# Model Notation

## Model Indices and Sets

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 $p \in \mathcal{P}^{CRM}$ |

## Decision Variables

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] |

## 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 $w \in \mathcal{W}$ |