SDOM Formulation#

Storage Deployment Optimization Model (SDOM): a mixed-integer linear program that co-optimizes investment in renewable generation, balancing units, and storage technologies while dispatching all resources to meet electrical demand at minimum annualized system cost subject to a clean-energy target.

Sets#

Notation	Index	Description
$\mathcal{P}$	$p$	PV plants
$\mathcal{W}$	$w$	Wind plants
$\mathcal{K}$	$k$	Balancing units (dispatchable)
$\mathcal{H}$	$h$	Hours (time steps)
$\mathcal{B}$	$b$	Hydro budget periods
$\mathcal{H}_b \subseteq \mathcal{H}$	$h$	Hours belonging to budget period $b$
$\mathcal{J}$	$j$	Storage technologies
$\mathcal{J}^{c} \subseteq \mathcal{J}$	$j$	Coupled storage technologies

Parameters#

Derived quantities#

The capital recovery factor annualizes a lump-sum investment over lifetime $l$ at discount rate $r$:

\[ \text{CRF}(l) = \frac{r\,(1+r)^{l}}{(1+r)^{l} - 1} \]

System#

Notation	Description	Units	Domain
$d_h$	Electrical demand at hour $h$	MW	$\mathbb{R}_+$
$r$	Discount rate	—	$\mathbb{R}_+$
$\tau$	Minimum clean-energy generation share	—	$[0, 1]$

Fixed generation profiles#

Exogenous time series scaled by activation parameters $\alpha$.

Notation	Description	Units	Domain
$\rho_h$	Run-of-river hydro generation	MW	$\mathbb{R}_+$
$\nu_h$	Nuclear generation profile	MW	$\mathbb{R}_+$
$\omega_h$	Other renewable generation profile	MW	$\mathbb{R}_+$
$\alpha^{nuc}$	Nuclear activation scalar	—	$\{0, 1\}$
$\alpha^{hyd}$	Hydro activation scalar	—	$\{0, 1\}$
$\alpha^{oth}$	Other renewables activation scalar	—	$\{0, 1\}$

Renewable investment#

Notation	Description	Units	Domain
$\sigma_{ph}$	Solar capacity factor for plant $p$ at hour $h$	—	$[0, 1]$
$\zeta_{wh}$	Wind capacity factor for plant $w$ at hour $h$	—	$[0, 1]$
$\bar{p}^{pv}_p$	Max allowed PV capacity for plant $p$	MW	$\mathbb{R}_+$
$\bar{p}^{wind}_w$	Max allowed wind capacity for plant $w$	MW	$\mathbb{R}_+$
$\kappa^{pv}_p$	PV CAPEX	$/MW	$\mathbb{R}_+$
$\kappa^{wind}_w$	Wind CAPEX	$/MW	$\mathbb{R}_+$
$\kappa^{tc,pv}_p$	PV transmission capital cost	$/MW	$\mathbb{R}_+$
$\kappa^{tc,wind}_w$	Wind transmission capital cost	$/MW	$\mathbb{R}_+$
$\phi^{pv}_p$	PV fixed O&M	$/MW-yr	$\mathbb{R}_+$
$\phi^{wind}_w$	Wind fixed O&M	$/MW-yr	$\mathbb{R}_+$
$l^{vre}$	VRE lifetime (shared across all PV and wind)	yr	$\mathbb{Z}_+$

Balancing units#

Notation	Description	Units	Domain
$\underline{p}^{bal}_k$	Min balancing capacity	MW	$\mathbb{R}_+$
$\bar{p}^{bal}_k$	Max allowed balancing capacity	MW	$\mathbb{R}_+$
$\kappa^{bal}_k$	Balancing unit CAPEX	$/MW	$\mathbb{R}_+$
$\gamma_k$	Marginal fuel cost (fuel price $\times$ heat rate)	$/MWh	$\mathbb{R}_+$
$\phi^{bal}_k$	Balancing fixed O&M	$/MW-yr	$\mathbb{R}_+$
$\psi^{bal}_k$	Balancing variable O&M	$/MWh	$\mathbb{R}_+$
$l^{bal}_k$	Balancing unit lifetime	yr	$\mathbb{Z}_+$

Hydro#

Notation	Description	Units	Domain
$\underline{g}^{hyd}_h$	Hourly hydro lower bound	MW	$\mathbb{R}_+$
$\overline{g}^{hyd}_h$	Hourly hydro upper bound	MW	$\mathbb{R}_+$
$\epsilon_b$	Energy budget for period $b$ (from time series)	MWh	$\mathbb{R}_+$

Trade#

Notation	Description	Units	Domain
$\bar{\iota}_h$	Max import capacity at hour $h$	MW	$\mathbb{R}_+$
$\bar{\xi}_h$	Max export capacity at hour $h$	MW	$\mathbb{R}_+$
$c^{imp}_h$	Import price at hour $h$	$/MWh	$\mathbb{R}_+$
$c^{exp}_h$	Export price at hour $h$	$/MWh	$\mathbb{R}_+$

Storage#

Notation	Description	Units	Domain
$\kappa^{P}_j$	CAPEX power	$/MW	$\mathbb{R}_+$
$\kappa^{E}_j$	CAPEX energy	$/MWh	$\mathbb{R}_+$
$\eta_j$	Roundtrip efficiency (charge $\times$ discharge)	—	$(0, 1]$
$\underline{\delta}_j$	Min duration	hr	$\mathbb{R}_+$
$\overline{\delta}_j$	Max duration	hr	$\mathbb{R}_+$
$\bar{p}^{stor}_j$	Max installable power capacity	MW	$\mathbb{R}_+$
$\alpha_j$	Cost ratio: fraction of power cost on charge side	—	$[0, 1]$
$\phi^{stor}_j$	Fixed O&M	$/MW-yr	$\mathbb{R}_+$
$\psi^{stor}_j$	Variable O&M (applied to discharge only)	$/MWh	$\mathbb{R}_+$
$l^{stor}_j$	Lifetime	yr	$\mathbb{Z}_+$
$\kappa^{cyc}_j$	Max lifetime cycles	—	$\mathbb{R}_+$

Coupled vs decoupled storage. A coupled technology (e.g. Li-Ion battery, pumped hydro) uses shared equipment for charge and discharge, so the power ratings are forced equal: $P^{ch}_j = P^{dis}_j$. A decoupled technology (e.g. CAES, hydrogen) has separate input and output equipment, so $P^{ch}_j$ and $P^{dis}_j$ can be independently sized.

The cost ratio $\alpha_j$ distributes power-related costs (CAPEX, FOM) between the charge side (fraction $\alpha_j$) and the discharge side (fraction $1 - \alpha_j$). For coupled storage where $P^{ch}_j = P^{dis}_j$, the split is immaterial.

Variables#

Renewable (investment and dispatch)#

Variable	Description	Units	Domain
$F^{pv}_p$	Fraction of max PV capacity to build	—	$[0, 1]$
$F^{wind}_w$	Fraction of max wind capacity to build	—	$[0, 1]$
$G^{pv}_h$	Aggregate PV generation	MW	$\mathbb{R}_+$
$G^{wind}_h$	Aggregate wind generation	MW	$\mathbb{R}_+$
$C^{pv}_h$	PV curtailment	MW	$\mathbb{R}_+$
$C^{wind}_h$	Wind curtailment	MW	$\mathbb{R}_+$

Balancing units (investment and dispatch)#

Variable	Description	Units	Domain
$P^{bal}_k$	Installed balancing capacity	MW	$\mathbb{R}_+$
$G^{bal}_{kh}$	Balancing unit generation	MW	$\mathbb{R}_+$

Storage (investment and operations)#

Both charge and discharge power capacity variables exist for all storage technologies.

Variable	Description	Units	Domain
$P^{dis}_j$	Installed discharge power capacity	MW	$\mathbb{R}_+$
$P^{ch}_j$	Installed charge power capacity	MW	$\mathbb{R}_+$
$E_j$	Installed energy capacity	MWh	$\mathbb{R}_+$
$D^{ch}_{jh}$	Charge power	MW	$\mathbb{R}_+$
$D^{dis}_{jh}$	Discharge power	MW	$\mathbb{R}_+$
$S_{jh}$	State of charge	MWh	$\mathbb{R}_+$
$U_{jh}$	Charge indicator ($1$ = charging)	—	$\{0, 1\}$

Hydro#

Variable	Description	Units	Domain
$G^{hyd}_h$	Dispatchable hydro generation	MW	$\mathbb{R}_+$

Trade#

Variable	Description	Units	Domain
$M_h$	Imports	MW	$\mathbb{R}_+$
$X_h$	Exports	MW	$\mathbb{R}_+$
$V_h$	Net-load sign indicator ($1$ if net load $> 0$)	—	$\{0, 1\}$

Objective Function#

\[ \min \; Z \;=\; Z^{pv} + Z^{wind} + Z^{bal} + Z^{stor} + Z^{trade} \]

PV cost#

Annualized CAPEX (including transmission) and FOM. VRE shares a single lifetime $l^{vre}$:

\[ Z^{pv} = \sum_{p \in \mathcal{P}} \Bigl[ \bigl(\text{CRF}(l^{vre}) \cdot (\kappa^{pv}_p + \kappa^{tc,pv}_p) + \phi^{pv}_p\bigr) \;\bar{p}^{pv}_p \; F^{pv}_p \Bigr] \]

Symbol	Type	Description
$\mathcal{P}$	Set	PV plants
$p$	Index	PV plant
$l^{vre}$	Parameter	VRE lifetime (yr)
$\kappa^{pv}_p$	Parameter	PV CAPEX ($/MW)
$\kappa^{tc,pv}_p$	Parameter	PV transmission capital cost ($/MW)
$\phi^{pv}_p$	Parameter	PV fixed O&M ($/MW-yr)
$\bar{p}^{pv}_p$	Parameter	Max allowed PV capacity (MW)
$F^{pv}_p$	Variable	Fraction of max PV capacity to build

Wind cost#

\[ Z^{wind} = \sum_{w \in \mathcal{W}} \Bigl[ \bigl(\text{CRF}(l^{vre}) \cdot (\kappa^{wind}_w + \kappa^{tc,wind}_w) + \phi^{wind}_w\bigr) \;\bar{p}^{wind}_w \; F^{wind}_w \Bigr] \]

Symbol	Type	Description
$\mathcal{W}$	Set	Wind plants
$w$	Index	Wind plant
$l^{vre}$	Parameter	VRE lifetime (yr)
$\kappa^{wind}_w$	Parameter	Wind CAPEX ($/MW)
$\kappa^{tc,wind}_w$	Parameter	Wind transmission capital cost ($/MW)
$\phi^{wind}_w$	Parameter	Wind fixed O&M ($/MW-yr)
$\bar{p}^{wind}_w$	Parameter	Max allowed wind capacity (MW)
$F^{wind}_w$	Variable	Fraction of max wind capacity to build

Balancing unit cost#

Per-unit CRF based on individual lifetimes. Marginal cost includes fuel and VOM:

\[ Z^{bal} = \sum_{k \in \mathcal{K}} \Bigl[ \bigl(\text{CRF}(l^{bal}_k) \cdot \kappa^{bal}_k + \phi^{bal}_k\bigr) \; P^{bal}_k \;+\; \sum_{h \in \mathcal{H}} \bigl(\gamma_k + \psi^{bal}_k\bigr) \; G^{bal}_{kh} \Bigr] \]

Symbol	Type	Description
$\mathcal{K}$	Set	Balancing units (dispatchable)
$k$	Index	Balancing unit
$\mathcal{H}$	Set	Hours (time steps)
$h$	Index	Hour
$l^{bal}_k$	Parameter	Balancing unit lifetime (yr)
$\kappa^{bal}_k$	Parameter	Balancing unit CAPEX ($/MW)
$\phi^{bal}_k$	Parameter	Balancing fixed O&M ($/MW-yr)
$\gamma_k$	Parameter	Marginal fuel cost ($/MWh)
$\psi^{bal}_k$	Parameter	Balancing variable O&M ($/MWh)
$P^{bal}_k$	Variable	Installed balancing capacity (MW)
$G^{bal}_{kh}$	Variable	Balancing unit generation (MW)

Storage cost#

The cost ratio $\alpha_j$ distributes power CAPEX and FOM between the charge side (fraction $\alpha_j$) and the discharge side (fraction $1 - \alpha_j$). This applies uniformly to all storage technologies:

\[ Z^{stor} = \sum_{j \in \mathcal{J}} \Bigl[ \text{CRF}(l^{stor}_j) \Bigl( \kappa^{P}_j \bigl(\alpha_j \; P^{ch}_j + (1 - \alpha_j) \; P^{dis}_j\bigr) \;+\; \kappa^{E}_j \; E_j \Bigr) \;+\; \phi^{stor}_j \bigl(\alpha_j \; P^{ch}_j + (1 - \alpha_j) \; P^{dis}_j\bigr) \;+\; \sum_{h \in \mathcal{H}} \psi^{stor}_j \; D^{dis}_{jh} \Bigr] \]

VOM is charged on discharge only.

Symbol	Type	Description
$\mathcal{J}$	Set	Storage technologies
$j$	Index	Storage technology
$\mathcal{H}$	Set	Hours (time steps)
$h$	Index	Hour
$l^{stor}_j$	Parameter	Storage lifetime (yr)
$\kappa^{P}_j$	Parameter	CAPEX power ($/MW)
$\kappa^{E}_j$	Parameter	CAPEX energy ($/MWh)
$\alpha_j$	Parameter	Cost ratio: fraction of power cost on charge side
$\phi^{stor}_j$	Parameter	Storage fixed O&M ($/MW-yr)
$\psi^{stor}_j$	Parameter	Storage variable O&M ($/MWh)
$P^{ch}_j$	Variable	Installed charge power capacity (MW)
$P^{dis}_j$	Variable	Installed discharge power capacity (MW)
$E_j$	Variable	Installed energy capacity (MWh)
$D^{dis}_{jh}$	Variable	Discharge power (MW)

Trade cost#

Import cost minus export revenue:

\[ Z^{trade} = \sum_{h \in \mathcal{H}} \bigl( c^{imp}_h \; M_h \;-\; c^{exp}_h \; X_h \bigr) \]

Symbol	Type	Description
$\mathcal{H}$	Set	Hours (time steps)
$h$	Index	Hour
$c^{imp}_h$	Parameter	Import price at hour $h$ ($/MWh)
$c^{exp}_h$	Parameter	Export price at hour $h$ ($/MWh)
$M_h$	Variable	Imports (MW)
$X_h$	Variable	Exports (MW)

Constraints#

System constraints#

Energy supply balance#

\[ \forall\, h \in \mathcal{H}: \quad \underbrace{ G^{pv}_h + G^{wind}_h + \sum_{k \in \mathcal{K}} G^{bal}_{kh} + G^{hyd}_h + \alpha^{nuc} \nu_h + \alpha^{oth} \omega_h + \sum_{j \in \mathcal{J}} D^{dis}_{jh} + M_h }_{\text{supply}} \;=\; \underbrace{ d_h + \sum_{j \in \mathcal{J}} D^{ch}_{jh} + X_h }_{\text{demand + charging + exports}} \]

Curtailment does not appear in the energy balance. It is handled per-technology by the VRE balance constraints below.

Carbon-free generation target#

Total balancing-unit generation plus imports must not exceed $(1 - \tau)$ of adjusted demand (demand plus net storage loading). This ensures that both thermal generation and imported electricity count against the clean-energy target:

\[ \sum_{k \in \mathcal{K}} \sum_{h \in \mathcal{H}} G^{bal}_{kh} + \sum_{h \in \mathcal{H}} M_h \;\leq\; (1 - \tau) \sum_{h \in \mathcal{H}} \Bigl( d_h + \sum_{j \in \mathcal{J}} D^{ch}_{jh} - \sum_{j \in \mathcal{J}} D^{dis}_{jh} \Bigr) \]

Note: Imports are treated as non-clean energy in this constraint. If imports are not modeled (i.e., the Imports formulation is set to “NotModel”), only thermal generation is constrained.

Symbols used in system constraints:

Symbol	Type	Description
$\mathcal{H}$	Set	Hours (time steps)
$\mathcal{K}$	Set	Balancing units (dispatchable)
$\mathcal{J}$	Set	Storage technologies
$h$	Index	Hour
$k$	Index	Balancing unit
$j$	Index	Storage technology
$d_h$	Parameter	Electrical demand at hour $h$ (MW)
$\alpha^{nuc}$	Parameter	Nuclear activation scalar
$\alpha^{oth}$	Parameter	Other renewables activation scalar
$\nu_h$	Parameter	Nuclear generation profile (MW)
$\omega_h$	Parameter	Other renewable generation profile (MW)
$\tau$	Parameter	Minimum clean-energy generation share
$G^{pv}_h$	Variable	Aggregate PV generation (MW)
$G^{wind}_h$	Variable	Aggregate wind generation (MW)
$G^{bal}_{kh}$	Variable	Balancing unit generation (MW)
$G^{hyd}_h$	Variable	Dispatchable hydro generation (MW)
$D^{ch}_{jh}$	Variable	Charge power (MW)
$D^{dis}_{jh}$	Variable	Discharge power (MW)
$M_h$	Variable	Imports (MW)
$X_h$	Variable	Exports (MW)

VRE balance constraints#

Generation plus curtailment equals total available resource. Per-technology equality:

PV balance#

\[ \forall\, h \in \mathcal{H}: \qquad G^{pv}_h + C^{pv}_h \;=\; \sum_{p \in \mathcal{P}} \sigma_{ph} \;\bar{p}^{pv}_p \; F^{pv}_p \]

Wind balance#

\[ \forall\, h \in \mathcal{H}: \qquad G^{wind}_h + C^{wind}_h \;=\; \sum_{w \in \mathcal{W}} \zeta_{wh} \;\bar{p}^{wind}_w \; F^{wind}_w \]

Symbols used in VRE balance constraints:

Symbol	Type	Description
$\mathcal{H}$	Set	Hours (time steps)
$\mathcal{P}$	Set	PV plants
$\mathcal{W}$	Set	Wind plants
$h$	Index	Hour
$p$	Index	PV plant
$w$	Index	Wind plant
$\sigma_{ph}$	Parameter	Solar capacity factor for plant $p$ at hour $h$
$\bar{p}^{pv}_p$	Parameter	Max allowed PV capacity (MW)
$\zeta_{wh}$	Parameter	Wind capacity factor for plant $w$ at hour $h$
$\bar{p}^{wind}_w$	Parameter	Max allowed wind capacity (MW)
$G^{pv}_h$	Variable	Aggregate PV generation (MW)
$C^{pv}_h$	Variable	PV curtailment (MW)
$F^{pv}_p$	Variable	Fraction of max PV capacity to build
$G^{wind}_h$	Variable	Aggregate wind generation (MW)
$C^{wind}_h$	Variable	Wind curtailment (MW)
$F^{wind}_w$	Variable	Fraction of max wind capacity to build

Balancing unit constraints#

Dispatch limit#

\[ \forall\, k \in \mathcal{K} \;\; \forall\, h \in \mathcal{H}: \qquad G^{bal}_{kh} \;\leq\; P^{bal}_k \]

Capacity bounds#

\[ \forall\, k \in \mathcal{K}: \qquad \underline{p}^{bal}_k \;\leq\; P^{bal}_k \;\leq\; \bar{p}^{bal}_k \]

Symbols used in balancing unit constraints:

Symbol	Type	Description
$\mathcal{K}$	Set	Balancing units (dispatchable)
$\mathcal{H}$	Set	Hours (time steps)
$k$	Index	Balancing unit
$h$	Index	Hour
$\underline{p}^{bal}_k$	Parameter	Min balancing capacity (MW)
$\bar{p}^{bal}_k$	Parameter	Max allowed balancing capacity (MW)
$P^{bal}_k$	Variable	Installed balancing capacity (MW)
$G^{bal}_{kh}$	Variable	Balancing unit generation (MW)

Storage constraints#

Coupled power equality#

For coupled technologies, charge and discharge power capacity must be equal:

\[ \forall\, j \in \mathcal{J}^{c}: \qquad P^{ch}_j \;=\; P^{dis}_j \]

Power capacity bounds#

\[ \forall\, j \in \mathcal{J}: \qquad P^{ch}_j \;\leq\; \bar{p}^{stor}_j \;, \quad P^{dis}_j \;\leq\; \bar{p}^{stor}_j \]

Charge and discharge hourly limits#

\[ \forall\, j \in \mathcal{J} \;\; \forall\, h \in \mathcal{H}: \qquad D^{ch}_{jh} \;\leq\; P^{ch}_j \;, \quad D^{dis}_{jh} \;\leq\; P^{dis}_j \]

Charge or discharge only#

A storage unit cannot simultaneously charge and discharge. Enforced with binary indicator $U_{jh}$ using $\bar{p}^{stor}_j$ as a tight big-$\mathcal{M}$:

\[ \forall\, j \in \mathcal{J} \;\; \forall\, h \in \mathcal{H}: \qquad D^{ch}_{jh} \;\leq\; \bar{p}^{stor}_j \;\cdot\; U_{jh} \]

\[ \forall\, j \in \mathcal{J} \;\; \forall\, h \in \mathcal{H}: \qquad D^{dis}_{jh} \;\leq\; \bar{p}^{stor}_j \;\cdot\; (1 - U_{jh}) \]

State-of-charge balance#

\[ \forall\, j \in \mathcal{J} \;\; \forall\, h \in \mathcal{H} \setminus \{1\}: \qquad S_{jh} \;=\; S_{j(h-1)} \;+\; \sqrt{\eta_j}\; D^{ch}_{jh} \;-\; \frac{1}{\sqrt{\eta_j}}\; D^{dis}_{jh} \]

Cyclic boundary condition (SOC wraps around):

\[ \forall\, j \in \mathcal{J}: \qquad S_{j1} \;=\; S_{j|\mathcal{H}|} \;+\; \sqrt{\eta_j}\; D^{ch}_{j1} \;-\; \frac{1}{\sqrt{\eta_j}}\; D^{dis}_{j1} \]

State-of-charge limits#

\[ \forall\, j \in \mathcal{J} \;\; \forall\, h \in \mathcal{H}: \qquad 0 \;\leq\; S_{jh} \;\leq\; E_j \]

Duration limits#

The energy-to-power ratio must lie within the allowable duration window. Duration is defined relative to discharge power corrected for discharge efficiency:

\[ \forall\, j \in \mathcal{J}: \qquad \frac{\underline{\delta}_j \; P^{dis}_j}{\sqrt{\eta_j}} \;\leq\; E_j \;\leq\; \frac{\overline{\delta}_j \; P^{dis}_j}{\sqrt{\eta_j}} \]

Cycle limits#

Total discharge throughput is bounded by the annualized maximum number of full cycles (lifetime cycles divided by lifetime):

\[ \forall\, j \in \mathcal{J}: \qquad \sum_{h \in \mathcal{H}} D^{dis}_{jh} \;\leq\; \frac{\kappa^{cyc}_j}{l^{stor}_j} \;\cdot\; E_j \]

Symbols used in storage constraints:

Symbol	Type	Description
$\mathcal{J}$	Set	Storage technologies
$\mathcal{J}^{c}$	Set	Coupled storage technologies
$\mathcal{H}$	Set	Hours (time steps)
$j$	Index	Storage technology
$h$	Index	Hour
$\bar{p}^{stor}_j$	Parameter	Max installable power capacity (MW)
$\eta_j$	Parameter	Roundtrip efficiency
$\underline{\delta}_j$	Parameter	Min duration (hr)
$\overline{\delta}_j$	Parameter	Max duration (hr)
$\kappa^{cyc}_j$	Parameter	Max lifetime cycles
$l^{stor}_j$	Parameter	Storage lifetime (yr)
$P^{ch}_j$	Variable	Installed charge power capacity (MW)
$P^{dis}_j$	Variable	Installed discharge power capacity (MW)
$E_j$	Variable	Installed energy capacity (MWh)
$D^{ch}_{jh}$	Variable	Charge power (MW)
$D^{dis}_{jh}$	Variable	Discharge power (MW)
$S_{jh}$	Variable	State of charge (MWh)
$U_{jh}$	Variable	Charge indicator ($1$ = charging)

Hydro budget constraints#

Hourly bounds#

\[ \forall\, h \in \mathcal{H}: \qquad \alpha^{hyd} \; \underline{g}^{hyd}_h \;\leq\; G^{hyd}_h \;\leq\; \alpha^{hyd} \; \overline{g}^{hyd}_h \]

Energy budget (equality)#

Total dispatchable hydro generation within each budget period must equal the available energy budget:

\[ \forall\, b \in \mathcal{B}: \qquad \sum_{h \in \mathcal{H}_b} G^{hyd}_h \;=\; \epsilon_b \]

Symbols used in hydro budget constraints:

Symbol	Type	Description
$\mathcal{H}$	Set	Hours (time steps)
$\mathcal{B}$	Set	Hydro budget periods
$\mathcal{H}_b$	Set	Hours belonging to budget period $b$
$h$	Index	Hour
$b$	Index	Budget period
$\alpha^{hyd}$	Parameter	Hydro activation scalar
$\underline{g}^{hyd}_h$	Parameter	Hourly hydro lower bound (MW)
$\overline{g}^{hyd}_h$	Parameter	Hourly hydro upper bound (MW)
$\epsilon_b$	Parameter	Energy budget for period $b$ (MWh)
$G^{hyd}_h$	Variable	Dispatchable hydro generation (MW)

Import and export constraints#

Capacity limits#

\[ \forall\, h \in \mathcal{H}: \qquad M_h \;\leq\; \bar{\iota}_h \;, \quad X_h \;\leq\; \bar{\xi}_h \]

Net-load indicator#

Define the net load as demand minus all VRE availability, fixed clean generation, and dispatchable hydro:

\[ \Lambda_h \;=\; d_h - (G^{pv}_h + C^{pv}_h) - (G^{wind}_h + C^{wind}_h) - \alpha^{nuc} \nu_h - \alpha^{oth} \omega_h - G^{hyd}_h \]

A binary indicator $V_h$ encodes whether the system is a net importer ($\Lambda_h > 0$) or net exporter ($\Lambda_h \leq 0$). An $\varepsilon$ offset prevents numerical degeneracy at $\Lambda_h = 0$:

\[ \forall\, h \in \mathcal{H}: \qquad \Lambda_h \;\leq\; \mathcal{M} \;\cdot\; V_h \]

\[ \forall\, h \in \mathcal{H}: \qquad -\Lambda_h + \varepsilon \;\leq\; \mathcal{M} \;\cdot\; (1 - V_h) \]

Import allowed only when net load is positive#

\[ \forall\, h \in \mathcal{H}: \qquad M_h \;\leq\; d_h \;\cdot\; V_h \]

Export allowed only when net load is negative#

\[ \forall\, h \in \mathcal{H}: \qquad X_h \;\leq\; \Bigl(\max_{h' \in \mathcal{H}} \bar{\xi}_{h'}\Bigr) \;\cdot\; (1 - V_h) \]

Symbols used in import and export constraints:

Symbol	Type	Description
$\mathcal{H}$	Set	Hours (time steps)
$h$	Index	Hour
$d_h$	Parameter	Electrical demand at hour $h$ (MW)
$\bar{\iota}_h$	Parameter	Max import capacity at hour $h$ (MW)
$\bar{\xi}_h$	Parameter	Max export capacity at hour $h$ (MW)
$\alpha^{nuc}$	Parameter	Nuclear activation scalar
$\alpha^{oth}$	Parameter	Other renewables activation scalar
$\nu_h$	Parameter	Nuclear generation profile (MW)
$\omega_h$	Parameter	Other renewable generation profile (MW)
$\mathcal{M}$	Parameter	Big-M constant
$\varepsilon$	Parameter	Small offset to prevent numerical degeneracy
$\Lambda_h$	Derived	Net load at hour $h$ (MW)
$G^{pv}_h$	Variable	Aggregate PV generation (MW)
$C^{pv}_h$	Variable	PV curtailment (MW)
$G^{wind}_h$	Variable	Aggregate wind generation (MW)
$C^{wind}_h$	Variable	Wind curtailment (MW)
$G^{hyd}_h$	Variable	Dispatchable hydro generation (MW)
$M_h$	Variable	Imports (MW)
$X_h$	Variable	Exports (MW)
$V_h$	Variable	Net-load sign indicator ($1$ if net load $> 0$)

Notation	Index	Description
\(\mathcal{P}\)	\(p\)	PV plants
\(\mathcal{W}\)	\(w\)	Wind plants
\(\mathcal{K}\)	\(k\)	Balancing units (dispatchable)
\(\mathcal{H}\)	\(h\)	Hours (time steps)
\(\mathcal{B}\)	\(b\)	Hydro budget periods
\(\mathcal{H}_b \subseteq \mathcal{H}\)	\(h\)	Hours belonging to budget period \(b\)
\(\mathcal{J}\)	\(j\)	Storage technologies
\(\mathcal{J}^{c} \subseteq \mathcal{J}\)	\(j\)	Coupled storage technologies

Notation	Description	Units	Domain
\(\sigma_{ph}\)	Solar capacity factor for plant \(p\) at hour \(h\)	—	\([0, 1]\)
\(\zeta_{wh}\)	Wind capacity factor for plant \(w\) at hour \(h\)	—	\([0, 1]\)
\(\bar{p}^{pv}_p\)	Max allowed PV capacity for plant \(p\)	MW	\(\mathbb{R}_+\)
\(\bar{p}^{wind}_w\)	Max allowed wind capacity for plant \(w\)	MW	\(\mathbb{R}_+\)
\(\kappa^{pv}_p\)	PV CAPEX	$/MW	\(\mathbb{R}_+\)
\(\kappa^{wind}_w\)	Wind CAPEX	$/MW	\(\mathbb{R}_+\)
\(\kappa^{tc,pv}_p\)	PV transmission capital cost	$/MW	\(\mathbb{R}_+\)
\(\kappa^{tc,wind}_w\)	Wind transmission capital cost	$/MW	\(\mathbb{R}_+\)
\(\phi^{pv}_p\)	PV fixed O&M	$/MW-yr	\(\mathbb{R}_+\)
\(\phi^{wind}_w\)	Wind fixed O&M	$/MW-yr	\(\mathbb{R}_+\)
\(l^{vre}\)	VRE lifetime (shared across all PV and wind)	yr	\(\mathbb{Z}_+\)

Notation	Description	Units	Domain
\(\bar{\iota}_h\)	Max import capacity at hour \(h\)	MW	\(\mathbb{R}_+\)
\(\bar{\xi}_h\)	Max export capacity at hour \(h\)	MW	\(\mathbb{R}_+\)
\(c^{imp}_h\)	Import price at hour \(h\)	$/MWh	\(\mathbb{R}_+\)
\(c^{exp}_h\)	Export price at hour \(h\)	$/MWh	\(\mathbb{R}_+\)

Notation	Description	Units	Domain
\(d_h\)	Electrical demand at hour \(h\)	MW	\(\mathbb{R}_+\)
\(r\)	Discount rate	—	\(\mathbb{R}_+\)
\(\tau\)	Minimum clean-energy generation share	—	\([0, 1]\)

Notation	Description	Units	Domain
\(\rho_h\)	Run-of-river hydro generation	MW	\(\mathbb{R}_+\)
\(\nu_h\)	Nuclear generation profile	MW	\(\mathbb{R}_+\)
\(\omega_h\)	Other renewable generation profile	MW	\(\mathbb{R}_+\)
\(\alpha^{nuc}\)	Nuclear activation scalar	—	\(\{0, 1\}\)
\(\alpha^{hyd}\)	Hydro activation scalar	—	\(\{0, 1\}\)
\(\alpha^{oth}\)	Other renewables activation scalar	—	\(\{0, 1\}\)

Notation	Description	Units	Domain
\(\underline{p}^{bal}_k\)	Min balancing capacity	MW	\(\mathbb{R}_+\)
\(\bar{p}^{bal}_k\)	Max allowed balancing capacity	MW	\(\mathbb{R}_+\)
\(\kappa^{bal}_k\)	Balancing unit CAPEX	$/MW	\(\mathbb{R}_+\)
\(\gamma_k\)	Marginal fuel cost (fuel price \(\times\) heat rate)	$/MWh	\(\mathbb{R}_+\)
\(\phi^{bal}_k\)	Balancing fixed O&M	$/MW-yr	\(\mathbb{R}_+\)
\(\psi^{bal}_k\)	Balancing variable O&M	$/MWh	\(\mathbb{R}_+\)
\(l^{bal}_k\)	Balancing unit lifetime	yr	\(\mathbb{Z}_+\)

Notation	Description	Units	Domain
\(\underline{g}^{hyd}_h\)	Hourly hydro lower bound	MW	\(\mathbb{R}_+\)
\(\overline{g}^{hyd}_h\)	Hourly hydro upper bound	MW	\(\mathbb{R}_+\)
\(\epsilon_b\)	Energy budget for period \(b\) (from time series)	MWh	\(\mathbb{R}_+\)

Notation	Description	Units	Domain
\(\kappa^{P}_j\)	CAPEX power	$/MW	\(\mathbb{R}_+\)
\(\kappa^{E}_j\)	CAPEX energy	$/MWh	\(\mathbb{R}_+\)
\(\eta_j\)	Roundtrip efficiency (charge \(\times\) discharge)	—	\((0, 1]\)
\(\underline{\delta}_j\)	Min duration	hr	\(\mathbb{R}_+\)
\(\overline{\delta}_j\)	Max duration	hr	\(\mathbb{R}_+\)
\(\bar{p}^{stor}_j\)	Max installable power capacity	MW	\(\mathbb{R}_+\)
\(\alpha_j\)	Cost ratio: fraction of power cost on charge side	—	\([0, 1]\)
\(\phi^{stor}_j\)	Fixed O&M	$/MW-yr	\(\mathbb{R}_+\)
\(\psi^{stor}_j\)	Variable O&M (applied to discharge only)	$/MWh	\(\mathbb{R}_+\)
\(l^{stor}_j\)	Lifetime	yr	\(\mathbb{Z}_+\)
\(\kappa^{cyc}_j\)	Max lifetime cycles	—	\(\mathbb{R}_+\)

Variable	Description	Units	Domain
\(F^{pv}_p\)	Fraction of max PV capacity to build	—	\([0, 1]\)
\(F^{wind}_w\)	Fraction of max wind capacity to build	—	\([0, 1]\)
\(G^{pv}_h\)	Aggregate PV generation	MW	\(\mathbb{R}_+\)
\(G^{wind}_h\)	Aggregate wind generation	MW	\(\mathbb{R}_+\)
\(C^{pv}_h\)	PV curtailment	MW	\(\mathbb{R}_+\)
\(C^{wind}_h\)	Wind curtailment	MW	\(\mathbb{R}_+\)

Variable	Description	Units	Domain
\(P^{bal}_k\)	Installed balancing capacity	MW	\(\mathbb{R}_+\)
\(G^{bal}_{kh}\)	Balancing unit generation	MW	\(\mathbb{R}_+\)

Variable	Description	Units	Domain
\(P^{dis}_j\)	Installed discharge power capacity	MW	\(\mathbb{R}_+\)
\(P^{ch}_j\)	Installed charge power capacity	MW	\(\mathbb{R}_+\)
\(E_j\)	Installed energy capacity	MWh	\(\mathbb{R}_+\)
\(D^{ch}_{jh}\)	Charge power	MW	\(\mathbb{R}_+\)
\(D^{dis}_{jh}\)	Discharge power	MW	\(\mathbb{R}_+\)
\(S_{jh}\)	State of charge	MWh	\(\mathbb{R}_+\)
\(U_{jh}\)	Charge indicator (\(1\) = charging)	—	\(\{0, 1\}\)

Variable	Description	Units	Domain
\(M_h\)	Imports	MW	\(\mathbb{R}_+\)
\(X_h\)	Exports	MW	\(\mathbb{R}_+\)
\(V_h\)	Net-load sign indicator (\(1\) if net load \(> 0\))	—	\(\{0, 1\}\)

Symbol	Type	Description
\(\mathcal{J}\)	Set	Storage technologies
\(\mathcal{J}^{c}\)	Set	Coupled storage technologies
\(\mathcal{H}\)	Set	Hours (time steps)
\(j\)	Index	Storage technology
\(h\)	Index	Hour
\(\bar{p}^{stor}_j\)	Parameter	Max installable power capacity (MW)
\(\eta_j\)	Parameter	Roundtrip efficiency
\(\underline{\delta}_j\)	Parameter	Min duration (hr)
\(\overline{\delta}_j\)	Parameter	Max duration (hr)
\(\kappa^{cyc}_j\)	Parameter	Max lifetime cycles
\(l^{stor}_j\)	Parameter	Storage lifetime (yr)
\(P^{ch}_j\)	Variable	Installed charge power capacity (MW)
\(P^{dis}_j\)	Variable	Installed discharge power capacity (MW)
\(E_j\)	Variable	Installed energy capacity (MWh)
\(D^{ch}_{jh}\)	Variable	Charge power (MW)
\(D^{dis}_{jh}\)	Variable	Discharge power (MW)
\(S_{jh}\)	Variable	State of charge (MWh)
\(U_{jh}\)	Variable	Charge indicator (\(1\) = charging)

SDOM Formulation#

Sets#

Parameters#

Derived quantities#

System#

Fixed generation profiles#

Renewable investment#

Balancing units#

Hydro#

Trade#

Storage#

Variables#

Renewable (investment and dispatch)#

Balancing units (investment and dispatch)#

Storage (investment and operations)#

Hydro#

Trade#

Objective Function#

PV cost#

Wind cost#

Balancing unit cost#

Storage cost#

Trade cost#

Constraints#

System constraints#

Energy supply balance#

Carbon-free generation target#

VRE balance constraints#

PV balance#

Wind balance#

Balancing unit constraints#

Dispatch limit#

Capacity bounds#

Storage constraints#

Coupled power equality#

Power capacity bounds#

Charge and discharge hourly limits#

Charge or discharge only#

State-of-charge balance#

State-of-charge limits#

Duration limits#

Cycle limits#

Hydro budget constraints#

Hourly bounds#

Energy budget (equality)#

Import and export constraints#

Capacity limits#

Net-load indicator#

Import allowed only when net load is positive#

Export allowed only when net load is negative#

This Page