1. ISO/IEC Guide 98-3 (GUM) Uncertainty Propagation
For a measurement model \(Y = f(X_1, X_2, \dots, X_N)\), the combined standard uncertainty \(u_c(y)\) is the root-sum-square of partial derivatives multiplied by component uncertainties:
\[ u_c^2(y) = \sum_{i=1}^N \left( \frac{\partial f}{\partial x_i} \right)^2 u^2(x_i) = \sum_{i=1}^N c_i^2 u^2(x_i) \]
The expanded uncertainty at 95.45% confidence is \(U = k \cdot u_c(y)\), where coverage factor \(k = 2.0\).
2. Energy Billing & Financial Risk Quantification
Energy delivery is the product of standard volume and Gross Calorific Value: \(E = Q_{\text{std}} \cdot \text{GCV}\). The combined energy uncertainty is:
\[ U_{\text{energy}} = \sqrt{U_{\text{volume}}^2 + U_{\text{GCV}}^2} \]
The annual financial exposure is evaluated directly against contract throughput and tariff pricing:
\[ \text{Annual Financial Risk (\$/yr)} = \text{Annual Billing Turnover (\$/yr)} \times \frac{U_{\text{energy, 95\%}}}{100} \]