Sub-Module 1.2-A
Mathematical Foundations and Formal Notation
Indices and Sets
The following indices are used throughout the manuscript. All time series operate at hourly resolution unless otherwise stated.
| Symbol | Range | Description |
|---|---|---|
| \(t \in \mathcal{T}\) | \(1,\ldots,8760\) | Hourly time index [h] |
| \(y \in \mathcal{Y}\) | \(\{2020, 2025, 2028, 2035\}\) | Epoch/year index |
| \(u \in \mathcal{U}\) | — | Utility/technology unit index |
| \(s \in \mathcal{S}\) | \(\{\text{EB, BB}\}\) | Pathway/strategy index |
| \(\omega \in \Omega\) | \(0,\ldots,99\) | Future/uncertainty realisation index |
Demand Construction (DemandPack)
The DemandPack construction maps a normalised hourly shape to absolute demand via a scaling factor:
\[D_{t,e} = \alpha_e \cdot s_{t,e}\]
where \(s_{t,e}\) is the normalised shape (dimensionless) for end-use \(e\) at hour \(t\), and \(\alpha_e\) is the scaling factor derived from annual energy consistency:
\[E_e^{\text{ann}} = \sum_{t \in \mathcal{T}} D_{t,e} \cdot \Delta t\]
The scaling factor \(\alpha_e\) is determined by solving this consistency condition given the declared annual total \(E_e^{\text{ann}}\).
Site Dispatch
Proportional dispatch allocates total heat demand \(D_t\) across committed units in proportion to nameplate capacity:
\[q_{t,u} = D_t \cdot \frac{C_u}{\sum_{u' \in \mathcal{U}^{\text{on}}} C_{u'}}\]
where \(C_u\) is the nameplate thermal capacity of unit \(u\) and \(\mathcal{U}^{\text{on}}\) is the set of committed units.
Unserved heat is:
\[U_t = \max\left(0,\ D_t - \sum_{u \in \mathcal{U}} q_{t,u}\right)\]
GXP Interface Signals
The incremental electricity signal exported by the site module:
\[\Delta E_t = p_t^{\text{pathway}} - p_t^{\text{baseline}}\]
where \(p_t\) is the site’s electricity import at hour \(t\).
GXP headroom exceedance at time \(t\):
\[X_t = \max\left(0,\ \Delta E_t - H_t\right)\]
where \(H_t\) is the available GXP headroom, scaled per future \(\omega\) by the headroom multiplier \(U_{\text{head}}^\omega\):
\[H_t^\omega = H_t^{\text{ref}} \cdot U_{\text{head}}^\omega\]
Economic Notation
| Symbol | Units | Description |
|---|---|---|
| \(C_{\text{fuel}}\) | NZD/MWh | Delivered fuel cost |
| \(C_{\text{elec}}\) | NZD/MWh | Electricity tariff |
| \(C_{\text{capex}}\) | NZD | Capital cost |
| \(C_{\text{fix}}\) | NZD/yr | Fixed operation and maintenance |
| \(C_{\text{var}}\) | NZD/MWh | Variable operation and maintenance |
| \(\text{VOLL}\) | NZD/MWh | Value of lost load |
| \(r\) | — | Discount rate |
| \(\text{CRF}\) | — | Capital recovery factor |
| \(p_{\text{CO}_2}\) | NZD/tCO₂e | Carbon price (ETS) |
Total system cost for pathway \(s\) under future \(\omega\):
\[Z(s,\omega) = C_{\text{site}}(s,\omega) + C_{\text{grid}}(s,\omega)\]
where \(C_{\text{grid}}\) includes annualised upgrade cost and unserved energy penalty.
Robustness and Regret
Regret of pathway \(s\) in future \(\omega\):
\[\rho(s,\omega) = Z(s,\omega) - \min_{s' \in \mathcal{S}} Z(s',\omega)\]
Regret is zero for the minimum-cost pathway in each future and positive for all others.
Maximum regret:
\[R_{\max}(s) = \max_{\omega \in \Omega} \rho(s,\omega)\]
Satisficing rate at threshold \(\tau\):
\[\text{SR}(s,\tau) = \frac{|\{\omega \in \Omega : Z(s,\omega) \leq \tau\}|}{|\Omega|}\]
Win rate (fraction of futures where \(s\) is minimum-cost):
\[\text{WR}(s) = \frac{|\{\omega \in \Omega : \rho(s,\omega) = 0\}|}{|\Omega|}\]
Uncertainty Multipliers (Paired-Futures Ensemble)
Each future \(\omega\) is characterised by a vector of uncertain driver realisations. In the proof-of-concept ensemble of 100 paired futures:
| Multiplier | Applied to | P10 / Median / P90 |
|---|---|---|
| \(P_{\text{elec}}^\omega\) | Site electricity cost | 0.785 / 0.987 / 1.184 |
| \(P_{\text{bio}}^\omega\) | Site biomass cost | 1.133 / 1.486 / 2.038 |
| \(P_{\text{ETS}}^\omega\) | Carbon proxy cost | 0.950 / 1.275 / 1.654 |
| \(U_{\text{head}}^\omega\) | GXP headroom | 0.817 / 0.890 / 0.950 |
| \(U_{\text{inc}}^\omega\) | Incremental load | 0.927 / 1.003 / 1.081 |
| \(U_{\text{upgrade}}^\omega\) | Upgrade CAPEX | 0.920 / 1.071 / 1.285 |
| \(\text{VOLL}^\omega\) | Unserved penalty | 10,000–20,000 NZD/MWh |
All multipliers are applied multiplicatively to baseline values. The paired-futures contract requires that both EB and BB pathways are evaluated under identical \(\{\omega\}_{0}^{99}\) to ensure comparability.