# Multi-Objective Based Optimal Energy Management of Grid-Connected Microgrid Considering Advanced Demand Response

^{*}

## Abstract

**:**

## 1. Introduction

- The proposed optimal energy management implementing CBIDR could improve the reliability of the peak load when compared to the existing IDR strategy and increase the economic benefits of the MG, since the operating costs, utility benefit, and the stability of the peak load are effectively considered simultaneously. Therefore, the optimal operating solution for the MG is more reasonable and feasible.
- A peak load shaving factor (PLSF) was adopted to present the effectiveness of our proposed strategy in peak load curtailment. The proposed factor helps the MG operator to determine conditions when decision-making, during which DR strategy is more advantageous in peak load reduction and thus enhances the reliability.
- A confidence-based velocity-controlled PSO (CVCPSO) with the recommended fuzzy-clustering technique was also formulated in the proposed approach for the MG system. By using CVCPSO, the solution quality and diversity of the optimal Pareto set was improved with respect to the conventional PSO and the best compromise solution can be obtained through the fuzzy-clustering technique.

## 2. Overall Scheme of Grid-Connected Microgrid

#### 2.1. Grid-Connected MG Model

#### 2.2. Modeling of WT

_{r}), rated speed (v), cut-in speed (v

_{cut-in}), and cut-out speed (v

_{cut-out}) of the WT.

#### 2.3. Modeling of PV

_{s}) and size (A) of the PV panels as well as the solar irradiation (SI). Furthermore, β is generally denoted as a negative percentage per Kelvin or degree Celsius. The total output solar power (P

_{ST}) can be calculated as

_{S}represents the number of solar generators.

#### 2.4. Modeling of DE and Load

## 3. Proposed Demand Response Strategy

#### 3.1. Confidence-Based IDR

- Customer incentive

^{k}

^{−1}is a function of the peak intensity, where higher incentives will be paid to the customer for load reduction during the peak period. Figure 2 compares the CBIDR and conventional IDR in terms of the incentive payment scheme. When the amount of load reduction is the same (D

^{k}to D

^{k}

^{−1}, D

^{k}

^{−n}to D

^{k}

^{−(n−1)}, and D

^{3}to D

^{2}) as x

_{t}

^{k}

^{−n}, the incentives (ρ

^{k}

^{−1}, ρ

^{k}

^{−n}, and ρ

^{2}for peak, off-peak, and valley period, respectively) paid to the customers vary with the period in which the DR event occurs. As shown in Figure 2, the incentive payment ρ

^{k}

^{−1}is higher for the same load reduction when compared with the off-peak and valley periods. This concept is an advantage of the CBIDR over the general IDR, where there is no such distinction between the incentive payments over the periods. It must be improved by considering incentive payments as a function of the peak intensity. This means that customers participating in the CBIDR program will earn higher incentive payments during the peak period than during the off-peak or valley periods. Higher incentives for peak reduction can motivate customers to take part in the DR program in the peak period. To this end, the customers must be more willing to decrease their load in the peak period with respect to the conventional IDR program.

- Customer cost function

_{2}x

_{t}

^{k}θ

_{j}term is included to ensure that different values of θ

_{j}leads to different values of customer marginal cost. Equation (6) represents the monetary loss incurred by load reduction x

_{t}by a customer of type θ

_{j}.

- Customer benefit function

_{t}must be equal to or exceed zero in order to encourage DR action.

- Utility benefit function

- Load shedding constraints

_{t}

^{k}− D

_{t}

^{k,min}. Equation (10) represents a constraint for k = 2, 3, …, K, such that the load reduction in step k for time t must be less than D

_{t}

^{k}− D

_{t}

^{k}

^{−1}and greater than zero. Equation (11) represents the total reduction for all the participants for time t. Equation (12) ensures that the customers’ net benefit exceeds zero. Equation (13) implies that the customers should be appropriately compensated for their load reduction.

#### 3.2. Peak Load Shaving Factor

_{W}and APLF

_{WO}denote the average to peak ratio (APLF) with and without DR, respectively. PLSF is the ratio between APLF

_{W}and APLF

_{WO}. This index represents the peak load reduction without DR. APLF can be defined as

_{W}will increase and PLSF will improve accordingly. The higher the value of PLSF, the greater the enhancement of the system reliability during the peak period.

_{t}reaches zero. It represents the effectiveness in terms of peak load reduction, which can be analyzed as follows:

- Condition 1. PLSF > 1.
- Condition 2. PLSF < 1.
- Condition 3. PLSF = 1.

_{W}is greater than APLF

_{WO}, the value of PLSF will be greater than 1. This represents the amount of peak demand shaving achieved without DR.

_{W}is less than APLF

_{W}, the value of PLSF is less than 1. This represents the reduction in APLF without DR.

## 4. Multi-Objective Optimal Formulation

#### 4.1. Objective Function

_{1}(x), which includes the DE fuel cost, transaction cost, and pollutant treatment cost, by maximum use of renewable energies, and (ii) maximizing the utility benefit function f

_{2}(x) by implementing DR.

#### 4.1.1. Minimization of Operating Cost Function: f_{1}(x)

_{t}between the main grid and the MG [18].

#### 4.1.2. Maximization of Utility Benefit Function: f_{2}(x)

_{2}(x) as follows:

#### 4.2. Constraints

- Power balance constraints

- Generation limit constraints for DE, WT, and PV

- Transmission power constraints

_{tr}

^{max}).

- Demand response constraints

_{2}(x). Equations (27) and (28) can be substituted for Equations (12) and (13) over a day rather than a single time interval. Equation (27) ensures that a customer’s daily total incentives are greater than or equal to zero. Equation (28) implies that the greater the customers’ power consumption reduction, the greater the remuneration that they receive over the same period T. Equation (29) implies that the daily total incentives paid to the customer by the utility should be less than UTDB. Equation (30) ensures that the total daily power reduced by each customer does not exceed CM

_{j}.

## 5. Solution Method

#### 5.1. Confidence-Based Velocity-Controlled PSO

_{i}= (x

_{i}

_{1}, x

_{i}

_{2}, ⋯, x

_{iN}), V

_{i}= (v

_{i}

_{1}, v

_{i}

_{2}, ⋯, v

_{iN}), and P

_{g}= (p

_{g}

_{1}, p

_{g}

_{2}, ⋯, p

_{gN}), respectively. The position and velocity of each particle will change according to the best and global positions. At each time step, the updated velocity and position of each particle is given by the following expression:

_{1}and c

_{2}indicate the cognitive and social learning rates ranging from 0 to 1, respectively. In addition, r

_{1}and r

_{2}are random numbers ranging from 0 to 1, respectively.

_{1}, c

_{1}, c

_{2}) that are fixed. However, they should be adjusted several times to obtain the desired value. To overcome the limitations of a conventional PSO, we proposed the CVCPSO to improve the solution quality of the algorithm as follows [29,30]:

_{1}, is a critical constant that affects the convergence speed and performance of the algorithm. Therefore, the inertia weights should be set judiciously by considering a particle’s maximum movement distance to optimize its local and global exploration capabilities. In general, there is a trade-off between a particle’s search speed and its accuracy depending on the value of w

_{1}. The higher the inertia weight, the higher the particle’s search speed and the lower its search accuracy. According to the characteristic of the inertia value, Equation (33) implies that, instead of being fixed, the inertia weight should vary with time, decreasing linearly or non-linearly, to achieve better performance. Therefore, given the characteristic of the inertia weight above-mentioned, Equation (33) can simultaneously guarantee high search accuracy and convergence speed.

_{1}(m) and c

_{2}(m), given by

_{1}(m) decreases linearly and c

_{2}(m) increases linearly to achieve the global search ability in the early stages of iteration and local optimization ability in the later stages of iteration. Here, c

_{1,initial}, c

_{1,final}, c

_{2,initial}, and c

_{2,fianal}are the initial and final values of c

_{1}(m) and c

_{2}(m), which were set to 2.5, 0.5, 0.5, and 2.5, respectively [31].

_{2}and random variable r

_{3}. Accordingly, the particle’s trust differs in each generation (i.e., the effect of g

_{best,i,n}(b) varies). This improvement can reduce the density of the particles and thus maintain the particles’ diversity. To minimize the running time of the algorithm, we set w

_{1}= w

_{2}.

#### 5.2. Fuzzy-Clustering Technique

_{1}and x

_{2}, where x

_{1}= [x

_{1,1}, x

_{1,2}

_{⋯}x

_{1,y}], x

_{2}= [x

_{2,1}, x

_{2,2}

_{⋯}x

_{2,y}], and y is the number of objective functions. In a maximization problem, a solution x

_{2}dominates x

_{1}if neither of the following two constraints is violated:

_{2}dominates x

_{1}. A vector x

_{2}is called a non-dominated local set and the solutions that are non-dominated among the entire search space are called the non-dominated global set or the Pareto-optimal set. Accordingly, a multi-objective optimization problem leads to a set of optimal solutions called the Pareto-optimal set. In this study, a fuzzy-clustering technique [32] was used to extract the best solution that had the maximum value of the fuzzy membership function and provide it to the decision maker. The optimal-Pareto solution sets were converted into fuzzy membership functions as follows:

_{y}(a), considering all the objective functions, can be calculated as

#### 5.3. Solution Procedure

_{WO}, according to Equation (15).

_{1}, w

_{2}, c

_{1}(1), c

_{2}(1), B, NP, S, and U) required to initialize the algorithm.

_{1}, w

_{2}, c

_{1}(m), and c

_{2}(m) according to Equations (33)–(35).

_{W}and PLSF.

## 6. Simulation Results

_{2}(x), Equation (20), into the minimization problem by multiplying it with (−1). CVCPSO is used to calculate the fitness values of the objective functions individually to search for the Pareto-optimal solution set with the trade-off characteristic. Then, the fuzzy-clustering technique was adopted to obtain the best result among the Pareto-optimal set. Conventional PSO, velocity-controlled PSO (VCPSO) [24], and the proposed CVCPSO algorithm were compared to show the superiority of our energy management approach. The simulations were performed in MATLAB R2017b on a computer with the following specifications: CPU, 3.4 GHz; RAM, 8 GB; operating system, Windows 10 Pro 64-bit.

#### 6.1. MG Test System 1

_{W}decreased with respect to the APLF

_{WO}. This indicates a smaller contribution to the peak load curtailment when implementing the DR event, and the value of PLSF must be less than one. In contrast, the value of APLF

_{W}in Case 2 increased up to 22.97, which represents better performance in terms of peak restriction. Here, the value of PLSF must be greater than one. As shown in Figure 6, the amount of peak restriction was greater in Case 2, and the value of APLF

_{W}was higher accordingly. These quantitative results showed that the proposed IDR strategy provides more reliable operational conditions by reducing the load in the peak period.

_{tr}) between the main grid and the MG for MG test system 1. Here, we assumed that the renewable energies generated the maximum power output. In general, before the renewable energies are introduced, power is usually generated from the DEs or purchased from the main grid in both cases. However, when the renewable sources generate the maximum output power, the MG starts decreasing the DE output power or purchases less power from the main grid. Note that the power generated from the DE varied with the amount of DR for each case. In Figure 7a, DE 1 and 2 generated nearly the maximum power through the entire period, while the power generation by DE 3 varied with the amount of DR. As shown in Figure 7b, in the range of 8–18 h, the amount of power generated from the DEs was reduced when compared to Case 1. By reducing the load in the peak period and operating with minimal DE generation, the MG operator can alleviate the risk of the peak load and thus protect the MG from instability, system collapse, and other precarious situations such as loss of DE, renewable energies, or the main grid.

#### 6.2. MG Test System 2

_{W}value was greater than that of Case 1. Finally, this advanced IDR strategy led to an improvement in the peak period stability of the large-scale MG.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Variables | |

ρ^{k−1}_{t} | Incentive payments at time t for k − 1 step |

x^{k}^{−1}_{t} | Amount of DR at time t for k − 1step |

D^{k}^{−1}_{t} | Amount of load at time t for k − 1step |

y^{k}^{−1}_{t} | Customer incentive at time t for k − 1 step |

λ_{t} | Power interruptibility |

Bc,t | Customer benefit function |

Bu,t | Utility benefit function |

x_{Total} | Total reduction for all customers |

d_{peak} | Peak load without DR |

d’_{peak} | Peak load with DR |

d’_{t} | Load at t with DR |

d_{t} | Load at t without DR |

z | Number of DE |

t | time |

j | Number of customers |

k | Number of steps |

i | Number of particles |

n | Dimension of particle |

b | Current iteration number |

y | Number of objective functions |

a | Possible compromise solution among Pareto set |

${\gamma}_{t}$ | Power exchange rate at time t |

e | Pollutant type |

C_{e} | Treatment cost for type e |

δ_{z,e} | Pollutant emission coefficient of z DE for e type |

δ_{grid,e} | Pollutant emission coefficient of main grid for e type |

$c({\theta}_{j},{x}_{t}^{k})$ | Customer cost function |

f_{1}(x) | Operating cost function |

f_{2}(x) | Utility benefit function |

P_{WT} | Output power of WT |

P_{PV} | Output power of PV |

P_{z}^{min} | Minimum DE power |

P_{z}^{max} | Maximum DE power |

P_{z}(t) | DE power output at time t |

P_{tr}(t) | Transaction power at time t |

F_{Z}(P_{Z}(t)) | DE fuel cost function |

C_{tr}(P_{tr}(t)) | Transaction cost function |

C_{p}(P_{Z}(t)) | Pollutant treatment cost function |

v_{i,n}(b) | Current velocity of ith particle of n dimension for b iteration. |

x_{i,n}(b) | Current position of ith particle of n dimension for b iteration. |

c_{1}(m) | Acceleration parameter |

c_{2}(m) | Cognitive parameters |

r_{1}, r_{2}, r_{3} | Random number |

p_{best,i,n}(b) | Current optimal solution of ith particle of n dimension for b iteration |

g_{best,i,n}(b) | Current optimal solution of entire population |

w_{1} | Inertia weight of PSO |

w_{2} | Inertia weight of confidence term |

f^{min}_{y} | Minimum value of yth single objective function |

f^{max}_{y} | Maximum value of yth single objective function |

f_{y}(a) | Value of yth single objective function |

μ_{y}(a) | Satisfactory degree at yth single objective |

μ(a) | Overall satisfying degree |

Parameters | |

α | Power law exponent |

v_{1}, v_{2} | Wind speed of WT |

h_{1,}h_{2} | Reference hub height |

Pr | Rated power of WT |

v_{cut-in} | Cut-in speed of WT |

v_{cut-out} | Cut-out speed of WT |

β | Temperature coefficients of maximum power of PV |

t_{0} | External temperature of PV |

η_{s} | Efficiency of PV |

A | Size of PV |

a_{z}, b_{z} | DE fuel cost coefficient |

URz | Maximum rates of zth DE ramp up |

DRz | Maximum rates of zth DE ramp down |

P^{PV}_{max} | Maximum PV power |

P_{ST} | Total output power of PV |

P^{tr}_{max} | Maximum transactional power |

c_{1,initial} | Initial value of c_{1}(m) |

c_{1,final} | Final value of c_{1}(m) |

c_{2,initial} | Initial value of c_{2}(m) |

c_{2,final} | Final value of c_{2}(m) |

D | Total demand |

J | Total number of customers |

K | Total number of steps |

T | Total period |

U | Total number of Pareto-optimal solution sets |

S | Total number of objective functions |

NP | Maximum number of particles |

B | Maximum iteration number |

w_{y} | Coefficients of the yth objective function |

Indices | |

PLSF | Peak load shaving factor |

APLF_{W} | Average to peak ratio with DR |

APLF_{WO} | Average to peak ratio without DR |

Abbreviations | |

MG | Microgrid |

DSM | Demand side management |

DR | Demand response |

IDR | Incentive-based demand response |

CBIDR | Confidence-based incentive DR |

PSO | Particle swarm optimization |

CVCPSO | Confidence-based PSO |

ESS | Energy storage system |

RES | Renewable energy system |

PV | Photovoltaic |

WT | Wind turbine |

DE | Diesel engine |

SI | Solar irradiation |

LMP | Locational marginal prices |

UDTB | Utility’s daily total budget |

CM_{j} | Customer j’s daily interruptible limit |

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z, j | DE | Customer | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

a_{z} | b_{z} | P_{z,min} | P_{z,max} | DR_{z} | UR_{z} | K_{1,j} | K_{2,j} | θ_{j} | CM_{j} (kWh) | |

1 | 0.06 | 0.5 | 1 | 4 | 3 | 3 | 1.079 | 1.32 | 0 | 50 |

2 | 0.03 | 0.25 | 1 | 6 | 5 | 5 | 1.078 | 1.63 | 0.45 | 55 |

3 | 0.04 | 0.3 | 1 | 9 | 8 | 8 | 1.847 | 1.64 | 0.9 | 60 |

Time (h) | λ_{1,t} ($) | λ_{2,t} ($) | λ_{3,t} ($) |
---|---|---|---|

1 | 1.70 | 3.70 | 2.70 |

2 | 1.40 | 2.70 | 1.90 |

3 | 2.20 | 3.20 | 1.80 |

4 | 3.70 | 2.60 | 1.90 |

5 | 4.50 | 3.80 | 2.30 |

6 | 4.70 | 1.70 | 0.70 |

7 | 5.10 | 2.30 | 1.40 |

8 | 5.30 | 1.50 | 0.50 |

9 | 6.70 | 4.30 | 2.90 |

10 | 6.60 | 4.60 | 1.60 |

11 | 6.80 | 3.50 | 4.30 |

12 | 6.20 | 4.20 | 4.80 |

13 | 7.30 | 4.30 | 5.10 |

14 | 7.80 | 6.30 | 5.40 |

15 | 0.50 | 3.50 | 5.50 |

16 | 5.20 | 5.30 | 6.10 |

17 | 6.80 | 5.30 | 5.60 |

18 | 5.70 | 6.10 | 6.30 |

19 | 4.80 | 2.60 | 4.50 |

20 | 3.90 | 3.60 | 4.20 |

21 | 3.80 | 4.20 | 3.90 |

22 | 3.10 | 3.80 | 3.20 |

23 | 2.50 | 2.30 | 2.80 |

24 | 1.90 | 3.80 | 4.20 |

Case 1 | Case 2 | |
---|---|---|

PLSF | 0.94 | 1.12 |

APLF_{WO} | 20.55 | 20.55 |

APLF_{W} | 19.31 | 22.97 |

Algorithm | IDR | Operating Cost ($) | Utility Benefit ($) | Run Time (s) |
---|---|---|---|---|

PSO | Case 1 | 640 | 125 | 890 |

Case 2 | 543 | 109 | 888 | |

VCPSO | Case 1 | 621 | 131 | 693 |

Case 2 | 525 | 111 | 692 | |

CVCPSO | Case 1 | 568 | 147 | 452 |

Case 2 | 479 | 135 | 446 |

z | a_{z} | b_{z} | P_{z}, Min | P_{z}, Max | DR_{z} | UR_{z} |
---|---|---|---|---|---|---|

1 | 0.0007 | 23.9 | 30 | 150 | 80 | 80 |

2 | 0.00079 | 21.62 | 33 | 143 | 60 | 60 |

3 | 0.0048 | 23.23 | 27 | 120 | 60 | 60 |

5 | 0.10908 | 19.58 | 20 | 80 | 40 | 40 |

4 | 0.00056 | 17.87 | 37 | 60 | 40 | 40 |

6 | 0.00951 | 22.54 | 25 | 55 | 25 | 25 |

7 | 0.00211 | 16.51 | 20 | 30 | 10 | 10 |

j | K_{1,j} | K_{2,j} | θ_{j} | CM_{j} (MWh) |
---|---|---|---|---|

1 | 1.847 | 11.64 | 0 | 180 |

2 | 1.378 | 11.63 | 0.1734 | 230 |

3 | 1.079 | 11.32 | 0.4828 | 310 |

4 | 0.9124 | 11.5 | 0.7208 | 390 |

5 | 1.378 | 11.63 | 0.84 | 440 |

Time (h) | λ_{1,t} ($) | λ_{2,t} ($) | λ_{3,t} ($) | λ_{4,t} ($) | λ_{5,t} ($) |
---|---|---|---|---|---|

1 | 27.61 | 28.30 | 28.79 | 26.93 | 27.60 |

2 | 29.41 | 30.07 | 30.53 | 28.79 | 29.44 |

3 | 28.24 | 28.87 | 29.28 | 27.66 | 28.33 |

4 | 26.69 | 28.76 | 29.28 | 27.66 | 28.32 |

5 | 29.01 | 32.24 | 32.64 | 31.20 | 31.66 |

6 | 33.96 | 36.67 | 37.15 | 35.38 | 35.99 |

7 | 83.97 | 89.46 | 90.65 | 85.71 | 87.70 |

8 | 81.10 | 82.88 | 83.79 | 79.06 | 81.06 |

9 | 110.60 | 112.93 | 114.11 | 107.72 | 110.44 |

10 | 74.12 | 75.43 | 76.09 | 72.40 | 73.95 |

11 | 78.95 | 80.19 | 80.65 | 77.29 | 78.93 |

12 | 66.85 | 67.55 | 67.76 | 65.75 | 66.67 |

13 | 47.98 | 48.58 | 48.63 | 47.10 | 47.93 |

14 | 66.82 | 67.74 | 68.07 | 65.55 | 66.74 |

15 | 48.50 | 49.35 | 49.69 | 47.41 | 48.47 |

16 | 49.21 | 50.28 | 50.87 | 49.94 | 49.19 |

17 | 66.65 | 69.36 | 70.29 | 66.05 | 67.71 |

18 | 61.49 | 66.57 | 67.19 | 59.69 | 66.24 |

19 | 56.19 | 57.67 | 58.25 | 54.48 | 56.53 |

20 | 57.92 | 59.38 | 59.98 | 55.58 | 57.98 |

21 | 49.16 | 49.86 | 50.36 | 48.31 | 48.96 |

22 | 54.00 | 54.38 | 54.84 | 53.46 | 53.63 |

23 | 34.37 | 34.67 | 34.96 | 33.98 | 34.21 |

24 | 30.30 | 30.71 | 31.00 | 29.89 | 30.20 |

Case 1 | Case 2 | |
---|---|---|

PLSF | 1.03 | 1.05 |

APLF_{WO} | 16.93 | 16.93 |

APLF_{W} | 17.51 | 17.87 |

Algorithm | IDR | Operating Cost ($) | Utility Benefit ($) | Run Time (s) |
---|---|---|---|---|

PSO | Case 1 | 167,662 | 3246 | 1275 |

Case 2 | 165,571 | 2901 | 1276 | |

VCPSO | Case 1 | 166,971 | 3379 | 877 |

Case 2 | 165,454 | 2998 | 873 | |

CVCPSO | Case 1 | 166,287 | 3758 | 652 |

Case 2 | 164,001 | 3420 | 650 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kim, H.-J.; Kim, M.-K.
Multi-Objective Based Optimal Energy Management of Grid-Connected Microgrid Considering Advanced Demand Response. *Energies* **2019**, *12*, 4142.
https://doi.org/10.3390/en12214142

**AMA Style**

Kim H-J, Kim M-K.
Multi-Objective Based Optimal Energy Management of Grid-Connected Microgrid Considering Advanced Demand Response. *Energies*. 2019; 12(21):4142.
https://doi.org/10.3390/en12214142

**Chicago/Turabian Style**

Kim, Hyung-Joon, and Mun-Kyeom Kim.
2019. "Multi-Objective Based Optimal Energy Management of Grid-Connected Microgrid Considering Advanced Demand Response" *Energies* 12, no. 21: 4142.
https://doi.org/10.3390/en12214142