# Optimal Management of a Virtual Power Plant Consisting of Renewable Energy Resources and Electric Vehicles Using Mixed-Integer Linear Programming and Deep Learning

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background and Motivation

#### 1.2. Literature Survey

- Based on the current trends in the power systems and disadvantages of fossil fuel-based units, considering DR programs and EVs are necessary. In addition, the emission cost is a vital factor for increasing the penetration of RESs that should be investigated in the studies for optimal management of VPPs. Therefore, modeling a comprehensive study by considering all the components indicated in Table 1 is essential and has not been completely considered by any of the reviewed papers.
- Modeling uncertainties, especially EVs’ behavior owing to the fact that VPPs have a high percentage of stochastic resources can completely change the bids of VPPs in the market. In this regard, using methods with high accuracy based on data-driven approaches plays a decisive role to decrease the penalty cost of VPPs in the electricity market.

#### 1.3. Paper Contribution

- Forecasting all uncertainties involved in the planning of a VPP by bi-directional long short-term memory (BLSTM) networks—load, price, RES, and EV uncertainties. Furthermore, in this paper, EV samples are generated based on three features (arrival time, departure time, and travel distance) by considering dependency among these features. However, other works considered them separately or used the Monte Carlo method which cannot effectively model EVs’ behavior and increased the number of infeasible samples.
- Proposing a comprehensive mixed-integer linear programming (MILP) model for technical VPPs’ energy management, by emission cost, network and unit constraints, DR, and degradation cost of ES.
- Considering the VPP participation in DAM.

#### 1.4. Paper Organization

## 2. Problem Formulation

## 3. Numerical Results

#### 3.1. Input Data

#### 3.2. Uncertainty Modeling

#### 3.3. Case Study Results

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Indices | |

$t$ | Time index |

$i$ | Conventional distributed generation (DG) index |

$w$ | Wind turbine (WT) index |

$e$ | Emission type index |

$k$ | Index for linear pieces |

$s$ | Energy storages (ESs) index |

$b,l$ | Nodes indices |

$\mathrm{p}$ | Electric vehicles (EVs) index |

Parameters | |

$S{U}_{i}$ | Start-up cost ($) |

${\alpha}_{e}$ | Externality costs |

$E{F}_{i,e}$ | Emission factor |

${\eta}_{s}^{CH}$,${\eta}_{s}^{DCH}$ | ESs’ charging and discharging efficiency |

$C{R}_{p}$ | Rated charger capacity (kW) |

${\eta}_{p}^{ch}$ | EVs’ charging efficiency |

$SO{C}_{p}^{EV,INI}$ | Initial state of charge (SOC) of EVs (kWh) |

$B{C}_{p}$ | EVs’ battery capacity (kWh) |

$P{D}_{b,t}$ | Active power demand (kW) |

${G}_{b,l}$ | Conductance value of line bl (p.u) |

$Q{D}_{b,t}$ | Reactive power demand (kVar) |

${\xi}_{b,l,t}$ | Linear load flow constant |

$C{o}_{w}^{WT}$ | Coefficient for WT cost ($/kW) |

${P}_{w}^{wt,max}$ | Maximum output power of WTs (kW) |

${T}_{i}^{DG,on}$, ${T}_{i}^{DG,off}$ | Minimum up and down times (h) |

${P}_{i}^{min}$, ${P}_{i}^{max}$ | Minimum and maximum output power of DGs (kW) |

${a}_{i}$,${b}_{i}$,${C}_{i}$ | DG cost coefficients |

${\alpha}_{s}$, ${\beta}_{s}$ | ES cost coefficients |

$SO{C}_{s}^{MIN}$,$SO{C}_{s}^{MAX}$ | Minimum and maximum SOC of ESs (kWh) |

${\mathrm{R}}_{s}^{ES,CH}$, ${\mathrm{R}}_{s}^{ES,DCH}$ | Maximum and minimum charge rate of ESs (kW) |

${B}_{b,l}$ | Susceptance value of line bl (p.u) |

$S{L}_{b,l}^{MAX}$ | Maximum magnitude of apparent power flow (kVA) |

${V}_{b}^{min}$, ${V}_{b}^{max}$ | Minimum and maximum voltage limit (p.u) |

${\vartheta}_{w}^{cutin}$, ${\vartheta}_{w}^{cutout}$, ${\vartheta}_{w}^{rated}$ | Cut-in, cut-out, and rated speed of WTs |

$\mathrm{Sbase}$ | Base apparent power (kVA) |

$P{r}_{t}^{DAM}$ | Active power price in day-ahead market (DAM) at hour t ($/kw) |

${P}^{DIS,MAX}$ | Maximum allowable exchanged power with upstream network |

$S{D}_{i}$ | Shut down cost ($) |

$U{R}_{i}$, $D{R}_{i}$ | Ramp-up and ramp-down rate limits (kW) |

${\vartheta}_{w,t}$ | Wind speed |

${S}_{i}^{k}$ | Slope of piece-line k |

$M$ | Big value |

Variables | |

$C{o}_{i,t}^{DG}$ | DG cost ($) |

$P{o}_{t}^{DIS}$ | Amount of active exchanged power with the upstream network (kW) |

$C{o}_{i,t}^{POL}$ | Emission cost ($) |

${u}_{i,t}$, ${u}_{i,t}^{\prime}$ | Binary variable for the commitment of DGs |

${C}_{s,t}^{ST}$ | ES cost ($) |

${C}_{w,t}^{WT}$ | WT cost ($) |

$D{R}_{b,t}^{INC}$, $D{R}_{b,t}^{DEC}$ | Maximum allowable amount for increasing or decreasing load in each bus (kW) |

${X}_{i,t}^{DG,off},{X}_{i,t}^{DG,on}$ | OFF and ON time of DGs (h) |

${P}_{m,t}$, ${Q}_{m,t}$ | Generated active and reactive power by unit m |

${P}_{i,t}^{K}$ | Generated power in segment k (kW) |

${P}_{s,t}^{ES,CH}$,${P}_{s,t}^{ES,DCH}$ | Amount of ESs’ charging and discharging (kW) |

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**Figure 4.**Forecasted values and their comparison with the real data: (

**a**) electricity price; (

**b**) load demand; (

**c**) wind speed.

**Figure 7.**Generated power (kW) by DGs in 24 h for real data and BLSTM methods: (

**a**) DG1; (

**b**) DG2; (

**c**) DG3; (

**d**) DG4.

Ref. | VPP Resources | Objective Function | Load Flow | Emission Cost | Optimization Problem | ||||
---|---|---|---|---|---|---|---|---|---|

RES | DG | ES | EV | DR | |||||

[2] | ☑ | ☒ | ☑ | ☒ | ☑ | Min. Cost | ☑ | ☒ | MILP |

[3] | ☑ | ☑ | ☑ | ☑ | ☑ | Max. Profit | ☑ | ☑ | MILP |

[4] | ☑ | ☑ | ☑ | ☒ | ☑ | Min. Cost | ☑ | ☒ | MINLP |

[5] | ☑ | ☑ | ☑ | ☒ | ☒ | Min. Cost | ☒ | ☒ | MINLP |

[6] | ☑ | ☑ | ☒ | ☑ | ☑ | Multi-objective | ☒ | ☒ | MINLP |

[7] | ☑ | ☑ | ☑ | ☒ | ☑ | Multi-objective | ☒ | ☑ | MIP |

[8] | ☑ | ☑ | ☒ | ☒ | ☑ | Max. Profit | ☑ | ☒ | MINLP |

[9] | ☑ | ☑ | ☑ | ☒ | ☑ | Max. Profit | ☑ | ☒ | MINLP |

[10] | ☑ | ☑ | ☑ | ☑ | ☒ | Min. Cost | ☑ | ☑ | MILP |

[11] | ☑ | ☑ | ☑ | ☒ | ☑ | Max. Profit | ☒ | ☒ | MILP |

[12] | ☑ | ☑ | ☒ | ☒ | ☑ | Max. Profit | ☒ | ☒ | MILP |

[13] | ☑ | ☑ | ☑ | ☒ | ☑ | Max. Profit | ☑ | ☑ | MILP |

[14] | ☑ | ☑ | ☑ | ☑ | ☒ | Max. Profit | ☒ | ☒ | MILP |

[15] | ☑ | ☒ | ☑ | ☒ | ☑ | Max. Profit | ☒ | ☒ | MILP |

[16] | ☑ | ☒ | ☑ | ☒ | ☑ | Min. Cost | ☒ | ☒ | MILP |

[17] | ☑ | ☑ | ☒ | ☒ | ☒ | Max. Profit | ☒ | ☒ | MILP |

[18] | ☑ | ☑ | ☒ | ☑ | ☑ | Min. Cost | ☒ | ☑ | MILP |

[19] | ☑ | ☑ | ☒ | ☑ | ☒ | Min. Cost | ☒ | ☑ | MILP |

[20] | ☑ | ☒ | ☒ | ☑ | ☑ | Max. Profit | ☒ | ☒ | MILP |

[21] | ☑ | ☑ | ☒ | ☒ | ☑ | Min. Cost | ☒ | ☒ | MINLP |

This paper | ☑ | ☑ | ☑ | ☑ | ☑ | Min. Cost | ☑ | ☑ | MILP |

Ref. | Uncertainty Modeling | Method | |||
---|---|---|---|---|---|

Price | RES | Load | EV | ||

[8] | ☑ | ☑ | ☒ | ☒ | Normal, Weibull distribution functions |

[17,25] | ☑ | ☑ | ☑ | ☒ | Point estimate method |

[5] | ☑ | ☑ | ☑ | ☒ | 2-point estimate method |

[2,4,21] | ☑ | ☑ | ☑ | ☒ | Scenario-based method |

[18] | ☒ | ☒ | ☒ | ☒ | – |

[19] | ☒ | ☑ | ☒ | ☒ | Historical data and Markov |

[20] | ☑ | ☑ | ☒ | ☒ | ARMA and adaptive neuro-fuzzy inference system |

[6] | ☑ | ☑ | ☑ | ☑ | Weibull, Beta, and Normal distribution function |

[7,11,15] | ☒ | ☑ | ☑ | ☒ | Weibull, Beta, and Normal distribution function |

[26,27] | ☑ | ☑ | ☑ | ☒ | Weibull, Beta, and Normal distribution function |

[9] | ☒ | ☑ | ☒ | ☒ | Weibull function |

[28] | ☑ | ☑ | ☒ | ☒ | Scenario generation |

[23] | ☑ | ☒ | ☒ | ☑ | SARIMA and Copula methods |

[24] | ☒ | ☒ | ☒ | ☑ | Empirical probability density function |

[29] | ☑ | ☑ | ☑ | ☒ | Monte Carlo simulation |

[12] | ☑ | ☑ | ☑ | ☒ | Robust optimization |

[14] | ☑ | ☑ | ☑ | ☒ | Scenario-based method |

[3,10,13] | ☑ | ☑ | ☑ | ☑ | RANN and LSTM |

[16] | ☒ | ☒ | ☒ | ☒ | – |

This paper | ☑ | ☑ | ☑ | ☑ | BLSTM |

Bus No. | Type | $\mathit{S}\mathit{O}{\mathit{C}}_{\mathit{s}}^{\mathit{E}\mathit{S},\mathit{M}\mathit{A}\mathit{X}}$ (kWh) | $\mathit{S}\mathit{O}{\mathit{C}}_{\mathit{s}}^{\mathit{E}\mathit{S},\mathit{M}\mathit{I}\mathit{N}}$ (kWh) | $\mathit{S}\mathit{O}{\mathit{C}}_{\mathit{s}}^{\mathit{E}\mathit{S},\mathit{I}\mathit{N}\mathit{I}}$ (kWh) | ${\mathit{\alpha}}_{\mathit{s}}$ ($/kW) | ${\mathit{\beta}}_{\mathit{s}}$ ($) | ${\mathbf{R}}_{\mathit{s}}^{\mathit{E}\mathit{S},\mathit{C}\mathit{H}}$ (kW) | ${\mathbf{R}}_{\mathit{s}}^{\mathit{E}\mathit{S},\mathit{D}\mathit{C}\mathit{H}}$ (kW) | ${\mathit{\eta}}_{\mathit{s}}^{\mathit{C}\mathit{H}}$ | ${\mathit{\eta}}_{\mathit{s}}^{\mathit{D}\mathit{C}\mathit{H}}$ |
---|---|---|---|---|---|---|---|---|---|---|

14 | ES1 | 70 | 5 | 25 | 0.01 | 1/5 | 30 | 30 | 0.95 | 0.95 |

20 | ES2 | 90 | 10 | 40 | 0.012 | 1/7 | 40 | 40 | 0.95 | 0.95 |

Bus No. | Type | ${\mathit{a}}_{\mathit{g}}$ ($/kW ^{2}) | ${\mathit{b}}_{\mathit{g}}$ ($/kW) | ${\mathit{c}}_{\mathit{g}}$ ($) | ${\mathit{P}}_{\mathit{g}}^{\mathit{D}\mathit{G},\mathit{M}\mathit{I}\mathit{N}}$ (kW) | ${\mathit{P}}_{\mathit{g}}^{\mathit{D}\mathit{G},\mathit{M}\mathit{A}\mathit{X}}$ (kW) | ${\mathit{Q}}_{\mathit{g}}^{\mathit{D}\mathit{G},\mathit{M}\mathit{I}\mathit{N}}$ (kVar) | ${\mathit{Q}}_{\mathit{g}}^{\mathit{D}\mathit{G},\mathit{M}\mathit{A}\mathit{X}}$ (kVar) |
---|---|---|---|---|---|---|---|---|

4 | DG1 | $9\times {10}^{-6}$ | 0.019 | 0.055 | 50 | 600 | −180 | 400 |

18 | DG2 | $8.8\times {10}^{-6}$ | 0.0197 | 0.048 | 30 | 500 | −150 | 350 |

5 | DG3 | $9.76\times {10}^{-5}$ | 0.0055 | 0.0184 | 20 | 300 | −160 | 200 |

15 | DG4 | $8.6\times {10}^{-5}$ | 0.0048 | 0.0151 | 10 | 250 | −100 | 100 |

$\mathit{e}$ | $\mathit{E}{\mathit{F}}_{\mathit{g},\mathit{e}}\left(\frac{\mathbf{\$}}{\mathbf{l}\mathbf{b}}\right)$ | ${\mathit{\alpha}}_{\mathit{e}}\left(\frac{\mathbf{l}\mathbf{b}}{\mathbf{k}\mathbf{W}\mathbf{h}}\right)$ |
---|---|---|

$N{O}_{x}$ | 4.2 | $4.4\times {10}^{-4}$ |

$S{O}_{2}$ | 0.99 | $0.8\times {10}^{-6}$ |

$C{O}_{2}$ | 0.014 | $1.59\times {10}^{-3}$ |

Bus No. | Type | $\mathit{C}{\mathit{o}}_{\mathit{w}}^{\mathit{W}\mathit{T}}$ ($/kW) | ${\mathit{P}}_{\mathit{w}}^{\mathit{w}\mathit{t},\mathit{m}\mathit{a}\mathit{x}}$ (kW) | ${\mathit{\vartheta}}_{\mathit{w}}^{\mathit{c}\mathit{u}\mathit{t}\mathit{i}\mathit{n}}$ (m) | ${\mathit{\vartheta}}_{\mathit{w}}^{\mathit{c}\mathit{u}\mathit{t}\mathit{o}\mathit{u}\mathit{t}}$ (m/s) | ${\mathit{\vartheta}}_{\mathit{w}}^{\mathit{r}\mathit{a}\mathit{t}\mathit{e}\mathit{d}}$ (m/s) |
---|---|---|---|---|---|---|

10, 21 | WT1, WT2 | 0.03 | 500 | 3 | 25 | 9 |

Method | Error Criterion | Load Demand p.u | Electricity Price $/kWh | Wind Speed m/s |
---|---|---|---|---|

LSVM | MAE | 0.0358 | 0.0104 | 1.298 |

RMSE | 0.0422 | 0.0134 | 1.461 | |

GSVM | MAE | 0.014 | 0.0078 | 0.8948 |

RMSE | 0.0178 | 0.0105 | 1.091 | |

SANN | MAE | 0.0215 | 0.0087 | 1.022 |

RMSE | 0.027 | 0.0117 | 1.191 | |

LSTM | MAE | 0.009 | 0.0044 | 0.3911 |

RMSE | 0.0112 | 0.0053 | 0.5700 | |

BLSTM | MAE | 0.0059 | 0.0028 | 0.2155 |

RMSE | 0.0082 | 0.0033 | 0.2592 |

Method | Network Cost | DG Cost | ES Cost | WT Cost | Total Cost | Error |
---|---|---|---|---|---|---|

Real data | 1247.9 | 491.2 | 83.3 | 23.1 | 1802.80 | - |

BLSTM | 1248.9 | 465.1 | 83.3 | 20.9 | 1776.2 | 1.47 |

LSTM | 1120.8 | 436.3 | 83.4 | 12.9 | 1619.5 | 10.16 |

Short ANN | 1221.4 | 318.4 | 80.2 | 25.6 | 1611.1 | 10.63 |

Gaussian SVM | 1138.0 | 382 | 80.7 | 29.8 | 1595.8 | 11.47 |

Linear SVM | 1155.6 | 243.5 | 79.2 | 34.2 | 1484.5 | 17.65 |

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## Share and Cite

**MDPI and ACS Style**

Ahmadian, A.; Ponnambalam, K.; Almansoori, A.; Elkamel, A.
Optimal Management of a Virtual Power Plant Consisting of Renewable Energy Resources and Electric Vehicles Using Mixed-Integer Linear Programming and Deep Learning. *Energies* **2023**, *16*, 1000.
https://doi.org/10.3390/en16021000

**AMA Style**

Ahmadian A, Ponnambalam K, Almansoori A, Elkamel A.
Optimal Management of a Virtual Power Plant Consisting of Renewable Energy Resources and Electric Vehicles Using Mixed-Integer Linear Programming and Deep Learning. *Energies*. 2023; 16(2):1000.
https://doi.org/10.3390/en16021000

**Chicago/Turabian Style**

Ahmadian, Ali, Kumaraswamy Ponnambalam, Ali Almansoori, and Ali Elkamel.
2023. "Optimal Management of a Virtual Power Plant Consisting of Renewable Energy Resources and Electric Vehicles Using Mixed-Integer Linear Programming and Deep Learning" *Energies* 16, no. 2: 1000.
https://doi.org/10.3390/en16021000