# A Multi-Agent Based Optimization Model for Microgrid Operation with Hybrid Method Using Game Theory Strategy

^{*}

## Abstract

**:**

## 1. Introduction

- The proposed two-layer MAS model is constructed with an MGO agent, an RE agent, a Battery agent, and a Load agent. It achieves coordinated and efficient MG operation by considering BESS and DR. It enables autonomous game theory based on communication among agents to optimize the MG operation.
- The effectiveness of applying BESS allocation is evaluated by the sum of the MG operation cost and BESS cost considering depreciation, whereas the superiority of DR allocation is demonstrated by comparison with different DR strategies.
- The proposed optimal hybrid method operation can improve the utilization costs, including operation, BESS, and DR costs. It assists the MGO in formulating reasonable and economical decisions in the MAS model.
- MAG-PSO, which adjusts the best global position and position vector of the particle, is formulated in the proposed MG model to prevent the curtailment of DR participation. By using the MAG-PSO, the solution is improved with the optimal capacity of BESS and DR and exhibits better performance. The solution can be applied to larger power systems.

## 2. Overall Scheme of the Microgrid

#### 2.1. Grid-Connected Microgrid

#### 2.2. Modeling of PV

_{s}and A are the efficiency (%) and area of the panels (m

^{2}), respectively. β denotes the temperature coefficient of the maximum output power, and SI represents solar irradiation (kW/m

^{2}). β denotes a negative percentage per Kelvin or degree Celsius and is considered as −0.005/°C in this study. T

_{t}represents the output air temperature.

_{s}is the number of solar generators.

#### 2.3. Modeling of WT

_{r}and v

_{r}represent the rated electrical power and rated wind speed, respectively. v and η

_{w}denote wind speed and efficiency. v

_{c}and v

_{f}represent the cut-in and cut-off wind speeds, respectively.

_{w}is the number of wind generators.

#### 2.4. Modeling of BESS

_{b}(t) is negative if it is discharged or positive if it is charged [27]. The BESS power can be expressed as follows:

_{b}(t) is the discharging or charging power on the AC side of BESS. P

_{b}

^{dch,max}and P

_{b}

^{ch,max}denote the maximum discharging power and maximum charging power, respectively.

_{b}

^{DC}(t) is the power on the DC side of BESS and E

_{b}is the capacity of BESS. Δt and P

_{b}

^{loss}(t) denote the interval of the time period and power loss in the converter, respectively.

_{b}

^{dch}(t) and P

_{b}

^{ch}(t) are the discharging charging power at time t, respectively. SOC(t) is the battery SOC at each time step, t

_{step,}bounded by an upper limit, SOC

_{max}and lower limit, SOC

_{min}. E

_{ESS}denotes the battery capacity in kWh.

_{ch}and η

_{dch}are the charging and discharging efficiencies of the battery, respectively. u

_{b}is a binary variable denoting the charging (“1”) or discharging (“0”) status at each time step.

#### 2.5. Load and Utility

_{d}(t), is taken as a continuous function with a time step of 1 h. The electricity price information invigorates the consumer to be active in power trading and to manage power demand.

_{g}(t) represents the net load and P

_{DR}(t) is the capacity of DR in time slot t, respectively.

_{g}(t), which is positive if the power is imported from the grid because of deficient power generation or negative if the power is exported to the grid [8]. In our work, the power exported to the grid is zero, and the financial benefits caused by excess electricity are also zero. P

_{b}(t) is on the right side because BESS power is negative when it is discharging and positive when it is charging.

## 3. Proposed Optimal Operation for Multi-Agent System

#### 3.1. Two-Layer MAS Model

- RE agent includes PV and WT power generation. The agent recognizes the environment, such as solar radiation or wind speed, and delivers information on renewable power generation to the MGO agent in a time slot.
- Because BESS is mainly used to adjust the peak power, the Battery agent monitors the SOC and determines the optimal BESS sizing based on the game theory with the operation schedule of the MGO agent.
- The load agent recognizes the available capacity to reduce the peak load during the time slot. This agent makes a decision with differential incentives based on a pay-as-bid pricing mechanism using game theory.
- The MGO agent receives the market price, load demand, and operation information from agents in the lower layer. After receiving the communication, the agent solves the game theory, which is discussed in Section 3.2. MGO operates to minimize MG operation costs through determined capacities of the BESS and DR.

#### 3.2. Game Theory Strategy

#### 3.2.1. Non-Cooperative Game Formulation

- (1)
- Players: Agents in the set N participates in the game theory strategy;
- (2)
- Strategies: Each agent n∈N decides its strategy by determining the usable power capacity and setting the cost to maximize its payoff;
- (3)
- Payoff: P
_{n}(x_{n}, x_{-n}, y_{n}) is a cost function for user n.

_{−n}

^{*}, …, x

_{1}

^{*}, …, x

_{n}

^{*}) denote the Nash equilibrium, and the optimal output is y

_{n}

^{*}, then:

_{n}

^{*}and x

_{−n}

^{*}are represented as the BESS and MG operation cost or DR capacity and cost during the time slot at the Nash equilibrium. y

_{n}

^{*}indicates the optimal BESS sizing or DR incentive value, which are the intermediate solutions, after achieving the optimal point.

**Proposition**

**1.**

_{n}is continuously differentiable in x

_{n}. Thus, the space of agent cost function is a non-empty convex compact subset of Euclidean space in x

_{n}.

**Proof.**

_{n}(x

_{n}, x

_{−n}, y

_{n}), it is continuously differentiable in x

_{n}. Because the Hessian of P

_{n}(x

_{n}, x

_{−n}, y

_{n}) is a positive semi-definite, P

_{n}(x

_{n}, x

_{−n}, y

_{n}) is convex [32]. Proposition 1 is a prerequisite for Proposition 2. □

**Proposition**

**2.**

**Proof.**

_{n}is convex in x

_{n}, the Nash equilibrium is proved to be present and also unique [23]. □

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

_{n}as the BESS capacity or DR incentive, there is a unique y

_{n}that can minimize MG cost. It is comprehended that there is a specific cost value with a certain BESS capacity or DR incentive.

**Proof.**

_{Grid}(t) and k

_{h}denote the electricity price and parameter of discharging/charging of BESS, respectively.

_{n}is the primary function [23]. Therefore, the function P

_{n}is convex in y

_{n}and exists as an optimal BESS capacity, y

_{n}

^{*}. Conversely, in DR strategy, the cost function, P

_{n}can be written as follow:

_{h}represents the parameter of the expected DR incentive price considered in MGO.

_{n}is a secondary function. The optimal DR incentive value exists, hence, the proof is complete. □

#### 3.2.2. BESS Strategy

_{n}) is a function of the storage size in terms of rated power and energy.

_{p}and c

_{E}are the specific costs of the BESS adopted technology, depending on the rating power, P

_{n}

^{rated}and the nominal capacity, E

_{n}

^{rated}, of the nth BESS. c

_{fixedO&M}and c

_{variO&M}are the coefficients of fixed and variable operation and maintenance costs, respectively.

_{s}denotes a capital recovery factor with a value of 0.1, considering that the BESS lifetime is 10 years.

_{0}= 2731.7, β

_{1}= 0.679, β

_{2}= 1.614.

_{s}used in Equation (19), the battery constraint that the DOD should be 80% or less is satisfied. In the first part of the BESS strategy, the Battery agent presents the BESS maximum capacity, which is set as equal to five times the peak load.

#### 3.2.3. DR Strategy

_{t}

^{bat}and ΔE

_{DR}denote the cost of the battery in house as a function of stored energy and DR capacity, respectively. a

_{t}is a pricing coefficient determined by utility. B represents a parameter and also serves as the maximum value of |ΔE

_{DR}|.

_{DR}/B < 1, the quadratic equation can be understood from its Taylor expansion.

_{DR}, assuming that the load is constant for an hour. Thus, the DR’s payoff factor can be represented by a quadratic function.

#### 3.3. Optimization Technique

#### 3.3.1. MG Formulation

_{DR}(t) denotes the determined incentive in time slot t.

- Power balance constraint

_{b}on the right-hand side is that BESS discharging is negative, whereas charging is positive.

- Generation limit constraints for PV, WT, and BESS

#### 3.3.2. Multi-Agent Guiding PSO

_{k}= [x

_{k}

_{1}, x

_{k}

_{2}, …, x

_{kn}]

^{T}and velocity vector, V

_{k}= [v

_{k}

_{1}, v

_{k}

_{2}, …, v

_{kn}]

^{T}. The velocity and position of a particle, k, can be expressed at the (i + 1)th iteration.

_{1}and r

_{2}are two different random numbers of the uniform distribution within the range [0 1], and c

_{1}and c

_{2}represent learning rates with positive constants. pbest

_{i}is the best solution, and gbest

_{i}is the best global position at the ith iteration. iw

_{i+1}denotes the inertia weight for the (i + 1)th iteration to control the velocity in the PSO algorithm. iw

_{max}and iw

_{min}are the initial and final inertia weights, respectively, and i

_{max}is the maximum number of iterations.

_{max}represents the absolute maximum vector value of n-dimensional gbest

_{1}in the operation with the optimal BESS capacity and gbest

_{min}denotes the absolute minimum vector values of gbest

_{2}derived during operations with incentives determined by the game theory.

_{i+1}and r

_{3}are the weight and random number ranging from 0 to 1, respectively.

#### 3.3.3. Hybrid Optimization Process

- Step 1:
- Construct the two-layer MAS model and initialize the MG input parameters, such as PV, WT, BESS, and Load.
- Step 2:
- Define an objective function as (16) and (17) with the following constraints (24)–(29). Obtain the intermediate solutions derived from Method 1 and 2. Further details of the process are as follows:

Method 1: Game Theory Strategy for BESS |

1 Set y_{n} = y_{n}^{max} |

2 Calculate initial x_{n} and x_{-n} |

3 repeat |

4 Define objective function as Equation (16) and solve cost function in Equation (15) |

5 if x_{n} changes then |

6 Update and broadcast x_{n} |

7 end |

8 if A new update is received then |

9 Update x_{-n} accordingly |

10 end |

11 until No user changes its strategy |

12 Select minimal daily cost P_{n}(x_{n}^{*}, x_{-n}^{*}, y_{n}^{*}) |

Method 2: Game Theory Strategy for DR |

1 for h = 1:24 |

2 Set y_{n} = y_{n}^{initial} |

3 Calculate initial x_{n} and x_{-n} |

4 repeat |

5 Define objective function as Equation (17) and solve cost function in Equation (15) |

6 if y_{n} changes then |

7 Update and broadcast y_{n} |

8 end |

9 until No user changes its strategy 10 end |

11 Select minimal daily cost P_{n}(x_{n}^{*}, x_{-n}^{*}, y_{n}^{*}) at each time slot h |

- Step 3:
- Establish the objective function and constraints given by Equations (24–29).
- Step 4:
- In the MAG-PSO, set the required input parameters to initialize the algorithm.
- Step 5:
- Obtain the best global particle position sets at the optimal capacity of BESS and DR, respectively, and store them in the repository.
- Step 6:
- Start iteration
- Step 7:
- Implement Multi-Agent Guiding. Update the maximum and minimum values of the global particle positions.
- Step 8:
- Update the velocity and position of each particle according to Equations (30) and (34). Calculate the fitness value of each particle.
- Step 9:
- Determine whether the stopping criteria is satisfied. If the number of iterations reaches the maximum, go to the next step. If not, repeat the iteration.
- Step 10:
- Output the iteration results based on MAG-PSO.
- Step 11:
- Step 11: Obtain BESS charge/discharge operation and DR scheduling during the time slot.

## 4. Simulation Results

^{®®}Core i5-8400 CPU @ 2.80 GHz and 16 GB RAM.

#### 4.1. Data Description

#### 4.2. Optimal Operation Results

#### 4.2.1. Method 1: BESS Strategy

#### 4.2.2. Method 2: DR Strategy

#### 4.2.3. Hybrid Method

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Input data over a 24 h horizon. (

**a**) Wind velocity and solar irradiation. (

**b**) Power load and market price.

**Figure 6.**Optimal charging/discharging cycles and state of charge (SOC) of BESS. (

**a**) 32 kWh BESS. (

**b**) 200 kWh BESS.

Parameter | Value | Unit |
---|---|---|

PV generators | ||

Covered area, A | 25 | m^{2} |

Efficiency, η_{s} | 16 | % |

Maximum power | 4 | kW |

No. of PV panels | 5 | |

Wind generators | ||

Cut-in velocity, v_{c} | 3 | m/s |

Cut-off velocity, v_{f} | 25 | m/s |

Rated speed, v_{r} | 10 | m/s |

Efficiency, η_{w} | 95 | % |

Maximum power, P_{r} | 5 | kW |

No. of WT | 6 |

**Table 2.**Microgrid (MG) operating and battery energy storage system (BESS) costs according to BESS capacity.

0 kWh | 35 kWh | 100 kWh | 200 kWh | |
---|---|---|---|---|

Operating Cost (¢) | 3790.52 | 3429.54 | 3353.90 | 3268.80 |

BESS Cost (¢) | 0 | 292.445 | 835.556 | 1671.11 |

Conventional DR Program | Method 2 | |
---|---|---|

DR Capacity (kWh) | 13.94 | 22.96 |

Daily DR Cost (¢) | 262.5082 | 395.5845 |

Operating Cost (¢) | 3400.10 | 3126.13 |

Utilization Cost (¢) | 3662.6082 | 3521.7145 |

Base Method (w/o Method 1 and 2) | Method 1 | Method 2 | Hybrid Method | |
---|---|---|---|---|

DR Capacity (kWh) | 0 | 0 | 22.96 | 15.94 |

Utilization Cost (¢) | 3790.52 | 3754.238 | 3521.7145 | 3470.106 |

Algorithm | Best Solution (¢) | Worst Solution (¢) | Average (¢) | Run Time (s) |
---|---|---|---|---|

PSO | 3586.420 | 3612.051 | 3600.918 | 387.7077 |

DG-PSO | 3534.873 | 3544.219 | 3538.318 | 377.8955 |

MAG-PSO | 3468.955 | 3472.075 | 3470.106 | 342.0087 |

Algorithm | DR Capacity (kWh) | Average Incentive (¢/kWh) | Utilization Cost (¢) |
---|---|---|---|

PSO | 7.69 | 16.46 | 3600.918 |

DG-PSO | 11.77 | 16.7 | 3538.318 |

MAG-PSO | 15.94 | 16.5 | 3470.106 |

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**MDPI and ACS Style**

Lee, J.-W.; Kim, M.-K.; Kim, H.-J.
A Multi-Agent Based Optimization Model for Microgrid Operation with Hybrid Method Using Game Theory Strategy. *Energies* **2021**, *14*, 603.
https://doi.org/10.3390/en14030603

**AMA Style**

Lee J-W, Kim M-K, Kim H-J.
A Multi-Agent Based Optimization Model for Microgrid Operation with Hybrid Method Using Game Theory Strategy. *Energies*. 2021; 14(3):603.
https://doi.org/10.3390/en14030603

**Chicago/Turabian Style**

Lee, Ji-Won, Mun-Kyeom Kim, and Hyung-Joon Kim.
2021. "A Multi-Agent Based Optimization Model for Microgrid Operation with Hybrid Method Using Game Theory Strategy" *Energies* 14, no. 3: 603.
https://doi.org/10.3390/en14030603