule = HOPE --- |**Notation** | **Description**| | :------------ | :-----------| -|$t \in \mathcal{T}$ | where $t$ denotes an time step and $\mathcal{T}$ is the set of time steps over which grid operations are modeled| -|$\mathcal{T}^{interior} \subseteq \mathcal{T}^{}$ | where $\mathcal{T}^{interior}$ is the set of interior timesteps in the data series| -|$\mathcal{T}^{start} \subseteq \mathcal{T}$ | where $\mathcal{T}^{start}$ is the set of initial timesteps in the data series. $\mathcal{T}^{start}={1}$ when representing entire year as a single contiguous period; $\mathcal{T}^{start}=\{\left(m-1\right) \times \tau^{period}+1 \| m \in \mathcal{M}\}$, which corresponds to the first time step of each representative period $m \in \mathcal{M}$| -|$n \in \mathcal{N}$ | where $n$ corresponds to a contiguous time period and $\mathcal{N}$ corresponds to the set of contiguous periods of length $\tau^{period}$ that make up the input time series (e.g. load, variable renewable energy availability) to the model| -|$\mathcal{N}^{rep} \subseteq \mathcal{N}$ | where $\mathcal{N}^{rep}$ corresponds to the set of representative time periods that are selected from the set of contiguous periods, $\mathcal{M}$| -|$m \in \mathcal{M}$ | where $m$ corresponds to a representative time period and $\mathcal{M}$ corresponds to the set of representative time periods indexed as per their chronological ocurrence in the set of contiguous periods spanning the input time series data, i.e. $\mathcal{N}$| -$z \in \mathcal{Z}$ | where $z$ denotes a zone and $\mathcal{Z}$ is the set of zones in the network| -|$l \in \mathcal{L}$ | where $l$ denotes a line and $\mathcal{L}$ is the set of transmission lines in the network| -|$y \in \mathcal{G}$ | where $y$ denotes a technology and $\mathcal{G}$ is the set of available technologies | -|$\mathcal{H} \subseteq \mathcal{G}$ | where $\mathcal{H}$ is the subset of thermal resources| -|$\mathcal{VRE} \subseteq \mathcal{G}$ | where $\mathcal{VRE}$ is the subset of curtailable Variable Renewable Energy (VRE) resources| -|$\overline{\mathcal{VRE}}^{y,z}$ | set of VRE resource bins for VRE technology type $y \in \mathcal{VRE}$ in zone $z$ | -|$\mathcal{CE} \subseteq \mathcal{G}$ | where $\mathcal{CE}$ is the subset of resources qualifying for the clean energy standard policy constraint| -|$\mathcal{UC} \subseteq \mathcal{H}$ | where $\mathcal{UC}$ is the subset of thermal resources subject to unit commitment constraints| -|$s \in \mathcal{S}$ | where $s$ denotes a segment and $\mathcal{S}$ is the set of consumers segments for price-responsive demand curtailment| -|$\mathcal{O} \subseteq \mathcal{G}$ | where $\mathcal{O}$ is the subset of storage resources excluding heat storage and hydro storage | -|$o \in \mathcal{O}$ | where $o$ denotes a storage technology in a set $\mathcal{O}$| -|$\mathcal{O}^{sym} \subseteq \mathcal{O}$ | where $\mathcal{O}^{sym}$ corresponds to the set of energy storage technologies with equal (or symmetric) charge and discharge power capacities| -|$\mathcal{O}^{asym} \subseteq \mathcal{O}$ | where $\mathcal{O}^{asym}$ corresponds to the set of energy storage technologies with independently sized (or asymmetric) charge and discharge power capacities| -|$\mathcal{O}^{LDES} \subseteq \mathcal{O}$ | where $\mathcal{O}^{LDES}$ corresponds to the set of long-duration energy storage technologies for which inter-period energy exchange is permitted when using representative periods to model annual grid operations| -$\mathcal{W} \subseteq \mathcal{G}$ | where $\mathcal{W}$ set of hydroelectric generators with water storage reservoirs| -|$\mathcal{W}^{nocap} \subseteq \mathcal{W}$ | where $\mathcal{W}^{nocap}$ is a subset of set of $ \mathcal{W}$ and represents resources with unknown reservoir capacity| -|$\mathcal{W}^{cap} \subseteq \mathcal{W}$ | where $\mathcal{W}^{cap}$ is a subset of set of $ \mathcal{W}$ and represents resources with known reservoir capacity| -|$\mathcal{MR} \subseteq \mathcal{G}$ | where $\mathcal{MR}$ set of must-run resources| -|$\mathcal{DF} \subseteq \mathcal{G}$ | where $\mathcal{DF}$ set of flexible demand resources| -|$\mathcal{G}_p^{ESR} \subseteq \mathcal{G}$ | where $\mathcal{G}_p^{ESR}$ is a subset of $\mathcal{G}$ that is eligible for Energy Share Requirement (ESR) policy constraint $p$| -|$p \in \mathcal{P}$ | where $p$ denotes a instance in the policy set $\mathcal{P}$| -|$\mathcal{P}^{ESR} \subseteq \mathcal{P}$ | Energy Share Requirement type policies | -|$\mathcal{P}^{CO_2} \subseteq \mathcal{P}$ | CO$_2$ emission cap policies| -|$\mathcal{P}^{CO_2}_{mass} \subseteq \mathcal{P}^{CO_2}$ | CO$_2$ emissions limit policy constraints, mass-based | -|$\mathcal{P}^{CO_2}_{load} \subseteq \mathcal{P}^{CO_2}$ | CO$_2$ emissions limit policy constraints, load emission-rate based | -|$\mathcal{P}^{CO_2}_{gen} \subseteq \mathcal{P}^{CO_2}$ | CO$_2$ emissions limit policy constraints, generation emission-rate based | -|$\mathcal{P}^{CRM} \subseteq \mathcal{P}$ | Capacity reserve margin (CRM) type policy constraints | -|$\mathcal{P}^{MinTech} \subseteq \mathcal{P}$ | Minimum Capacity Carve-out type policy constraint | -|$\mathcal{Z}^{ESR}_{p} \subseteq \mathcal{Z}$ | set of zones eligible for ESR policy constraint $p \in \mathcal{P}^{ESR}$ | -|$\mathcal{Z}^{CRM}_{p} \subseteq \mathcal{Z}$ | set of zones that form the locational deliverable area for capacity reserve margin policy constraint $p \in \mathcal{P}^{CRM}$ | -|$\mathcal{Z}^{CO_2}_{p,mass} \subseteq \mathcal{Z}$ | set of zones are under the emission cap mass-based cap-and-trade policy constraint $p \in \mathcal{P}^{CO_2}_{mass}$ | -|$\mathcal{Z}^{CO_2}_{p,load} \subseteq \mathcal{Z}$ | set of zones are under the emission cap load emission-rate based cap-and-trade policy constraint $p \in \mathcal{P}^{CO_2}_{load}$ | -|$\mathcal{Z}^{CO_2}_{p,gen} \subseteq \mathcal{Z}$ | set of zones are under the emission cap generation emission-rate based cap-and-trade policy constraint $p \in \mathcal{P}^{CO2,gen}$ | -|$\mathcal{L}_p^{in} \subseteq \mathcal{L}$ | The subset of transmission lines entering Locational Deliverability Area of capacity reserve margin policy $p \in \mathcal{P}^{CRM}$ | -|$\mathcal{L}_p^{out} \subseteq \mathcal{L}$ | The subset of transmission lines leaving Locational Deliverability Area of capacity reserve margin policy $p \in \mathcal{P}^{CRM}$ | +|$D$ |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 $w/w’$| --- - - -## Decision Variables +## Subsets --- |**Notation** | **Description**| | :------------ | :-----------| -|$\Omega_{y,z} \in \mathbb{R}_+$ | Installed capacity in terms of the number of units (each unit, being of size $\overline{\Omega}_{y,z}^{size}$) of resource $y$ in zone $z$ [Dimensionless]| -|$\Omega^{energy}_{y,z} \in \mathbb{R}_+$ | Installed energy capacity of resource $y$ in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}$ [MWh]| -|$\Omega^{charge}_{y,z} \in \mathbb{R}_+$ | Installed charging power capacity of resource $y$ in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}^{asym}$ [MW]| -|$\Delta_{y,z} \in \mathbb{R}_+$ | Retired capacity of technology $y$ from existing capacity in zone $z$ [MW]| -|$\Delta^{energy}_{y,z} \in \mathbb{R}_+$ | Retired energy capacity of technology $y$ from existing capacity in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}$[MWh]| -|$\Delta^{charge}_{y,z} \in \mathbb{R}_+$ | Retired charging capacity of technology $y$ from existing capacity in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}^{asym}$[MW]| -|$\Delta_{y,z}^{total} \in \mathbb{R}_+$ | Total installed capacity of technology $y$ in zone $z$ [MW]| -|$\Delta_{y,z}^{total,energy} \in \mathbb{R}_+$ | Total installed energy capacity of technology $y$ in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}$ [MWh]| -|$\Delta_{y,z}^{total,charge} \in \mathbb{R}_+$ | Total installed charging power capacity of technology $y$ in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}^{asym}$ [MW]| -|$\bigtriangleup\varphi^{max}_{l}$ | Additional transmission capacity added to line $l$ [MW] | -|$\Theta_{y,z,t} \in \mathbb{R}_+$ | Energy injected into the grid by technology $y$ at time step $t$ in zone $z$ [MWh]| -|$\Pi_{y,z,t} \in \mathbb{R}_+$ | Energy withdrawn from grid by technology $y$ at time step $t$ in zone $z$ [MWh]| -|$\Gamma_{y,z,t} \in \mathbb{R}_+$ | Stored energy level of technology $y$ at end of time step $t$ in zone $z$ [MWh]| -|$\Lambda_{s,z,t} \in \mathbb{R}_+$ | Non-served energy/curtailed demand from the price-responsive demand segment $s$ in zone $z$ at time step $t$ [MWh] | -|$l_{l,t} \in \mathbb{R}_+$ | Losses in line $l$ at time step $t$ [MWh]| -|$\varrho_{y,z,t}\in \mathbb{R}_+$ | Spillage from a reservoir technology $y$ at end of time step $t$ in zone $z$ [MWh]| -|$f_{y,z,t}\in \mathbb{R}_+$ | Frequency regulation contribution [MW] for up and down reserves from technology $y$ in zone $z$ at time $t$\footnote{Regulation reserve contribution are modeled to be symmetric, consistent with current practice in electricity markets} | -|$r_{y,z,t} \in \mathbb{R}_+$ | Upward spinning reserves contribution [MW] from technology $y$ in zone $z$ at time $t$\footnote{we are not modeling down spinning reserves since these are usually never binding for high variable renewable energy systems}| -|$f^{charge}_{y,z,t}\in \mathbb{R}_+$ | Frequency regulation contribution [MW] for up and down reserves from charging storage technology $y$ in zone $z$ at time $t$ | -|$f^{discharge}_{y,z,t}\in \mathbb{R}_+$ | Frequency regulation contribution [MW] for up and down reserves from discharging storage technology $y$ in zone $z$ at time $t$ | -|$r^{charge}_{y,z,t} \in \mathbb{R}_+$ | Upward spinning reserves contribution [MW] from charging storage technology $y$ in zone $z$ at time $t$| -|$r^{discharge}_{y,z,t} \in \mathbb{R}_+$ | Upward spinning reserves contribution [MW] from discharging storage technology $y$ in zone $z$ at time $t$| -|$r^{unmet}_t \in \mathbb{R}_+$ | Shortfall in provision of upward operating spinning reserves during each time period $t \in T$ | -|$\alpha^{Contingency,Aux}_{y,z} \in \{0,1\}$ | Binary variable that is set to be 1 if the total installed capacity $\Delta^{\text{total}}_{y,z} > 0$ for any generator $y \in \mathcal{UC}$ and zone $z$, and can be 0 otherwise | -|$\Phi_{l,t} \in \mathbb{R}_+$ | Power flow in line $l$ at time step $t$ [MWh]| -|$v_{y,z,t}$ | Commitment state of the generation cluster $y$ in zone $z$ at time $t$| -|$\mathcal{X}_{y,z,t}$ | Number of startup decisions, of the generation cluster $y$ in zone $z$ at time $t$| -|$\zeta_{y,z,t}$ | Number of shutdown decisions, of the generation cluster $y$ in zone $z$ at time $t$| -|$\mathcal{Q}_{o,n} \in \mathbb{R}_+$ | Inventory of storage of type $o$ at the beginning of input period $n$ [MWh]| -|$\Delta\mathcal{Q}_{o,m} \in \mathbb{R}$ | Excess storage inventory built up during representative period $m$ [MWh]| -|$ON^{+}_{l,t} \in \{0,1\} $ | Binary variable to activate positive flows on line $l$ in time $t$| -|$TransON^{+}_{l,t} \in \mathbb{R}_+$ | Variable defining maximum positive flow in line $l$ in time $t$ [MW]| +|$D_{i}$ | 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 $W$| --- - - ## Parameters --- |**Notation** | **Description**| | :------------ | :-----------| -|$D_{z,t}$ | Electricity demand in zone $z$ and at time step $t$ [MWh]| -|$\tau^{period}$ | number of time steps in each representative period $w \in \mathcal{W}^{rep}$ and each input period $w \in \mathcal{W}^{input}$| -|$\omega_{t}$ | weight of each model time step $\omega_t =1 \forall t \in T$ when modeling each time step of the year at an hourly resolution [1/year]| -|$n_s^{slope}$ | Cost of non-served energy/demand curtailment for price-responsive demand segment $s$ [\$/MWh]| -|$n_s^{size}$ | Size of price-responsive demand segment $s$ as a fraction of the hourly zonal demand [%]| -|$\overline{\Omega}_{y,z}$ | Maximum capacity of technology $y$ in zone $z$ [MW]| -|$\underline{\Omega}_{y,z}$ | Minimum capacity of technology $y$ in zone $z$ [MW]| -|$\overline{\Omega}^{energy}_{y,z}$ | Maximum energy capacity of technology $y$ in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}$ [MWh]| -|$\underline{\Omega}^{energy}_{y,z}$ | Minimum energy capacity of technology $y$ in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}$ [MWh]| -|$\overline{\Omega}^{charge}_{y,z}$ | Maximum charging power capacity of technology $y$ in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}^{asym}$ [MW]| -|$\underline{\Omega}^{charge}_{y,z}$ | Minimum charging capacity of technology $y$ in zone $z$- only applicable for storage resources, $y \in \mathcal{O}^{asym}$ [MW]| -|$\overline{\Delta}_{y,z}$ | Existing installed capacity of technology $y$ in zone $z$ [MW]| -|$\overline{\Delta^{energy}_{y,z}}$ | Existing installed energy capacity of technology $y$ in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}$ [MW]| -|$\overline{\Delta^{charge}_{y,z}}$ | Existing installed charging capacity of technology $y$ in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}$ [MW]| -|$\overline{\Omega}_{y,z}^{size}$ | Unit size of technology $y$ in zone $z$ [MW]| -|$\pi_{y,z}^{INVEST}$ | Investment cost (annual amortization of total construction cost) for power capacity of technology $y$ in zone $z$ [\$/MW-yr]| -|$\pi_{y,z}^{INVEST,energy}$ | Investment cost (annual amortization of total construction cost) for energy capacity of technology $y$ in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}$ [\$/MWh-yr]| -|$\pi_{y,z}^{INVEST,charge}$ | Investment cost (annual amortization of total construction cost) for charging power capacity of technology $y$ in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}$ [\$/MW-yr]| -|$\pi_{y,z}^{FOM}$ | Fixed O&M cost of technology $y$ in zone $z$ [\$/MW-yr]| -|$\pi_{y,z}^{FOM,energy}$ | Fixed O&M cost of energy component of technology $y$ in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}$ [\$/MWh-yr]| -|$\pi_{y,z}^{FOM,charge}$ | Fixed O&M cost of charging power component of technology $y$ in zone $z$ - only applicable for storage resources, $y \in \mathcal{O}$ [\$/MW-yr]| -|$\pi_{y,z}^{VOM}$ | Variable O&M cost of technology $y$ in zone $z$ [\$/MWh]| -|$\pi_{y,z}^{VOM,charge}$ | Variable O&M cost of charging technology $y$ in zone $z$ - only applicable for storage and demand flexibility resources, $y \in \mathcal{O} \cup \mathcal{DF}$ [\$/MWh]| -|$\pi_{y,z}^{FUEL}$ | Fuel cost of technology $y$ in zone $z$ [\$/MWh]| -|$\pi_{y,z}^{START}$ | Startup cost of technology $y$ in zone $z$ [\$/startup]| -|$\upsilon^{reg}_{y,z}$ | Maximum fraction of capacity that a resource $y$ in zone $z$ can contribute to frequency regulation reserve requirements| -|$\upsilon^{rsv}_{y,z}$ | Maximum fraction of capacity that a resource $y$ in zone $z$ can contribute to upward operating (spinning) reserve requirements| -|$\pi^{Unmet}_{rsv}$ | Cost of unmet spinning reserves in [\$/MW]| -|$\epsilon^{load}_{reg}$ | Frequency regulation reserve requirement as a fraction of forecasted demand in each time step | -|$\epsilon^{vre}_{reg}$ | Frequency regulation reserve requirement as a fraction of variable renewable energy generation in each time step | -|$\epsilon^{load}_{rsv}$ | Operating (spinning) reserve requirement as a fraction of forecasted demand in each time step | -|$\epsilon^{vre}_{rsv}$ | Operating (spinning) reserve requirement as a fraction of forecasted variable renewable energy generation in each time step | -|$\epsilon_{y,z}^{CO_2}$ | CO$_2$ emissions per unit energy produced by technology $y$ in zone $z$ [metric tons/MWh]| -|$\epsilon_{y,z,p}^{MinTech}$ | Equals to 1 if a generator of technology $y$ in zone $z$ is eligible for minimum capacity carveout policy $p \in \mathcal{P}^{MinTech}$, otherwise 0| -|$REQ_p^{MinTech}$ | The minimum capacity requirement of minimum capacity carveout policy $p \in \mathcal{P}^{MinTech}$ [MW]| -|$\epsilon_{y,z,p}^{CRM}$ | Capacity derating factor of technology $y$ in zone $z$ for capacity reserve margin policy $p \in \mathcal{P}^{CRM}$ [fraction]| -|$RM_{z,p}^{CRM}$ | Reserve margin of zone $z$ of capacity reserve margin policy $p \in \mathcal{P}^{CRM}$ [fraction]| -|$\epsilon_{z,p,mass}^{CO_2}$ | Emission budget of zone $z$ under the emission cap $p \in \mathcal{P}^{CO_2}_{mass}$ [ million of metric tonnes]| -|$\epsilon_{z,p,load}^{CO_2}$ | Maximum carbon intensity of the load of zone $z$ under the emission cap $p \in \mathcal{P}^{CO_2}_{load}$ [metric tonnes/MWh]| -|$\epsilon_{z,p,gen}^{CO_2}$ | Maximum emission rate of the generation of zone $z$ under the emission cap $p \in \mathcal{P}^{CO_2}_{gen}$ [metric tonnes/MWh]| -|$\rho_{y,z}^{min}$ | Minimum stable power output per unit of installed capacity for technology $y$ in zone $z$ [%]| -|$\rho_{y,z,t}^{max}$ | Maximum available generation per unit of installed capacity during time step t for technology y in zone z [%]| -|$VREIndex_{y,z}$ | Resource bin index for VRE technology $y$ in zone $z$. $VREIndex_{y,z}=1$ for the first bin, and $VREIndex_{y,z}=0$ for remaining bins. Only defined for $y\in \mathcal{VRE}$ | -|$\varphi^{map}_{l,z}$ | Topology of the network, for line l: $\varphi^{map}_{l,z}=1$ for zone $z$ of origin, - 1 for zone $z$ of destination, 0 otherwise. | -|$\eta_{y,z}^{loss}$ | Self discharge rate per time step per unit of installed capacity for storage technology $y$ in zone $z$ [%]| -|$\eta_{y,z}^{charge}$ | Single-trip efficiency of storage charging/demand deferral for technology $y$ in zone $z$ [%]| -|$\eta_{y,z}^{discharge}$ | Single-trip efficiency of storage (and hydro reservoir) discharging/demand satisfaction for technology $y$ in zone $z$ [%]| -|$\mu_{y,z}^{stor}$ | ratio of energy capacity to discharge power capacity for storage technology (and hydro reservoir) $y$ in zone $z$ [MW/MWh]| -|$\mu_{y,z}^{\mathcal{DF}}$ | Maximum percentage of hourly demand that can be shifted by technology $y$ in zone $z$ [%]| -|$\kappa_{y,z}^{up}$ | Maximum ramp-up rate per time step as percentage of installed capacity of technology y in zone z [%/hr]| -|$\kappa_{y,z}^{down}$ | Maximum ramp-down rate per time step as percentage of installed capacity of technology y in zone z [%/hr]| -|$\tau_{y,z}^{up}$ | Minimum uptime for thermal generator type y in zone z before new shutdown [hours].| -|$\tau_{y,z}^{down}$ | Minimum downtime or thermal generator type y in zone z before new restart [hours].| -|$\tau_{y,z}^{advance}$ | maximum time by which flexible demand resource can be advanced [hours] | -|$\tau_{y,z}^{delay}$ | maximum time by which flexible demand resource can be delayed [hours] | -|$\eta_{y,z}^{dflex}$ | energy losses associated with shifting the flexible load [%]| -|$\mu_{p,z}^{\mathcal{ESR}}$ | share of total demand in each model zone $z \in \mathcal{ESR}^{p}$ that must be served by qualifying renewable energy resources $y \in \mathcal{G}^{ESR}_{p}$| -|$f(n)$ | Mapping each modeled period $n \in \mathcal{N}$ to corresponding representative period $w \in \mathcal{W}$| ---- \ No newline at end of file +|$ALW_{t,w}$ | 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$/t| +|$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, $/sum_{t /in T} N_{t} |H_{t}| = 8760$| +|$NI_{i.h}$ | 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| +--- +## Variables +--- +|**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| +|$p_{g,t,h}$ | 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| +--- +