# Model Notation

## Model Indices and Sets

NotationDescription
$t \in \mathcal{T}$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 $p \in \mathcal{P}^{CRM}$

## Decision Variables

NotationDescription
$\Omega_{y,z} \in \mathbb{R}_+$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]

## Parameters

NotationDescription
$D_{z,t}$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 $w \in \mathcal{W}$