# Optimal Coordinated Bidding of a Profit Maximizing, Risk-Averse EV Aggregator in Three-Settlement Markets Under Uncertainty

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## Abstract

**:**

## 1. Introduction

- Development of a two-stage SD-MILP optimal coordinated bidding model for an aggregator who manages numerous storage units (stationary and EVs) and trades electric power in three-settlement markets, taking into account uncertainties in market price and fleet characteristics, as well as existing market rules. This model can be used for market exchange irrespective of (1) production or consumption technology and (2) mobile or stationary storage unit.
- Derivation of optimal coordinated charge (discharge) bids for day-ahead, intra-day and real-time markets with reasonable computation time using scenario-reduction techniques.
- Uncertainty modeling in all three market prices as well as of EV mobility parameters.
- Incorporation of the hourly CVaR (T-CVaR) to focus on the lower tails of the profits on hourly bases.

## 2. Decision Process Framework

- The daily bidding in day-ahead markets under the volatile day-ahead market prices, meanwhile allocating capacity in adjustment and regulating markets.
- The hourly/quarter-hourly intra-day selling/buying and balancing up/down-regulating positions driven by intra-day and real-time volatile market prices, and the strict requirements to satisfy driving needs of its fleet of EVs. These adjustments are necessary to address the errors stemming from the availability and price forecast.

## 3. Scenario Generation

#### 3.1. Market Places

#### 3.2. Market Price Scenario Generation and Reduction

#### 3.3. Availability Simulation

#### 3.4. Rolling Planning

## 4. Mathematical Problem Formulation

#### Two-Stage Stochastic Optimal Strategy of an EV Aggregator

## 5. Case Study

#### 5.1. Market Price Series

#### 5.2. General Parameters

#### 5.3. Simulation Results

#### 5.3.1. Tractability of the Solution

#### 5.3.2. Rolling Planning Horizon

#### 5.3.3. Controlling the Risk Measure T-CVaR

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Abbreviations | |

MILP | mixed integer linear program; |

SD-MILP | stochastic dynamically updated MILP; |

EV | electric vehicle; |

T-CVaR | hourly CVaR; |

DER | distributed energy resources; |

PV | photovoltaic panel; |

TCL | thermostatically controlled loads; |

TSO | transmission system operator; |

EES | electric energy storage; |

RES | renewable energy source |

Indices | |

k | storage indices, $k=1,\dots ,K$; |

t | planning periods, $t=1,\dots ,T$; |

s | scenarios, $s=1,\dots ,S$; |

i | index for possible bid prices $i=1,\dots ,I$ |

Parameters | |

${\omega}_{s}$ | probabilities associated with the scenarios; |

${\overline{P}}_{k}/{{\displaystyle \underline{P}}}_{k}$ | max/min storage rate of discharge/charge [kW]; |

${\overline{E}}_{k}$ | max capacity of a storage [kWh]; |

${\overline{\gamma}}_{k}/{{\displaystyle \underline{\gamma}}}_{k}$ | scalar to calculate max/min SoC; |

${\eta}_{k}^{ch/dch}$ | charge, discharge efficiency of a storage; |

$So{C}_{k,t=0}$ | start storage level [kWh]; |

$So{C}_{k,}^{end}$ | end storage level [kWh]; |

${\lambda}_{s,t}$ | day-ahead market price scenario [€/MWh]; |

${\lambda}_{s,t}^{sell/buy}$ | intra-day market price scenario [€/MWh]; |

${\lambda}_{s,t}^{up/dn}$ | real-time market price scenario [€/MWh]; |

${\rho}_{i}$ | fixed bid price for day-ahead market [€/MWh]; |

${\rho}_{i}^{sell/buy}$ | fixed bid price for intra-day market [€/MWh]; |

${\rho}_{i}^{up/dn}$ | fixed bid price for real-time market [€/MWh]; |

${c}_{t}$ | aggregator’s offer to storage owner [€/MWh]; |

${c}_{k}^{cap}$ | capital cost of a storage [€]; |

${\mu}_{k}$ | the slope of the linear approximation of the battery life as a function of the cycles; |

${A}_{s,k,t}$ | availability matrix indicating whether EV is available or not; |

${D}_{s,k,t}$ | average hourly driving distance of an EV [km]; |

${\eta}_{k}^{dr}$ | driving efficiency of an EV; |

${\nu}_{s,k}$ | total energy spent on driving [kWh]; |

$\chi $ | risk aversion level; |

$\delta $ | significance level; |

${\Pi}_{s,t}$ | profit in hour t and scenario s [€]; |

${\Gamma}_{1},{\Gamma}_{2},{\Gamma}_{3}$ | sufficiently big constants |

Variables | |

${p}_{s,k,t}^{DAch/DAdch}$ | charging/discharging dispatch level for kth storage in day-ahead market [kWh]; |

${p}_{s,k,t}^{IDch/IDdch}$ | charging/discharging dispatch level for k-th storage in intra-day market [kWh]; |

${p}_{s,k,t}^{Bch/Bdch}$ | charging/discharging dispatch level for k-th storage in real-time market [kWh]; |

${\mathbb{P}}_{s,t}^{DAch/DAdch}$ | energy as day-ahead buying/selling position [kWh]; |

${\mathbb{P}}_{s,t}^{IDch/IDdch}$ | buying/selling dispatch volume in intra-day market [kWh]; |

${\mathbb{P}}_{s,t}^{Bch/Bdch}$ | down/up-regulating dispatch volume in real-time market [kWh]; |

$C{\mathbb{P}}_{s,t}^{DAch/DAdch}$ | total cost of charging/discharging in day-ahead market [€]; |

$C{\mathbb{P}}_{s,t}^{IDch/IDdch}$ | total cost of charging/discharging in intra-day market [€]; |

$C{\mathbb{P}}_{s,t}^{Bch/Bdch}$ | total cost of charging/discharging in real-time market [€]; |

${x}_{i,t}^{DAch/DAdch}$ | charging/discharging bid volume in day-ahead market [kWh]; |

${x}_{i,t}^{IDch/IDdch}$ | charging/discharging bid volume in intra-day market [kWh]; |

${x}_{i,t}^{Bch/Bdch}$ | charging/discharging bid volume in real-time market [kWh]; |

$So{C}_{s,k,t}$ | storage level at the end of time step t [kWh]; |

$\xi $ | value-at-risk for a significance level $\delta $; |

${r}_{s,t}$ | auxiliary variable employed to calculate hourly CVaR |

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**Figure 2.**Illustration of the rolling planning approach implemented in the two-stage optimal bidding model.

**Figure 3.**Flowchart of the proposed algorithm for deriving the optimal coordinated bidding curves to day-ahead, intra-day, and real-time markets.

**Figure 5.**The optimal coordinated bidding in the day-ahead market for four iterations of rolling planning ($\chi =0.1$).

**Figure 6.**The optimal coordinated bidding in the intra-day market for four iterations of rolling planning ($\chi =0.1$).

**Figure 7.**The optimal coordinated bidding in the real-time market for four iterations of rolling planning ($\chi =0.1$).

**Figure 9.**Day-ahead trading strategies; hourly conditional value at risk (T-CVaR) risk aversion shift in day-ahead bids of 24 h of planning day (upper figure-discharging bids, lower figure-charging bids).

**Figure 10.**Intra-day trading strategies; T-CVaR risk aversion shift in intra-day bids of 24 h of planning day (upper figure-discharging bids, lower figure-charging bids).

**Figure 11.**Real-time trading strategies; T-CVaR risk aversion shift in real-time bids of 24 h of planning day (upper figure-discharging bids, lower figure-charging bids).

Par. | Value | Units | Ref. | Par. | Value | Units | Ref. |
---|---|---|---|---|---|---|---|

$\overline{E}$ | 50 | KWh | [28] | ${\eta}_{k}^{ch}$ | 90% | [29] | |

${\overline{\gamma}}_{k}$ | 20 % | [30] | ${\eta}_{k}^{dch}$ | 93% | [29] | ||

${{\displaystyle \underline{\gamma}}}_{k}$ | 100% | [30] | $So{C}_{k,}^{end}$ | 60% | |||

${\overline{P}}_{k}$ | 6 | KW | [28] | ${c}_{k}^{cap}$ | 200 | € | [15] |

${{\displaystyle \underline{P}}}_{k}$ | 6 | KW | [28] | ${\mu}_{k}$ | −[0.0013] | [15] |

It. 1 | It. 2 | It. 3 | It. 4 | |
---|---|---|---|---|

$\mathbb{E}[{\Pi}^{Tot}]$ (€) | 110.31 | 101.87 | 77.1 | 76.26 |

Comp. time (second) | 68.34 | 58.3 | 59.56 | 63.76 |

Number of EVs | 10 | 100 | 500 | 1000 | 5000 |

Continuous variables | 73,177 | 526,777 | 2,542,777 | 5,062,777 | 25,150,802 |

Discrete variables | 12,960 | 12,960 | 12,960 | 12,960 | 12,960 |

Constraints | 115,921 | 569,521 | 2,585,521 | 5,105,521 | 25,205,521 |

Comp. time (second) | 0.78 | 5.3 | 40.25 | 73.76 | 387.84 |

**Table 4.**Day-ahead, intra-day and real-time market trading with increased volatility of market prices; $\chi $ = 0.

Base Prices | Volatile Prices | Difference in % | |
---|---|---|---|

Day-ahead trading (€) | 183 | 9 | 95% decrease |

Intra-day trading (€) | 64 | 413 | 85% increase |

Real-time trading (€) | 1924 | 1983 | 3% increase |

Hours | T-CVaR (€) | Hours | T-CVaR (€) | Hours | T-CVaR (€) |
---|---|---|---|---|---|

1 | $-56.6$ | 9 | 0 | 17 | 0 |

2 | $-21.2$ | 10 | $-37.86$ | 18 | 0 |

3 | $-16.4$ * | 11 | $-52.4$ | 19 | $-12.3$ * |

4 | $-21.6$ | 12 | $-64.8$ | 20 | $-29$ |

5 | $-33.3$ | 13 | $-26.3$ | 21 | $-42$ |

6 | $-34$ | 14 | $-8.8$ * | 22 | $-23.6$ |

7 | 0 | 15 | $-16$ * | 23 | $-18.5$ |

8 | 0 | 16 | $-74.8$ | 24 | $-23$ |

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**MDPI and ACS Style**

Vardanyan, Y.; Madsen, H.
Optimal Coordinated Bidding of a Profit Maximizing, Risk-Averse EV Aggregator in Three-Settlement Markets Under Uncertainty. *Energies* **2019**, *12*, 1755.
https://doi.org/10.3390/en12091755

**AMA Style**

Vardanyan Y, Madsen H.
Optimal Coordinated Bidding of a Profit Maximizing, Risk-Averse EV Aggregator in Three-Settlement Markets Under Uncertainty. *Energies*. 2019; 12(9):1755.
https://doi.org/10.3390/en12091755

**Chicago/Turabian Style**

Vardanyan, Yelena, and Henrik Madsen.
2019. "Optimal Coordinated Bidding of a Profit Maximizing, Risk-Averse EV Aggregator in Three-Settlement Markets Under Uncertainty" *Energies* 12, no. 9: 1755.
https://doi.org/10.3390/en12091755