# Real-Time Peak Valley Pricing Based Multi-Objective Optimal Scheduling of a Virtual Power Plant Considering Renewable Resources

^{*}

## Abstract

**:**

## 1. Introduction

- (a)
- A VPP diminishes the need for the conventional generation to provide the provision of dynamic ancillary services.
- (b)
- It controls a cluster of heterogeneous renewable energy resources (RERs).
- (c)
- The intermittency and uncertainty caused by renewables such as solar and wind power, which are highly weather dependent, can be reduced to a certain extent.
- (d)
- It maintains favorable grid conditions for real-time management and supervision in emergencies.

- To ensure efficient management of the grid, sources such as fuel cells and CHP can be considered for optimal scheduling to reduce the cost of power generation along with emissions in a VPP system.
- To handle a non-convex problem such as VPP efficiently, advanced, and recently developed soft computing (SC) techniques can be implemented or modified by choosing related constraints.
- To incentivize the participants, peak-valley pricing mechanisms with the incorporation of (15 min) interval scheduling is introduced and compared with day-ahead scheduling.

- One centrally controlled VPP system comprised of multiple resources including solar PV modules, WT, fuel cells, electric loads, heat-only units, and CHP units has been attempted to solve the multi-objective optimal scheduling problem.
- The multi-objective optimal scheduling of the VPP considering renewable resources has been solved using the weighting factor method to simultaneously maximize profit and minimize emissions.
- Peak valley’s power pricing strategy is introduced in the multi-objective optimal scheduling of the VPP problem.
- The new price-based multi-objective black widow optimization (MOBWO) is presented and implemented by considering constraint handling.
- Statistical analysis was performed for both single and multi-objective optimal scheduling of the VPP problems and quality solution sets were obtained from the MOBWO algorithm after 100 different independent trials.
- Pareto optimal solutions were obtained specifically for multi-objective optimal scheduling of the VPP problem for the maximization of profit along with simultaneously minimizing the emissions for both scenarios I and II, respectively.
- Results obtained by the proposed MOBWO algorithm were also compared with the latest published works.

## 2. VPP Concept

#### Methodology

## 3. Problem Formulation

#### 3.1. Objective Function

#### 3.1.1. Net Profit

- p and s are set of plants and scenarios, t is time ranges from 1 to 24 in Day-ahead scheduling, followed by 1 to 96 in 15-min scheduling.
- c
_{ph}and c_{se}are the tariffs for purchasing and selling power from the grid system. - π
_{s}is the probability of scenarios. ^{p}_{em}is the energy market price; c_{ens}is the cost of energy not served.- c
_{chp}and c_{hou}are the cost function of CHP and heat-only units. ^{p}_{hou}is the price of heat-only units.

#### 3.1.2. Emission

- e
_{chp}and e_{hou}are the emissions by CHP and heat-only units, respectively. - e
_{ph}and e_{se}are the emissions by the grid system.

#### 3.1.3. Multi-Objective Framework

- w is considered as 0.5 for giving equal weightage to both objectives.

#### 3.2. Constraints Handling

#### 3.2.1. Power Balancing

- ${P}_{s,t,p}^{eqv}$ is equivalent to power scenario s, time t, and plant p.
- ${P}_{ex}^{i}$ is the exchanging power between the main grid and the CHP system at interval i (MW).
- ${P}_{fl}^{i}$ is the power of the fuel cell at interval i (MW).
- ${P}_{wt}^{i}$ is the power of the wind turbine at interval i (MW).
- ${P}_{pv}^{i}$ is the power of the solar photovoltaic at interval i (MW).
- ${P}_{el}^{i}$ is the electrical load at interval i (MW).

#### 3.2.2. Heat Balancing

- ${r}_{fl}^{i}$ is the ratio of heat to the electricity of the fuel cell at interval i (MW).
- ${\eta}_{hr\_bl}$ is the efficiency of the heat rate boiler (MW).
- ${P}_{gb}^{i}$ is the power of a gas boiler at interval i (MW).
- ${P}_{th}^{i}$ is thermal power balance at interval i (MW).

#### 3.3. Power Switching between Main Grid and CHP Units

- ${P}_{ex}^{min}$ is the minimum exchange of power between the main grid and the CHP system.
- ${P}_{ex}^{max}$ is the maximum exchange of power between the main grid and the CHP system.

#### 3.4. Constraints of Waste Heat and Gas Boiler

- ${P}_{bl}^{min}$ is the minimum limit of the waste heat boiler.
- ${P}_{bl}^{max}$ is the maximum limit of the waste heat boiler.
- ${P}_{gb}^{min}$ is the minimum limit of the gas boiler.
- ${P}_{gb}^{max}$ is the maximum limit of the gas boiler.

#### 3.5. Fuel Cells

- ${\eta}_{fl}^{i}$ is the fuel cell efficiency at interval i (p.u.) and
- ${r}_{fl}^{i}$ is the ratio of heat to the electricity of the fuel cell at interval i (MW).

#### Ramp Rate Limit of Fuel Cell

- $\Delta {P}_{f{l}_{up}}$T is the up-ramp limit and $\Delta {P}_{fl\_down}$ is the down-ramp limit of the fuel cell.
- ${P}_{fl}^{i}$ is the power generated by the fuel cell at interval i (kW).
- ${P}_{fl}^{i-1}$ is the power generated by the fuel cell at the previous interval (kW).

#### 3.6. CHP Units

- ${\mathrm{P}}_{\mathrm{t},\mathrm{p}}^{\mathrm{chp}}$ is the electrical output power of the CHP and
- ${\mathrm{V}}_{\mathrm{t},\mathrm{p}}^{\mathrm{chp}}$ is the commitment status of the CHP.

#### 3.7. Solar PV Modules

- ${P}_{s,t,p}^{pv}$ is output power, ${N}_{p}^{pv}$ is the number of PV modules, and ${sor}_{s,t}$ is the solar radiation (KW/${\mathrm{m}}^{2}$).
- ${V}_{OC}$ is the open-circuit voltage and ${I}_{sc}$ is the short circuit current.
- ${K}_{i}$ is the current temperature and ${K}_{V}$ is the voltage temperature coefficient.
- ${T}_{Cs,t}$ is the solar cell temperature and FF denotes the Fill Factor of the PV module.

#### 3.8. Wind Turbine

- ${P}_{s,t,p}^{wt}$ is the output power of the wind turbine (MW).
- ${P}_{rated}^{wt}$ is the nominal power of the wind turbine (MW).
- ${N}_{p}^{wt}$ is the number of wind turbines.

## 4. Optimization Algorithm

#### 4.1. Population Initialization

- ${k}_{1},{k}_{2}\dots {k}_{m}$ are the floating numbers in the form of variables.

#### 4.2. Procreation, Cannibalism, Mutation

_{Var}two times, in which duplication should be avoided to enhance the accuracy of the fitness value among the pairs. The next step is the cannibalism rate which ensures better performance for the exploitation and guarantees faster convergence at the same time for BWO. Every operator corresponds to a contender solution to the given problem. The mutation stage is considered to bring the balance between both exploration and exploitation.

## 5. Results

- Case I: Single-objective scheduling for profit maximization.
- Case II: Single-objective scheduling for emission minimization.
- Case III: Multi-objective scheduling for profit/emission, i.e., maximization followed by minimization.

#### 5.1. Scenario Generation

#### 5.1.1. Scenario I (Day-ahead Scheduling)

**Case I:**The statistical results obtained after 100 independent runs from MOBWO converge to an optimum value of 27,785.6723 $ which outperforms the existing value of 23,302.8271 $ obtained from multi-objective particle swarm optimization (MOPSO) Ref. [22]. Additionally, the numerical results for other techniques are mentioned in Table 7.

**Case II:**On the other hand, in Case II, the purpose is to reduce emissions and the level of emission is reduced significantly by a considerable margin. The unit of the emission is in Kg and it is the combination of three emission factors, namely, ${\mathrm{NO}}_{\mathrm{X}},{\mathrm{CO}}_{2},{\mathrm{and}\mathrm{SO}}_{2}$. The statistical results obtained after 100 independent runs from MOBWO are shown in Table 8 as converging to an optimum value of 57,532.2738 Kg which outperforms the existing value of 64,432.3217 Kg obtained from MOPSO Ref. [22]. In Case II, the computational time for the proposed MOBWO is 81.3745 (for 100 runs), much less when compared with 171.4826 (for 20 runs) from MOPSO. In addition, the convergence characteristics of the proposed algorithm for Case II are shown in Figure 13.

**Case III:**The multi-objective optimal scheduling is carried out and the convergence of both objectives is displayed together in Figure 14, followed by Figure 15 and Figure 16 with other techniques. It is worth mentioning that including emission as an objective function has an immense effect on the overall working of the system since the CHP unit is considered one of the resources. Here, the peak valley electricity pricing is considered which is very essential in the peak load shifting to enhance the economic benefit. For every time period, the sale and purchase prices are given in Table 9. The effect of this pricing scheme on the operating costs has been given proper consideration in this paper.

#### 5.1.2. Scenario II (15-min Interval Scheduling)

- (a)
- Buying and selling of electricity one day before the following day.
- (b)
- The VPP acts as a price taker in the day-ahead market.
- (c)
- Ease of unit commitment and power dispatching.

- (a)
- This scheduling helps to determine the imbalance in settlement prices.
- (b)
- Offers the purchase and selling of electricity during the functioning day.
- (c)
- Real-time scheduling stabilizes the differences between day-ahead and real-time demand and production of electricity.
- (d)
- Operating systems that work in real-time can execute quickly without any delay, resulting in a nearly immediate output.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Con sent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

TPP | Traditional power plant |

DERs | Distributed energy resources |

VPP | Virtual power plant |

MOBWO | Multi-objective black widow optimization |

MOOS | Multi-objective optimal scheduling |

MOSS | Multi-objective scheduling strategy |

SP | Spot pricing |

TOU | Time of use |

EVs | Electric vehicles |

ES | Energy storage |

RERs | Renewable energy resources |

ICT | Information communication technology |

IRP | Integrated resource planning |

P2P | Peer to peer |

NM | Net metering |

BTM | Behind the meter |

PSO | Particle swarm optimization |

GA | Genetic algorithm |

DR | Demand response |

V2H | Vehicle to home |

BSS | Battery storage system |

CEM | Consecutive energy management |

SOC | State of charge |

TNPC | Total net present cost |

UC | Unit commitment |

SC | Soft computing |

PV | Photovoltaics |

WT | Wind turbine |

FC | Fuel cells |

CHP | Combined heat and power |

EL | Electric load |

EM | Energy market |

PLR | Part load ratios |

FOR | Feasible regions of operation |

Probability distribution function | |

FF | Fill factor |

ACO | Ant colony optimization |

ABC | Artificial bee colony |

ANN | Artificial neural network |

CR | Cannibalism rate |

MR | Mutation rate |

RP | Reproduction rate |

PVPP | Peak valley power pricing |

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Refs. No. | Nature of Problem | Control Method | Features of Control Method |
---|---|---|---|

[22,23] | Heuristic | PSO/MOPSO | Fewer parameters. Ease of implementation. Local entrapment. |

[24,25] | Stochastic | ABC | Poor in exploitation stage. Limited population diversity. |

[26,27] | Computational | ANN | More precise predictions. Good computational efficiency. |

[28,29] | Heuristic | GA | Can determine multiple solutions simultaneously. |

[30,31] | Meta-heuristic | ACO | Can discover good solutions rapidly. |

[32,33] | Mathematical | Fuzzy Logic | Improved prediction accuracy. Use of Fuzzy sets. |

[34,35] | Mathematical | Game Theory | Computational load increases as the no. of participants increases. |

This paper | Meta-heuristic | BWO/MOBWO | High searching accuracy. Better updating strategy. Converge to the global optimum in lesser iterations. |

Emissions | Heat-Only Unit | CHP Unit |
---|---|---|

SO_{2} | 0.0027 | 0.0036 |

NO_{X}CO _{2} | 0.3145 401.43 | 0.1995 723.94 |

${\mathit{P}}_{\mathit{r}\mathit{a}\mathit{t}\mathit{e}\mathit{d}}^{\mathit{w}\mathit{t}}$(MW) | ${\mathit{v}}_{\mathit{i}\mathit{n}}^{\mathit{c}}$(m/s) | ${\mathit{v}}_{\mathit{o}\mathit{u}\mathit{t}}^{\mathit{c}}$(m/s) | ${\mathit{v}}_{\mathit{r}\mathit{a}\mathit{t}\mathit{e}\mathit{d}}$(m/s) | ${\mathit{N}}_{\mathit{p}}^{\mathit{w}\mathit{t}}$ |
---|---|---|---|---|

150 | 3.5 | 25 | 13.5 | 3 |

${\mathit{V}}_{\mathit{O}\mathit{C}}$(V) | ${\mathit{I}}_{\mathit{S}\mathit{C}}$(A) | ${\mathit{K}}_{\mathit{i}}$(I/°C) | ${\mathit{K}}_{\mathit{V}}$(V/°C) | ${\mathit{N}}_{\mathit{O}\mathit{T}}$(°C) | ${\mathit{I}}_{\mathit{M}\mathit{P}\mathit{P}\mathit{T}}$ (A) | ${\mathit{V}}_{\mathit{M}\mathit{P}\mathit{P}\mathit{T}}$ (V) | ${\mathit{N}}_{\mathit{p}}^{\mathit{p}\mathit{v}}$ |
---|---|---|---|---|---|---|---|

21.98 | 5.32 | 0.003 | 0.0144 | 43 | 4.76 | 17.32 | 2240 |

${\mathit{H}}_{\mathit{m}\mathit{a}\mathit{x}\mathbf{,}\mathit{p}}^{\mathit{h}\mathit{o}\mathit{u}}$(MWth) | ${\mathit{a}}_{\mathit{p}}$($/MWth^{2}) | ${\mathit{b}}_{\mathit{p}}$($/MWth) | ${\mathit{c}}_{\mathit{p}}$($) |
---|---|---|---|

1.2 | 0.052 | 3.0651 | 4.8 |

${\mathit{g}}_{\mathit{p}}$($/MW^{2}) | ${\mathit{h}}_{\mathit{p}}$($/MW) | ${\mathit{i}}_{\mathit{p}}$($) | ${\mathit{j}}_{\mathit{p}}$($/MWth^{2}) | ${\mathit{k}}_{\mathit{p}}$($/MWth) | ${\mathit{i}}_{\mathit{p}}$($/MW.MWth) | ${\mathit{C}}_{\mathit{p}}^{\mathit{s}\mathit{u}}$($) | ${\mathit{C}}_{\mathit{p}}^{\mathit{s}\mathit{d}}$($) |
---|---|---|---|---|---|---|---|

0.0345 | 44.5 | 26.5 | 0.03 | 4.2 | 0.031 | 20 | 20 |

Output | Ref. [46] | MOPSO [22] | ABC | ACO | Proposed MOBWO |
---|---|---|---|---|---|

Maximum Profit ($) | 19,737 | 23,302.8271 | 24,191.8221 | 24,950.7372 | 27,785.6723 |

Minimum Profit ($) | - | 22,600.1679 | 19,636.7483 | 20,190.8183 | 21,400.3254 |

Mean Profit ($) | - | 22,955.3462 | 21,914.2852 | 22,570.7776 | 24,592.9985 |

Computational time (Seconds) | - | 148.095 (For 20 runs) | 139.3737 (For 100 runs) | 135.4932 (For 100 runs) | 123.058 (For 100 runs) |

Output | Ref. [47] | MOPSO [22] | ABC | ACO | Proposed MOBWO |
---|---|---|---|---|---|

Minimum Emission (Kg) | 56,270 | 64,432.3217 | 62,467.8291 | 61,346.4838 | 57,532.2738 |

Maximum Emission (Kg) | 77,430 | 67,077.3937 | 72,383.7292 | 71.463.2612 | 67,342.3798 |

Mean Emission (Kg) | - | 66,070.1682 | 67,425.7792 | 66,404.8725 | 62,437.3268 |

Computational Time (Sec) | - | 171.4826 (For 20 runs) | 153.4826 (For 100 runs) | 131.3633 (For 100 runs) | 81.3745 (For 100 runs) |

Period | Detail Time (Hr.) | Purchase Price ($/MWh) | Sale Price ($/MWh) |
---|---|---|---|

Peak | 9,12,17,22 | 0.0079 | 0.0044 |

Intermediate | 13,16 | 0.0070 | 0.0035 |

Valley | 1,8,23,24 | 0.0062 | 0.0026 |

Objective Functions | Profit ($) | Emissions (Kg) | ||||||
---|---|---|---|---|---|---|---|---|

Parameters | MOPSO [22] | ABC | ACO | MOBWO | MOPSO [22] | ABC | ACO | MOBWO |

MaxF^{profit} | 23,302.83 | 24,286.82 | 25,183.74 | 26,167.78 | 122,963.46 | 119,789.29 | 119,432.37 | 116,400.85 |

MinF^{emission} | 9883.69 | 10,320.38 | 11,723.47 | 11,808.47 | 64,432.32 | 62,467.83 | 61,346.48 | 58,785.34 |

Objective Functions | Profit ($) | Emissions (Kg) | ||||
---|---|---|---|---|---|---|

Parameters | ABC | ACO | MOBWO | ABC | ACO | MOBWO |

MaxF^{profit} | 27,392.5631 | 26,312.3523 | 28,415.3525 | 120,913.4532 | 120,325.463 | 119,843.4532 |

MinF^{emission} | 10,404.7262 | 11,123.4253 | 11,929.7262 | 56,402.4216 | 57,342.4235 | 59,921.3248 |

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**MDPI and ACS Style**

Pandey, A.K.; Jadoun, V.K.; Sabhahit, J.N.
Real-Time Peak Valley Pricing Based Multi-Objective Optimal Scheduling of a Virtual Power Plant Considering Renewable Resources. *Energies* **2022**, *15*, 5970.
https://doi.org/10.3390/en15165970

**AMA Style**

Pandey AK, Jadoun VK, Sabhahit JN.
Real-Time Peak Valley Pricing Based Multi-Objective Optimal Scheduling of a Virtual Power Plant Considering Renewable Resources. *Energies*. 2022; 15(16):5970.
https://doi.org/10.3390/en15165970

**Chicago/Turabian Style**

Pandey, Anubhav Kumar, Vinay Kumar Jadoun, and Jayalakshmi N. Sabhahit.
2022. "Real-Time Peak Valley Pricing Based Multi-Objective Optimal Scheduling of a Virtual Power Plant Considering Renewable Resources" *Energies* 15, no. 16: 5970.
https://doi.org/10.3390/en15165970