Appendix: illustrative transition studies

This appendix is a laboratory notebook for the retained annual-electricity studies. Its comparison map identifies a perturbation, an expected invariant and an observable failure before inspecting a result. That ordering makes the studies useful for diagnosing the calculation. A match with frozen output establishes reproduction; a match with an independently derived identity tests a different property. All inputs remain illustrative. The complete Aster Works dispositions and authoritative responsibility register are in The committee needs balanced evidence. This appendix uses that case by reference and concentrates on the experiments behind the numerical exhibits.

Construct a comparison sheet from the complete input objects. Match global fields by name and measures by their stable names, then mark additions, deletions, field changes and changes of array position separately. A pair described as price-only must have identical demand, factors, measure terms and valuation convention. An order-only pair must contain the same measures even though their positions differ. A standalone efficiency investment is a different package when supply capital and continuing costs are removed. This field-level classification prevents a result difference from being assigned to a cause that the actual input comparison cannot isolate.

The result register is a compact index into complete output objects, not the entire test evidence. Use its physical and cash columns to select a comparison, then read the corresponding measure records, warnings and reversed-order diagnostic. Record a predicted relationship before running the comparison: equality, a direction under stated conditions or an algebraic difference. Keep an unexpected result visible until its cause is located. Changing the prediction after seeing the result would remove the experiment's ability to expose an error in either the model or the reader's interpretation.

Reconciliation can locate a defect more precisely than a headline comparison. Let E denote final demand, G grid energy, R admitted renewable energy, Q accumulated renewable supply and C unused supply. The physical residuals are E minus G minus R and Q minus R minus C. Both should vanish in the annual boundary, with all quantities nonnegative. The emissions residual is reported emissions minus the sum of G times the grid factor and R times the renewable factor. The cash residual is reported net saving minus baseline electricity cost plus final electricity cost plus accumulated continuing expenditure. A nonzero residual identifies the layer that requires examination.

Apply these checks after each state transformation as well as at package close. An early residual that disappears later can be hidden by compensating mistakes. Separately check telescoping: successive emissions differences must sum to the baseline-to-final difference, and successive energy-cost differences must sum to the baseline-to-final cost saving. Sum signed increments even when an increment's cost-per-tonne field is unavailable. Dropping a nonpositive-abatement row from those sums would change the physical package. An average of the displayed available ratios is consequently an unsuitable reconciliation; compare the summed numerators and denominators first.

Financial reconciliation uses the same constant annual convention as the engine. Capital divided by the annuity factor, less net annual saving, must equal the package annualised cost. Net annual saving times that factor, less capital, must equal NPV. These identities do not independently establish the chosen horizon or discount rate. They do detect a displaced sign, duplicated continuing cost or a mixed convention between rows and totals. Carry full canonical quantities into the check. Displayed decimals are for reading; subtraction of separately rounded values may create an apparent residual that is solely a display effect.

The factor and relative-price pairs allow a stronger test than inspecting whether the headline improves. With B denoting baseline demand and the final mix held fixed, a grid-factor change alters abatement by B minus G, multiplied by the factor change. A renewable-factor change alters abatement in the opposite direction, by R times its change. Neither term enters the supplied bill. The factor pair therefore tests the dependency graph: attribution changes while energy and cash remain fixed. It also shows why a more negative ratio cannot be interpreted without checking the reduction quantity that divides the unchanged annual cost.

For the same fixed mix, a grid-price change alters net annual saving by B minus G, multiplied by its price change. A renewable-price change alters saving in the opposite direction, by R times its change. NPV changes by the annuity factor times the resulting saving change when capital and continuing costs are held. This gives an independent signed prediction for the relative-price comparison. A simultaneous common price multiplier answers a different question: it scales gross energy-cost saving while leaving measure continuing expenditure unscaled. The common-price diagnostic cannot be substituted for an adverse change in the relative prices.

Turn these relations into an extension design without manufacturing a new observation. Select a supported input field, state an admissible direction of change and name all fields held constant. Derive the expected difference symbolically, then compute a complete successor fixture if a numerical experiment is needed. Compare the derivative prediction only while its physical regime is unchanged. When a changed supply quantity reaches the annual cap, the regime changes and the next section supplies the appropriate prediction. A field change that also changes service, capital or a tariff obligation requires another comparison design rather than a claim of isolated sensitivity.

The order experiment should distinguish an altered final state from an altered allocation along the path. Define an added demand a and an efficiency fraction e applied to the demand present at its step. Adding a before applying efficiency leaves less final demand than adding a afterwards, by a times e. This difference follows by expanding the two transformations; it does not require the app's reversed-order output as its premise. It is also an engineering question. The transformation is appropriate only if the efficiency mechanism can affect that added load. The input order encodes that scope assumption and cannot establish it physically.

Efficiency and supply accumulation behave differently. In the overlap pair, the efficiency step changes demand while the supply step changes accumulated Q; neither alters the other's input parameter. Their final demand and accumulated supply therefore coincide, and the annual cap produces the same final mix. Intermediate states need not coincide. A reduction assigned to the first step can be reassigned to the second when the path changes. The practical verification object is a pair of endpoint and path checks: compare the complete final state for equality, then reconcile each path's signed increments to that common endpoint. Do not interpret a changed row as a new standalone benefit.

The annual cap creates a useful stress test of its own. Below the cap, an added supply quantity displaces grid energy; its cash effect depends on the price difference, and its attributed reduction depends on the factor difference. Above the cap, additional Q increases C without changing E, G or R. Under the retained used-energy charging convention it adds neither a bill saving nor an emissions reduction. Keep capital and continuing expenditure unchanged for this quantity-only diagnostic. If those costs change with Q, their financial effect is a separate perturbation. Purchase obligations for C are examined symbolically in the following worksheet appendix.

The small-positive-abatement study separates an undefined denominator from a small one. Annual baseline demand 1e-09 MWh, factor 1e-09 tCO2e/MWh and 10% efficiency create positive calculated reduction 1e-19 tCO2e/year. Zero investment and continuing cost, a one-year zero-rate convention and the supplied energy price produce package marginal cost -1000000000 USD/tCO2e. The ratio is mathematically defined because the computed denominator is strictly positive. The extreme inputs diagnose arithmetic rather than represent measurable industrial abatement.

The canonical decimal annual balance determines whether reduction is positive, zero or negative before each quantity is converted for display. A small genuine positive reduction remains eligible for a cost ratio, without an absolute floor. Separately rounding baseline and final totals can obscure that difference, so subtracting displayed totals does not reproduce the unrounded comparison. The reversed-order physical diagnostic compares the two canonical annual emissions balances exactly. A positive software denominator establishes neither detection capability, materiality, reporting eligibility nor a certified reduction. Those need the baseline boundary, measurement evidence and claim rules developed in the preceding chapters.