Test Case 23
Test Case 23 asserts that piece-wise quadratic cost functions for input procurement work as desired. It is structurally similar to Test Case 3, only that the first block of input procurement only goes to 25% of the total capacity of 1. After a quantity of 0.2, an increase in marginalized cost slope can be identified. Below, a graphical representation of the optimal solution is provided. Hence, this test passes if the equilibrium price is $P=1.5$, with an objective value of 0.45 and corresponding social welfare of 0.575 (see the grey area in the graphic).
Sets
| Set Name | Set Value |
|---|---|
| $\mathcal{A}$ | $\{DEU\_to\_DEU\}$ |
| $\mathcal{AC}$ | $\{(DEU\_to\_DEU, CNG)\}$ |
| $\mathcal{C}$ | $\{CNG\}$ |
| $\mathcal{DSB}$ | $\{Block 1\}$ |
| $\mathcal{I}$ | $\{Natural Gas\}$ |
| $\mathcal{IOB}$ | $\{Block 1,Block 2\}$ |
| $\mathcal{M}$ | $\{OnlyTimeStep\}$ |
| $\mathcal{N}$ | $\{DEU\}$ |
| $\mathcal{O}$ | $\{FES\}$ |
| $\mathcal{P}$ | $\{P\_DEU\}$ |
| $\mathcal{RA}$ | ∅ |
| $\mathcal{RS}$ | ∅ |
| $\mathcal{RV}$ | ∅ |
| $\mathcal{S}$ | ∅ |
| $\mathcal{T}$ | $\{T\_DEU\}$ |
| $\mathcal{V}$ | ∅ |
| $\mathcal{VT}$ | ∅ |
| $\mathcal{Y}$ | $\{2020\}$ |
Parameters
| Parameter | y=2020 |
|---|---|
| $\frac{1}{ | \Delta |_{y}}$ | $1$ |
| ${1}^{NC}_{T\_DEU,DEU,CNG}$ | $1$ |
| $r_{y}$ | $1$ |
| $d_{OnlyTimeStep}$ | $1$ |
| $c^{P}_{P\_DEU,CNG,FES,y}$ | $0.3$ |
| $c^{\Delta P}_{P\_DEU,CNG,FES,y}$ | $1$ |
| $fi^{P}_{CNG,Natural Gas,FES}$ | $1$ |
| $L^{P}_{CNG,FES}$ | $50$ |
| $\Lambda^{P}_{P\_DEU,CNG,FES,y}$ | $10$ |
| $\Lambda^{I}_{P\_DEU,Natural Gas,Block 1,y}$ | $0.25$ |
| $\Lambda^{I}_{P\_DEU,Natural Gas,Block 2,y}$ | $0.75$ |
| $\Omega^{I}_{P\_DEU,Natural Gas,Block 1,y}$ | $0$ |
| $\Omega^{I}_{P\_DEU,Natural Gas,Block 2,y}$ | $0$ |
| $c^{\Delta^{I}}_{P\_DEU,Natural Gas,Block 1,y}$ | $0$ |
| $c^{\Delta^{I}}_{P\_DEU,Natural Gas,Block 2,y}$ | $0$ |
| $\Lambda^{T}_{T\_DEU,DEU,CNG,FES,y}$ | $10$ |
| $\Omega^{P}_{P\_DEU,CNG,FES,y}$ | $10$ |
| $l^{A}_{DEU\_to\_DEU,CNG}$ | $0$ |
| $c^{A}_{DEU\_to\_DEU,CNG,y}$ | $0$ |
| $c^{\Delta A}_{DEU\_to\_DEU,CNG,y}$ | $0$ |
| $\Lambda^{A}_{DEU\_to\_DEU,CNG,y}$ | $0$ |
| $L^{A}_{CNG}$ | $50$ |
| $c^{I_{l}}_{P\_DEU,Natural Gas,Block 1,OnlyTimeStep,y}$ | $0.1$ |
| $c^{I_{q}}_{P\_DEU,Natural Gas,Block 1,OnlyTimeStep,y}$ | $0.4$ |
| $av^{I}_{P\_DEU,Natural Gas,Block 1,OnlyTimeStep}$ | $1$ |
| $c^{I_{l}}_{P\_DEU,Natural Gas,Block 2,OnlyTimeStep,y}$ | $0.2$ |
| $c^{I_{q}}_{P\_DEU,Natural Gas,Block 2,OnlyTimeStep,y}$ | $2.0$ |
| $av^{I}_{P\_DEU,Natural Gas,Block 2,OnlyTimeStep}$ | $1$ |
| $\alpha^{D}_{DEU,CNG,Block 1,OnlyTimeStep,y}$ | $2$ |
| $\beta^{D}_{DEU,CNG,Block 1,OnlyTimeStep,y}$ | $-1$ |
Test Criteria
| Expression | Result y=2020 |
|---|---|
| $\tilde{P}^{T \rightarrow D}_{DEU,CNG,Block 1,OnlyTimeStep,y}$ | $1.5$ |
| Objective Value | $0.45$ |
| Social Welfare | $0.575$ |
