# A Backwards Induction Framework for Quantifying the Option Value of Smart Charging of Electric Vehicles and the Risk of Stranded Assets under Uncertainty

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## Abstract

**:**

## 1. Introduction

## 2. Planning under Uncertainty

- Demonstration of the application of the BIF, for the first time in the context of power system investment planning under uncertainty.
- Quantification of the Option Value of Smart Charging of EV, for the first time, through the BIF.
- Quantification of the risk of stranded assets, for the first time through the BIF.
- Comparison of BIF and SOF, for the first time in the literature.
- Sensitivity analyses on key factors that are driving the Option Value of Smart Charging of EV, for the first time in the literature via the BIF.

## 3. The Backwards Induction Framework

## 4. Case Study

#### 4.1. Description

#### 4.2. Results

#### 4.2.1. Decision ${D}_{1}$

#### 4.2.2. Decision ${D}_{2}$

#### 4.2.3. Decision ${D}_{3}$

#### 4.2.4. Decision ${D}_{4}$

## 5. Sensitivity Analysis

#### 5.1. Sensitivity Analysis on Flexibility of Smart Charging

#### 5.2. Sensitivity Analysis on Social Cost

#### 5.3. Sensitivity Analysis on Scenario Probabilities

## 6. Comparison of the Backwards Induction Framework (BIF) with the Stochastic Optimization Framework (SOF)

#### 6.1. Basic Case Study

#### 6.2. Sensitivity Analysis

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Scenario tree, describing the investment decision making process according to the BIF, consisting of M scenarios (each with a probability${p}_{i}$) and N candidate investment decisions (${D}_{1}$, … ${D}_{N}$ ), one of which will be selected as the optimal.

**Figure 2.**Schematic diagram of the HV electricity distribution network under study. All values shown correspond to the first stage of the problem; peak loads are shown under the black arrows, line thermal limits are shown in black, and the power flows are shown in blue. This schematic diagram applies to the first epoch, during the peak hourly period; it also applies to the second epoch only under scenario 3 realization (i.e., no EV integration, and baseload remains the same).

**Figure 3.**Illustration of the uncertainty through a scenario tree that consists of two stages, where the first stage is when the investment decision is taken (“decision point”) and the second stage is when the uncertainty resolves (i.e., the planner learns which scenario realizes).

**Figure 4.**Two-stage scenario tree where stage1 includes the decision point (i.e., time when the investment decision is made) and stage 2 includes the uncertainty resolution, involving calculations on total system costs ${C}_{ij}$ and the corresponding expected values ${{\rm E}}_{j}$, where $i\in \left\{1,2,3\right\}$ is the scenario index, and $j\in \left\{1,2,3,4\right\}$ is the decision index, for each of the four investment decisions ${D}_{1},{D}_{2},{D}_{3},{D}_{4}$.

**Figure 5.**Schematic diagram of the HV electricity grid showing the first-stage power flows (in blue), the initial line thermal limits (in black), and the second-stage peak load (downward black arrows) assuming that in the second stage scenario ${S}_{1}$ has occurred (i.e., 100% load growth at every bus). Moreover, the second stage power flows are shown (in yellow) ignoring thermal line limits and investments (so that we can see the magnitude of the 2nd stage flows). Summing all second stage peak load yields 32 MW, consisting of 16 MW peak baseload (equal to that in the first stage i.e., no change in baseload) plus 16 MW peak EV load (all connected in the second stage). E.g., the total peak load at bus 5 is equal to 2 MW, where 1 MW is the peak baseload (same as in Figure 2) and the remaining 1 MW is the peak EV load (connected in the 2nd stage). The amount of capacity needed to be invested per line so that the second-stage flows are fully accommodated is equal to the difference between the values in yellow (power flows in the second stage) and those in black (first-stage thermal limits).

**Figure 6.**Schematic diagram of the HV electricity grid showing the first-stage power flows (in blue), the initial line thermal limits (in black), and the second-stage peak load (downward black arrows) assuming that in the second stage scenario ${S}_{2}$ has occurred (i.e., 50% load growth at every bus). Moreover, the second stage power flows are shown (in yellow) ignoring thermal line limits and investments (so that we can see the magnitude of the 2nd stage flows). Summing all second stage peak load yields 24 MW, consisting of 16 MW peak baseload (equal to that in the first stage i.e., no change in baseload) plus 8 MW peak EV load (all connected in the second stage).

**Figure 7.**Schematic diagram of the HV electricity grid showing the initial line thermal limits (in black), and the second-stage peak load (downward black arrows) assuming that in the second stage scenario ${S}_{1}$ has occurred (i.e., 100% load growth at every bus) and that EV smart charging, with flexibility = 40%, has been deployed for all EVs in the system i.e., at every bus in the system. Moreover, the second stage power flows are shown (in yellow) ignoring thermal line limits.

**Figure 8.**Schematic diagram of the HV electricity grid showing the initial line thermal limits (in black), and the second-stage peak load (downward black arrows) assuming that in the second stage scenario ${S}_{2}$ has occurred (i.e., 50% load growth at every bus) and that EV smart charging, with flexibility = 40%, has been deployed for all EVs in the system i.e., at every bus in the system. Moreover, the second stage power flows are shown (in yellow) ignoring thermal line limits.

**Figure 9.**Optimal Value of the Objective Function as a function of the Smart Charging flexibility, using BIF.

**Figure 10.**Option Value of Smart Charging of EV as a function of the Smart Charging flexibility, using BIF.

**Figure 13.**Optimal Value of the Objective Function as a function of scenario probabilities, using BIF.

**Figure 15.**Optimal total expected system cost as a function of smart charging flexibility, under SOF and BIF.

**Figure 17.**Optimal total expected system cost as a function of scenario probabilities, under SOF and BIF.

**Figure 18.**Option Value of investing in Smart Charging of EV (vertical axis) as a function of its investment cost (horizontal axis), for different values of Smart Charging flexibility, conducted via BIF.

**Table 1.**Description of the source of uncertainty in the problem through three scenarios for EV penetration.

Source of Uncertainty | Scenarios | Probabilities | |
---|---|---|---|

Future EV penetration growth | ${S}_{1}$ | High (i.e., 100% load growth per bus) | 40% |

${S}_{2}$ | Medium (i.e., 50% load growth per bus) | 35% | |

${S}_{3}$ | No change | 25% |

**Table 2.**Description of the availability of technologies for each of the candidate investment decisions.

${D}_{1}$ | Conventional network reinforcement is the only technology available to the planner. |

${D}_{2}$ | The planner does not make any investments at all (Do-nothing approach). |

${D}_{3}$ | Smart Charging of EV is the only technology available to the planner. |

${D}_{4}$ | Both conventional network reinforcement and Smart Charging of EV are available to the planner. |

Flexibility of Smart Charging | $\mathbf{Optimal}\mathbf{Value}(\pounds )\mathbf{of}\mathbf{Objective}\mathbf{Function}(\mathbf{i}.\mathbf{e}.,\mathbf{min}{\mathit{{\rm E}}}_{\mathit{j}})$ | Optimal Decision | Option Value of Smart Charging (£) |
---|---|---|---|

10% | 166,225 | 1 | 0 |

20% | 165,844 | 4 | 381 |

40% | 131,344 | 4 | 34,881 |

60% | 84,571 | 3 | 81,654 |

80% | 52,571 | 3 | 113,654 |

100% | 52,571 | 3 | 113,654 |

Social Cost (£/MWh) | $\mathbf{Optimal}\mathbf{Value}(\pounds )\mathbf{of}\mathbf{Objective}\mathbf{Function}(\mathbf{i}.\mathbf{e}.,\mathbf{min}{\mathit{{\rm E}}}_{\mathit{j}})$ | Optimal Decision | Option Value of Smart Charging (£) |
---|---|---|---|

1000 | 5600 | 2 | 0 |

10,000 | 56,000 | 2 | 0 |

25,000 | 100,571 | 3 | 65,654 |

50,000 | 131,344 | 4 | 34,881 |

100,000 | 166,225 | 1 | 0 |

500,000 | 166,225 | 1 | 0 |

1,000,000 | 166,225 | 1 | 0 |

${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{p}}_{3}$ | $\mathbf{Optimal}\mathbf{Value}\mathbf{of}\mathbf{Objective}\mathbf{Function}\mathbf{i}.\mathbf{e}.,{\mathit{E}}_{\mathit{j}}$ | Optimal Decision | Option Value of Smart Charging (£) |
---|---|---|---|---|---|

0.40 | 0.35 | 0.25 | 131,344 | 4 | 34,880 |

0.10 | 0.10 | 0.80 | 37,714 | 3 | 128,510 |

0.25 | 0.25 | 0.50 | 94,285 | 3 | 71,939 |

0.40 | 0.40 | 0.20 | 132,201 | 4 | 34,023 |

0.10 | 0.80 | 0.10 | 69,714 | 3 | 96,510 |

0.25 | 0.50 | 0.25 | 105,714 | 3 | 60,510 |

0.40 | 0.20 | 0.40 | 128,772 | 4 | 37,452 |

0.80 | 0.10 | 0.10 | 166,225 | 1 | 0 |

0.50 | 0.25 | 0.25 | 142,058 | 4 | 24,166 |

0.20 | 0.40 | 0.40 | 84,571 | 3 | 81,653 |

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**MDPI and ACS Style**

Giannelos, S.; Borozan, S.; Strbac, G.
A Backwards Induction Framework for Quantifying the Option Value of Smart Charging of Electric Vehicles and the Risk of Stranded Assets under Uncertainty. *Energies* **2022**, *15*, 3334.
https://doi.org/10.3390/en15093334

**AMA Style**

Giannelos S, Borozan S, Strbac G.
A Backwards Induction Framework for Quantifying the Option Value of Smart Charging of Electric Vehicles and the Risk of Stranded Assets under Uncertainty. *Energies*. 2022; 15(9):3334.
https://doi.org/10.3390/en15093334

**Chicago/Turabian Style**

Giannelos, Spyros, Stefan Borozan, and Goran Strbac.
2022. "A Backwards Induction Framework for Quantifying the Option Value of Smart Charging of Electric Vehicles and the Risk of Stranded Assets under Uncertainty" *Energies* 15, no. 9: 3334.
https://doi.org/10.3390/en15093334