# An Effective Algorithm for MAED Problems with a New Reliability Model at the Microgrid

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## Abstract

**:**

## 1. Introduction

## 2. MAED Optimization Problems

#### 2.1. Objective Functions

#### 2.1.1. Minimizing Operational Cost Considering Reliability Issues

- 1:
- ${F}_{n}({P}_{n})=\left(\right)open="\{">\begin{array}{l}{a}_{n1}{P}_{n}^{2}+{b}_{n1}{P}_{n}+{c}_{n1}+\left|{e}_{n1}\times \mathrm{sin}({f}_{n1}\times ({P}_{n,\mathrm{min}}-{P}_{n}))\right|,\text{}fuel1,\text{}{P}_{n,\mathrm{min}}\le {P}_{n}\le {P}_{n1}\hfill \\ \dots \hfill \\ {a}_{nk}{P}_{n}^{2}+{b}_{nk}{P}_{n}^{}+{c}_{nk}+\left|{e}_{nk}\times \mathrm{sin}({f}_{nk}\times ({P}_{n,\mathrm{min}}-{P}_{n}^{}))\right|,\text{}fuel\text{}k,\text{}{P}_{nk-1}\le {P}_{n}\le {P}_{nk}\hfill \\ \dots \hfill \\ {a}_{nk}{P}_{n}^{2}+{b}_{nk}{P}_{n}^{}+{c}_{nk}+\left|{e}_{nk}\times \mathrm{sin}({f}_{nk}\times ({P}_{n,\mathrm{min}}-{P}_{n}^{}))\right|,\text{}fuel\text{}k,\text{}{P}_{nk-1}\le {P}_{n}\le {P}_{n,\mathrm{max}}\hfill \end{array}$
- 2:
- n is the index of available generation units and N is the number of available generation units.
- 3:
- k is the index fuel type and K is the number of fuel types.
- 4:
- P
_{n}is the output power of the nth unit and P_{n,}_{max}and P_{n,}_{min}are maximum and minimum output power limits of the nth unit, respectively. - 5:
- ${a}_{nk}{P}_{n}^{2}+{b}_{nk}{P}_{n}+{C}_{nk}$ is a quadratic generation cost function for fuel type k of the nth unit.
- 6:
- a
_{nk}, b_{nk}, and c_{nk}are cost function coefficients of the nth unit for fuel type k. - 7:
- $\left|{e}_{nk}\times \mathrm{sin}({f}_{nk}\times ({P}_{n,\mathrm{min}}-{P}_{n}^{}))\right|$ is sinusoidal and the non-smooth fuel cost function due to the VPL effects for fuel type k of the n
^{th}unit. - 8:
- e
_{nk}and f_{nk}are cost function coefficients of the VPL effects model of the nth unit for fuel type k.

_{j}is the cost function associated with the jth line, and T

_{j}is the active power flow through the jth line.

#### 2.1.2. Minimizing Emissions

- 1:
- ${E}_{n}({P}_{n})=\left(\right)open="\{">\begin{array}{l}{a}_{n1}{P}_{n}^{2}+{\beta}_{n1}{P}_{n}+{\gamma}_{n1},\text{}fuel1,\text{}{P}_{n,\mathrm{min}}\le {P}_{n}\le {P}_{n1}\hfill \\ \dots \hfill \\ {a}_{nk}{P}_{n}^{2}+{\beta}_{n1}{P}_{n}^{}+{\gamma}_{n1},\text{}fuel\text{}k,\text{}{P}_{nk-1}\le {P}_{n}\le {P}_{nk}\hfill \\ \dots \hfill \\ {a}_{nk}{P}_{n}^{2}+{\beta}_{nk}{P}_{n}^{}+{\gamma}_{nk},\text{}fuel\text{}k,\text{}{P}_{nk-1}\le {P}_{n}\le {P}_{n,\mathrm{max}}\hfill \end{array}$
- 2:
- ${a}_{nk}{P}_{n}^{2}+{\beta}_{nk}{P}_{n}^{}+{\gamma}_{nk}$ is emission generated by the nth unit for fuel type k.
- 3:
- a
_{nk}, β_{nk}, and γ_{nk}are the emission coefficients of the nth unit for fuel type k.

#### 2.2. Constraints

#### 2.2.1. Area Total Active Power Balance

_{q}is the number of committed generating units for the qth area, P

_{Loadq}is the power demand in the qth area, and M

_{q}is the set of all areas connected to the qth area via a tie-line.

#### 2.2.2. Generator Output Power Limits

#### 2.2.3. Ramp-Rate Limits

_{n}and Un

_{i}are ramp-up and ramp-down rate limits of the nth thermal generator, respectively. This constraint determines the lower and upper bounds of the objective variables.

#### 2.2.4. Prohibited Operating Zones (POZ) Due to Physical Operational Limitations

#### 2.2.5. Maximum and Minimum Power Transfer Through Tie-Lines

_{qw}) must not violate the maximum tie-line power transfer capacity limit (${T}_{qw,\mathrm{max}}$) [8].

#### 2.2.6. Spinning Reserve (SR) Requirement in Each Area

#### 2.2.7. Limitation on Power Transfers Considering SR Contribution

_{Load}is the entire real load demand in the system. g

_{1}is max(|T

_{qw}| − T

_{qw}

_{,max}, 0) in MAED problem and max(max{|T

_{qw}| − T

_{qw}+ RC

_{qw}} − T

_{qw}

_{,max}, 0) in RCMAED and RCMAEED problems. g

_{2}is 0 for the MAED problem and $\sum _{q=1}^{NA}\mathrm{max}({S}_{q,req}-{\displaystyle \sum _{n=1}^{{N}_{q}}{S}_{nq}-{\displaystyle \sum _{w\in {M}_{q}}R{C}_{wq},0),\text{}w\in {M}_{q}}}$ for the RCMAED and RCMAEED problems (NA denotes the number of areas).

## 3. Phasor Particle Swarm Optimization (PPSO) Technique

#### 3.1. Background of Different Variants of PSO

_{i}, and the best position all particles have experienced so far is the best global position vector, Gbest.

_{1}and c

_{2}are acceleration control coefficients, which can be chosen by the designer, r1

_{id}and r2

_{id}are uniform random coefficients in the range of (0, 1), and Iter is the number of the current iteration.

_{i}, are constrained to the range defined to prevent particles [8] from travelling out of the issue search room.

_{max}(= ω

^{iter}

^{=1}) (initial value) to ω

_{min}(= ω

^{iter}

^{max}) (final value) during the optimization process as follows [23]:

_{max}and ω

_{min}values, respectively.

_{1}= c

_{2}= 2.05, where the control parameter χ is set to 0.729 using Equation (28) [38]:

#### 3.2. Parameter Setting in PPSO

_{i}, is defined for each particle so that, for example, the ith particle could be modelled by a magnitude vector $\overrightarrow{{X}_{i}}$ with angle θ

_{i}and represented as $\overrightarrow{{X}_{i}}\angle \theta $.

_{i}, is defined for each particle such that, for example, the ith particle could be modeled by a magnitude vector $\overrightarrow{{X}_{i}}$ with angle θ

_{i}and represented as $\overrightarrow{{X}_{i}}\angle \theta $.

#### 3.3. Flowchart of PPSO

_{Pop}random particles (initial population) $\overrightarrow{{X}_{i}}=|{X}_{i}|\text{}{\theta}_{i}$ (i = 1: N

_{Pop}) are generated in the D-dimensional space of the problem with their phasor angle θ

_{i}with uniform distribution ${\theta}_{i}^{Iter=1}=U(0,\text{}2\pi )$ and with the initial speed limit ${v}_{\mathrm{max},i}^{Iter=1}$. Then, the velocity of each particle in each iteration of the algorithm is updated with the following equation [40]:

_{best}and G

_{best}are determined, similar to the original PSO algorithm.

## 4. Results and Discussion

#### 4.1. Optimization Results

#### 4.1.1. The MAED Problems Optimization Process Using PPSO

**Step 1:**Setting the control parameters and the required data of the power system and generation units in the multi-area network.

**Step 2:**Producing the initial random phasor particle swarm of the PPSO optimizer as follows [9]:

**Step 3:**Calculating the objective function values of the MAED problem, while imposing constraints of the generation units and multi-area network.

**Step 4:**Producing a new particle phasor swarm of the PPSO optimizer using Equation (32) to (35).

**Step 5:**Calculating the objective of the MAED problem.

**Step 6:**Repeating steps 4 and 5 until the iterations are finished.

#### 4.1.2. Practical MAED Optimization Problems

_{Pop}= 80. This table represents the results of PPSO, APSO, CLPSO, SPSO2011, FPSO, FIPS, PSO-TVAC [8], Hopfield neural network (HNN) method [46], and direct search method (DSM) [47]. Table 1 shows that the minimum operation and fuel cost obtained by the PPSO optimizer was 10,604.6741 ($/H), which was less than that of HNN [46], DSM [47], and PSO-TVAC [8].

_{Pop}= 80, respectively. The best global optimal solution obtained by the proposed PPSO optimizer and the best global optimal solutions reported in the literature for the medium-scale test system are presented in Table 2. The global optimal solution to which the proposed PPSO optimizer reached is feasible ($\sum {P}_{g}$ = 1250.0 MW), but the best results reported in previous studies using other algorithms, e.g., PSO algorithms, the classical evolutionary programming (CEP) method [51], the hybrid harmony search (HHS) algorithm [48], the network flow programming (NFP) method [52], and the pattern search (PS) algorithm [53] are not feasible. Additionally, the solution obtained by the PPSO optimizer is better than that of the hybridizing sum-local search optimizer (HSLSO ) [10] algorithm.

_{Pop}= 80. Table 3 shows that the new PPSO optimizer can converge to a better-quality solution in solving a large-scale MAED problem with different practical constraints, whose cost is 125,100.2436 ($/H).

#### 4.1.3. RCMAEED and RCMAED Problems

_{Pop}= 120, respectively, and ϕ was also set to 120 for the RCMAEED optimization problem. The global optimal results for the RCMAED and RCMAEED problems obtained using all the PSO optimizers are given in Table 5 and Table 6, respectively. The tables show that the best solutions to the RCMAED and RCMAEED problems were obtained by the proposed PPSO optimizer; the optimizer was found to be superior to all other variants of PSO.

#### 4.1.4. Reliability-Oriented MAED

#### 4.2. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Indices: | |

$lp:$ | Load point index |

$n:$ | Committed generation units $\u03f5\text{}\left[1,\dots ,\text{}\mathrm{N}\right]$ |

$k:$ | Input fuel types $\u03f5\text{}\left[1,\dots ,\text{}\mathrm{K}\right]$ |

$j:$ | Transmission lines $\u03f5\text{}\left[1,\dots ,\text{}\mathrm{M}\right]$ |

$q,w:$ | Area’s index |

$d:$ | Decision variable’s index |

$D:$ | The number of decision variables |

$h:$ | The ${h}^{\mathrm{th}}$ prohibited operating zones (POZ) |

$H$/${H}^{\prime}$ | The $H/{H}^{\prime \mathrm{th}}$ available/unavailable generation units |

${M}_{q}:$ | Set of all areas which are connected to qth area |

Parameters: | |

${N}_{q}:$ | The number of committed generating units in the qth area |

$Std:$ | Standard deviation |

$zn:$ | The number of POZs in the nth thermal unit power curve |

${a}_{n},{b}_{n},{c}_{n},{e}_{n},{f}_{n}:$ | The fuel cost coefficients of nth thermal unit |

${a}_{ik},{b}_{ik},{c}_{ik},{e}_{ik},{f}_{ik}:$ | The fuel cost coefficients of nth thermal unit for kth fuel type |

$D{R}_{n}:$ | The down ramp rate-limit of nth thermal unit |

$U{R}_{n}:$ | The up ramp rate-limits of nth thermal unit |

${P}_{n}^{0}:$ | The power output of nth thermal unit in the first stage |

${P}_{n,min}:$ | The minimum power output of nth thermal unit |

${P}_{n,max}:$ | The maximum power output of nth thermal unit |

${P}_{nk,min}:$ | The minimum power output of nth thermal unit for kth fuel type |

${P}_{nk}:$ | The maximum power output of nth thermal unit for kth fuel type |

${P}_{nh}^{l}:$ | The lower bound for prohibited zone k of nth thermal unit |

${P}_{nh}^{u}:$ | The upper bound for prohibited zone k of nth thermal unit |

${P}_{Load}:$ | System total load demand |

${P}_{Loadq}:$ | The power demand in qth area |

${T}_{qw,max}:$ | The maximum capacity of the tie-line between qth and wth areas |

${\alpha}_{nk},{\beta}_{nk},{\gamma}_{nk},:$ | The emission coefficients of nth thermal unit for kth fuel type |

${S}_{q,req}:$ | The spinning reserve requirement in the qth area |

$\phi :$ | User defined weighting factor for emission cost (in this study: 120) |

$\lambda :$ | Penalty coefficient value |

${N}_{Pop}:$ | Number of initial population |

${X}_{i}:$ | The current position vector of ith particle |

${V}_{i}:$ | The current velocity vector of ith particle |

$Ite{r}_{max}:$ | Maximum number of iterations for PSO algorithm |

$Pbes{t}_{i}:$ | The best personal position vector of ith particle |

$Gbest:$ | The global best position vector |

$\theta :$ | The phase angle |

${c}_{ens}:$ | The cost of energy not supplied ($7\text{}\$/\mathrm{kWh}$) |

$EPN{S}_{lp}:$ | Expected power not supplied at ${\mathrm{lp}}^{\mathrm{th}}$ load point |

$T{D}_{lp}:$ | The time duration of ${\mathrm{lp}}^{\mathrm{th}}$ load point |

$Probabilit{y}_{lp}$ | The probability of availability and unavailability of generation |

$X$/$Y$ | The set of available (unavailable) generation units |

$F{R}_{n}$ | The failure rate of nth generator |

$R{R}_{n}$ | The repair rate of nth generator |

$MTT{R}_{n}$ | Mean time to repair nth generator |

$MTT{F}_{n}$ | Mean time to failure of nth generator |

${T}_{Failure,n}$ | Failure time of nth generator |

${T}_{Repair,n}$ | Repair time of nth generator |

${U}_{n}$ | Unavailability (force outage rate) of nth generator |

Functions and Variables: | |

${F}_{T}$ | Objective function |

${F}_{n}\left({P}_{n}\right):$ | The fuel cost function of nth thermal unit |

${P}_{n}:$ | The power output of nth thermal unit |

${f}_{j}:$ | Cost function associated with jth transmission line |

${T}_{j}:$ | Active power flow through jth transmission line |

${T}_{qw}:$ | The power flow from qth area to wth area |

${E}_{n}\left({P}_{n}\right):$ | The emission function of nth thermal unit |

$R{C}_{wq}:$ | The amount of reserve contributed between qth and wth areas |

${S}_{n}$ | The reserve provided by all thermal units in the nth area |

Abbreviations: | |

APSO: | Adaptive PSO |

CEP: | Classical evolutionary programming |

CENS | Cost of energy not supplied |

CLPSO: | Comprehensive learning PSO |

DEC2: | Chaotic DE/2 algorithm |

DSM: | Direct search method |

EENS: | Expected energy not supplied |

ELD: | Economic load dispatch |

EP: | Evolutionary programming |

FIPS: | Fully informed particle swarm |

FPSO: | Frankenstein’s PSO |

HNN: | Hopfield neural network |

HS: | Harmony search |

HSLSO: | Hybridizing sum-local search optimizer |

LOLP: | Loss of Load Probability |

MAED: | Multi-area economic dispatch |

NFP: | Network flow programming |

POZ: | Prohibited operating zones |

PPSO: | Phasor particle swarm optimization |

PS: | Pattern search |

PSO: | Particle swarm optimization |

PSO-cf: | Modified PSO by constriction factor |

PSO-TVAC: | Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients |

PSO-ω: | Modified PSO by the inertia weight |

RCMAED: | Reserve constrained multi-area economic dispatch |

RCMAEED: | Reserve constrained multi-area environmental/economic dispatch |

SPSO2011: | The improved standard PSO 2011 |

SR: | Spinning reserve |

VPL: | Valve-point loading |

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Method | P_{1} (MW) | P_{2} (MW) | P_{3} (MW) | P_{4} (MW) | T_{12} (MW) | $\sum {\mathit{P}}_{\mathit{g}}$ | Cost ($/H) |
---|---|---|---|---|---|---|---|

HNN [46] | - | - | - | - | - | - | 10,605 |

DSM [47] | - | - | - | - | - | - | 10,605 |

PSO-TVAC [8] | 444.8047 | 139.1953 | 211.0609 | 324.9391 | −200 | 1120 | 10,604.68 |

PPSO | 445.1223 | 138.8778 | 212.0426 | 323.9573 | −199.9999 | 1120 | 10,604.67 |

APSO | 445.3207 | 138.6794 | 212.2054 | 323.7945 | −199.9999 | 1120 | 10,604.67 |

CLPSO | 445.1213 | 138.8788 | 212.0413 | 323.9586 | −199.9999 | 1120 | 10,604.67 |

SPSO2011 | 445.1223 | 138.8778 | 212.0426 | 323.9573 | −199.9999 | 1120 | 10,604.67 |

FPSO | 445.0654 | 138.9347 | 211.9258 | 324.0741 | −199.9999 | 1120 | 10,604.67 |

FIPS | 445.2274 | 138.7727 | 211.9977 | 324.0022 | −199.9999 | 1120 | 10,604.67 |

Area No. | PSO [10] | HHS [48] | NFP [52] | CEP [51] | PS [53] | HSLSO [10] | PPSO | |
---|---|---|---|---|---|---|---|---|

1 (400 MW) | P_{1} (MW) | 150 | 150 | 150 | 150 | 150 | 150 | 150 |

P_{2} (MW) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |

P_{3} (MW) | 67.366 | 66.86 | 66.97 | 68.826 | 66.971 | 67.3848 | 67.31016 | |

P_{4} (MW) | 100 | 100 | 100 | 99.985 | 100 | 100 | 100 | |

2 (200 MW) | P_{5} (MW) | 56.613 | 57.04 | 56.97 | 56.373 | 56.9718 | 57.0625 | 57.07953 |

P_{6} (MW) | 95.474 | 96.22 | 96.25 | 93.519 | 96.2518 | 96.1749 | 96.34877 | |

P_{7} (MW) | 41.617 | 41.74 | 41.87 | 42.546 | 41.8718 | 41.8472 | 41.86785 | |

P_{8} (MW) | 72.356 | 72.5 | 72.52 | 72.647 | 72.5218 | 72.4505 | 72.53403 | |

3 (350 MW) | P_{9} (MW) | 50 | 50 | 50 | 50 | 50.002 | 50 | 50 |

P_{10} (MW) | 35.973 | 36.24 | 36.27 | 36.399 | 36.272 | 36.319 | 36.28298 | |

P_{11} (MW) | 38.21 | 38.39 | 38.49 | 38.323 | 38.492 | 38.5911 | 38.50812 | |

P_{12} (MW) | 37.162 | 37.2 | 37.32 | 36.903 | 37.322 | 37.3719 | 37.26609 | |

4 (300 MW) | P_{13} (MW) | 150 | 150 | 150 | 150 | 150 | 150 | 150 |

P_{14} (MW) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |

P_{15} (MW) | 57.83 | 56.9 | 57.05 | 56.648 | 57.051 | 56.9272 | 56.9218 | |

P_{16} (MW) | 97.349 | 96.2 | 96.27 | 95.826 | 96.271 | 95.8709 | 95.88068 | |

Tie-line power flow | T_{12} (MW) | 0 | 0 | 0 | −0.018 | 0 | 0 | 0 |

T_{13} (MW) | 22.588 | 16.86 | 18.18 | 19.587 | 18.181 | 17.4643 | 17.42629 | |

T_{14} (MW) | −5.176 | 0 | −1.21 | −0.758 | −1.21 | −0.0795 | −0.116132 | |

T_{23} (MW) | 66.064 | 70.61 | 69.73 | 68.861 | 69.73 | 70.2537 | 70.51652 | |

T_{24} (MW) | −0.004 | −3.11 | −2.11 | −1.789 | −2.111 | −2.7186 | −2.686341 | |

T_{34} (MW) | −100 | −100 | −100 | −99.927 | −100 | −100 | −100 | |

$\sum {P}_{g}\left(MW\right)$ | 1249.95 | 1249.29 | 1249.98 | 1247.995 | 1249.998 | 1250 | 1250 | |

Cost ($/H) | 7336.93 | 7329.85 | 7337 | 7337.75 | 7336.98 | 7337.03 | 7337.026 |

Area 1 (PD = 7500 MW) | Area 2 (PD = 3000 MW) | ||||||
---|---|---|---|---|---|---|---|

Output (MW) | DEC2 [8] | HSLSO [10] | PPSO | Output (MW) | DEC2 [8] | HSLSO [10] | PPSO |

P_{1} | 112.8292 | 110.8012 | 110.8012 | P_{21} (MW) | 343.7598 | 523.2792 | 523.2794 |

P_{2} | 114 | 113.9997 | 113.9998 | P_{22} (MW) | 433.5196 | 523.2791 | 523.2794 |

P_{3} | 97.3999 | 120 | 120 | P_{23} (MW) | 523.2794 | 523.2794 | 523.2795 |

P_{4} | 179.7331 | 179.7331 | 179.7332 | P_{24} (MW) | 550 | 523.2794 | 523.2794 |

P_{5} | 97 | 95.551 | 95.5504 | P_{25} (MW) | 550 | 523.2795 | 523.2793 |

P_{6} | 68.0001 | 140 | 140 | P_{26} (MW) | 254 | 254 | 254 |

P_{7} | 300 | 300 | 300 | P_{27} (MW) | 10 | 10.0001 | 10 |

P_{8} | 284.5997 | 284.5997 | 284.5997 | P_{28} (MW) | 10.0001 | 10 | 10 |

P_{9} | 284.5997 | 284.5997 | 284.5997 | P_{29} (MW) | 10 | 10 | 10 |

P_{10} | 130 | 270 | 270 | P_{30} (MW) | 47 | 87.7997 | 87.7997 |

P_{11} | 360 | 94 | 94.0002 | P_{31} (MW) | 159.7331 | 188.5959 | 188.5954 |

P_{12} | 94.0001 | 300 | 300 | P_{32} (MW) | 190 | 159.7331 | 159.7331 |

P_{13} | 304.5196 | 304.5195 | 304.5195 | P_{33} (MW) | 163.7269 | 159.733 | 159.7331 |

P_{14} | 500 | 394.2797 | 394.2793 | P_{34} (MW) | 164.7998 | 164.8002 | 164.8 |

P_{15} | 484.0392 | 484.0395 | 484.0395 | P_{35} (MW) | 200 | 164.7998 | 164.7998 |

P_{16} | 500 | 484.0391 | 484.0391 | P_{36} (MW) | 164.7998 | 164.7998 | 164.7992 |

P_{17} | 489.2794 | 489.2794 | 489.2797 | P_{37} (MW) | 110 | 89.1143 | 89.1143 |

P_{18} | 500 | 489.2796 | 489.2794 | P_{38} (MW) | 57.0571 | 89.114 | 89.1142 |

P_{19} | 550 | 549.9998 | 549.9998 | P_{39} (MW) | 25 | 89.1134 | 89.1142 |

P_{20} | 550 | 511.2791 | 511.2794 | P_{40} (MW) | 511.2794 | 242.0001 | 242 |

T_{12} (MW) | −1500 | −1500 | −1500 | Cost ($/H) | 127,344.9 | 125,100.3 | 125,100.2 |

Test System | Index | FIPS | FPSO | SPSO2011 | CLPSO | APSO | PPSO |
---|---|---|---|---|---|---|---|

small-scale system | Best | 10,604.6742 | 10,604.67 | 10,604.6741 | 10,604.67 | 10,604.67 | 10,604.67 |

Mean | 10,605.3272 | 10,604.92 | 10,604.8543 | 10,604.68 | 10,604.73 | 10,604.67 | |

Std | 1.5275 | 1.1547 | 0.5774 | 0.7022 | 0.4407 | 5.75 × 10^{−5} | |

Mean time (s) | 4.56 | 4.78 | 4 | 3.16 | 6.82 | 2.93 | |

medium-scale system | Best | 7341.7942 | 7340.455 | 7340.2795 | 7344.357 | 7341.714 | 7337.026 |

Mean | 7559.7788 | 7487.087 | 7637.4443 | 7486.892 | 7605.919 | 7338.115 | |

Std | 74.2674 | 61.8126 | 71.271 | 84.3494 | 53.07 | 0.629 | |

Mean time (s) | 20.95 | 20.67 | 19.19 | 18.31 | 25.57 | 17.84 | |

large-scale system | Best | 128,554.2844 | 128,128.2 | 127,085.5386 | 127,008.9 | 128,514 | 125,100.2 |

Mean | 130,615.4572 | 129,486 | 129,414.4588 | 128,315.4 | 129,495.4 | 125,263.2 | |

Std | 1.19 × 10^{3} | 1.02 × 10^{3} | 9.84 × 10^{2} | 2.16 × 10^{2} | 6.93 × 10^{2} | 85.3092 | |

Mean time (s) | 54.74 | 55.85 | 48.51 | 48.35 | 75.3 | 47.88 |

**Table 5.**Optimization results obtained by the PSO optimizers for the RCMAED problem for the four-area power system.

Output (MW) | FIPS | FPSO | SPSO2011 | CLPSO | APSO | PPSO |
---|---|---|---|---|---|---|

P_{1} | 8.5605 | 8.7146 | 8.1347 | 9.9192 | 8.468 | 11.1868 |

P_{2} | 9.9835 | 10 | 8.011 | 7.259 | 8.011 | 9.9596 |

P_{3} | 11.26 | 12.3534 | 13 | 8.7913 | 12.7035 | 6.6997 |

P_{4} | 0.2616 | 0.05 | 1.9639 | 4.8491 | 1.605 | 2.982 |

P_{5} | 18.807 | 22.1526 | 19.5629 | 23.6046 | 20.1371 | 22.8244 |

P_{6} | 11.261 | 9.4047 | 10.9592 | 9.2998 | 8.8157 | 8.8553 |

P_{7} | 6.1084 | 3.833 | 2.1895 | 3.9163 | 4.7532 | 3.9286 |

P_{8} | 15.2128 | 17.2551 | 17.925 | 16.3556 | 17.925 | 17.2481 |

P_{9} | 4.6622 | 6.0147 | 2.9329 | 6.1264 | 22.0733 | 5.8222 |

P_{10} | 10.5192 | 6.1858 | 8.5093 | 0.05 | 5.3173 | 0.1993 |

P_{11} | 20.4782 | 9.1548 | 23.2033 | 9.8982 | 7.4855 | 9.6888 |

P_{12} | 4.0058 | 17.2236 | 5.1026 | 22.3468 | 5.1026 | 22.8171 |

P_{13} | 11 | 8.1669 | 9.8282 | 9.1629 | 9.8282 | 8.0614 |

P_{14} | 20 | 19.7135 | 17.7366 | 19.9614 | 17.7366 | 19.8021 |

P_{15} | 26.8153 | 28.7234 | 30 | 28.1791 | 30 | 29.8714 |

P_{16} | 0.1384 | 1.2864 | 0.05 | 0.0869 | 0.05 | 0.0534 |

P_{17} | 0.1275 | 0.9452 | 0.5512 | 0.1 | 0.5512 | 0.486 |

P_{18} | 0.3016 | 0.2525 | 0.2146 | 0.2701 | 0.1433 | 0.204 |

P_{19} | 0.1 | 0.18 | 0.3539 | 0.3648 | 0.143 | 0.138 |

P_{20} | 0.1 | 1.4402 | 0.1105 | 1.4421 | 0.1105 | 1.4566 |

P_{21} | 1.1012 | 1.8441 | 1.1357 | 1.84 | 1.8756 | 1.8858 |

P_{22} | 0.1 | 0.1 | 0.2364 | 0.1034 | 0.2364 | 0.188 |

RC_{12} | 1.6842 | 2.2985 | 0.3671 | 0.6544 | 2.5336 | 1.5336 |

RC_{13} | 0.2249 | 0.1 | 0.1 | 0.8206 | 0.1 | 0.5945 |

RC_{14} | 1.8788 | 1.3163 | 2.1507 | 0.2931 | 1.05 | 0.1674 |

RC_{23} | 0.4049 | 0.2073 | 0.1 | 0.1 | 1.0058 | 0.1001 |

RC_{24} | 0.1645 | 2.4179 | 1.0183 | 1.6712 | 1.0183 | 2.1473 |

RC_{34} | 0.4582 | 0.1 | 0.602 | 0.2867 | 0.602 | 0.3759 |

Reserve area 1 | 18.9344 | 17.882 | 17.8904 | 18.1814 | 18.2125 | 18.1719 |

Reserve area 2 | 23.6108 | 22.3546 | 24.3634 | 21.8237 | 23.369 | 22.1436 |

Reserve area 3 | 80.3346 | 81.4211 | 80.2519 | 81.5786 | 80.0213 | 81.4726 |

Reserve area 4 | 33.0463 | 33.1098 | 33.3852 | 33.6097 | 33.3852 | 33.2117 |

Best Cost ($) | 2187.418 | 2178.6024 | 2188.247 | 2171.0535 | 2193.541 | 2166.377 |

Mean Cost ($) | 2700.367 | 2634.0676 | 2461.538 | 2494.3471 | 2510.7 | 2185.794 |

Std | 363.4401 | 325.6323 | 204.1498 | 182.2067 | 250.0191 | 13.7298 |

Time (s) | 69.34 | 66.06 | 54.48 | 52.27 | 77.31 | 51.93 |

**Table 6.**Optimization results obtained by PSO optimizers for the RCMAEED problem for the four-area power system.

Output (MW) | FIPS | FPSO | SPSO2011 | CLPSO | APSO | PPSO |
---|---|---|---|---|---|---|

P_{1} | 8.2773 | 10 | 8.2773 | 0.05 | 10.0006 | 10.0005 |

P_{2} | 5.444 | 5.3234 | 5.444 | 5.2116 | 5.3258 | 5.3259 |

P_{3} | 6.9757 | 7.0561 | 6.9757 | 12.3788 | 7.049 | 7.0491 |

P_{4} | 7.4634 | 11.998 | 7.4634 | 11.2161 | 11.9979 | 11.9978 |

P_{5} | 21.3822 | 9.9619 | 21.3822 | 16.8448 | 9.9143 | 12.2994 |

P_{6} | 7.3638 | 11.3019 | 7.3638 | 2.117 | 11.2907 | 11.3626 |

P_{7} | 7.6367 | 14.5624 | 12.0773 | 12.9769 | 14.5612 | 14.6209 |

P_{8} | 18 | 12.6441 | 14.0655 | 13.6205 | 12.7082 | 10.191 |

P_{9} | 17.3563 | 13.2916 | 17.3563 | 16.4181 | 13.2926 | 13.2921 |

P_{10} | 0.05 | 0.0891 | 0.05 | 0.05 | 0.0917 | 0.0919 |

P_{11} | 13.6655 | 13.0317 | 13.6655 | 3.4361 | 12.6414 | 12.618 |

P_{12} | 8.1572 | 14.2427 | 8.1572 | 19.1587 | 14.6307 | 14.6544 |

P_{13} | 9.2933 | 4.7521 | 9.2933 | 8.7209 | 4.753 | 4.7537 |

P_{14} | 12.6671 | 15.4968 | 11.2406 | 12.7486 | 15.4936 | 15.4934 |

P_{15} | 17.339 | 11.813 | 17.339 | 11.364 | 11.8142 | 11.8143 |

P_{16} | 20.0235 | 24.4354 | 20.0235 | 30 | 24.4352 | 24.435 |

P_{17} | −1.8289 | 1.19 | −1.8289 | 4.3668 | 1.1836 | 1.1836 |

P_{18} | −2.035 | −0.3652 | −2.035 | −1.7739 | −0.3653 | −0.3653 |

P_{19} | 1.7937 | 3.5526 | 1.0664 | −1.7123 | 3.555 | 3.555 |

P_{20} | 3.2182 | 0.1308 | 3.0699 | 0.6117 | 0.1288 | 0.1288 |

P_{21} | −2.1972 | −0.47 | −0.0632 | −0.925 | −0.4709 | −0.4713 |

P_{22} | 0.842 | 0.4207 | 0.842 | 0.8286 | 0.4199 | 0.4199 |

RC_{12} | −2.6122 | 0.4371 | −2.6122 | −1.1046 | 0.4376 | 0.4355 |

RC_{13} | 1.8735 | −3.5706 | 1.3435 | 0.4457 | −3.4213 | −3.396 |

RC_{14} | 3.3365 | −5.3328 | −10.8828 | −7.5176 | −5.4329 | −4.638 |

RC_{23} | −2.5379 | 0.2241 | −0.901 | 2.0858 | 0.2103 | 0.2071 |

RC_{24} | 0.6483 | −2.7136 | 0.6483 | −1.8365 | −2.0113 | −3.0807 |

RC_{34} | −0.5127 | −0.6214 | −0.5127 | −0.2932 | −0.7097 | −0.6741 |

Reserve area 1 | 20.8396 | 14.6225 | 20.8396 | 20.1435 | 14.6267 | 14.6267 |

Reserve area 2 | 20.6173 | 26.5297 | 20.1112 | 29.4408 | 26.5256 | 26.5261 |

Reserve area 3 | 80.771 | 79.3449 | 80.771 | 80.9371 | 79.3436 | 79.3436 |

Reserve area 4 | 31.6771 | 34.5027 | 33.1036 | 28.1665 | 34.504 | 34.5036 |

Cost ($) | 2197.8688 | 2185.4666 | 2194.0611 | 2189.6647 | 2185.0785 | 2184.0477 |

Emission (ton) | 3.6756 | 3.4257 | 3.5176 | 4.2518 | 3.4288 | 3.4097 |

**Table 7.**The obtained values for the proposed reliability-oriented MAED problem for the large-scale test system.

Unit | Output (MW) | Unit | Output (MW) |
---|---|---|---|

P_{1} | 114 | P_{21} | 513.66 |

P_{2} | 114 | P_{22} | 513.66 |

P_{3} | 120 | P_{23} | 513.66 |

P_{4} | 180 | P_{24} | 513.66 |

P_{5} | 97 | P_{25} | 513.66 |

P_{6} | 140 | P_{26} | 302.097 |

P_{7} | 300 | P_{27} | 10 |

P_{8} | 266 | P_{28} | 10 |

P_{9} | 266 | P_{29} | 10 |

P_{10} | 270 | P_{30} | 90 |

P_{11} | 126.2842 | P_{31} | 190 |

P_{12} | 300 | P_{32} | 188 |

P_{13} | 300 | P_{33} | 140.55 |

P_{14} | 393.66 | P_{34} | 152.1113 |

P_{15} | 482.666 | P_{35} | 148.345 |

P_{16} | 490.1 | P_{36} | 148.345 |

P_{17} | 481.76 | P_{37} | 91.55 |

P_{18} | 484.55 | P_{38} | 91.55 |

P_{19} | 550 | P_{39} | 91.55 |

P_{20} | 523.9797 | P_{40} | 267.6017 |

T12 (MW) | −1500 | ||

Operation Cost ($) | 127,893.5 | ||

CENS ($) | 7469 | ||

Total Cost | 135,362.5 |

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## Share and Cite

**MDPI and ACS Style**

Naderipour, A.; Kalam, A.; Abdul-Malek, Z.; Faraji Davoudkhani, I.; Mustafa, M.W.B.; Guerrero, J.M.
An Effective Algorithm for MAED Problems with a New Reliability Model at the Microgrid. *Electronics* **2021**, *10*, 257.
https://doi.org/10.3390/electronics10030257

**AMA Style**

Naderipour A, Kalam A, Abdul-Malek Z, Faraji Davoudkhani I, Mustafa MWB, Guerrero JM.
An Effective Algorithm for MAED Problems with a New Reliability Model at the Microgrid. *Electronics*. 2021; 10(3):257.
https://doi.org/10.3390/electronics10030257

**Chicago/Turabian Style**

Naderipour, Amirreza, Akhtar Kalam, Zulkurnain Abdul-Malek, Iraj Faraji Davoudkhani, Mohd Wazir Bin Mustafa, and Josep M. Guerrero.
2021. "An Effective Algorithm for MAED Problems with a New Reliability Model at the Microgrid" *Electronics* 10, no. 3: 257.
https://doi.org/10.3390/electronics10030257