# Application of a Continuous Particle Swarm Optimization (CPSO) for the Optimal Coordination of Overcurrent Relays Considering a Penalty Method

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Overcurrent Relay Problem

_{j}is the primary pickup current and PSM stands for the Plug Setting Multiplier:

#### Constraints

_{j}should operate later than the primary relay R

_{i}[32]. This is critical for satisfying the requirement for selectivity of the primary and backup relays. The coordination constraint is defined as follows:

_{i}and T

_{j}are the operating times of the primary and backup relays, respectively, for a fault occurring in front of the primary relay. The value of the CTI could vary from 0.2 to 0.5 s, depending upon different circumstances and factors.

## 3. Continuous Particle Swarm Optimization

_{best}). The tour will find the best position among all the possible solutions and this is referred to as the “global best position” (g

_{best}). Some features of the continuous particle swarm optimization are found in the literature [38,39,40]. In the literature survey, PSO in its standard form has been widely used for unconstrained optimization projects. In this paper two modifications have been added to the authentic PSO algorithm; the penalty method and the initialization of PSO with a local search. As CPSO basically solves the unconstrained optimization problem, to convert the relay coordination problem into an unconstrained optimization problem a new objective function is defined via the penalty method. It is probable that the PSO executes such a bearded exploration that it generates immature results, which is an insufficient solution. To produce a more satisfactory solution, it is necessary to insert a local search algorithm into the original PSO. In this paper, the author inserted a local search alongside the global best position vector. The CPSO method proposed for the coordination of the overcurrent relay problem deals with each particle position on three key vectors; velocity (v

_{i}), position (x

_{i}) and open facility (y

_{i}), where v

_{i}expresses the ith velocity vector in the swarm, x

_{i}represents the ith position vector in swarm, and y

_{i}expresses the opening facilities determined based on the position vector (x

_{i}). For N number of facility problems, each particle contains N number of dimensions so the position vector x

_{i}approaches the continuous value for N facilities, x

_{i}= [x

_{i}

_{1}, x

_{i}

_{2}, …, x

_{in}], although it does not describe a candidate solution to calculate the total cost. To create a candidate solution, the position vector is reciprocated to a binary variable, y

_{i}←x

_{i}. Specifically, a discrete set is formed from the continuous set for generating a candidate solution. The fitness of the ith particle is calculated with the help of the open facility vector (y

_{i}). The personal best fitness value of the ith particle p

_{i}is expressed by fi

^{bp}. At the beginning the personal best vector is computerized with the position vector (p

_{i}= x

_{i}), where p

_{i}is the position vector and the fitness values of the personal bests are equal to the fitness of the positions, fi

^{k}= f(x

_{i}

^{k}). Then, the best particle in the whole swarm with respect to the fitness value is selected with the named global best and expressed as g

_{i}. The global best, fb

^{k}= f(y←g) can be achieved by finding the best of the personal bests over the whole swarm, fi

^{k}= min{f(x

_{i}

^{k})}, with its corresponding position vector xg which is to be used for g = xg and yg = y where yg express the y

_{i}vector of the global best. Then, the velocity of the individual particle is updated based on its personal best and the global best in the following way (10):

_{1}and c

_{2}are the inertia weight and learning factors, also known as the social and cognitive parameters respectively, while r

_{1}and r

_{2}are random numbers with limits between [0, 1]. The job of w is to control the influence of the preceding velocity on the present one. The next step is to update the positions that are given as follows:

Algorithm 1 scale equations to the same size as the rest of the text |

1. Set parameter w_{min}, w_{max}, c_{1}, c_{2} and r_{1}, r_{2} of PSO2. Initialize population of particles as having positions X and velocities V 3. Set iteration k = 1 4. Calculate fitness of particles ${F}_{i}^{k}=f({x}_{i}^{k})$ ∀i and find the index of the best particle b 5. Select $Pbes{t}_{i}^{k}={x}_{i}^{k},\forall i$ and $Gbes{t}^{k}={x}_{b}^{k}$ 6. w = w _{max} − k × (w_{max} − w_{min})/Maxite7. Update velocity and position of particles ${v}_{ik}^{(t+1)}=(w.{v}_{ik}^{t}+{c}_{1}{r}_{1}({p}_{ik}^{t}-{x}_{ik}^{t})+{c}_{2}{r}_{2}({g}_{k}^{t}-{x}_{ik}^{t}))$ ${x}_{ik}^{(t+1)}={x}_{ik}^{t}+{v}_{ik}^{t+1}$ 8. Update Pbest population 9. If ${F}_{i}^{k+1}<{F}_{i}^{k}$ then $Pbes{t}_{i}^{k+1}={x}_{i}^{k+1}$ Or else $Pbes{t}_{i}^{k+1}=Pbes{t}_{i}^{k}$ 10. If $F{b}_{b1}^{k+1}<{F}_{b}^{k}$ then $Gbest{b}^{k+1}=Pbes{t}_{b1}^{k+1}$ and set b = b1 Or else $Gbest{b}^{k+1}=Pbes{t}^{k}$ 11. If K < Maxite then K = K + 1 and go to step 6 12. End while 13. End PSO-LS Or go to step 14 14. Display optimum solution as Gbest ^{k} |

## 4. Results and Discussion

#### 4.1. Case I

_{1}and R

_{4}are non-directional while relays R

_{2}and R

_{3}have a directional feature. Two faults are taken into consideration: A and B. At bus 2 the maximum load current, including overload, is 600 A. The current transformer (CT) and plug setting ratio for each relay is 300:1 and 1, respectively. The maximum fault current is 4000 A. For each relay the minimum operating time (MOP) is 0.1 s. The primary and backup relation of the relays is shown in Table 1. Table 2 provides the detail of the a

_{ρ}constant and current seen by the relays for different fault points. In this case the total number of constraints is six; four constraints emerge as a result of the boundaries of the relay operation and two constraints emerge as a result of the coordination condition. The TMS range is 0.025–1.2. The CTI is 0.3 s. The TMSs of all four relays are x

_{1}–x

_{4}. The optimal operations of the relays obtained by the proposed algorithm are given in Table 3, which also provides the comparative results of the proposed algorithm with a previously published algorithm described in the literature. According to Table 3, the proposed algorithm is a better solution for the current case.

_{1}, c

_{2}) is 2. In addition, r

_{1}and r

_{2}are between [0, 1]. As can be seen in Table 3, the proposed method works and it performs better as compared to other evolutionary techniques. The proposed algorithm gives an optimal solution and lower total operating time $({{\displaystyle \sum}}^{\text{}}{T}_{op})$ and can solve the overcurrent relay problem faster and in a superior way. The values of the TMSs obtained are found to satisfy all the constraints. They give the minimum operating time of the relays for any fault location and also ensure proper coordination. Figure 4 depicts the graphical representation of the optimized Time Multiplier Setting in the literature.

#### 4.2. Case II

_{1}–x

_{5}.

_{1}to initiate its operation is lowest for a fault at point A (0.214 s) and will require extra time for a fault at point D (0.43 s) and E (0.3 s). Hence, at fault point A the relay R

_{1}will operate first while at fault points D and E, relays R

_{4}and R

_{5}should operate first. If the expected relay fails to activate, then relay R

_{1}should take over the tripping action. The graphical representation of the optimized TMS is shown in Figure 7, and demonstrates that the TMS is optimized up to the optimum value. Figure 8 shows the convergence characteristic graph obtained during the simulation. According to Table 8 and Table 9, the proposed method finds a better solution for this case.

#### 4.3. Case III

_{3}to relay R

_{2}while for the fault at B the backup will be provided by relay R

_{1}to R

_{4}, and for the fault at C back up will be provided by R

_{1}, R

_{3}to R

_{5}. In this case, the total number of constraints is nine; five constraints arise as a result of the boundaries of the relay operation and four constraints emerge as a result of the coordination condition. The MOP of each relay is 0.1 s. The CTI is 0.2 s. The TMSs of all the relays is x

_{1}–x

_{5}. The currents seen by the relays and ${a}_{\rho}$ constants for different fault locations are given in Table 10.

_{1}is first to operate, whereas for the fault at point B relay R

_{4}will operate, and for the fault at point C the relay R

_{5}should get the first chance to operate. The total net gain in time achieved by the proposed algorithm is tabulated in Table 12.

#### 4.4. Case IV

_{1}which is 0.027. The CTI is 0.3 s. The TMSs of all six relays are x

_{1}–x

_{6}. The optimal operation of the relays as achieved by the proposed algorithm is given in Table 17, which also provides the comparative results of the proposed algorithm with a previous optimization algorithm explained in the literature. According to Table 18, the proposed algorithm achieves a better solution for this case.

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Fault Point | Primary Relay | Backup Relay |
---|---|---|

A | 2 | 4 |

B | 3 | 1 |

Fault Point | 1 | 2 | 3 | 4 | |
---|---|---|---|---|---|

A | I_{relay} | 10 | 3.33 | - | 3.33 |

${a}_{\rho}$ | 2.97 | 5.749 | - | 5.749 | |

B | I_{relay} | 3.33 | - | 3.33 | 10 |

${a}_{\rho}$ | 5.749 | - | 5.749 | 2.97 |

TMS | GA 1 [42] | GA 2 [42] | SM [42] | DSM [43] | CPSO |
---|---|---|---|---|---|

TMS 1 | 0.081 | 0.168 | 0.07718 | 0.15 | 0.078 |

TMS 2 | 0.025 | 0.0250 | 0.0250 | 0.041 | 0.0250 |

TMS 3 | 0.025 | 0.0250 | 0.0250 | 0.041 | 0.0250 |

TMS 4 | 0.081 | 0.168 | 0.07718 | 0.15 | 0.078 |

T_{op} z (s) | 1.70 | 3.23 | 1.64 | 3.09 | 1.65 |

**Table 4.**Comparison of the total net gain in time achieved by the proposed algorithm with the literature for Case I.

Net Gain | CPSO/GA | CPSO/GA | CPSO/DSM |
---|---|---|---|

∑∆(t)s | 0.05 | 1.58 | 1.44 |

Fault Point | Primary Relay | Backup Relay |
---|---|---|

A | 1 | - |

B | 3 | - |

C | 1, 2 | -, 3 |

D | 3, 4 | -, 1 |

E | 5 | 1, 3 |

Relay | CT Ratio | Plug Setting |
---|---|---|

1 | 300/1 | 1 |

2 | 300/1 | 1 |

3 | 300/1 | 1 |

4 | 300/1 | 1 |

5 | 100/1 | 1 |

Fault Point | Relay | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | ||

A | I_{relay} | 42.34 | - | - | - | - |

${\mathit{a}}_{\mathbf{\rho}}$ | 1.799 | - | - | - | - | |

B | I_{relay} | - | 42.34 | - | - | - |

${\mathit{a}}_{\mathbf{\rho}}$ | - | 1.799 | - | - | - | |

C | I_{relay} | 4.876 | 4.876 | 4.876 | - | - |

${\mathit{a}}_{\mathbf{\rho}}$ | 4.348 | 4.348 | 4.348 | - | - | |

D | I_{relay} | 4.876 | - | 4.876 | 4.876 | - |

${\mathit{a}}_{\mathbf{\rho}}$ | 4.348 | - | 4.348 | 4.348 | - | |

E | I_{relay} | 4.876 | - | 4.876 | - | 29.25 |

${\mathit{a}}_{\mathbf{\rho}}$ | 4.348 | - | 4.348 | - | 2.004 |

TMS | CGA [9] | CPSO |
---|---|---|

TMS 1 | 0.08 | 0.0690 |

TMS 2 | 0.026 | 0.0230 |

TMS 3 | 0.08 | 0.0690 |

TMS 4 | 0.026 | 0.0230 |

TMS 5 | 0.052 | 0.0499 |

T_{op} (z) | 2.52 | 2.21 |

**Table 9.**Total net gain in time achieved by the proposed algorithm compared with the literature for Case II.

Net Gain | ∑∆(t)s |
---|---|

CPSO/CGA | 0.31 |

Fault Point | Relay | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | ||

A | I_{relay} | 9.059 | 3.019 | 3.019 | - | - |

${\mathit{a}}_{\mathbf{\rho}}$ | 3.106 | 6.265 | 6.265 | - | - | |

B | I_{relay} | 3.019 | - | 9.059 | 3.019 | - |

${\mathit{a}}_{\mathbf{\rho}}$ | 6.265 | - | 3.106 | 6.265 | - | |

C | I_{relay} | 4.875 | - | 4.875 | - | 29.25 |

${\mathit{a}}_{\mathbf{\rho}}$ | 4.348 | - | 4.348 | - | 2.004 |

TMS | TPSM [44] | CPSO ^{1} | FA [24] | CFA [24] | CPSO ^{2} |
---|---|---|---|---|---|

TMS 1 | 0.069 | 0.069 | 0.032 | 0.032 | 0.069 |

TMS 2 | 0.025 | 0.0160 | 0.0160 | 0.047 | 0.0160 |

TMS 3 | 0.069 | 0.069 | 0.121 | 0.091 | 0.069 |

TMS 4 | 0.025 | 0.0160 | 0.0160 | 0.0160 | 0.0160 |

TMS 5 | 0.0499 | 0.0499 | 0.104 | 0.094 | 0.0499 |

T_{op} z (s) | 2.27 | 2.17 | 1.73 | 1.63 | 0.7291 |

^{1}For the objective function mentioned in Equation (31);

^{2}for the objective function mentioned in Equation (32).

**Table 12.**Comparison of the total net gain in time achieved by the proposed algorithm compared with the literature for Case III.

Net Gain | ∑∆(t)s |
---|---|

CPSO ^{1}/TPSM | 0.10 |

CPSO ^{2}/FA | 1.01 |

CPSO ^{2}/CFA | 0.91 |

^{1}For the objective function mentioned in Equation (31);

^{2}for the objective function mentioned in Equation (32).

Line | Impedance (Ω) |
---|---|

1-2 | 0.08j1 |

2-3 | 0.08 + j1 |

1-3 | 0.16 + j2 |

Fault Point | Primary Relay | Backup Relay |
---|---|---|

A | 1, 2 | -, 4 |

B | 3, 4 | 1, 5 |

C | 5, 6 | -, 3 |

D | 3, 5 | 1, - |

Relay | CT Ratio | Plug Setting |
---|---|---|

1 | 1000/1 | 1 |

2 | 300/1 | 1 |

3 | 1000/1 | 1 |

4 | 600/1 | 1 |

5 | 600/1 | 1 |

6 | 600/1 | 1 |

Fault Point | Relay | ||||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | ||

A | I_{relay} | 6.579 | 3.13 | - | 1.565 | 1.565 | - |

${a}_{\rho}$ | 3.646 | 6.065 | - | 15.55 | 15.55 | - | |

B | I_{relay} | 2.193 | - | 2.193 | 2.193 | 2.193 | - |

${a}_{\rho}$ | 8.844 | - | 8.844 | 8.844 | 8.844 | - | |

C | I_{relay} | 1.096 | - | 1.096 | - | 5.482 | 1.827 |

${a}_{\rho}$ | 75.91 | - | 75.91 | - | 4.044 | 11.539 | |

D | I_{relay} | 1.644 | - | 1.644 | - | 2.741 | - |

${a}_{\rho}$ | 13.99 | - | 13.99 | - | 6.872 | - |

TMS | CGA [9] | FA [24] | CFA [24] | CPSO |
---|---|---|---|---|

TMS 1 | 0.0765 | 0.027 | 0.027 | 0.0589 |

TMS 2 | 0.034 | 0.130 | 0.221 | 0.0250 |

TMS 3 | 0.0339 | 0.025 | 0.025 | 0.0250 |

TMS 4 | 0.036 | 0.025 | 0.025 | 0.0290 |

TMS 5 | 0.0711 | 0.489 | 0.363 | 0.0630 |

TMS 6 | 0.0294 | 0.0285 | 0.029 | 0.0250 |

T_{op} z (s) | 15.88 | 16.25 | 14.39 | 11.87 |

**Table 19.**Comparison of the total net gain in time achieved by the proposed algorithm compared with the literature for Case IV.

Net Gain | ∑∆(t)s |
---|---|

CPSO/CGA | 3.242 |

CPSO/FA | 4.348 |

CPSO/CFA | 2.82 |

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**MDPI and ACS Style**

Wadood, A.; Kim, C.-H.; Khurshiad, T.; Farkoush, S.G.; Rhee, S.-B. Application of a Continuous Particle Swarm Optimization (CPSO) for the Optimal Coordination of Overcurrent Relays Considering a Penalty Method. *Energies* **2018**, *11*, 869.
https://doi.org/10.3390/en11040869

**AMA Style**

Wadood A, Kim C-H, Khurshiad T, Farkoush SG, Rhee S-B. Application of a Continuous Particle Swarm Optimization (CPSO) for the Optimal Coordination of Overcurrent Relays Considering a Penalty Method. *Energies*. 2018; 11(4):869.
https://doi.org/10.3390/en11040869

**Chicago/Turabian Style**

Wadood, Abdul, Chang-Hwan Kim, Tahir Khurshiad, Saeid Gholami Farkoush, and Sang-Bong Rhee. 2018. "Application of a Continuous Particle Swarm Optimization (CPSO) for the Optimal Coordination of Overcurrent Relays Considering a Penalty Method" *Energies* 11, no. 4: 869.
https://doi.org/10.3390/en11040869