2. Problem formulation

economic dispatch (ED) [1, 2]. With the practical constraints on the generators, finding opti-

In addition, as the occurrence of failures and faults in the power grid is inevitable, the entire power system must be protected. The relay protection scheme is designed to detect faults and isolate the faulty parts of the grid from the healthy sections in order to mitigate the consequences of the faults and maintain continuity of service. If a fault occurs, the nearest corresponding relays must operate as fast as possible to clear the fault. If due to any reason these primary relays fail to react, their backup relays must operate and accomplish the task. Directional overcurrent relays (DOCRs) are a suitable and economical protection scheme for distribution systems [3]. The protection design of DOCRs is based on two parameters, time multiplier setting (TMS) and plug setting (PS). Proper settings of TMS and PS allow a primary relay to clear the faults in its protection zone as fast as possible and in case of failure, its backup relay operates immediately after a time interval to clear the fault. TMS and PS values of each relay must be coordinated with other backup relays, where again relays act with different current settings, which make the coordination a complex task. Each pairs of relays include four variables (TMS, PS) and the complexity of coordination will be intense in bigger systems with more relays and constraints. Due to the complex interconnection of the distribution systems and also nonlinear characteristics of operation time of relays, finding best relay settings could

Considering the non-convex and nonlinear nature of these problems, traditional methods fail to feasibly or optimally solve them. Therefore, evolutionary algorithms have gained more attentions as solutions to such optimization problems. Some of the recent related works on ED problem have been studied with the metaheuristic methods such as Genetic Algorithm (GA) [4], Particle Swarm Optimization (PSO) [4], Imperialist Competitive Algorithm (ICA) [5], Artificial Bee Colony (ABC) [6], Bacterial Foraging Optimization (BF) [7], Hybrid Harmony Search with Arithmetic Crossover (ACHS) [8], GA with a special class of ant colony optimization (GAAPI) [9] and so on. The modified and hybrid models of PSO such as Modified PSO (MPSO) [10], guaranteed convergence PSO (GCPSO) [10], Species-based Quantum Particle Swarm Optimization (SQPSO) [11], Iteration PSO (IPSO) [12], Parallel PSO with Modified Stochastic Acceleration Factors (PSO-MSAF) [13], Distributed Sobol PSO and Tabu Search Algorithm (DSPSO-TSA) [14], Self-Organizing Hierarchical PSO (SOH-PSO) [15], Passive Congregation-based PSO (PC-PSO) [15] and Simple PSO (SPSO) [15] have also been employed

Application of metaheuristic algorithms on power system protection and particularly, DOCR coordination in distribution networks has been introduced in literature such as PSO [3, 16], Harmony Search Algorithm (HSA) [17], Cuckoo Algorithm [18], chaotic firefly algorithm [19],

In this chapter, PSO is applied as a solution to the introduced power system operation problems, namely ED and DOCR coordination. The rest of the chapter is organized as follows: in Section 2, these power system problems are defined and formulated as optimization problems. PSO algorithm is explained in Section 3. In Section 4, PSO is applied in two case study systems to conduct the performance and feasibility of this method. Finally, Section 5 concludes the

mum power outputs with minimum fuel costs is challenging.

26 Particle Swarm Optimization with Applications

be very difficult.

to address the ED problem.

differential evolution [20] and so on.

chapter with the results in pervious sections.

In this section, overcurrent relay coordination and economic dispatch problems are formulated separately as optimization problems.

## 2.1. Relay coordination problem

In a protection scheme, each primary relay should operate as fast as possible to clear the fault in a system. If the operation time of the relay takes longer than an acceptable time, the damage on the faulty equipment would be severe with serious consequences. In other words, minimizing the total operation time of relays decreases the risk and stress on the protected apparatus, which can be depicted as an optimization objective function:

$$OF = \min \sum\_{i=1}^{n} w\_i t\_i \tag{1}$$

where ti is the operation time of relay Ri , wi is the probability of the occurrence of fault on transmission line in the zone of protection, and it is normally set to 1; n is the total number of relays in the system.

Generally, the operation time of DOCRs is defined in (2):

$$t\_i = \frac{\lambda \times TMS\_i}{\left(\frac{I\_{Fi}}{PS\_i}\right)^{\eta} - 1} + L \tag{2}$$

where IFi is the fault current seen by the appropriate relay Ri after being transformed through the secondary winding of corresponding current transformer (CT). Depending on the type of relays, the characteristic constants λ, L and η are selected [16]. In this chapter, continuous form of TMS and PS is considered with relay type of standard inverse definite minimum time (IDMT). Based on that, all the relays in the system are assumed identical with a common characteristic function approximated by:

$$t\_i = \frac{0.14 \times TMS\_i}{\left(\frac{I\_i}{PS\_i}\right)^{0.02} - 1} \tag{3}$$

To ensure that the operation time of an individual relay is proper enough to mitigate the damage impact of faults on the apparatus, the time must be within an acceptable range:

$$t\_{i\text{min}} \le t\_i \le t\_{i\text{max}}; i = 1, \dots, n \tag{4}$$

where, respectively, timin and timax are the minimum and maximum operating time of the relay Ri. Each overcurrent relay has a manufactured TMS range to provide controllability of response to faults with different speeds. As shown in (3), ti is proportional to TMS values. Also, the PS has nonlinear effect on the operating time. Within a security margin and to avoid maloperation of an individual relay with normal load or slight overload current, the minimum pickup current setting is selected bigger than the maximum load current. The maximum plug setting is chosen not greater than the minimum fault current [19]. Therefore, there are constraints on TMS and PS as follows:

$$\text{TMS}\_{\text{imin}} \le \text{TMS}\_{\text{i}} \le \text{TMS}\_{\text{imax}}; \text{i = 1, \dots, n} \tag{5}$$

$$PS\_{\text{imin}} \le PS\_i \le PS\_{\text{irmax}}; i = 1, \dots, n \tag{6}$$

In a practical ED optimization, the generator constraints and network limits such as ramp rate limit, the prohibited zones of operation, generation capacity constraints and valve point effects are considered. Single quadratic equation is used to formulate the ED optimization problem:

> Xm i¼1

where αi, β<sup>i</sup> and γ<sup>i</sup> are the cost equation coefficients of uniti, Pi is the output power in MWand F Pð Þ<sup>i</sup> is the cost function of that unit in \$/h. The index m denotes the number of generators in a system.

The fuel cost function in (8) as the objective function of ED problem here is associated with practical constraints. Considering the ramp rate limits, the momentary output power of a generator cannot exceed its previously generated power more than a certain amount of URi, the up-ramp rate limit and neither can it be less than that of the previously generated power by more than a certain amount of DRi, the down-ramp rate limit of the generator. Therefore:

where Pi is the current power output and Pi<sup>0</sup> refers to the previous power output of generator i. URi and DRi represent the up-ramp limit and down-ramp limit of the generator i, respectively,

The input-output curves of the generation units have separate operation zones. The prohibited zones of operations are due to the operation of steam valve or the shaft bearing vibration of the generators. Therefore, the generated power is within the feasible zones of operation and

i,1

i,j�1

and maximum power capacity, respectively, of a generator that produces. From (11) and (12), it can be deduced that the operational power of a generator must be within the constraint in (13)

Total delivered power from the units needs to meet the power demand and transmission loss

� � ≤ Pi ≤ min Pmax

i,j�<sup>1</sup> <sup>≤</sup> Pi <sup>≤</sup> Pl

i,n ≤ Pi ≤ Pmax i

α<sup>i</sup> þ β<sup>i</sup>

Pi þ γ<sup>i</sup> Pi

Particle Swarm Optimization Solution for Power System Operation Problems

Pi � Pi<sup>0</sup> ≤ URi (9)

Pi<sup>0</sup> � Pi ≤ DRi (10)

Pi<sup>0</sup> � DRi ≤ Pi ≤ Pi<sup>0</sup> � URi (11)

i,j and Pu

<sup>i</sup> and Pmax

� � (13)

, j ¼ 2, 3, …, n; i ¼ l, …, m

<sup>i</sup> ; Pi<sup>0</sup> þ URi

<sup>2</sup> (8)

29

http://dx.doi.org/10.5772/intechopen.72409

(12)

i,j are the lower and

<sup>i</sup> are minimum

minFt <sup>¼</sup> <sup>X</sup><sup>m</sup>

i¼1

in which the current generated power has the following constraint in MW/t:

outside the prohibited zones. For a generator i:

Pi ∈

and also be out of the prohibited zones in (12):

in the grid.

Pmin <sup>i</sup> ≤ Pi ≤ P<sup>l</sup>

Pu

8 >><

>>:

max Pmin

Pu

where j is the number of prohibited zones of operation for unit i, Pl

upper boundaries, respectively, of prohibited zone j of generator i, Pmin

<sup>i</sup> ; Pi<sup>0</sup> � DRi

F Pð Þ¼<sup>i</sup>

where TMSimax, PSimax, TMSimin and PSimin are the maximum and minimum values of TMS and PS of relay Ri . Although the constraints in Eqs. (4)–(6) seem to provide satisfactory limits on performance of each relay, they are not enough to guarantee correct performance of the protection scheme as the coordination between primary-backup relays has not been considered.

The constraints can only ensure the operation of an individual relay not a primary-backup pair. To coordinate adjacent relays as primary and backup relays, the primary relay should operate as fast as possible within its acceptable boundaries. If it fails to act, its backup relay needs to take over the tripping action with a minimum time. The minimum operating time of the backup relay must be small but yet bigger than the operating time of primary relay. Therefore, a coordination time interval (CTI) is added to the constraints to satisfy the proper coordination scheme.

$$t\_{j\text{\\_backup}} - t\_{i\text{\\_ir}\text{\\_ir}} \gtrsim \text{CTI} \tag{7}$$

where ti and tj are the operation time of primary and backup relays, respectively. CTI depends upon the relay type, circuit breaker speed, relay over-travel time and the safety factor time for CT saturation, setting errors, contact gaps and so on. According to the IEEE standard [21], CTI is set to 0.2 s for the digital relays. Figure 1 shows the backup-primary pair of relays in a radial network.

#### 2.2. Economic dispatch

How to allocate the available generators in the grid to respond to the load demand with minimum fuel costs is an economical aspect of power dispatch in the electric power system that annually costs millions of dollars to operate.

Figure 1. Primary-backup relation between relays in a distribution system.

In a practical ED optimization, the generator constraints and network limits such as ramp rate limit, the prohibited zones of operation, generation capacity constraints and valve point effects are considered. Single quadratic equation is used to formulate the ED optimization problem:

setting is chosen not greater than the minimum fault current [19]. Therefore, there are con-

where TMSimax, PSimax, TMSimin and PSimin are the maximum and minimum values of TMS and

performance of each relay, they are not enough to guarantee correct performance of the protection scheme as the coordination between primary-backup relays has not been considered.

The constraints can only ensure the operation of an individual relay not a primary-backup pair. To coordinate adjacent relays as primary and backup relays, the primary relay should operate as fast as possible within its acceptable boundaries. If it fails to act, its backup relay needs to take over the tripping action with a minimum time. The minimum operating time of the backup relay must be small but yet bigger than the operating time of primary relay. Therefore, a coordination time interval (CTI) is added to the constraints to satisfy the proper

where ti and tj are the operation time of primary and backup relays, respectively. CTI depends upon the relay type, circuit breaker speed, relay over-travel time and the safety factor time for CT saturation, setting errors, contact gaps and so on. According to the IEEE standard [21], CTI is set to 0.2 s for the digital relays. Figure 1 shows the backup-primary pair of relays in a radial

How to allocate the available generators in the grid to respond to the load demand with minimum fuel costs is an economical aspect of power dispatch in the electric power system

. Although the constraints in Eqs. (4)–(6) seem to provide satisfactory limits on

TMSimin ≤ TMSi ≤ TMSimax; i ¼ 1, …, n (5)

PSimin ≤ PSi ≤ PSimax; i ¼ 1, …, n (6)

tj backup � ti primary ≥CTI (7)

straints on TMS and PS as follows:

28 Particle Swarm Optimization with Applications

PS of relay Ri

coordination scheme.

2.2. Economic dispatch

that annually costs millions of dollars to operate.

Figure 1. Primary-backup relation between relays in a distribution system.

network.

$$\text{minF}\_{t} = \sum\_{i=1}^{m} F(P\_i) = \sum\_{i=1}^{m} \alpha\_i + \beta\_i P\_i + \gamma\_i P\_i^{\,2} \tag{8}$$

where αi, β<sup>i</sup> and γ<sup>i</sup> are the cost equation coefficients of uniti, Pi is the output power in MWand F Pð Þ<sup>i</sup> is the cost function of that unit in \$/h. The index m denotes the number of generators in a system.

The fuel cost function in (8) as the objective function of ED problem here is associated with practical constraints. Considering the ramp rate limits, the momentary output power of a generator cannot exceed its previously generated power more than a certain amount of URi, the up-ramp rate limit and neither can it be less than that of the previously generated power by more than a certain amount of DRi, the down-ramp rate limit of the generator. Therefore:

$$P\_i - P\_{i0} \le \text{LIR}\_i \tag{9}$$

$$P\_{i0} - P\_i \le DR\_i \tag{10}$$

where Pi is the current power output and Pi<sup>0</sup> refers to the previous power output of generator i. URi and DRi represent the up-ramp limit and down-ramp limit of the generator i, respectively, in which the current generated power has the following constraint in MW/t:

$$P\_{i0} - DR\_i \le P\_i \le P\_{i0} - LR\_i \tag{11}$$

The input-output curves of the generation units have separate operation zones. The prohibited zones of operations are due to the operation of steam valve or the shaft bearing vibration of the generators. Therefore, the generated power is within the feasible zones of operation and outside the prohibited zones. For a generator i:

$$P\_i \in \begin{cases} P\_i^{\min} \le P\_i \le P\_{i,1}^l \\\ P\_{i,j-1}^u \le P\_i \le P\_{i,j-1}^l \ne j = 2, 3, \dots, n; i = l, \dots, m \\\ P\_{i,n}^u \le P\_i \le P\_i^{\max} \end{cases} \tag{12}$$

where j is the number of prohibited zones of operation for unit i, Pl i,j and Pu i,j are the lower and upper boundaries, respectively, of prohibited zone j of generator i, Pmin <sup>i</sup> and Pmax <sup>i</sup> are minimum and maximum power capacity, respectively, of a generator that produces. From (11) and (12), it can be deduced that the operational power of a generator must be within the constraint in (13) and also be out of the prohibited zones in (12):

$$\max\{P\_i^{\min}, P\_{i0} - DR\_i\} \le P\_i \le \min\left(P\_i^{\max}, P\_{i0} + LR\_i\right) \tag{13}$$

Total delivered power from the units needs to meet the power demand and transmission loss in the grid.

$$\sum\_{i=1}^{m} P\_i = P\_D + P\_L; i \in m \tag{14}$$

<sup>w</sup> <sup>¼</sup> <sup>w</sup>max � wmax � <sup>w</sup>min

The general steps of PSO on solving an optimization problem are as follows:

2. Generate initial populations having initial positions p and velocities v.

<sup>i</sup> <sup>¼</sup> f pk i

<sup>k</sup> <sup>þ</sup> <sup>c</sup><sup>1</sup> � rand<sup>1</sup> � pbesti � pi

<sup>i</sup> then pbest<sup>k</sup>þ<sup>1</sup>

<sup>b</sup> then gbest<sup>k</sup>þ<sup>1</sup> <sup>¼</sup> pbestkþ<sup>1</sup>

11. If k < Itermax then k ¼ k þ 1 and go to step 6 else go to step 12.

b.

itermax � k.

<sup>i</sup> <sup>¼</sup> pkþ<sup>1</sup>

Application of PSO on solving the defined problems in previous sections is validated here. In the first case study system, the overcurrent relay coordination problem in a distribution network is solved. The size of population and iteration numbers of PSO are 60 and 100, respectively. In the second case study system, the ED problem is addressed with population size of 100 and 500 iterations. c1, c2, wmin and wmax in both case study systems are the same and

<sup>i</sup> , <sup>∀</sup><sup>i</sup> and set gbest<sup>k</sup> <sup>¼</sup> pk

simulations, respectively.

3. Set iteration k ¼ 1.

5. Select pbestk

vi

pi <sup>k</sup>þ<sup>1</sup> <sup>¼</sup> pi

8. Calculate fitness Fkþ<sup>1</sup>

4. Calculate fitness of particles F<sup>k</sup>

<sup>i</sup> <sup>¼</sup> pk

<sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>w</sup> � vi

9. Update pbest for all particles.

10. Update gbest of the population

12. Print gbest<sup>k</sup> as optimum solution.

set to 2, 2, 0.4 and 0.9, respectively.

if F<sup>k</sup>þ<sup>1</sup> <sup>b</sup><sup>1</sup> < F<sup>k</sup>

4. Simulation results

1. Set initial parameters wmin, wmax, c<sup>1</sup> and c2.

6. Update inertia factor: <sup>w</sup> <sup>¼</sup> <sup>w</sup>max � wmax�wmin

<sup>k</sup> <sup>þ</sup> vi kþ1 ; ∀i

> if Fkþ<sup>1</sup> <sup>i</sup> < Fk

7. Update the velocity and position of particles.

<sup>i</sup> <sup>¼</sup> f p<sup>k</sup>þ<sup>1</sup> i

where k and itermax are the current iteration and the maximum number of iteration during

itermax

, ∀i and find the index of the best particle b.

Particle Swarm Optimization Solution for Power System Operation Problems

<sup>k</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup> � rand<sup>2</sup> � gbest � pi

, ∀i and obtain the index of the best particle b1.

<sup>i</sup> elsepbest<sup>k</sup>þ<sup>1</sup>

<sup>i</sup> <sup>¼</sup> pbest<sup>k</sup>

<sup>b</sup><sup>1</sup> and <sup>b</sup> <sup>¼</sup> <sup>b</sup>1 else gbestkþ<sup>1</sup> <sup>¼</sup> gbestk

i

� k (19)

http://dx.doi.org/10.5772/intechopen.72409

31

<sup>k</sup> ; ∀i

where PD and PL represent the demand power and the power loss in the gird, respectively. The overall power loss of the committed units is based on the output power, which is formulated by B matrix coefficients known as Kron's formula:

$$P\_L = \sum\_{i=1}^{m} \sum\_{j=1}^{m} P\_i B\_{i\bar{j}} P\_{\bar{j}} + \sum\_{i=1}^{m} B\_{0i} P\_i + B\_{00} \tag{15}$$

The power loss itself cannot be more than some permissible values:

$$\mathbb{P}\left|P\_{\mathcal{L}f,k}\right| \le P\_{\mathcal{L}f,k}^{\max}; k = j, \dots, L\tag{16}$$

where the real power flow of line j is represented with PLf , <sup>k</sup> and k is the number of transmission lines in a system. The power loss cannot be more than a maximum value of Pmax Lf , <sup>k</sup> .

## 3. Particle swarm optimization algorithm

PSO algorithm is a nature inspired method from social behavior of bird flocking and fish schooling, which is first introduced by Erberhart and Kennedy in 1995 [22]. As a populationbased stochastic optimization technique using swarm intellects in the search space, this technique is based on interaction of swarm of particles. Every particle includes two values of position and velocity that are updated during the iteration runs by considering each particle's best experience (best position) and the best achieved experience (global position) of all particles.

The update of position and velocity of the particles must be processed, and it has to follow Eqs. (17)–(19) for satisfying the constraint of an optimization problem. Each particle movement is based on the changes in its position and its velocity:

$$p\_i^{k+1} = p\_i^k + \upsilon\_i^{k+1} \tag{17}$$

where pkþ<sup>1</sup> <sup>i</sup> and pk <sup>i</sup> are the position of particle <sup>i</sup> in the iteration <sup>k</sup> <sup>þ</sup> 1 and <sup>k</sup>, respectively, vkþ<sup>1</sup> <sup>i</sup> is the velocity of the particle in k þ 1 iteration. A particle's velocity is defined as follows:

$$\boldsymbol{\upsilon}\_{i}^{k+1} = \boldsymbol{\upsilon} \times \boldsymbol{\upsilon}\_{i}^{k} + \boldsymbol{c}\_{1} \times \boldsymbol{r} \text{and}\_{1} \times \left( \boldsymbol{pbest}\_{i} - \boldsymbol{p}\_{i}^{k} \right) + \boldsymbol{c}\_{2} \times \boldsymbol{r} \text{and}\_{2} \times \left( \boldsymbol{gbest} - \boldsymbol{p}\_{i}^{k} \right) \tag{18}$$

where pbesti is the best so far position of the particle i as the best experience, while gbest is the best position among the whole swarm with all the particles in movement as global experience. c<sup>1</sup> and c<sup>2</sup> are the weighting factors, while rand<sup>1</sup> and rand<sup>2</sup> are two random numbers between zero and one. The parameter w is the inertia factor varying between ½ � wmin; wmax , as shown in (19), which is a linear decreasing inertia weight in this chapter.

Particle Swarm Optimization Solution for Power System Operation Problems http://dx.doi.org/10.5772/intechopen.72409 31

$$w = w\_{\text{max}} - \frac{w\_{\text{max}} - w\_{\text{min}}}{iter\_{\text{max}}} \times k \tag{19}$$

where k and itermax are the current iteration and the maximum number of iteration during simulations, respectively.

The general steps of PSO on solving an optimization problem are as follows:


Xm i¼1

> Xm j¼1

PL <sup>¼</sup> <sup>X</sup><sup>m</sup> i¼1

The power loss itself cannot be more than some permissible values:

PLf , <sup>k</sup> � � �

lines in a system. The power loss cannot be more than a maximum value of Pmax

� <sup>≤</sup> <sup>P</sup>max

where the real power flow of line j is represented with PLf , <sup>k</sup> and k is the number of transmission

PSO algorithm is a nature inspired method from social behavior of bird flocking and fish schooling, which is first introduced by Erberhart and Kennedy in 1995 [22]. As a populationbased stochastic optimization technique using swarm intellects in the search space, this technique is based on interaction of swarm of particles. Every particle includes two values of position and velocity that are updated during the iteration runs by considering each particle's best experience (best position) and the best achieved experience (global position) of all particles.

The update of position and velocity of the particles must be processed, and it has to follow Eqs. (17)–(19) for satisfying the constraint of an optimization problem. Each particle movement

<sup>i</sup> <sup>þ</sup> vkþ<sup>1</sup>

<sup>i</sup> are the position of particle <sup>i</sup> in the iteration <sup>k</sup> <sup>þ</sup> 1 and <sup>k</sup>, respectively, vkþ<sup>1</sup>

<sup>k</sup> � � <sup>þ</sup> <sup>c</sup><sup>2</sup> � rand<sup>2</sup> � gbest � pi

pkþ<sup>1</sup> <sup>i</sup> <sup>¼</sup> pk

the velocity of the particle in k þ 1 iteration. A particle's velocity is defined as follows:

where pbesti is the best so far position of the particle i as the best experience, while gbest is the best position among the whole swarm with all the particles in movement as global experience. c<sup>1</sup> and c<sup>2</sup> are the weighting factors, while rand<sup>1</sup> and rand<sup>2</sup> are two random numbers between zero and one. The parameter w is the inertia factor varying between ½ � wmin; wmax , as shown in

<sup>k</sup> <sup>þ</sup> <sup>c</sup><sup>1</sup> � rand<sup>1</sup> � pbesti � pi

(19), which is a linear decreasing inertia weight in this chapter.

by B matrix coefficients known as Kron's formula:

30 Particle Swarm Optimization with Applications

3. Particle swarm optimization algorithm

is based on the changes in its position and its velocity:

where pkþ<sup>1</sup>

<sup>i</sup> and pk

vi

<sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>w</sup> � vi

where PD and PL represent the demand power and the power loss in the gird, respectively. The overall power loss of the committed units is based on the output power, which is formulated

PiBijPj <sup>þ</sup>X<sup>m</sup>

i¼1

Pi ¼ PD þ PL; i∈ m (14)

B0iPi þ B<sup>00</sup> (15)

<sup>i</sup> (17)

<sup>i</sup> is

<sup>k</sup> � � (18)

Lf , <sup>k</sup> .

Lf , <sup>k</sup> ; k ¼ j, …, L (16)


$$\begin{aligned} \upsilon\_i^{k+1} &= \upsilon \times \upsilon\_i^k + c\_1 \times rand\_1 \times \left( pbest\_i - p\_i^k \right) + c\_2 \times rand\_2 \times \left( gbest - p\_i^k \right); \forall i \\\ p\_i^{k+1} &= p\_i^k + \upsilon\_i^{k+1}; \forall i \end{aligned}$$


$$\text{if } F\_i^{k+1} < F\_i^k \text{ then } pbest\_i^{k+1} = p\_i^{k+1} \text{ else } pbest\_i^{k+1} = pbest\_i^k$$

10. Update gbest of the population

$$\text{if } F\_{b1}^{k+1} < F\_b^k \text{ then } gb \text{est}^{k+1} = pb \text{est}\_{b1}^{k+1} \text{ and } b = b1 \text{ else } gb \text{est}^{k+1} = gb \text{est}^k$$


## 4. Simulation results

Application of PSO on solving the defined problems in previous sections is validated here. In the first case study system, the overcurrent relay coordination problem in a distribution network is solved. The size of population and iteration numbers of PSO are 60 and 100, respectively. In the second case study system, the ED problem is addressed with population size of 100 and 500 iterations. c1, c2, wmin and wmax in both case study systems are the same and set to 2, 2, 0.4 and 0.9, respectively.

## 4.1. Case study 1: relay protection coordination

A 15-node radial network system including a total of 13 loads is considered, and PSO is employed to determine the optimal settings of all 28 digital over-current relays shown in Figure 2. Identical digital relays are used with same current transformer (CT) ratio 500:1. The constraint values of the relays are as follows: timin and timax for each relay is set to 0.1 and 4 s, respectively; TMSmin and TMSmax constraints are 0.1 and 1.1, respectively; PSmin and PSmax are 0.5 and 2.5, respectively [16].

The primary-backup relationships of the relays in the system are shown in Table 1. According to the objective function in (3), the maximum fault currents sensed by the relays are also required. Therefore, the case study system in Figure 2 has been modeled in DigSILENT PowerFactory software with simulating three-phase faults occurring in front of each relay. The collected data have been shown in Table 2. The load parameters of the radial network are shown in Table 3.

After data collection in DigSILENT PowerFactory software, the optimal settings (TMS and PS) of all 28 overcurrent relays are obtained by solving (3) subject to the constraints in (4)–(7) using the PSO in MATLAB software. The algorithm was executed 100 times to achieve accurate results. Table 4 shows the best TMS and PS settings of the relays to provide a reliable protection scheme

in the distribution system. The total operation time of the relays in this system is 26.189 s. Figure 3

Load Active power (MW) Reactive power (Mvar) Load Active power (MW) Reactive power (Mvar)

1 4 1.5 8 0.5 0.2 2 1 0.2 9 2 0.8 3 2.5 1 10 1 1 4 3 2 11 2.5 0.9 5 1 0.5 12 1 0.5 6 3 0.7 13 3 2

Primary Backup Primary Backup Primary Backup Primary Backup — 8 — 15 13 22 — — 9 1 16 — 23 21 1 10 — 17 13 24 — — 11 9 18 — 25 5 3 12 — 19 3 26 — — 13 1 20 — 27 5 5 14 — 21 19 28 —

Particle Swarm Optimization Solution for Power System Operation Problems

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33

Relay Fault current (kA) Relay Fault current (kA) Relay Fault current (kA) Relay Fault current (kA)

1 22.778 8 4.636 15 7.7 22 4.636 2 11.532 9 11.509 16 5.79 23 4.632 3 11.509 10 7.71 17 7.7 24 3.866 4 7.71 11 7.7 18 5.79 25 5.785 5 7.7 12 5.79 19 7.7 26 4.636 6 5.79 13 11.509 20 5.79 27 5.785 7 5.785 14 7.71 21 5.785 28 4.636

A 15-unit test system is used to investigate the feasibility of PSO in solving the nonsmooth economic dispatch considering transmission losses, ramp rate limits and the prohibited

illustrates the convergence curve of PSO in solving the objective function.

4.2. Case study 2: economic dispatch

Table 1. Primary-backup pair relationship of the relays.

Table 2. Maximum fault currents.

7 1 0.5

Table 3. Load parameters in case study one.

Figure 2. A 15-node distribution system as case study one.


Table 1. Primary-backup pair relationship of the relays.


Table 2. Maximum fault currents.

4.1. Case study 1: relay protection coordination

Figure 2. A 15-node distribution system as case study one.

0.5 and 2.5, respectively [16].

32 Particle Swarm Optimization with Applications

shown in Table 3.

A 15-node radial network system including a total of 13 loads is considered, and PSO is employed to determine the optimal settings of all 28 digital over-current relays shown in Figure 2. Identical digital relays are used with same current transformer (CT) ratio 500:1. The constraint values of the relays are as follows: timin and timax for each relay is set to 0.1 and 4 s, respectively; TMSmin and TMSmax constraints are 0.1 and 1.1, respectively; PSmin and PSmax are

The primary-backup relationships of the relays in the system are shown in Table 1. According to the objective function in (3), the maximum fault currents sensed by the relays are also required. Therefore, the case study system in Figure 2 has been modeled in DigSILENT PowerFactory software with simulating three-phase faults occurring in front of each relay. The collected data have been shown in Table 2. The load parameters of the radial network are

After data collection in DigSILENT PowerFactory software, the optimal settings (TMS and PS) of all 28 overcurrent relays are obtained by solving (3) subject to the constraints in (4)–(7) using the PSO in MATLAB software. The algorithm was executed 100 times to achieve accurate results. Table 4 shows the best TMS and PS settings of the relays to provide a reliable protection scheme


Table 3. Load parameters in case study one.

in the distribution system. The total operation time of the relays in this system is 26.189 s. Figure 3 illustrates the convergence curve of PSO in solving the objective function.

#### 4.2. Case study 2: economic dispatch

A 15-unit test system is used to investigate the feasibility of PSO in solving the nonsmooth economic dispatch considering transmission losses, ramp rate limits and the prohibited


Table 4. Obtained TMS and PS values of relays by PSO.

This system has many local minima with high dimensionality that draws realistic analysis for practical applications. PSO is applied on the 15-unit test system, and the results are compared with best results in literature on the same system: ACHS [8], SQPSO [11], ABC [6], IPSO [12], PSO-MSAF [13], DSPSO-TSA [14], ICA [5], GAAPI [9], MPSO [10], SOH-PSO [15], GCPSO

12 20 80 40 230 9.90 0.005513 80 80 [30 40] [55 65]

<sup>i</sup> Pi<sup>0</sup> α<sup>i</sup> β<sup>i</sup> γ<sup>i</sup> URi DRi Prohibited zones (MW)

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2 150 455 300 574 10.2 0.000183 80 120 [185 225] [305 335] [420 450]

5 150 470 90 461 10.40 0.000205 80 120 [180 200] [305 335] [390 420] 6 135 460 400 630 10.10 0.000301 80 120 [230 255] [365 395] [430 455]

The simulation is tested for 100 times to ensure reliable analysis. Table 6 shows the optimum results of each method for the 15-unit system. Figure 4 illustrates the convergence of PSO method in 100 different trials, while the final fuel costs in 100 trials are shown in Figure 5. Table 7 shows the best economic dispatch of power using PSO for ED optimization in the 15-

The optimum cost of 15-unit system with the proposed PSO solution is 32701.282 (\$), which has better result than other methods. GA has the most deviating results compared with other hybrid and improved methods in this test system. Also, the mean value of final fuel cost of generators using PSO over 100 trials is less than minimum values obtained by other method, which indicates higher quality of solution and better performance of PSO compared with other algorithms on the same test system. The power loss in the grid obtained from PSO is less than other algorithms, which shows better dispatching scheme using PSO. The best convergence of PSO to the minimum fuel cost of generators in the grid while satisfying the constraints is

In case study two, PSO achieves better results when compared with other hybrid or improved methods. It is worth mentioning that the maximum iteration number is 500 and the population

[10], PC-PSO [15], BF [7], SPSO [15], PSO [4] and GA [4].

1 150 455 400 671 10.1 0.000299 80 120

3 20 130 105 374 8.80 0.001126 130 130 4 20 130 100 374 8.80 0.001126 130 130

7 135 465 350 548 9.8 0.000364 80 120 8 60 300 95 227 11.2 0.000338 65 100 9 25 162 105 173 11.2 0.000807 60 100 10 25 160 110 175 10.7 0.001203 60 100 11 20 80 60 186 10.2 0.003586 80 80

13 25 85 30 225 13.1 0.000371 80 80 14 15 55 20 309 12.1 0.001929 55 55 15 15 55 20 323 12.4 0.004447 55 55

Table 5. Generation unit characteristics of a 15-unit system in case study two.

unit system.

Unit Pmin

<sup>i</sup> <sup>P</sup>max

shown in Figure 6.

Figure 3. Convergence of PSO solution in case study one.

operation zones of the thermal generators. The cost curves data and operation limits of the 15-unit system are shown in Table 5. The B loss coefficients to calculate the power loss can be found in [4].


Table 5. Generation unit characteristics of a 15-unit system in case study two.

operation zones of the thermal generators. The cost curves data and operation limits of the 15-unit system are shown in Table 5. The B loss coefficients to calculate the power loss can be

Relay TMS PS Relay TMS PS 0.74394 0.54999 15 0.42632 0.79222 0.32216 1.2551 16 0.54396 0.67839 0.3416 1.5153 17 0.21794 0.61987 0.55245 2.4141 18 0.12037 1.1896 0.36342 0.60859 19 0.31997 0.66671 0.4309 1.0805 20 0.56566 1.1323 0.11188 1.2611 21 0.26376 0.60754 0.2726 1.822 22 0.14313 1.8741 0.45395 1.1148 23 0.12496 0.7978 0.15359 1.3933 24 0.55302 1.3848 0.32133 0.50089 25 0.22254 0.65529 0.26418 2.4727 26 0.2904 1.6316 0.32088 1.9075 27 0.13471 0.77885 0.73899 2.1497 28 0.10277 2.4925

found in [4].

OF (s) 26.189

34 Particle Swarm Optimization with Applications

Table 4. Obtained TMS and PS values of relays by PSO.

Figure 3. Convergence of PSO solution in case study one.

This system has many local minima with high dimensionality that draws realistic analysis for practical applications. PSO is applied on the 15-unit test system, and the results are compared with best results in literature on the same system: ACHS [8], SQPSO [11], ABC [6], IPSO [12], PSO-MSAF [13], DSPSO-TSA [14], ICA [5], GAAPI [9], MPSO [10], SOH-PSO [15], GCPSO [10], PC-PSO [15], BF [7], SPSO [15], PSO [4] and GA [4].

The simulation is tested for 100 times to ensure reliable analysis. Table 6 shows the optimum results of each method for the 15-unit system. Figure 4 illustrates the convergence of PSO method in 100 different trials, while the final fuel costs in 100 trials are shown in Figure 5. Table 7 shows the best economic dispatch of power using PSO for ED optimization in the 15 unit system.

The optimum cost of 15-unit system with the proposed PSO solution is 32701.282 (\$), which has better result than other methods. GA has the most deviating results compared with other hybrid and improved methods in this test system. Also, the mean value of final fuel cost of generators using PSO over 100 trials is less than minimum values obtained by other method, which indicates higher quality of solution and better performance of PSO compared with other algorithms on the same test system. The power loss in the grid obtained from PSO is less than other algorithms, which shows better dispatching scheme using PSO. The best convergence of PSO to the minimum fuel cost of generators in the grid while satisfying the constraints is shown in Figure 6.

In case study two, PSO achieves better results when compared with other hybrid or improved methods. It is worth mentioning that the maximum iteration number is 500 and the population


Figure 5. Cost distribution with PSO over 100 different trials.

Unit (MW) PSO ICA [5] GCPSO [10] MPSO [10] GA [4] PSO [4] ABC [6] SOH-PSO [15]

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P1 454.9963 455 449.89252 455 415.3108 439.1162 455 455 P2 379.9998 380 366.99066 380 359.7206 407.9727 380 380 P3 130 130 130 130 104.4250 119.6324 130 130 P4 129.9954 130 130 130 74.9853 129.9925 130 130 P5 169.9999 167.4174 170 170 380.2844 151.0681 169.9997 170 P6 459.9999 460 460 460 426.7902 459.9978 460 459.96 P7 430 430 430 430 341.3164 425.5601 430 430.00 P8 66.1794 113.4737 75.88460 92.7278 124.7867 98.5699 71.9698 117.53 P9 64.9485 25.1555 50.22689 43.0282 133.1445 113.4936 59.1798 77.90 P10 159.2255 155.3478 160 140.1938 89.2567 101.1142 159.8004 119.54 P11 79.9996 80 80 80 60.0572 33.9116 80 54.50 P12 79.9901 80 77.87063 80 49.9998 79.9583 80 80.00 P13 25.0001 25 25 27.6403 38.7713 25.0042 25.0024 25.00 P14 15.0005 15 15.8312 20.7610 41.9425 41.4140 15.0056 17.86 P15 15.0029 15 39.66146 22.2724 22.6445 35.6140 15.0014 15 Total power 2660.338 2661.394 2661.35806 2661.6235 2668.4 2262.4 2735.959 2662.29 Power loss 30.338 31.291 30.86593 29.978 38.2782 32.4306 30.9591 32.28 Cost (\$) 32701.282 32715.4305 32764.4616 32738.41778 33113 32858 32707.85 32751.39

Table 7. Optimum solution of PSO compared with other methods in case study two.

Table 6. Comparison of PSO results with other methods in case study two.

Figure 4. Convergence curve of PSO over 100 different trials.

Figure 5. Cost distribution with PSO over 100 different trials.

Method Best cost (\$/h) Mean (\$/h) PSO 32701.282 32704.10578 ACHS [8] 32706.6500 32706.65 SQPSO [11] 32706.6740 32708.4457 ABC [6] 32707.85 32707.95 IPSO [12] 32709.00 32784.5 PSO-MSAF [13] 32713.09 32759.64 DSPSO-TSA [14] 32715.06 32724.63 ICA [5] 32715.4305 NA GAAPI [9] 32732.95 NA MPSO [10] 32738.4177 NA SOH-PSO [15] 32751.39 32,878 GCPSO [10] 32764.4616 NA PC-PSO [15] 32775.36 NA BF [7] 32784.5024 32796.81 SPSO [15] 32798.69 NA PSO [4] 32,858 33,039 GA [4] 33,113 33,228

36 Particle Swarm Optimization with Applications

Table 6. Comparison of PSO results with other methods in case study two.

Figure 4. Convergence curve of PSO over 100 different trials.


Table 7. Optimum solution of PSO compared with other methods in case study two.

Author details

References

Mostafa Kheshti\* and Lei Ding

Ministry of Education, Jinan, China

2010;38(7):786-803

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\*Address all correspondence to: mostafa\_kheshti@yahoo.com

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Key Laboratory of Power System Intelligent Dispatch and Control, Shandong University,

Particle Swarm Optimization Solution for Power System Operation Problems

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Figure 6. Convergence of PSO with best fuel cost result for ED problem in case study two.

size is 100. In most of the quoted methods, the iteration and population sizes vary that can affect the final results. For example, the PSO in [4] has population size and iteration number of 100 and 500, respectively. The population size, iteration, crossover rate, mute rate and crossover parameter of GA [4] are 100, 200, 0.8, 0.01 and 0.5, respectively.

Different system configuration and programming language frameworks can also influence the results in which MATLAB 2015Ra was used for programming.

## 5. Conclusions

Distribution network relay coordination and the economic dispatch of generators in the electric power system were modeled as optimization problems. Particle swarm optimization (PSO) was successfully employed to solve the defined problems where two case study systems were conducted to validate the results. In the first case study system, PSO provided proper relay settings that allow all the relays in a system to perform with high reliability and accuracy. In the second case, the optimal power outputs of thermal generators in the grid were scheduled to satisfy the load demands and other practical constraints on the generators and the grid with minimum fuel costs. The compared results with other methods demonstrated higher quality of solution, and less fuel costs obtained by PSO. The general performance of PSO in this chapter indicates applicability of this method on practical power system-related problems that are difficult to be handled by conventional methods.
