# Transactive-Market-Based Operation of Distributed Electrical Energy Storage with Grid Constraints

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

#### Subject to power flow, load and DG units, line flow, and EES unit constraints

## 3. Simulation Results

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## Abbreviations and Nomenclature

Abbreviations: | |

EES | Electrical Energy Storage |

DG | Distributed Generation |

LMP | Locational Marginal Price |

DLMP | Distribution LMP |

MCP | Market Clearing Price |

DER | Distributed Energy Resource |

RES | Renewable Energy Resource |

LA | Load Aggregator |

Functions: | |

$\mathcal{C}$ | Operation cost function |

Sets: | |

$\mathcal{T}$ | Set of timeslots |

$\mathcal{B}$ | Set of buses |

$\mathcal{L}$ | Set of line segments |

$\mathcal{R}$ | Set of load blocks |

$\mathcal{Q}$ | Set of generation blocks |

$\mathcal{G}$ | Set of buses with DG units |

$\mathcal{E}$ | Set of buses with EES units |

$\mathcal{A}$ | Set of buses with LA units |

Indices: | |

i | Index of distribution bus |

ij | Distribution line index connecting i to j |

q | Generation block index |

r | Load block index |

Variables: | |

${p}_{i,q,t}^{\mathrm{G}}$ | Real power output at block q of dispatchable unit at bus i at time t |

${p}_{i,r,t}^{\mathrm{D}}$ | Real power demand at block r of load at bus i at time t |

$({p}_{i,t}^{\mathrm{G}},{q}_{i,t}^{\mathrm{G}})$ | Net real/reactive output of dispatchable generator unit at bus i at time t |

$({p}_{i,t}^{\mathrm{D}},{q}_{i,t}^{\mathrm{D}})$ | Real/reactive power demand of load at bus i at time t |

$({p}_{i,t}^{\mathrm{E}},{q}_{i,t}^{\mathrm{E}})$ | Net real/reactive power output of EES unit at bus i at time t |

$({\widehat{p}}_{i,t}^{\mathrm{E}},{\stackrel{\u02c7}{p}}_{i,t}^{\mathrm{E}})$ | Extraction (^{∧})/Injection (^{∨}) of EES unit at bus i at time t |

$({p}_{i,t}^{\mathrm{FR}},{q}_{i,t}^{\mathrm{FR}})$ | Forecasted real/reactive generation of renewable unit at bus i at time t |

$({p}_{i,t},{q}_{i,t})$ | Net real/reactive power injection at bus i at t |

${P}_{ij,t},{Q}_{ij,t}$ | Real/reactive power flow in line ij at time t |

${v}_{i,t}$ | Per unit voltage of bus i at time t |

${c}_{i,t}^{\mathrm{E}}$ | State of charge of EES at bus i at time t |

Binary and Integer Variables: | |

${\stackrel{\u02c7}{\U0001d4cf}}_{i,t}$ | EES unit charging state (1 for charging, 0 for not charging) at time t |

${\widehat{\U0001d4cf}}_{i,t}$ | EES unit discharging state (1 for discharging, 0 for not discharging) at time t |

${\stackrel{\u02c7}{\U0001d4cfm}}_{i,t}$ | EES unit charging indicator (Unity ‘1’ only where it starts charging) at time t |

${\stackrel{\u02c7}{\U0001d4cfn}}_{i,t}$ | EES unit not charging indicator (Unity ‘1’ only where it stops charging) at time t |

${\widehat{\U0001d4cfm}}_{i,t}$ | EES unit discharging indicator (Unity ‘1’ only where it starts discharging) at time t |

${\widehat{\U0001d4cfn}}_{i,t}$ | EES unit not discharging indicator (Unity ‘1’ only where it stops discharging) at time t |

${\stackrel{\u02c7}{\U0001d4cf\U0001d4b8}}_{i,t}$ | EES unit charging counter (counts the number of times ${\stackrel{\u02c7}{\U0001d4cf}}_{i,t}$ is ON) at time t |

${\widehat{\U0001d4cf\U0001d4b8}}_{i,t}$ | EES unit discharging counter (counts the number of times ${\widehat{\U0001d4cf}}_{i,t}$ is ON) at time t |

Parameters: | |

${\mathrm{V}}_{1,t}$ | Substation voltage at time t |

$\mathrm{T}$ | Scheduling horizon |

${\overline{\mathrm{S}}}_{ij}$ | MVA limit of line ij |

${\mathrm{r}}_{ij}$ | Resistance of line ij |

${\mathrm{x}}_{ij}$ | Reactance of line ij |

${\underset{\xaf}{\mathrm{E}}}_{i}$ | Minimum energy withdraw rate of EES unit |

${\overline{\mathrm{E}}}_{i}$ | Maximum energy withdraw rate of EES unit |

${\underset{\xaf}{\mathrm{C}}}_{i}$ | Minimum state of charge of EES unit |

${\overline{\mathrm{C}}}_{i}$ | Maximum state of charge of EES unit |

${\mathsf{\alpha}}_{i,t}$ | Self-discharge coefficient of EES unit |

${\hat{\mathsf{\alpha}}}_{i,t}$ | Discharge efficiency coefficient of EES unit |

${\stackrel{\u02c7}{\mathsf{\alpha}}}_{i,t}$ | Charge efficiency coefficient of EES unit |

${\widehat{\mathcal{M}}}_{i}$ | Minimum number of consecutive discharging hours of EES unit at bus i |

${\stackrel{\u02c7}{\mathcal{M}}}_{i}$ | Minimum number of consecutive charging hours of EES unit at bus i |

${\overline{\mathrm{P}}}_{i,q}^{\mathrm{G}}$ | Maximum generation in each block q of DG unit i |

${\overline{\mathrm{P}}}_{i}^{\mathrm{G}}$ | Maximum real power output of a DG unit at bus i |

${\underset{\xaf}{\mathrm{P}}}_{i}^{\mathrm{G}}$ | Minimum real power output of a DG unit at bus i |

${\overline{\mathrm{P}}}_{i,r}^{\mathrm{D}}$ | Maximum load in each block r of load at bus i |

${\overline{\mathrm{P}}}_{i}^{\mathrm{D}}$ | Maximum demand of load at bus i |

${\underset{\xaf}{\mathrm{P}}}_{i}^{\mathrm{D}}$ | Minimum demand of load at bus i |

${\mathsf{\delta}}_{i}^{\mathrm{D}}$ | Fraction of real power of load as reactive power |

${\mathsf{\delta}}_{i}^{\mathrm{G}}$ | Fraction of real power of DG as reactive power |

${\stackrel{\u02c7}{\mathrm{c}}}_{i,t}^{\mathrm{E}}$ | Bidding price for charging of EES unit at bus i at time t |

${\hat{\mathrm{c}}}_{i,t}^{\mathrm{E}}$ | Bidding price for discharging of EES unit at bus i at time t |

${\mathrm{c}}_{i,q,t}^{\mathrm{G}}$ | Selling bid at block q of DG unit at bus i at time t |

${\mathrm{c}}_{i,r,t}^{\mathrm{D}}$ | Buying bid in block r of load at bus i at time t |

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**Figure 1.**Piecewise traditional supply-demand curve with supply blocks labeled ‘q’, demand blocks labeled ‘r’, and prices labeled ‘cr’ and ‘cq’. Demand curve is shifted due to LA’s base load block (r1).

**Figure 2.**Radial IEEE 13-bus distribution system with different types of distributed energy resources (DERs) on each bus.

Bus | Unit | (Min, Max SOC) (MWh) | (Min, Max Power Rate) (MW) | (Min Charge, Discharge Time) (h) | (Injection, ExRaction Cost ($/MWh) | Initial SOC (MWh) |
---|---|---|---|---|---|---|

3 | EES3 | (0.05,0.5) | (0.02,0.05) | (3,3) | (20,25) | 0.1 |

7 | EES7 | (0.05,0.4) | (0.02,0.05) | (3,3) | (20,25) | 0.25 |

**Table 2.**Energy bids and maximum DG supply (${\mathsf{\delta}}_{i}^{\mathrm{G}}=\pm 30\%$) for each block at 2:00 p.m.

Bus No, Unit | DG Bids | |||||
---|---|---|---|---|---|---|

Block 1 | Block 2 | Block 3 | ||||

Bid ($/MWh) | Max Supply (MW) | Bid ($/MWh) | Max Supply (MW)) | Bid ($/MWh) | Max Supply (MW) | |

1, DG1 | 39.80 | 10 | N/A | N/A | N/A | N/A |

7, DG7 | 30.60 | 1.8 | 32.40 | 0.5 | 35.82 | 0.5 |

13, DG13 | 33.75 | 0.8 | 36.72 | 0.4 | 39.15 | 0.3 |

**Table 3.**LA bid price and maximum demand (${\mathsf{\delta}}_{i}^{\mathrm{D}}=30\%$) for each block at 2:00 p.m.

Bus No, Unit | LA Bid | |||||
---|---|---|---|---|---|---|

Block 1 | Block 2 | Block 3 | ||||

Bid ($/MWh) | Demand (MW) | Bid ($/MWh) | Demand (MW) | Bid ($/MWh) | Demand (MW) | |

3, LA3 | N/A | 0.28 | 28.05 | 0.08 | 23.68 | 0.10 |

4, LA4 | N/A | 0.38 | 29.29 | 0.09 | 23.75 | 0.17 |

6, LA6 | N/A | 0.11 | 25.25 | 0.22 | 20.87 | 0.16 |

7, LA7 | N/A | 0.64 | 30.38 | 0.41 | 25.79 | 0.48 |

9, LA9 | N/A | 0.28 | 27.91 | 0.14 | 21.05 | 0.17 |

10, LA10 | N/A | 0.11 | 27.66 | 0.21 | 22.40 | 0.18 |

11, LA11 | N/A | 0.11 | 28.79 | 0.22 | 23.77 | 0.28 |

12, LA12 | N/A | 0.46 | 30.73 | 0.30 | 25.93 | 0.19 |

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## Share and Cite

**MDPI and ACS Style**

Faqiry, M.N.; Edmonds, L.; Zhang, H.; Khodaei, A.; Wu, H.
Transactive-Market-Based Operation of Distributed Electrical Energy Storage with Grid Constraints. *Energies* **2017**, *10*, 1891.
https://doi.org/10.3390/en10111891

**AMA Style**

Faqiry MN, Edmonds L, Zhang H, Khodaei A, Wu H.
Transactive-Market-Based Operation of Distributed Electrical Energy Storage with Grid Constraints. *Energies*. 2017; 10(11):1891.
https://doi.org/10.3390/en10111891

**Chicago/Turabian Style**

Faqiry, M. Nazif, Lawryn Edmonds, Haifeng Zhang, Amin Khodaei, and Hongyu Wu.
2017. "Transactive-Market-Based Operation of Distributed Electrical Energy Storage with Grid Constraints" *Energies* 10, no. 11: 1891.
https://doi.org/10.3390/en10111891