24 Sub-Module 4.4-A
System-Level Regret, Formal Derivation
24.1 SM-4.4-A: System-Level Regret, Formal Derivation
Let \(a \in \mathcal{A}\) denote a pathway alternative, \(\omega \in \Omega\) a future, and \(Z(a, \omega)\) the total system cost consequence of choosing alternative \(a\) under future \(\omega\). The framework defines three cost functions corresponding to the three evaluation frames of §4.4.
The site-only cost \(C_{\text{site}}(a, \omega)\) is the total annual cost as experienced by the facility operator:
\[ C_{\text{site}}(a, \omega) = C_{\text{capex}}(a) + C_{\text{opex}}(a, \omega) + C_{\text{ets}}(a, \omega) + C_{\text{unserved}}(a, \omega) \]
where \(C_{\text{capex}}\) is the annualised capital cost of the pathway’s assets, \(C_{\text{opex}}\) is the annual operating cost including energy and maintenance, \(C_{\text{ets}}\) is the annual ETS cost computed from direct combustion emissions and the carbon price declared in the FutureArtefact, and \(C_{\text{unserved}}\) is the penalty cost for any unserved heat demand.
The infrastructure-conditional cost \(C_{\text{system}}(a, \omega)\) adds the regional cost adder from the SignalsPack:
\[ C_{\text{system}}(a, \omega) = C_{\text{site}}(a, \omega) + C_{\text{regional}}(a, \omega) \]
where \(C_{\text{regional}}(a, \omega)\) is the annualised regional infrastructure cost attributable to the pathway under the future’s hosting capacity conditions. For the biomass pathway, \(C_{\text{regional}}(a, \omega)\) reflects any biomass supply scarcity premium declared in the biomass SignalsPack. For the electrification pathway, it reflects the upgrade cost adder associated with the upgrade class determined by the regional electricity screening module. For futures in which no upgrade is required, \(C_{\text{regional}}(a, \omega) = 0\) for the electrification pathway.
Site-perspective regret is defined relative to the best alternative under the site-only cost function:
\[ r_{\text{site}}(a, \omega) = \min_{a' \in \mathcal{A}} C_{\text{site}}(a', \omega) - C_{\text{site}}(a, \omega) \]
Since lower cost is better, regret is negative for the preferred alternative and non-positive for all others. Alternatively, when cost is expressed as a loss (larger is worse), regret is the amount by which the chosen alternative’s cost exceeds the minimum available:
\[ r_{\text{site}}(a, \omega) = C_{\text{site}}(a, \omega) - \min_{a' \in \mathcal{A}} C_{\text{site}}(a', \omega) \geq 0 \]
System-perspective regret is defined analogously using the infrastructure-conditional cost:
\[ r_{\text{system}}(a, \omega) = C_{\text{system}}(a, \omega) - \min_{a' \in \mathcal{A}} C_{\text{system}}(a', \omega) \geq 0 \]
The divergence function is:
\[ \Delta r(a, \omega) = r_{\text{system}}(a, \omega) - r_{\text{site}}(a, \omega) \]
The divergence is zero when the site-perspective and system-perspective assessments agree on which alternative has the lower cost under future \(\omega\). The divergence is positive when the system-perspective cost assessment, which includes \(C_{\text{regional}}\), assigns a different preferred alternative from the site-perspective assessment.
Conditions under which \(\Delta r > 0\) are analytically predictable from the structure of the cost functions. For the electrification pathway under future \(\omega\):
\[ \Delta r(a_{\text{EB}}, \omega) > 0 \iff C_{\text{regional}}(a_{\text{EB}}, \omega) > C_{\text{regional}}(a_{\text{BB}}, \omega) + [C_{\text{site}}(a_{\text{EB}}, \omega) - C_{\text{site}}(a_{\text{BB}}, \omega)] \]
In plain terms: the divergence is positive when the regional cost adder associated with electrification exceeds both the biomass resource cost adder and the private cost advantage of electrification over biomass. This occurs in futures where GXP hosting capacity is exceeded (triggering a major upgrade class) and where the ETS carbon price advantage of electrification is insufficient to offset the infrastructure cost. These are precisely the 23 futures in the Edendale ensemble where the divergence finding is reported in §6.9.
Policy interpretation. The divergence function \(\Delta r(a, \omega)\) provides a quantitative measure of the incentive misalignment created by current network pricing arrangements. A future in which \(\Delta r(a_{\text{EB}}, \omega) > 0\) is a future in which the site operator, making a private investment decision on the basis of \(C_{\text{site}}\), would choose or prefer the electrification pathway, while a system planner evaluating all costs would prefer the biomass pathway. The magnitude of \(\Delta r\) in that future represents the minimum network cost adder that would need to be charged to the site operator to align private and system incentives. A policy instrument that charges this adder, whether through connection agreement terms, a capacity charge, or a reformed network tariff, would eliminate the divergence in that future.
The divergence analysis does not prescribe which policy instrument is most appropriate. It does prescribe, with analytical precision, what the incentive misalignment is, in which futures it occurs, and what magnitude of cost reallocation would correct it. This precision is what distinguishes the framework’s policy contribution from a qualitative observation about incentive problems in the electricity sector.